Extreme laser-matter interactions: kinetic modeling ofrelativistic, ultrarelativistic and radiation dominated plasmaKUMULATIVE HABILITATIONSSCHRIFTzurErlangung des akademischen Gradesdoctor rerum naturalium habilitata (Dr. rer. nat. habil.)der Mathematisch-Naturwissenschaftlichen FakultätDER UNIVERSITÄT ROSTOCKvorgelegt vonTatyana V. LISEYKINAgeb. am 7. Juni 1973in NowosibirskRostock, 2019https://doi.org/10.18453/rosdok_id00002875Gutachter:Prof. Dr. Dieter Bauer, Universität Rostock, Institut für PhysikProf. Caterina Riconda, Paris, Sorbonne Université, Französische RepublikProf. Igor Kostyukov, Nizhny Novgorod, Institute of Applied Physics,Russian Academy of Sciences, Russische FöderationJahr der Einreichung: 2019Jahr der Verteidigung: 2020ContentsI Abstract iII Abbreviations iiiIII Notations vIVErklärung vii1 Introduction 11.1 Intense laser-plasma interaction scenario . . . . . . . . . . . . . . . . . . . 21.2 Collisionless absorption and hot electron generation:a sharp-boundary plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Ion acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Droplets in intense laser field . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.5 Dynamics of charged particles and radiation in laser-plasma interac-tions at ultrahigh intensities: collective effects . . . . . . . . . . . . . . . . 171.6 Particle-in-Cell simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Radiation pressure acceleration of ions 313 Hot electron generation and Brunel’s model 934 Radiation friction effect in laser-plasma interactions 1155 He-Droplets in intense laser field 1476 Numerical support to the experimental studies 173i ABSTRACTI. AbstractThis cumulative thesis presents a summary of contributions made by the author over the past twelveyears and dedicated to the theory of relativistic plasma driven by intense electromagnetic radiation.Starting from the early 90s of the last century, the physics of plasmas has made several considerablesteps forward, forming new immense research areas. Physics of laser-driven plasma is one of thecentral constituents of this progress. The fast and very fruitful development of this research fieldwas triggered by three considerable advances of the last decades. Below we briefly describe thesefundamentals of the laser plasma physics.Firstly, the progress of laser sources boosted by the invention of the chirped pulse amplification tech-nique [1], made routinely possible the generation of electromagnetic fields of ultrahigh intensitiesexceeding the value of 1018 W/cm2, which makes the motion of electrons relativistic. Such electro-magnetic fields are created by laser sources of muti-terawatt (TW) and petawatt (PW) power. Nowa-days they are capable of reaching intensities∼ 1021÷1022 W/cm2. These extreme laser intensities, incombination with short pulse duration typically equal to tenths of femtoseconds (1fs=10−15s), madepossible laboratory studies of relativistic and ultra-relativistic collisionless plasmas, which were notaccessible at lower intensities and longer pulse duration. As a result, two decades have been essen-tially dedicated to the experimental and theoretical exploration of the physics of relativistic classicalcollisionless plasma. This exploration brought such phenomena as laser acceleration of charged par-ticles, laser-induced generation of giant electric and magnetic fields and secondary radiation fromlaser-driven plasma into the scope of the research community. Studies of these effects deliver bothnew knowledge in fundamental physics and a number of promising applications [2, and referencestherein]. The current progress in the development and construction of muti-PW laser facilities in-cluding APRI [3] (Korea), SULF [4] and CAEP [5] (China), ELI [6] (Europe) and Apollon [7] opensa way to the new level of electromagnetic intensities ∼ 1023 W/cm2 and higher. This next genera-tion of laser technology will make possible experimental endeavors in the presently unexplored areaof relativistic radiation-dominated and quantum plasma physics including such effects as ignitionof quantum cascades of elementary particles, creation of coupled electron-positron-photon plasmaand a variety of single-particle or collective dynamical phenomena resulting from strong classical orquantum radiation of laser-driven ultra-relativistic plasma. This progress of laser sources has createda great demand of theoretical work and calls for a further development of analytical and numericalmethods in this field of physics.Secondly, the progress in observational astronomy has led to the possibility of quantitative studies ofsuch astrophysical phenomena as jets and shocks and thus, triggered the creation of a new multidisci-plinary research field. Known as laboratory astrophysics [8, 9], [10, and references therein] it aims atmodeling space phenomena in plasma labs using strong lasers and magnetic fields. From the exper-imental side, the field of laboratory astrophysics employs femtosecond to nanosecond (1ns=10−9s)pulses of high and moderate intensities and strong static or pulsed magnetic fields created by magnetsiior by specially designed plasma targets ignited by strong laser radiation. From the theoretical side,it is based on the scalings and similarity parameters, which allow to correctly map micro- and meso-scopic plasma phenomena to those developing on astrophysical and even cosmological scales. Thus,the recently achieved mutual correspondence between the capabilities of the observational astronomyand those of the laser-plasma laboratory experiment provided another strong motivation for studiesof the plasma dynamics induced by intense laser radiation.Finally, the enormous progress in high-performance computing made possible the numerical simu-lations of complex phenomena, which were totally out of reach twenty or more years ago. For themodeling of classical collisionless plasma dynamics in external fields of femtosecond and picosec-ond (1ps=10−12s) duration, the Particle-in-Cell method [12, 13], numerically realizing the system ofthe self-consistent Maxwell-Vlasov equations, remains the main workhorse of the whole field. Theorders-of-magnitude acceleration of computer algorithms and a comparable growth of the availablememory allowed to extend the area of numerical experiments from the class of toy models to fullyrealistic three-dimensional interaction setups capable of precisely reproducing experimental data andpredicting new nontrivial effects. PIC simulations are being more and more frequently considered asan trustworthy and cheap alternative to laboratory experiments. Experimental observations of newphenomena at ultrahigh laser intensities and, even to a higher extent, the expectations to probe thenew physics at intensities exceeding 1023 W/cm2 to be reached at the multi-PW laser facilities, havestimulated a considerable modification of numerical approaches to include new effects in to the scopeof numerical experiments. This led to the development of several new PIC- or molecular dynamics-based numerical codes [14, 15, 16, 17, 18, 19, and others] capable of incorporating effects of radiationreaction (classical and quantum) and quantum electrodynamical (QED) processes on plasma dynam-ics and radiation, including creation of electron-positron pairs. Simultaneously, several PIC codes,which have been tested on a broad variety of plasma problems, have been modified to incorporateradiation reaction and field ionization of gases. These achievements resulted in the creation of anefficient distributed network of advanced open-source numerical codes that are currently being mas-sively used to analyze experimental data, as well as to plan new experiments.The theoretical work of this cumulative thesis combines these three major subfields of the physics oflaser plasma. Its essence in the application of advanced numerical methods realized in the PIC codeUMKA originated from the study in [20]. The code has been modified or extended according to theneeds of the particular problem, for studies of laser-plasma interactions both at presently accessibleparameters and under conditions expected at multi-PW laser facilities of the near future. These stud-ies are devoted to four research topics:(i) laser acceleration of ions;(ii) collisionless absorption of laser radiation in plasma and generation of hot electrons;(iii) interaction of intense laser radiation with microdroplets;(iv) interaction of laser radiation of extreme intensity with plasma in the radiation-dominated regime.The first two items belong to the field of research that is currently extensively studied in experiment.Correspondingly, results on (i) and (ii) presented in this thesis have been experimentally verified.Item (iii) also falls into the domain of parameters available for experimental research; possible lab-oratory realizations of laser-droplet interactions are therefore discussed. Finally, results of (iv) arerather looking into the near and not so near future so that there the discussion of purely theoreticalaspects dominates experimental particulars.iii ABBREVIATIONSII. Abbreviations1D/2D/3D one/two/three-dimensionalCP circular polarizationCSA collisionless shock accelerationEM electromagnetic (wave, energy)HB hole-boring RPA regimeIFE inverse Faraday effectLAD Lorentz–Abraham–Dirac (equation)LL Landau-Lifshitz (force, equation)LP linear polarizationLS light sail RPA regimePIC particle-in-cell (method, code)QED quantum electrodynamicsQM quantum mechanicsRF radiation frictionRPA radiation pressure accelerationTNSA target normal sheath accelerationXUV extreme ultravioletv NOTATIONSIII. NotationsE Electric fieldB Magnetic inductionc speed of lighth̄ Planck constantme, e electron mass and chargemi, qi ion mass and chargene electron number densityωL, λL, TL, τL laser pulse carrier frequency, wavelength, period, durationE0 amplitude of the electric field of the laser pulseIL = cE20/4π laser pulse intensity (linear polarization)S Poynting vectorηabs absorption degreencr = meω2L/4πe2 critical plasma densityωp =√4πnee2/me Langmuir (plasma) frequencyjs current density for species sΛD Debye lengthδe = c/ωp collisionless skin-deptha0 = eE0/meωLc dimensionless amplitude of the laser fieldvg = ∂ω/∂k group velocity of the waveES = m2ec3/eh̄ Schwinger field (critical field of QED)λc = h̄/mec Compton wavelengthvosc = eE0/meωL oscillation (quiver) velocityrosc = eE0/meω2L oscillation amplitude (quiver radius)vth thermal velocitycs sound velocityrc = e2/mc2 classical electron radiusEosc = mec2(√1 + a20/2− 1) electron mean oscillation energyνei collisional frequency∆ foil thicknessThot hot electron temperatureγ = 1/√1− v2/c2 relativistic γ-factorχ =eh̄m3c4√(−(Fµνpν)2 relativistically invariant quantum parametervii ERKLÄRUNGIV. ErklärungHiermit erkläre ich, dass ich die vorliegende Habilitationsschrift selbständig verfasst und nur dieangegebenen Hilfsmittel verwendet habe. Ich habe bisher noch keinen Habilitationsversuch unter-nommen.Unterschrift:Rostock, den1 INTRODUCTION1. IntroductionPlasma appears in the Universe in a staggering variety of forms, which are determined by the col-lective dynamics of charged particles and electromagnetic (EM) fields. This highly non equilibrium1matter, dominated by nonlinearities, can produce and manipulate2 enormous energy density EM fieldsand electromagnetic radiation on mesoscopic scales in the laboratory as well as on huge spatial scalesin the Universe. The possibility to obtain mesoscopic amounts of relativistic, ionized matter in lab-oratory [21, 22] is due to the recent developments in the generation of laser pulses with ultra-highpower. The opportunity to study relativistic plasmas in relatively compact size experiments opensthe way to new advances in fields that range [24] from nonlinear dynamics and nonlinear relativis-tic optics to (i) novel powerful radiation sources at high energy, (ii) alternative particle accelerationtechniques, (iii) high energy astrophysics, including relativistic shocks, cosmic ray physics and ra-diative magnetic reconnection, and finally, (iv) plasma dominated by QED effects, such as quantumrecoil and creation of electron-positron pairs. Each of these topics leads to important technologicalachievements or/and to the creation of interdisciplinary knowledge [25, 26, 27, 28, 29, 30, 31, 32].The control the dynamics of a relativistic plasma interacting with superintense laser pulses may allowfor the spatial and temporal tailoring of the EM field, and thus, achieving frequency up-shift and verytight focusing. In this way, the fields intense enough to probe QED effects even up to the Schwingerlimit [33] (ES = m2ec3/eh̄ ' 1.3× 1016V/cm, corresponds to the intensity I ' 1029 W/cm2) mightbe obtained.The generally adopted framework for the description of dilute laboratory and space plasmas is basedon coupling Vlasov’s equation to Maxwell’s equations for the EM fields. The sources of these fieldsare not the discrete particles that compose the plasma but a continuous distribution of charges andcurrents. In this asymptotic mean field description particle correlations and dynamical effects due toincoherent radiation are neglected. At the extreme optical laser intensities, as those foreseen in next-generation experiments (I > 1023 W/cm2), electrons experience extreme accelerations and emitrelatively large amounts of high frequency incoherent radiation. Thus, even within the non quan-tum description, i.e., neglecting quantum recoil and development of electron-positron pair cascades,which may lead to a considerable depletion of a laser pulse [34], the dynamical effects of the radi-ation friction (RF) force must be addressed. Large projects have been conceived (ELI[6], Apollon[7], SULF [4], CAEP [5], APRI [3]) with the declared objective of reaching even higher EM en-ergy densities inside a macroscopic relativistic medium and probing the physics of plasma regimesdominated by QED effects, in particular, by the generation of electron-positron pair plasmas as theSchwinger field is approached. Besides, the correct treatment of the RF is not only relevant to intenselaser plasma interactions. It is also important in astrophysics, e.g., for the description of pulsar winds,1in terms of local thermodynamic equilibrium2Relativistic self–focusing of the laser pulse prediced by G. A. Askar’yan in 1962 [23] is one example of such a manipu-lation. In intense laser-plasma interaction experiments this phenomenon is nowadays regularly observed.2gamma-ray bursts, jets of active galactic nuclei, radiation-dominated relativistic current sheets [35],and the modeling, e.g., of the flaring Crab nebula with high-frequency γ-ray emission [36].1.1 Intense laser-plasma interaction scenarioOne of the most attractive applications of short intense laser pulses is related to the development ofnew methods of accelerating charged particles. Conventional accelerators are expensive and haveapproached the maximum reasonable size3. These difficulties were evident already more than a half-century ago, yet before the invention of lasers, when V. I. Veksler, G. I. Budker and Ya. B. Fainbergproposed to use collective electric fields excited in a plasma to accelerate charged particles [37, 38,39]. The three key features of the collective acceleration are (i) the accelerating field on each par-ticle is proportional to the number of accelerated particles, (ii) the field is localized in space andsynchronized in time with the accelerated particles, (iii) in the acceleration process globally charge-neutralized bunches are produced.In this work we refer to a particular acceleration regime occurring in the interaction of laser lightwith dense targets. The laser pulse is assumed to be intense enough to ionize target material al-most instantaneously. The interaction typically leads to the generation of energetic electrons (moredetails are given in Section 1.2), which tend to escape from the target, producing the electrostaticcharge separation fields at target boundaries (”sheath” regions). These fields in turn accelerate ionsand drive the expansion of the plasma. Under suitable conditions, the combination of heating andradiation pressure, acting on the target, can drive nonlinear shock wave which provides an additionalacceleration mechanism. The basic mechanisms of ion acceleration, briefly described in Section 1.3,originate from the dominance of each of these effects. In experiments they generally coexist, leadingto a complex acceleration scenario [40, 41].Basic laser-plasma interaction parametersThe interaction geometry is characterized by the angle of incidence of the laser radiation on theplasma surface and by the field polarization, which can be linear, circular or elliptic. In the case oflinear polarization and for oblique incidence one distinguishes between S- and P-polarized waves.Let us consider a laser pulse of duration τL, with electric field amplitude E0, and carrier frequencyωL. The dimensionless parameter corresponding to the oscillation momentum of a single electron inthe electric field of the laser is related to the radiation intensity IL and the wavelength λL = 2πc/ωLbya0 =eE0mωLc=(IL[W/cm2]λ2L[µm2]1.35× 1018 W/cm2µm2)1/2. (1.1)When a0 1, the electron oscillation velocity vosc = eE0/meωL is small with respect to the speedof light in vacuum, and its oscillation amplitude rosc = eE0/meω2L is much shorter than λL. In thiscase relativistic effects can be safely neglected and the use of the non-relativistic classical approachto describe the interaction is justified. In the opposite limit, when a0 1, the laser-matter interactionmust be described in the framework of the relativistic theory. If the laser field amplitude a0 >3The main restriction on the accelerator size is imposed by the critical value of the electric field strength of the order of afew tens of MV/cm. If this value is exceeded an electric discharge forms on the acceleration chamber wall.3 INTRODUCTION√mi/me, with mi being the mass of plasma ions, the processes due to relativistic ion dynamics alsocome into play.The plasma density can be characterized by a dimensionless parameter α = ne/ncr, where ncr is theso-called critical density, defined as ncr = ω2Lme/4πe2. In a plasma with ne = ncr, the frequencyof the laser radiation equals the plasma frequency, ωL/ωp =√ncr/ne = 1. When ωL/ωp 1, thegroup velocity of the EM waves in such a plasma, vg = ∂ω/∂k = c√1− (ωp/ωL)2, is close tothe speed of light, and the plasma itself is called ”underdense”. If instead, ωL/ωp < 1 (”overdense”plasma), the EM wave can only penetrate into such plasma as far as the evanescence length δ ∼ c/ωp,and the efficiency of the laser field to plasma coupling strongly depends on the inhomogeneity scaleof the density. If the density profile is steep, the interaction takes place at the plasma-vacuum inter-face, where the laser radiation can easily extract the electrons from the plasma and accelerate themtowards the vacuum region. This process provides an effective mechanism of laser light absorption(see Sec.1.2). Moreover, at a steep plasma-vacuum interface the effective generation of high-orderharmonics is possible due to the so called ”oscillating mirror” mechanism or due to the nonlinearmotion of the electrons in the narrow region near the plasma boundary [42]. Several authors put for-ward the plasma-mirror concept as the promising tool to enhance the potential of attosecond sourcesat relativistic, i.e. a0 1, laser intensities [43]. When the density distribution is smooth, the plasmafrequency ωp(r) turns to be a function of the coordinates, and the processes that occur in the vicinityof the critical surface ne = ncr, in the region of the plasma resonance ωL = ωp(r), play a key role[44, 45]. In this region the EM wave resonantly excites strong electric field, localized in a tight areawhere the laser radiation is absorbed and fast particles are generated.For a0 > 1 the transmission of the laser pulse through a plasma is modified by relativistic effects andan overdense plasma with ne > ncr can become transparent. This phenomenon is known as ”rela-tivistic transparency”. Details depend on the laser and target parameters [46]. According to a simplemodel applicable in the relativistic regime to targets with thicknesses ∆ > λL, the value of criticaldensity increases as ncr → γncr with γ =√1 + a20/2 [47]. This is equivalent to assume4 an effectiveelectron mass equal to meγ. If the ∆ < λL, the transparency threshold also depends on ∆. For a foilwith a Dirac delta-like density profile [20, 48, 49] the nonlinear reflectivity can be calculated analyti-cally. The onset of relativistic transparency in this case is found to occur for a0 > πne∆/(ncrλL). Theefficient generation of carbon ion beams with average energies exceeding 10MeV/nucleon has beenrecently observed in the laser-plasma interaction experiment in the regime of relativistically inducedtransparency at the LANL Trident Laser [50].4The effective mass γme is a function of time and position since it depends on the local amplitude a0(r, t) of the EMfield.5 INTRODUCTION1.2 Collisionless absorption and hot electron generation:a sharp-boundary plasmaEven very intense laser radiation cannot penetrate into the solid density regions of the irradiated targetand the absorbed energy is transported to these regions mostly by hot electrons. Here, we define elec-trons as ”hot” if their energy exceeds the mean oscillation energy Eosc in the electric field in vacuum,E > Eosc = mec2(√1 + a20/2− 1). Hot electrons entering solid targets have been observed inmany experiments at high intensities for different interaction conditions [51, and references therein].They play a fundamental role in applications such as laser-driven photonuclear physics, fast ignitionof fusion targets, and new radiation sources in the extreme ultraviolet (XUV) domain. Moreover,in most of the laser-plasma experiments reported so far, acceleration of protons and heavier ions isdriven by hot electrons. Turning to the possible mechanisms of their production in the absorption ofintense light beams, at high intensities collisional absorption is ineffective because the collisional fre-quency and the inverse Bremsstrahlung rate quickly decrease with increasing electron energy and/orlaser intensity as νei ∼ v−3e , ve ' max(vth, vosc), vth ∼ T 1/2e , vosc ∼ (ILλ2L)1/2. Thus, in order toensure a sizable absorption at relativistic and subrelativistic intensities measured in experiments anddemonstrated in simulations [51], [52, and references therein], other effects of non-collisional naturehave to become active. The experimental and theoretical results on the absorption degrees collectedup to the year 20095 [51] in the regime of intensities ILλ2L ∈ [1018, 1021] W/cm2µm2 report absorp-tion degrees ranging from 5% to 95%. Absorption between 35% and 85% is reported at the constantirradiance of ILλ2L = 6 × 1018 W/cm2µm2. These huge differences can be partially explained bythe high sensitivity of hot electron generation to the laser and plasma parameters. For this reasonabsorption values and charateristics of the hot electron population are often taken into account in aphenomenological way [41], e.g., the hot electron distribution is often assumed to be Maxwellianwith a temperature Thot ' Eosc as a function of the laser intensity.In terms of basic physics collisionless absorption is understood as a result of an interplay of the os-cillating laser field with the space charge field produced in the plasma. In fact, under quasi-steadyconditions Poynting’s theorem averaged over one laser cycle reduces to〈∇S〉 = −〈j · E〉. (1.2)Here S =c4πE × B is the Poynting vector, which relates to the intensity as IL = 〈S〉. The currentdensity is j = −enev, where v is the mean electron flow velocity. Equation (1.2) decribes bothcollisional and non-collisional absorption. In the laser field with a mean frequency ωL the electricfield evolves in time as E ∼ sinωLt, the current density follows j ∼ cos(ωLt+ φ) and〈∇S〉 = −〈cos(ωLt+ φ) sinωLt〉 = −12sinϕ. (1.3)In collisional absorption the dephasing ϕ between current and driver field is due to the collisionsbetween electrons and ions,〈j · E〉collisional = ω2pνeiω2p + ν2ei|E|2 > 0.5See also [53], where the collection of hot electron temperature measurements obtained for subpicosecond laser pulsesup to the year 2000 is presented.6As it vanishes at νei = 0, any finite phase shift ϕ for high field intensities can only be of collisionlessorigin. For the optical laser field with IL ∼ 1019−1021 W/cm2 this origin is in space charge inducedby ∇v 6= 0 [54]. The compulsory dephasing between current and driver field, that determines thedegree of absorption, is provided by the longitudinal space charge field between the oscillating elec-trons and the quasistatic ions.Difficulties and complications arise when the degree of absorption has to be quantified. Absorptionmodels that have been proposed so far include vacuum heating [55], stochastic electron accelerationin fluctuating fields [56], wave breaking [57], linear and nonlinear Landau damping [58], anomalousskin effect [59], Brunel (sharp-egde) absorption [60], ponderomotive heating (longitudinal [61] andtrasverse [62]), j × B heating [63], surface plasmon excitation [64, 65] and ”laser dephasing heat-ing” [66]. These models are either based on simulations only or they are in disagreement with basicproperties observed in experiments and simulations [51, 53, 67]. For example, the anomalous skin-effect at high laser intensities yields an absorption of a few percent only. Absorption by excitation ofsurface plasmons requires corrugated [65] or grating targets [68], while in flat targets [64] the cou-pling is very delicate and unlikely to occur. The best know non-collisional candidate, the resonanceabsorption at oblique incidence6[44], is sensitive to the density scale length ne/|∇ne| because thedriving force is evanescent in the resonance region, but in the interaction of the high intensity laserpulses with good contrast ratios with solid targets, there is no time to form a preplasma in front of anirradiated sample that could couple to a resonantly excited plasma wave7.The first successful proposal able to explain a sizable absorption in a sharp boundary solid-densitytarget was the so-called j × B heating. The results of PIC simulations [63] have shown that appre-ciable absorption, target heating, and production of superthermal electrons at any density above ncrfor normal laser pulse incidence is due to the the Lorentz force, which induces non-resonant electronoscillations at frequency 2ωL normal8 to the target surface. However, at that point no attempt wasmade to explain how the observed absorption comes into play. A notable step in understanding high-power collisionless absorption (offering within limits of a laminar electron flow dynamics the firstphysical explanation of the j × B heating at 2ωL) was made by F. Brunel [60]. Brunel proposed anelectrostatic model of electron heating at a step-boundary plasma and has shown that the resonanceabsorption concept [44] could be adapted to steep, highly overdense plasma profiles. In this casesignificant absorption is achieved under oblique incidence despite total absence of plasma resonanceat ωL = ωp and no possibility for a plasma wave to propagate into a shallow preplasma in front of thetarget. Until recently Brunel’s model has been the only successful attempt to explain efficient col-lisionless absorption on physical grounds under a minimum of assumptions and simplest geometry[69].An accurate analysis of Brunel’s model [69] has revealed essential aspects of collisionless absorp-tion, such as prompt generation of fast electrons during one laser cycle or a fraction of it underP -polarization, insensitivity of absorption with respect to the target density well above ncr, simplic-ity, and robustness. The splitting of the electron energy spectrum into the two groups of slow andof fast electrons, observed in experiments and simulations, also comes about in the most natural6The resonance absorption at oblique incidence consists of the direct conversion of laser light into an electron plasmawave resonantly excited at the critical electron density where ωL = ωp.7It is worth mentioning that in most high intensity experiments the main interaction pulse is preceded by prepulses whichcause early plasma formation and expansion so that the short-pulse interaction does not occur with a sharp-boundary,solid-density plasma.8Electron motion parallel to the laser field defines a perfectly reversible process. Absorption is due to the motion perpen-dicular to the target surface induced by the magnetic field B.7 INTRODUCTIONFigure 1.1: Collisionless absorption – (a) Electron spectrum f(E) from Brunel’s nonrelativistic model [60],(b) – F (E) =∫ E0 f(E′)dE′. Energy E in units of mean oscillation energy in vacuum, energy cutoff is atE = 9.1Eosc. (c)-(d) Electron energy spectra f(E) in logarithmic scale obtained in 1D simulations of theinteraction of laser pulse irradiating a dense plasma slab (initial enectron density ne = 100 ncr) at 45◦ incidenceangle. The spectra are calculated at t = 30 × 2π/ωL and t = 45 × 2π/ωL cycles after the beginning of theinteraction. Vertical dashed lines mark the mean oscillation energies. The hot electrons follow a Maxwelliandistribution. (e) Total absorption (triangles) and the absorption by electrons (blue diamonds) as a function ofa0 obtained from 1D PIC simulations. Figure is adapted from [54].way. However, in the geometry of Brunel’s model, PIC simulations unambiguously show that thehot electron spectrum is broad and can be reasonably well described with an exponential functionf(Ehot) ∝ exp(−Ehot/kBThot), extended up to a cutoff of a few times kBThot, (see Fig. 1.1(c)-(d)showing the energy spectra of the electrons for different laser pulse amplitudes), and that the fractionof hot electrons is a few percent only. On the contrary, the spectrum of Brunel electrons is non-Maxwellian with a pronounced maximum at E = 9.1Eosc, followed by a sharp cutoff, Fig.1.1(a)-(b),and the resulting percentage of fast electrons is nearly two orders of magnitude bigger than the val-ues obtained in PIC simulations. The scaling of the absorption coefficient with intensity in Brunel’smodel is A ∼ (ILλ2L)1/2. It reaches unity at incidence θ = 86◦ (no solution exist beyond) and atintermediate angles of incidence it is considerably smaller, than the values measured in experiments[70] so that the absorption in Brunel’s mechanism is inefficient below very oblique incidence. Fi-nally, the average energy of the hot electrons in Brunel’s model is proportional to the intensity, whichis dramatically different from the experiment, performed under Brunel’s conditions [70], as well asfrom relevant results in the literature [71, 72, 73, 74, 75, 76, 77].The further investigation of statistical ensembles of electron orbits, extracted from 1D PIC simu-lations [54], showed that the absorption process is localized at the ion-vacuum interface and in theskin layer. Contrary to the laminar dynamics assumed in Brunel’s model, the electron flow becomeshighly turbulent. Single electrons enter into resonance with the laser field thereby undergoing a phaseshift which causes orbit crossing and breaking of Brunel’s laminar flow. This anharmonic resonance[77] is responsable for the formation of a Maxwellian tail in the electron energy spectrum. Mostremarkable results of the study reported in [54] are the Brunel-like spectra of hot electrons at therelativistic threshold, the minimum of absorption, Fig.1.1 (e), at IL ∼ (0.3 − 1.2) × 1021 W/cm2in plasma targets of ne ' 100 ncr, the drastic reduction of the number of hot electrons in the sameintensity range and their reappearance in the highly relativistic domain.As for the most controversially discussed subject in the relevant literature, the scaling of the hot elec-8tron energy with the laser intensity, early 1D PIC simulations [71] obtained a scaling Ehot ∼√IL,which has been reconfirmed by independent simulations [78] and by experiments [79], but seems tobe in contrast with other experimental data [72] and more sophisticated analysis [75]. Analogousscaling laws have been proposed by other authors [70, 73, 76]. Nevertheless, there is no convergencetowards a definite scaling yet [41, 49, 54, 80].Note that for S-polarization or/and normal incidence the laser electric field vector does not have acomponent perpendicular to the target surface. However, for high intensities the magnetic force termv × B becomes important and may drive the electron oscillation along the density gradient also fornormal incidence. In this case the generation of hot electrons will occur twice per laser period ifthe polarization is linear. For circular polarization (CP) and normal incidence the generation of hotelectrons might be strongly suppressed because the perpendicular to the surface oscillating j × Bcomponent vanishes for CP (see also discussion in Sec. 1.3). A more detailed analysis shows thatelectron heating is quenched when the ellipticity exceeds some threshold value [81, 82].Although we believe that the reported phenomena will survive in 2D and 3D also, the 2D simulationsreveal additional effects, such as the deformation of the plasma surface due to ”hole-boring” drivenby the radiation pressure, which changes the local angle of incidence [71]. This effect leads to anincreased absorption and provides a dynamic effect of collimating the electron flow inside the target[83]. A similar effect takes place in microcone targets [84, 85, 86, 87, 88, 89]. Absorption and fastelectron generation in solid density targets are also sensitive to sub-wavelength density gradients andstructuring of the interaction surface [90]. However, in order to exploit this effect experimentally,e.g., to enhance and optimize the ion acceleration, the use of high contrast laser systems is strictlynecessary.9 INTRODUCTION1.3 Ion accelerationThe progress in laser technology has led already to light sources delivering pulses of focused inten-sities exceeding 1021W/cm2 [91]. Even higher intensities are foreseen with the new generation oflaser sources expected to come into operation in the nearest years [6, 7]. When such a pulse impingeson a solid target9 both electrons [92] and ions [93, 94, 95, 96] can be accelerated up to hundreds ofMeV or even GeV energies.The first succesful experiments on laser acceleration of protons with beam-like properties and multi-MeV energies were reported almost two decades ago [26, 96, 97]. Since then multi-Mev proton andion beams with ultrashort burst emission, high brilliance and low emittance have been obtained over awide range of laser10 and target parameters. The recently reported maximum proton energy approach100 MeV [98]. Widely accepted basic scenarios of ion acceleration are the “target normal sheathacceleration” (TNSA) (see e.g. [96, 97, 99, 100] and references therein) and the “radiation pressureacceleration” (RPA) [101]. In the former, laser-driven fast electrons escape from the rear side of athin foil and drag ions behind, in the latter electrons and ions are accelerated together. Under suitableconditions, the combination of heating and radiation pressure can drive nonlinear shock waves andlead to ”collisionless shock acceleration” (CSA).Most of the observations of ion acceleration cannot be fully explained in terms of TNSA, RPA orCSA. Depending on the laser and target parameters, the acceleration mechanism may be of hybridnature and combine aspects of all the three ”basic” acceleration concepts. In addition, in the litera-ture there are many proposals of particular schemes, employing, e.g., complex target configurations.Although these proposals are ordinarily supported by numerical simulations, only very few of themhave been realized in the laboratory so far. Extensive surveys on the underlying physical principleof the main ion acceleration mechanisms along with related experimental indications are given in thereviews [40, 41, 102, 103].Target Normal Sheath Acceleration – The most successful, although qualitative, description of theTNSA [99] was proposed to explain the first experimental results on multi MeV proton accelerationfrom solid targets [26, 96, 97]. In the intensity regime of relevance, i.e. for ILλ2L > 1018 W/cm2µm2,the laser pulse can efficiently accelerate electrons. The average energy of the hot electrons is of theorder of MeV, their collisional range is much larger than the target thickness, so that they can propa-gate to the rear side of the target and drive the acceleration of ions from surface layers via the chargeseparation field. While only a limited number of very energetic electrons will definitely leave thetarget, most of hot electrons will be held back within it by the space charge. These electrons willform a sheath of a Debye length from the initially unperturbed rear surface. According to the modeldeveloped in [97], the accelerating sheath field Esheath is defined by the temperature Thot and densitynhot of the hot electrons and is given byEsheath ∼kBThoteΛD∼√nhotkBThot.If a scaling Thot ∼√ILλ2L [53, 60, 69] and a roughly 10% fractional absorption in hot electrons areassumed, the strength of the sheath field will be Esheath ∼ 1010V/cm for ILλ2L = 1019 W/cm2µm2.9In this work we refer to ion acceleration occurring in the interaction of intense laser light with the solid targets, wherethe electron density significantly exceeds the critical density ncr.10Experiments on ion acceleration use ultrashort (few tens of fs) laser systems, typically based on solid state Ti:Satechnology, as well as high energy picosecond laser systems (Nd:glass). Some experiments employ also C02 lasers.10Figure 1.2: Hole-boring and light sail acceleration by a circularly polarized laser pulse – (a) Snapshots ofthe ion density from a 2D PIC simulation for a linear density profile rising from 0 to 10ncr over 2λL. Thewhite dotted line indicates the initial position of the ne = ncr surface. The arrow indicates the location of theshort dense ion bunch. The divergence of the bunch is about 4◦. Picture is adapted from [110]; b) Snapshots (att = 75 fs and t = 100 fs) of ion density from a 3D PIC simulation of a thin hydrogen foil ∆ = 0.4λ, irradiatedby a tightly focused CP laser pulse with IL = 3.4× 1019 W/cm2. Pictures are adapted from [126].This large field will backhold most of the escaping electrons, ionize atoms at the rear surface of thetarget, and accelerate ions up to energy in the MeV range. Under appropriate conditions, includingthe right combination of target thickness and laser pulse duration, the energy of the ions can be en-hanced by the recirculation of the hot electrons through the target during the ion acceleration process[104]. The energy of the TNSA accelerated ions scales with the laser pulse intensity roughly11 as∼√IL. Concerning the properties of the accelerated ion beam, in both simulations and experimentsthe energy spectrum is found to be quasi-thermal with a cutoff at a maximum energy. The angulardistribution of the ions is also quite broad, the average divergence is in the range 10◦-20◦ with thehigh energy ions having smaller divergence. Note that while TNSA can accelerate any ion speciespresent in the surface layer, in most experimental settings it results in preferential acceleration of lightions from contaminant layers, with protons being the dominant component of the TNSA-acceleratedion beams. Removing hydrocarbon contaminants, e.g., by resistive heating [105], may suppress thedominant proton acceleration. It should be noted that even though in ultra-high contrast laser pulsesa symmetric acceleration from the front side (in backward direction) has also been observed [107],the front-TNSA efficiency is drastically reduced if the front preplasma formation is not inhibited.Radiation Pressure Acceleration – The possibility of an ion contribution originating from the frontsurface of the target was considered already in the first reported measurements of MeV protons ac-celerated in the forward direction [96, 97]. The mechanisms of acceleration, alternative or comple-mentary to TNSA and efficient in the front target regions, have been extensively investigated. Theincidence of an EM wave of intensity IL on a plane target leads to absorption of its linear momentumand produces at the front surface a pressure Prad. The maximum of this pressure (for normal inci-11The power-law scalings of the ion energy ∝ IνL, with ν ∈ [1/3, 1] depending on the laser pulse duration have beeninferred by reviewing the data from different laser laboratories [40] and performing parametric studies on a single lasersystem [106].11 INTRODUCTIONdence) is obtained for an ideally reflecting target surface and reads Prad = 2IL/c. For sufficientlyhigh intensity this radiation pressure may overcome the thermal pressure and push an overdense tar-get inward, leading to the steepening of the density profile, bending of the surface and recession of theinteraction surface, see Fig.1.2a). This process is called ”hole-boring” and the velocity of the surfaceis known as the ”hole boring” (HB) velocity vHB. It can be estimated, assuming the balance betweenthe flows of EM and kinetic momentum at the surface. For vHB/c 1 the resulting expression [108]reads vHB =√IL/ρc, where ρ ' nmi is the mass density12. The balance of mass and momentumflows at the moving surface implies that there must exist a flow of ions ”reflected” from the recessionfront at twice vHB. This results in an ion population with energy per nucleon equal toEHB =12mp · (2vHB)2 = 2mpILρc= 2mec2ZncrAne· a20. (1.4)Here Z andA are the charge and mass numbers of the plasma ions. Since IL ∝ a20, the scaling of EHBwith the intensity (1.4) is more favorable than the√IL scaling for TNSA. The scaling with the den-sity EHB ∝ n−1e implies, however, that higher energies are achievable if the target density is reducedto relatively small values slightly exceeding ncr. For values typical for solid targets (ne > 100 ncr)only modest energies may be obtained. Combining low-density targets with laser pulses at fore-seeable intensities IL > 1022 W/cm2 may allow reaching > 100 MeV energies, as investigatedtheoretically [111]. Note that for moderate laser intensities the hole-boring efficiency is strongly re-duced because of the generation of fast electrons. They decrease the reflectivity of the plasma andproduce a strong thermal pressure which counteracts the radiation pressure. Only at high intensitiesIL > 5 × 1021 W/cm2 RPA starts to dominate over TNSA for any laser polarization, as was shownfirst by PIC simulations [101]. A dominance of RPA over TNSA may be, however, obtained at muchlower intensity if a laser pulse with circular polarization at normal incidence, instead of linear polar-ization (LP), is used [108, 110]. In fact, under such conditions the acceleration of ”fast” electronsat the laser-plasma interface is suppressed, ruling out TNSA. The suppression of fast electrons forCP can be understood by recalling that models of electron acceleration at a sharp plasma surfacesrequire the driving force having an oscillating component along the density gradient. For normalincidence such component is given by the part of the term j × B, ocillating at 2ωL, which vanishesfor CP. The numerical studies [108] confirmed this prediction and showed that in the interaction ofa CP laser pulse with IL < 1019 W/cm2 all the ions in the skin layer get accelerated and the fastestones produce a very dense bunch directed in the forward direction. In Fig.1.2a) a single, ultrashortion bunch generated in the interaction of a CP laser pulse of intensity IL = 5.5 × 1018 W/cm2 andτL = 33 fs duration with slightly overdense plasma is shown. The ion acceleration initially occursnear the cutoff layer where ne = ncr and produces a dense, ni > 10 ncr, ion bunch with a narrowenergy spectrum.The first successful experiment of HB acceleration has been performed at the Accelerator Test Facil-ity of Brookhaven National Laboratory using a CO2 infrared laser with λ = 10µm (ncr ' 1019cm−3)and IL ' 6.4 × 1015 W/cm2 (corresponding to a0 ' 0.5) and a hydrogen jet target with the den-sity a few times ncr. In this experiment a spectral peak at energy E ' 1.2 MeV have been obtained[112]. The hole-boring velocity has been measured later in a similar experiment [113]. Experimentalevidence of HB acceleration in solid targets is at present less clear [41]. But signatures of such ac-celeration have been associated with the interferometric observation of collimated plasma jets in PW12The relativistic expression [109] reads vHB/c =√IL/ρcc+√IL/ρc.12laser pulse interactions with a few µm thick targets [114].HB-PRA applies to targets with thicknesses much larger than the skin layer, in which ion accelerationoccurs. If the target is thin, such that ∆ vHBτL, the HB front reaches the rear side of the targetbefore the end of the pulse duration and the whole target is accelerated13, Fig.1.2b). In this case thelaser pulse can further accelerate ions since they are not screened by the background plasma anymore[115, 116, 117]. This regime in its simplest form can be modeled as a thin mirror, boosted by radiationpressure, and is called ”light sail” (LS) acceleration. The characteristic features of LS-RPA, such ashigh conversion efficiency in the relativistic limit and the possibility to reach very high energies withforeseeable laser and target technologies, may be qualitatively understood within the simple model ofa perfectly reflecting mirror accelerated by a normally incident plane wave. Due to the Doppler effectthe frequency ωL of each incoming photon reflected by the mirror moving with the velocity V = βcin the laboratory frame is downshifted to ω′, according to the relation ω′ = ωL(1 − β)/(1 + β), sothat almost all of the energy h̄ωL of the photon is delivered to the target in the limit β → 1. Since thenumber of photons is conserved for a perfect mirror, a complete conversion of the wave energy intomechanical energy is obtained for β → 1. For a given surface density ρ ·∆ of the sail, its reflectivitydrops down due to the onset of relativistic transparency for the field amplitude a0 > πne∆/ncrλL.This effect suggests a0 = πne∆/ncrλL as an optimal condition for LS acceleration [48, 118, 119].The interest in the LS-RPA acceleration regime was greatly stimulated by the results of the 3DPIC simulations of a thin (∆ = λL) dense (ne = 49 ncr) hydrogen foil boosted by an ultra short(8×2π/ωL) ultra intense (IL = 1.4×1023 W/cm2) laser pulse [101]. These simulations demonstratedthat most of the ions in the foil were accelerated coherently up to energies E ∼ 1.5 GeV and that thetemporal dependence and typical values of the ion energy were well described by the LS model. Latertheoretical studies addressed the issue of target stability during LS-RPA. 2D [110, 117, 120] and 3D[121] PIC simulations in the ultraintense regime showed a bending instability, Fig.1.2b), which hasbeen interpreted to be of the Rayleigh–Taylor (RT) type. Some experimental evidence of radiation-pressure-driven RT instability in thin targets has been also reported [122]. The first experimentalinvestigations of the LS-RPA regime [123] showed several issues, such as non monoenergetic ionspectra, weak dependence of the energy cutoff on polarization, and non-uniformity of the acceleratedbeam. An experiment performed recently on the GEMINI laser with a short intense pulse (τL = 45 fs,IL = 6× 1020 W/cm2) and ultrathin (∆ = 0.01− 0.1 µm) amorphous Carbon targets has given ev-idence of much higher Carbon ion energies for CP, where the ions energy cutoff 25 MeV has beenobserved. For LP the cutoff energy has been measured at 10 MeV [124].Although some authors have proposed the RPA of a thin foils as a way to generate high-energy pro-ton beams, this approach is most interesting for the acceleration of higher-Z ions. While it seemstechnologically unfeasible to have an ultrathin target made of hydrogen only, in a target made of mul-tiple species all the ions will be accelerated to the same velocity, resulting in higher energies for theheavier species. However, an analysis of the LS dynamics beyond the rigid mirror model [48] showsthat for pulses of finite duration only part of the target ions are accelerated as a monoenergetic bunch,and under proper conditions this part may contain only the target protons. In addition, the formationof a region of an accelerating field ahead of the target related to ”leaking” transmission of the laserpulse may also accelerate protons [125].Collisionless Shock Acceleration – This mechanism of ion acceleration, often associated with as-trophysical scenarios, in laser-plasma interaction is related to the collisionless shock generated inthe plasma due to the HB process. Its investigation is still relatively preliminary compared to that13The transverse size of the accelerated region corresponds roughly to the focal spot-size of the laser pulse.13 INTRODUCTIONFigure 1.3: Electrostatic shock with reflected ions – (a) The distribution of ions on the (x, vx) phase plane.An electrostatic shock wave with M = vsh/cs ' 2.7 propagates to the left. Strong reflection (left frame) leadsto the broad ion spectrum. True shock is formed in the presence of a sufficiently warm ion distribution and thesteady ion reflection occurs. 1D PIC non relativistic simulations have been performed by A. Hanusch.of TNSA, though collisionless shock experiments using laser produced plasmas have a long history[127]. In [128], a collisionless shock was produced by irradiating a solid target mounted inside apinch device filled with low-density plasma. In [129], the supersonic flow of a laser ablated plasma,which got past a stationary obstacle, was investigated. Several recent experiments [130] have detailedthe propagation of collisionless electrostatic shocks driven by the laser ablation of thin metal foils in alow density plasma. As a mechanism of ion acceleration, potentially leading to the generation of ionbeams with exceptionally narrow spectral distribution in superintense laser interaction with µm-scaleplasma targets, CSA was proposed on the basis of PIC simulations [131, 132]. These simulationshave demonstrated the generation of shock waves with sonic Mach number M = vsh/cs = 2 − 3and the production of energetic ions of velocity ∼ 2vsh. The sound speed cs was estimated ascs =√ZThot/Amp, with the temperature of the hot electrons being Thot ' Eosc. The charge sepa-ration field at the shock front acts as a potential barrier for the ions in the plasma, accelerating someof them by reflection up to twice the shock velocity. If the velocity of the shock is kept constant, thisresults in monoenergetic ion beam with energy per nucleonECS =12mp · (2vsh)2 = 2mpM2c2s = 2ZThotA M2. (1.5)In the experiment reported in [133], CSA has been indicated as the mechanism responsible for theproton acceleration in the interaction of a CO2 laser pulse with a gas-jet target. A proton beam with∼20 MeV energy per particle and only 1% energy spread and low emittance has been obtained.The number of accelerated protons in this experiment was, however, approximately three orders ofmagnitude lower than the number of protons accelerated via the HB mechanism in similar laser andtarget conditions [112]. It should be noted here that the number of CSA accelerated ions must below if one aims for a monoenergetic ion beam (see Fig. 1.3). In fact, the particle reflection stronglydepends on the amplitude of the shock (overshoot of electrostatic potential), which depends on theshock speed [134, 135, 136] in the first place. The reflected particles alter the shock amplitudeand speed (momentum conservation), and thus the reflection threshold, causing the ion spectrum tobroaden. The demonstration of CSA with optical lasers, which allow for a much higher intensity butrequire for the efficient acceleration higher target density, has been achieved so far by simulationsonly. The recent 3D PIC parametric studies [137, 138] have shown that the control of the targetdensity profile is crucial to achieve multi MeV protons with PW laser systems.14The extensive research interest in laser-accelerated ion beams is stimulated by their exceptional prop-erties, including high brightness, high spectral cutoff, high directionality, laminarity, and ultrashortduration. These properties make laser-driven ion beams suitable for any technological applicationwhich requires an extremely localized energy deposition [139, 140]. Moreover, if significant im-provements over present laser plasma accelerators were achieved, they could become attractive formedical purposes both for oncological ion beam therapy [26, 28] and short-lived radioisotope produc-tion [27, 102, 103, 141]. Nowadays ions in the energy range of hundreds of MeV are routinely usedfor cancer treatment in several14 medical facilities in the world. In the medical centers for hadron-therapy, the ion beams with the required parameters are obtained with conventional accelerators. Anecessary and the most expensive element in such centers is the gantry system, i.e. a huge device forthe irradiation of the lying patient from different directions. Compared to conventional acceleratorsof charged particles the laser ion acceleration would be more advantageous due to compactness andrelatively low cost of the accelerator itself, and due to the possibility to develop an optical scheme totransport and rotate photon beams instead of magnetic transportation lines and gantry for high energyions [28]. However, before plasma based accelerators could compete with traditional accelerators inthis field, several important improvements seem to be necessary. In particular, the cutoff energy ofthe laser-accelerated ions is still too low to be of medical interest, and their spectra are too broad. Butthe major difficulty is related to the reliability of the plasma based acceleration systems leaving muchto be desired. In this context the application of long wavelength CO2 laser as a driver enables the useof a flowing gas target, which may simplify high repetition rate operation compared to a solid target,which needs to be mechanically replaced or displaced in a very short time. However, in order to reachthe ion energy range suitable for medical applications, the CO2 laser intensity should be increased bytwo orders of magnitude. Some development projects and required technical advances are discussedin the literature [143].The low emittance of laser-acceleted protons favored their application for imaging [144]. The broadenergy spectrum combined with the short duration of the proton bunch generated via TNSA has en-abled the development of a single-shot detection of EM fields generated in laser-plasma interaction,with picosecond temporal resolution [145]. This innovative application has provided much valuableinformation on the nonlinear dynamics of plasmas, including the proton acceleration mechanismitself [146].14As of 2014, there were 40 centers worldwide and about the same number under construction. Now (November 2019)about 100 centers operate and more than 30 are under construction [142].15 INTRODUCTION1.4 Droplets in intense laser fieldWhile laser-based particle acceleration shows an unprecedented efficiency as far as particle energyper acceleration length is concerned, the beam quality is not yet sufficient for many applications.In order to be useful specifically for medical purposes the ion spectrum should be monochromatic,tunable, and reproducible. For any useful application one needs moreover a high repetition rate anda debris free target system. In this context, finite-size mass-limited targets are of particular interestbecause they can be used to confine the absorbed energy, obtaining thus higher temperatures. In awell know example the interaction of ultrashort moderate intensity laser pulses with subwavelengthclusters resulted in the acceleration of ions up to energies sufficient to drive deuterium–deuteriumnuclear fusion reactions [147]. An obvious limitation on the use of such clusters as ion sources isthe isotropic ion emission and the resulting low brilliance [41]. Enhancement of the conversion effi-ciency and proton cutoff energy has been observed recently in isolated droplets [148, 149, 150] andin foils with limited (tens of microns) transverse extension [151]. In a recent experiment [152], aplasma produced from a 1 µm diameter levitated plastic sphere positioned with micrometer precisionin the focus of a PW laser pulse has been used to generate a single proton bunch with narrow energyspread. Central energies for consecutive laser shots were found in the range from 20 to 40 MeV.Among various technique to generate ultrashort electron bunches, which in turn have potential ap-plications in attosecond electron diffraction experiments or as sources of attosecond X-ray pulses,mass-limited targets in general and micro-droplets in particular may offer a number of advantages.Attosecond control of the collective electron motion and directional emission from isolated dielec-tric nanoparticles with phase stabilized two-cycle laser pulses have been demonstrated experimen-tally, supported by the microscopic analysis of the electron emission [153]. Electron accelera-tion from micro and nanodroplets has been extensively investigated numerically. Narrow electronbunches have been observed in 3D PIC simulations of the interaction of a short (τL = 30 fs), intense(IL = 2 − 8 × 1018 W/cm2), linearly polarized plane-wave laser pulse with helium micro-droplets[154]. The spatial width of the generated bunches is much smaller than the wavelength, thereforetheir temporal structure is already in the attosecond domain. The bunches are emitted each half-cycle,Fig. 1.4, under plus or minus a certain angle in the polarization plane. The preferred electron emissionangles and the kinetic energies in the multi-MeV range arising due to local field enhancements at thedroplet surface can be calculated using Mie theory. A parametric study with respect to the droplet sizefor a similar setup but much shorter laser pulses (τL = 5 fs) has been performed in [155]. Significantdisagreement with Mie theory has been reported for electron bunch emission from droplets, whoseradii R satisfy the condition δr < R < 10δr, where δr =√γδe is the relativistic skin depth. This ef-fect has been attributed to the induced relativistic transparency. PIC simulations with the synchrotronradiation module incorporated [156] demonstrated the generation of bright fs X-rays by irradiating aliquid helium micro-droplet with an intense CP laser pulse at intensity IL = 6× 1019 W/cm2 [157].The possibility to obtain tunable X-rays with required brightness and energy by an appropriate choiceof the laser and droplet parameter has been discussed.Heating mechanisms and ionization in laser-illuminated clusters have been systematically investi-gated in the last two decades [147, 158, 159, and references therein]. The ionization dynamics inwavelength-sized targets irradiated by LP laser pulses of intensity IL ' 5 × 1017 W/cm2 has beenaddressed in [160] by parametric 3D PIC simulations with field ionization included, and the frac-tional ionization degrees for various droplet and laser parameters have been obtained. The results ofthe simulations have shown that a strong near-infrared or optical laser pulse interacting with an ini-16Figure 1.4: Helium droplet in intense laser fields – Kinetic electron energy in MeV (a) and density (b) contourplots of the He droplet in a laser pulse of intensity IL = 8×1018 W/cm2. In the bunch ejected around the pulsemaximum, the maximum electron energy is∼ 6 MeV. (c) - Spectral distribution of the radiation, emitted by thetwo bunches in the black rectangle in frame (b). (d) - Angles of the electron emission vs droplet radius R: PICsimulations (colored symbols) and Mie results. Lengths are normalized to λL. Frames (a) and (b) are adaptedfrom [160].tially neutral, wavelength-sized spherical He-droplet may generate a charge density distribution thatneither is homogeneous throughout the droplet nor created only within a thin skin layer at the surface.The time-dependent field and density distributions inside the droplet are not accessible to standardMie theory and fall into the realm of extreme nonlinear optics. At higher laser intensities a qualita-tively similar ionization dynamics is expected for higher-Z materials. The resulting inhomogeneouscharge distribution may be probed via scattering of short-wavelength radiation [161].17 INTRODUCTION1.5 Dynamics of charged particles and radiation in laser-plasmainteractions at ultrahigh intensities: collective effectsWithin the immense research field of the interaction of extreme fields with plasma, elementary par-ticles and even with vacuum [2, 162, 163], a new area of classical and quantum dynamics in theradiation-dominated regime has recently emerged. Here we refer to the radiation-dominated regimewhenever radiation of photons by accelerated elementary particles significantly affects the individualdynamics of these particles, as well as the collective dynamics of the plasma [164, and referencestherein]. Apparently, the distortion of the particles’ dynamics influences the process of radiation it-self, making the whole problem self-consistent and, due to the extreme intensities of the applied laserfields highly nonlinear and even nonperturbative. Until recently, the theoretical studies of elementaryparticles and plasma strongly coupled to their radiation were remote from possible experimental veri-fication of the related effects and remained on the level of academic interest. With the new generationof laser sources expected to come into operation in the nearest years [3, 4, 5, 6, 7] or in the foreseeablefuture [165, 166, 167, 168], this research became of topical interest and already resulted in a numberof new fundamental findings. Below we briefly mention the reasons, justifying a constantly growinginterest in effects of radiation friction (RF)15 in the interaction of ultra-intense laser fields with matter.Firstly, the influence of RF on the dynamics of individual charged particles or plasma becomes moresignificant as the laser intensity increases. Therefore, a number of laser-plasma phenomena, as wellas single-particle effects for ultra relativistic electrons in strong EM fields, have to be revisited tak-ing RF into account. In the last 15 years, this reconsideration has resulted in extensive work on RFeffects in nonlinear Thomson- [170] and Compton scattering [171], in Raman spectra [172], in ra-diative trapping [173], in electron and ion acceleration in vacuum [174], and in X-ray emission fromplasma targets [175]. Recent experiments based on Thomson scattering of a superintense laser pulseby a high-energy electron bunch [176, 177, 178] have provided a first (albeit weak) evidence fordeviations from classical predictions in the radiation spectrum. In the modeling of the experiments,a so-called ”semiclassical” approach based on a modification of the Landau-Lifshitz RF force [179]appeared to provide a better agreement than a ”quantum” approach where the emission of radiation isdescribed stochastically. This finding has been explained by the possible breakdown of approxima-tions underlying the quantum model, as also suggested by a different experiment [180], and attemptsare being pursued to overcome the approximations in the quantum model [181, 182, 183]. This sce-nario suggests that identifying different RF signatures and test their sensitivity to the onset of quantumeffects is important in order to improve the theoretical and numerical modeling.Second, at present the understanding of RF is far from being complete, because some principal ques-tions of the theory of this phenomenon are still not answered. Even within the classical treatment,several not exactly equivalent expressions for the RF force are discussed [184]. The situation becomeseven more complicated for very energetic particles and very high field strengths when classical elec-trodynamics do not apply. One has to admit that presently, a comprehensive QED theory of RF doesnot exist.Finally, there are qualitative effects whose very existence results from the RF force. Such phenomena,if observed experimentally, can provide an unambiguous proof of the crucial role RF plays at highintensities of laser fields and high energies of plasma electrons. In a recent paper [185], it has been15RF arises from the back-action on the accelerated electron of the EM field generated by the electron itself [169].18Figure 1.5: Inverse Faraday effect driven by radiation friction – (a) The distribution of ion density at t =14 × 2π/ωL ' 37 fs and (b) axial magnetic field Bx (normalized to B0 = 1.34 × 108 Gauss) at t = 32 ×2π/ωL ' 100 fs after the beginning of the interaction (left: without RF, right: with RF included). Thedistributions are shown in the (x, y) plane, very similar patterns are observed in the (x, z) plane. The CP laserpulse of IL = 7.6× 1023 W/cm2 (a0 = 600) is incident along the x-axis from the left side on the target withdensity ne = 1.55× 1023 cm−3. The thin black lines denote the boundaries of the target. The coordinates arenormalized to λL. Picture on frame b) is adapted from [185].shown that RF induces a specific form of the inverse Faraday effect (IFE), i.e., the generation of mag-netic fields due to absorption of EM angular momentum into a plasma. From a classical viewpoint,in the presence of dissipative effects, the EM angular momentum carried by a circularly polarizedlaser pulse is transferred to the plasma electrons, which acquire a torque and produce an azimuthalcurrent and an axial magnetic field. From a quantum viewpoint, the RF-induced absorption of an-gular momentum is due to the annihilation of N 1 polarized laser photons needed to generate asingle high-energy photon so that an angular momentum amount equal to (N − 1)h̄ ' Nh̄ (h̄ beingthe photon spin independent of the photon energy) is transferred to the orbital motion of electrons.In a regime of interaction with high-density plasmas, the conversion efficiency of laser energy intoincoherent radiation may be a few ten percent, which results in axial quasistatic magnetic fields upto gigagauss values at intensities ' 1024 W/cm2, as observed in 3D PIC simulations with classicalRF included [185], see. Fig.1.5. Such huge magnetic fields, besides affecting the plasma dynamics,provide an unambigous signature of RF effects and may be measured by polarimetry methods.Based on the equations of macroscopic electrodynamics and conservation laws, a description of IFEin the field of an intense laser pulse [185] predicts the maximal amplitude of the quasistatic longitu-dinal magnetic field excited on the axis of a laser beam to be linear with respect to the laser magneticfield amplitude BL and to the fraction of the laser energy η associated with the irreversible transfer ofangular momentum from the laser field to the plasma:Bxm = Cηa0B0 ≡ CηBL. (1.6)Herea0 =E0B0, B0 =mcωLe= 1.34× 108 G (1.7)are the dimensionless laser field amplitude and the characteristic magnetic filed, respectively. Thedimensionless coefficient C is determined by the shape of the laser pulse envelope and has typicalvaluesC ' 0.1÷0.2. The structure of Eq. (1.6) is consistent with the general theory of IFE [186, 187].The absorption coefficient η readsη =ωLLabsUL, (1.8)19 INTRODUCTIONwhere Labs is the angular momentum absorbed by the plasma and UL = Aλ3La20B20 is the energystored in the laser pulse. The dimensionless coefficient A is determined by the pulse duration, timeenvelope and focusing. Equations (1.6)–(1.8) are insensitive to a particular physical mechanism of theangular momentum transfer. In particular, Eq. (1.6) applies independently of the impact of quantumeffects on the plasma dynamics. In the high-field regime, radiation by plasma electrons is the onlymechanism for energy absorption, thus ωLLabs = Urad where Urad is the radiation energy emitted bythe electrons, and Eq.(1.8) readsη ≡ ηrad =UradUL≡∫d3r∫dtPrad(r, t)ne(r, t)UL≤ 1. (1.9)Here, Prad is the emission power for a single electron moving under the action of the local EM fieldwith a given time- and space-dependent envelope, and ne is the electron density. The conversionefficiency ηrad is the key value for the determination of magnetic field amplitude.In [185], the scaling of ηrad with the laser intensity agreed reasonably with the results of 3D PICsimulations of the laser-plasma interaction, up to intensities approaching 7 × 1023 W/cm2. Beyondthis limit, the model predicts values of ηrad > 1 because neither the modification of the radiatingelectron trajectories due to RF nor the depletion of the laser pulse is taken into account. In addition,at those intensities the classical description becomes questionable and quantum effects are expectedto become relevant.The further development of the classical model for the calculation of radiation losses [188] included aself-consistent picture of the IFE, accounting for RF effects on the electron motion via the Zeldovichmodel [189], for the plasma motion driven by radiation pressure in the hole boring regime, and foran inhomogeneous distribution of the laser intensity. This improved model predicts values of ηrad ingood agreement with the PIC simulations and it is found to be robust with respect to assumptionson the particular shape of electron and ion density distributions in the radiating layer despite ofits analytic simplicity. The effect of the RF force, in combination with the global HB motion ofthe plasma and of the laser field in space and time resulted in a slower increase of the conversionefficiency with the laser intensity, compared to predictions made in [185], so that ultimately it appearsto be on the level of ηrad ' 0.2 for IL = 1024 W/cm2, leading to an upper limit Bxm ≈ 3.2× 109 Gat such intensities.The significant relative suppression of the radiation losses leads to a specific freezing of the electronlateral motion for a0 > acr = (4πrc/3λL)−1/3 so that the relativistic γ factor grows much slowerγ ∼ a1/40 than in the perturbative domain, where the RF force is negligible. Here, rc ≡ e2/mc2 '2.8× 10−9 µm is the classical electron radius. This in turn shifts the border between the classical andthe quantum regime of interaction to considerably higher intensities. The significance of QED effectsis determined by the value of the relativistically invariant quantum parameterχ =eh̄m3c4√(−(Fµνpν)2 =E ′LES, (1.10)which equals the ratio of the external (laser) electric field in the electron rest frame E ′L to the criticalfield of quantum electrodynamics, ES = m2c3/eh̄. Here, Fµν is the EM field tensor and pν is thefour-momentum vector. The asymptotic value16 of χ is found to be [188]χa0→∞ '32α(a0ξ22√µ)1/4,16in the strong field limit a0 →∞, when the hole-boring velocity approaches the speed of light20where µ = Zncrme/Anemp, ξ = 4πrc/3λL = a−3cr and α = e2/h̄c = 1/137 is the fine-structureconstant. In the weak field regime, a0 acr, a quantum parameters χ turns to be equal toχ =32αξa20.In the range of a0 ∈ [200, 800], corresponding to the range of intensities IL ∈ [1.7, 27]×1023 W/cm2,and for the parameters of the simulations performed in [185, 188] (λL = 800 nm, ne = 90ncr, A/Z =2) the value of χ increases from 0.092 to 0.436, and even for an ”extreme” amplitude of a0 = 2000the value χ ' 0.636 is obtained. This shows that the onset of a full radiation-dominated regime isprevented and a classical description of the dynamics and radiation of electrons is applicable, at leaston a qualitative level. This prediction may be tested by simulations with QED effects included.21 INTRODUCTION1.6 Particle-in-Cell simulationsThe complexity of the laser-plasma interaction related to the high dimensionality of the problem, tothe disparity of the involved scales, to the lack of symmetry and to the importance of nonlinear andkinetic effects prevent analytical methods from providing a detailed description of the phenomenathat take place in one or another interaction setup. The unfolding of rich and complex dynamics ofsuch an interaction, involving collective and self-organization effects, requires therefore the use ofself-consistent numerical simulations. The interaction of intense and super-intense coherent radiationwith plasmas poses some of the most challenging problems for numerical physics. This is because themost important and interesting phenomena are multiscale and cannot be described hydrodynamically.The first feature imposes the use of very large numerical grids since the spatial size and/or evolutiontime of the system are much larger than the smallest spatial and temporal scales to be resolved17.The second one implies that a hydrodynamic description in real space must be abandoned and muchmore demanding kinetic equations in phase space have to be solved. To this end, the PIC method[12, 13] is by far the most commonly used approach because it allows a great saving of memory inallocating the momentum space with respect to a purely Eulerian approach. However, even with aPIC method a realistic simulation would require a three-dimensional spatial grid with thousands ofgrid-points in each spatial direction, resulting in a total number of grid-points exceeding some tensof billions. Moreover, thousands of particles per cell may be needed to properly resolve high en-ergy tails in the distribution [190] as well as sharp density gradients, which always occur in intenselaser-plasma interactions. Thus, the progress in this area is directly related to the possibility to accesslarger high performance computer resources. Nowadays, large-scale multidimensional PIC simula-tions running on parallel supercomputers are an effective support for the design and interpretation oflaser-plasma acceleration experiments, although fully realistic simulations in three spatial dimensionsand for experimentally achievable laser pulse and target parameters are most of the time still beyondcomputational capabilities [49, 191].The PIC approach is a means to solve the Vlasov-Maxwell system of equationsdfidt=∂fi∂t+ v · ∇rfi +qimi(E +vc×B)· ∇pfi = 0, (1.11)1c∂B∂t= −∇× E, 1c∂E∂t+4πcJ = ∇×B, (1.12)∇ · E = 4πρ, ∇ ·B = 0, (1.13)using pseudo particles of the same charge-to-mass ratio as the real particles in the plasma. Thedistribution function is sampled asf̃(r,p, t) =J∑j=1R(r, rj(t))δ(p− pj) (1.14)so that the charge and current densities can be obtained fromρ̃(r, t) =J∑j=1qjR(r, rj(t)), j̃(r, t) =J∑j=1qjvjR(r, rj(t)). (1.15)17In the laser-plasma acceleration setup the smallest scales are the laser and/or plasma wavelength (∼ 0.1÷ 100µm), andthe largest scale is the acceleration distance, which may range from a few centimeters to meters or even kilometers.22The weighting function R(r, r′) obeys the normalization condition∫VR(r, r′)dr = 1. For symmetricR(r, r′) = R(r′, r) the trajectories of the pseudo particlesdrjdt=pγmj,dpjdt= qj(E +vc×B), (1.16)coincide with the characteristics of the Vlasov equation. The electric and magnetic fields in a PICcode are calculated on a numerical grid. For propagation, each time-step the current-density on thegrid and the field values at the particle positions need to be calculated through a position-to-grid map-ping procedure.The number of pseudo particles in a ”numerical plasma” is usually significantly smaller than the num-ber of particles in a real plasma. Therefore, it is important to bear in mind that the characteristic prop-erties of these two plasmas can also be very different. Let a value α = n/n′ 1 be the ratio of realand model plasma densities. Hereinafter, the symbol ′ indicates the values of the model plasma. Fromthe conservation of total charge and mass it follows that q′ = αq,m′ = αm, wherem, q andm′, q′ arethe masses and charges of individual particles in real and model plasma, respectively. In this case, theparticle dynamics indeed does not change, q′/m′ = q/m. The electron and ion plasma frequenciesare also conserved ω′p =√4πq′n′2/m′ = ωp. If we assume that the Debye length is invariant [13]Λ′D =√T ′e/4πe′2n′e = ΛD then the model and real plasma temperatures will be different T′ = αTthough the thermal velocities remain the same v′T =√T ′/m′ = vT . Finally, for the mean interparticledistance, mean free path, number of particle in the Debye sphere and the collisional frequency of themodel plasma we have 〈r′〉 = n′−1/3 = α1/3 〈r〉 , 〈r′col〉 = 〈rcol〉 /α, N ′ = n′Λ′D = N/α, ν ′ = αν.Since α 1, the collisional frequency in the numerical plasma can be significantly bigger andthe mean free path, in turn, significantly smaller then the corresponding values in a real plasma.This brings the model plasma closer a state where the particle collisions should be rather taken intoaccount. Fortunately, in many simulation setups the inequality 〈r′col〉 L′, where L′ is the charac-teristic spatial scale of the model plasma, still can hold, and a collisionless simulation is possible.Most of the numerical studies discussed in this Thesis were performed with the Supercomputers atthe John von Neumann Institute for Computing (NIC, Jülich, Germany) and at the CINECA Super-computer facility (Bologna, Italy) using the fully electromagnetic PIC code UMKA originated fromthe study reported in [20]. UMKA employs state of the art, widely used numerical algorithms suchas an Yee Cartesian lattice for EM fields, the time-reversible, semi-implicit Boris pusher to integratethe Lorentz force and the current reconstruction scheme [192] to satisfy the continuity equation. Inthe following we briefly present the peculiarities of the state of the art version of this code.Ionization dynamics – The “standard” PIC codes start with a preformed plasma of a certain densityprofile and temperature, which means that ionization both by the electric field and by collisions isneglected. This approach is adequate for relativistic laser-field intensities and relatively light targetmaterial so that it turns into a dilute plasma with a well defined ionization degree within a frac-tion of the laser-pulse period. In a more general situation the charge state and density distributionsare expected to be sensitive to the ionization dynamics and have to be calculated self-consistently.Moreover, in targets that turn overdense in the course of ionization the skin-effect prevents the laserfrom penetrating further so that a complicated space and time-dependent charge distribution develops[160, 193]. Such interactions of laser pulses with rapidly self-generated plasmas have already found23 INTRODUCTIONapplications, e.g., as ”plasma mirrors”, which are routinely used to increase the pulse contrast for in-tense laser-matter experiments [194]. Note that molecular dynamics is a powerful tool that is widelyused to describe the ionization dynamics in small laser-driven targets, e.g., clusters [195]. However,for wavelength-sized or bigger targets the influence of the target on the propagation of the incidentEM wave needs to be taken into account self-consistently. This requires the solution of Maxwell’sequations together with the equations of motion for the charged particles. In the case of weakly cou-pled plasmas, the problem can be reduced to the solution of the Vlasov-Maxwell system of equations,which is efficiently achieved using PIC codes.In the UMKA code, the ionization of an ion with charge state Z − 1 and ionization potential I due tothe electric field E is implemented using the tunneling ionization rate formula [196]w(E) =(2Ech|E|)2n∗k2h̄m|E|Echexp(−2Ech3|E|)(1.17)with k =√2mIh̄, Ech =h̄2k3me, n∗ = Z√IHI. Here, m is the electron mass and IH is the ionizationpotential of atomic hydrogen. The ionization probability for the ith ion during time τ is then calcu-lated18 from Pi = τw(E). When an ionization event takes place a free electron at rest is created atthe position of the ion. The energy needed for ionization is taken out of the field via an ”ionizationcurrent” jion parallel to the electric field at the ion location. The value of jion is such that jion · E isthe work spent on ionization per time step [198]. Energy conservation is accounted for during thewhole process; if the remaining field energy in a cell is insufficient for further ionization, this cell isnot considered anymore during the current time step [199].In order to address the process of impact ionization, we use tabulated data of impact ionization cross-sections σi(ve) [200]. The total collision ionization frequency of an electron with velocity ve isthen calculated from νtot(ve) = veZ−1∑i=0σi(ve)ni, and the total collisional ionization probability isPe = τνtot. When an ionization event takes place, the charge state of the ion under consideration ischanged, and a free electron is created at rest at the position of the ion. The value of the impactingelectron’s velocity is reduced in order to ensure energy conservation. Numerically, we use the Monte-Carlo null collision method [201] for the treatment of electron impact ionization. Excited electronicstates are not considered, neither are recombination processes. Self-consistency requires that col-lisional absorption is taken into account along with the collisional ionization, as in the PIC codesdescribed in Refs. [17, 18, 202]. A microscopic PIC code [19, 203] bridges the gap between PICand molecular dynamics and is also capable of incorporating collisional ionization and collisionalabsorption, albeit so far only for small targets.Classical radiation friction – At optical laser intensities exceeding IL ∼ 1023 W/cm2, the plasmaelectrons become ultrarelativistic within a fraction of the laser wave period. They experience a verystrong acceleration and emit relatively large amounts of EM radiation, rendering radiation frictioneffects [169] more important as the laser intensity increases. First PIC simulation that includedRF [204] on the level of the Lorentz–Abraham–Dirac (LAD) equation with the RF force that isexact for a point particle [205], showed that RF effects become important at intensities exceeding5 × 1022 W/cm2, and are thus expected to play a crucial role in experiments soon. It is therefore18This method has the drawback of restricting the time step to very small values in order to properly describe the fieldionization in sufficiently strong fields. A multiple ionization scheme proposed in [197] may help to relax this constraint.24important to incorporate RF in PIC simulations of laser-plasma by an appropriate modeling, takingthe essential RF effects into account while retaining at the same time the capability to perform large-scale simulations. In order to do so one should solve the LAD equation. This equation, however,suffers from inconsistencies such as the existence of the so-called ”runaway” solutions, in which theelectron momentum grows exponentially in the absence of external fields. In the realm of classicalelectrodynamics, i.e., neglecting quantum effects, the LAD equationmcduµdτ= eF µνuν − eτ0(d2uµdτ 2+ uµuνd2uνdτ 2), τ0 = 2e2/(3mc3) (1.18)can be consistently approximated by the Landau–Lifshitz (LL) equationmcduµdτ= eF µνuν + eτ0[uνuα∂αFµν +emcF µνFναuα +emc(F νβuβFναuα)uµ],which is obtained by inserting the unperturbed Lorentz acceleration in Eq. (1.18) [206]. This removesthe third order derivative of the position and thus runaway solutions, and gives a more conventionalphase space description. In 3D notation, the above equation readsdpdt=e(E +vc×B)+ eτ0γ[(∂∂t+ v · ∇)E +vc×(∂∂t+ v · ∇)B]+τ0e2mc{[(E +vc×B)×B +(vc· E)E]− γ2[(E +vc×B)2−(vc· E)2] vc}.As long as a classical description is adequate, RF effects are relevant, and quantum effects are sub-dominant, the expression for the RF force given in textbooks [206] can be safely used,FRF ' −23r2c[γ2(F2L −(vc· E)2) vc− FL ×B− (vc· E)E]. (1.19)Here, FL = E +vc× B, and the small terms containing the temporal derivatives of the fields aredropped .To account self-consistently for the effect of radiation emission on the electron dynamics we imple-mented in UMKA the radiation reaction via the LL approach, using a numerical scheme [208] basedon the assumption that the acceleration of particles is dominated by the Lorentz force, with the RFforce giving a smaller, albeit non-negligible contribution19. The numerical implementation [208, 209]allows the addition of RF effects to any PIC code, which uses the standard Boris pusher algorithm forthe acceleration of the particles, at a small computational cost. We emphasize that the inclusion ofthe radiation loss as a dissipative process via the RF force requires the following assumptions: (i) thedominant frequencies in the escaping radiation are much higher than the highest frequency that canbe resolved on the numerical grid, (ii) the radiation at such frequencies is incoherent, (iii) the plasmais transparent to such frequencies [185, 208].The above presented approach discards the effect of quantum recoil on the spectrum of emitted radia-tion. One one hand, the quantum parameter χ, Eq. (1.10), remains smaller than unity up to intensities∼ 1025 W/cm2, making a classical description of dynamics and radiation of electrons applicable at19For a benchmark with other approaches see [207] M. Vranic, J. L. Martins, R. A. Fonseca, L. O. Silva, ClassicalRadiation Reaction in Particle-In-Cell Simulations, Computer Physics Communications 204, 141-151 (2016).25 INTRODUCTIONleast on a qualitative level. On the other hand, the spectrum of emitted photons appears consider-ably modified by quantum effects already for χ ≈ 0.1 [210, 211]. For the targets with ne ∼ 100ncrthis is achieved for IL ' 1.9 × 1023 W/cm2 so that quantum corrections to the radiated power maybecome numerically important. To account for the suppression of the radiation power due to theoff-set in the emission spectrum we introduce20 the quantum factor g(χ). To that end we have furthermodified our code by incorporating the parameter χ in the expression for the radiation friction forceF̃RF = g(χ)FRF. The quantum parameter χ is calculated at each time step by taking the values ofelectric and magnetic fields at the position of each electron.Finally, the PIC code UMKA allows simulations in 1D, 2D and 3D Cartesian geometries. The simu-lation grid is rectangular, and the grid spacings ∆x,∆y,∆z may be different. It is possible to choosebetween periodic and open boundary conditions. UMKA allows to perform simulations with a ”mov-ing window”, which may be useful in several cases (laser wakefield acceleration, RPA accelerationby long laser pulses). The implementation of complex target geometries, as well as complex targetcompositions having different ion species with a given A/Z ratios, is sraightforward. For a non-standard initial geometry, such as foam-targets used in [213], it is possible to load an external file,containing the coordinates and radii of a collection of spheres.The locality of the PIC methods allows for an efficient parallelization obtained via a domain decom-position technique. In the UMKA code, parallelization is implemented by a decomposition in the(y, z)-discretized coordinates, perpendicular to the propagation of the laser pulse in x direction. Thischoice provides an initially balanced partition of the particles for problems where the particle densityis initially uniform in the (y, z)-plane. A dynamical load-balancing keeps the data partition balancedas particles move across different domains. The distribution of numerical data among the process-ing units and the mutual exchanges are handled by MPI routines. No special libraries or packagesare needed by the code. Recent simulations performed with UMKA on JURECA (NIC, Jülich) andMarconi (CINECA, Bologna) show a fully parallel performance up to 13600 cores.The material presented below in this work is organized as follows. In Chapter 2, the papers dis-cussing the analytical and simulation work on both the hole-boring (HB) and light sail (LS) regimesof radiation-pressure-dominant acceleration of ions by circularly polarized laser pulses are collected.These include a parametric study of the optimal target thickness in the LS case, evaluation of pre-plasma effects, investigation of the absorption of the angular momentum of the laser pulse by denseplasma. Papers devoted to the study of collisionless absorption of the laser radiation and generationof hot electrons are presented in Chapter 3. The contributions to the research of the radiation frictioneffects that emerge in the interaction of super intense laser radiation with plasma are collected inChapter 4. These include a comprehensive presentation of an approach to a kinetic description oflaser-plasma interaction with radiation reaction effects being included via the Landau-Lifshitz force,a detailed study of polarization and radiation reaction effects in the radiation pressure dominatedacceleration regime and a proposal to an alternative approach to investigate radiation friction in thelaboratory, namely to search for the regimes where collective effects in the laser-plasma interactionboost radiation friction losses and the radiation friction signatures become strong and unambiguous.In Chapter 5, the results of the study of the interplay between ionization, pulse propagation and par-ticle acceleration in intense laser-droplet interaction are shown. In Chapter 6, the papers are includedthat arose from the collaboration with the experimental team of the Queen’s University of Belfast(UK).20General formulas for the quantum factor g(χ) can be found in the literature [210]. For practical calculations we use afit [212] g(χ) = (1 + 12χ+ 31χ2 + 3.7χ3)−4/9.27 INTRODUCTIONAuthor contribution statementRADIATION PRESSURE ACCELERATION OF IONS1 T. V. Liseikina and A. MacchiFeatures of ion acceleration by circularly polarized laser pulsesAppl. Phys. Lett. 91, 171502 (2007)T. V. L. contributed to the design and implementation of the research, designed the model andcomputational framework, performed the numerical calculations, contributed to the analysis ofthe results and to the writing of the manuscript.2 A. Macchi, A. Bigongiari, F. Ceccherini, F. Cornolti, T. V. Liseikina, M. Borghesi, S. Kar andL. RomagnaniIon dynamics and coherent structure formation following laser pulse self-channelingPlasma Phys. Contr. Fusion , 49, B71 (2007)T. V. L. contributed to the design and implementation of the theoretical part of research, de-signed the numerical model and computational framework, performed the numerical calcula-tions, contributed to the analysis of the results and to the writing of the manuscript.3 T. V. Liseikina, D. Prellino, F. Cornolti, A. MacchiPonderomotive Acceleration of Ions: Circular Versus Linear PolarizationIEEE Transaction of Plasma Science 36, 1866-1871 (2008)T. V. L. contributed to the design and implementation of the research, designed the model andcomputational framework, performed 2D numerical calculations, contributed to the analysis ofthe results and to the writing of the manuscript.4 T. V. Liseykina, M. Borghesi, A. Macchi, S. TuveriRadiation pressure acceleration by ultraintense laser pulsesPlasma Phys. Contr. Fusion , 50, 124033 (2008)T. V. L. contributed to the design and implementation of the research, designed the model andcomputational framework, performed 2D and 3D numerical simulations, contributed to theanalysis of the results and to the writing of the manuscript.5 A. Macchi, T. V. Liseikina, S. Tuveri, S. VeghiniTheory and simulation of ion acceleration with circularly polarized laser pulsesC. R. Physique 10, 207–215 (2009)T. V. L. contributed to the design and implementation of the research, designed the model andcomputational framework, performed 3D numerical simulations, contributed to the analysis ofthe results and to the writing of the manuscript.286 A. Macchi, S. Veghini, T. V. Liseykina, F. PegoraroRadiation pressure acceleration of ultrathin foilsNew J. Phys. , 12, 045013 (2010)T. V. L. contributed to the design and implementation of the research, designed the model andcomputational framework, contributed to the analysis of the results and to the writing of themanuscript.7 M. Tamburini, A. Di Piazza, T. V. Liseykina, C. H. KeitelPlasma-based generation and control of a single few-cycle high-energy ultrahigh-intensitylaser PulsePhys. Rev. Lett. , 113, 025005 (2014)T. V. L. designed the numerical model and computational framework for the 2D simulations,contributed to the analysis of the results and to the writing of the manuscript.• HOT ELECTRON GENERATION8 P. Mulser, S.M. Weng, T. LiseykinaAnalysis of the Brunel model and resulting hot electron spectraPhys. Plasmas , 19, 043301 (2012)T. V. L. designed the numerical model and computational framework for the PIC simulations,contributed to the analysis of the results and to the writing of the manuscript.9 T. Liseykina, P. Mulser, M. MurakamiCollisionless absorption, hot electron generation, and energy scaling in intense laser-targetinteractionPhys. Plasmas , 22, 033302 (2015)T. V. L. contributed to the design and implementation of the research, designed the numeri-cal model and computational framework, performed numerical simulations, contributed to theanalysis of the results and to the writing of the manuscript.• RADIATION FRICTION EFFECT IN LASER-PLASMA INTERACTIONS10 M. Tamburini, F. Pegoraro, A. Di Piazza, C. H. Keitel, T. V. Liseykina, A. MacchiRadiation reaction effects on electron nonlinear dynamics and ion acceleration in laser–solidinteractionNuclear Instruments & Methods in Physics ResearchA 653, 181–185 (2011)T. V. L. contributed to the design and implementation of the research, designed the numericalmodel and computational framework for 2D PIC simulations, contributed to the analysis of theresults and to the writing of the manuscript.11 M. Tamburini, T. V. Liseykina, F. Pegoraro, A. MacchiRadiation-pressure-dominant acceleration: Polarization and radiation reaction effects and en-ergy increase in three-dimensional simulationsPhys. Rev. E , 85, 916407 (2012)T. V. L. contributed to the design and implementation of the research, designed the numerical29 INTRODUCTIONmodel and computational framework for 2D and 3D PIC simulations, carried out the imple-mentation of the radiation friction force in the 3D PIC code, contributed to the analysis of theresults and to the writing of the manuscript.12 T. Liseykina, S. Popruzhenko, A. MacchiInverse Faraday effect driven by radiation frictionNew J. Phys. , 18 072001 (2016)T. V. L. contributed to the design and implementation of the research, designed the model andcomputational framework, performed the numerical calculations, contributed to the analyticalcalculations, to the analysis of the results and to the writing of the manuscript.13 S. Popruzhenko, T. Liseykina, A. MacchiEfficiency of radiation friction losses in laser-driven ”hole boring” of dense targetsNew J. Phys. , 21, 033009 (2019)T. V. L. contributed to the design and implementation of the research, designed the model andcomputational framework, performed the numerical calculations, contributed to the analyticalcalculations, to the analysis of the results and to the writing of the manuscript.• HE-DROPLETS IN INTENSE LASER FIELD14 T.V. Liseykina, S. Pirner, D. BauerRelativistic Attosecond Electron Bunches from Laser-Illuminated DropletsPhys. Rev. Lett. , 104, 092002 (2010)T. V. L. contributed to the design and implementation of the research, designed the model andcomputational framework for the 3D PIC simulations, performed 3D simulations, contributedto the analysis of the results and to the writing of the manuscript.15 T.V. Liseykina, D. BauerPlasma-Formation Dynamics in Intense Laser-Droplet InteractionPhys. Rev. Lett. , 110, 145003 (2013)T. V. L. contributed to the design and implementation of the research, designed the model andcomputational framework for the 3D PIC simulations, carried out the implementation of thefield ionization in 3D PIC code, performed 3D simulations, contributed to the analysis of theresults and to the writing of the manuscript.16 P. Sperling, T. Liseykina, D. Bauer, R. RedmerTime-resolved Thomson scattering on high-intensity laser-produced hot dense helium plasmasNew J. Phys. , 15, 025041 (2013)T. V. L. design the model and computational framework for the 3D PIC simulations, contributedto the analysis of the results and to the final version the manuscript.• NUMERICAL SUPPORT TO THE EXPERIMENTAL STUDIES17 S. Kar, M. Borghesi, C. A. Cecchetti, L. Romagnani, F. Ceccherini, T. V. Liseykina, A. Macchi,R. Jung, J. Osterholz, O. Willi, L. A. Gizzi, A. Schiavi, M. Galimberti, & R. HeathcoteDynamics of charge-displacement channeling in intense laser-plasma interactions30New J. Phys. , 9, 402 (2007)T. V. L. contributed to the modeling (designed the numerical framework and performed the2D PIC simulations) and interpretation of laser beam self-channeling experimentally observedthanks to the use of laser-driven proton beams as a time and space–resolved probe of electro-magnetic fields, contributed to the writing the manuscript.18 K. Quinn, P. A. Wilson, C. A. Cecchetti, B. Ramakrishna, L. Romagnani, G. Sarri, L. Lancia,J. Fuchs, A. Pipahl, T. Toncian, O. Willi, R. J. Clarke, D. Neely, M. Notley, P. Gallegos, D. C.Carroll, M. N. Quinn, X. H. Yuan, P. McKenna, T. V. Liseykina, A. Macchi, & M. BorghesiLaser-driven ultrafast field propagation on solid surfacesPhys. Rev. Lett. , 102, 194801 (2009)T. V. L. contributed to the modeling (designed the numerical framework and carried out 2DPIC simulations) of giant current pulses of k-Ampere intensity and picosecond duration, ob-served after the high-intensity irradiation of metal targets, contributed to the final version of themanuscript.19 L. Romagnani, A. Bigongiari, S. Kar, S. V. Bulanov, C. A. Cecchetti, T. Zh. Esirkepov, M.Galimberti, R. Jung, T. V. Liseykina, A. Macchi, J. Osterholz, F. Pegoraro, O. Willi, & M.BorghesiObservation of magnetized soliton remnants in the wake of intense laser pulse propagationthrough plasmasPhys. Rev. Lett. , 105, 175002 (2010)T. V. L. contributed to the modeling (designed the numerical framework and performed the2D PIC simulations) and interpretation of coherent electromagnetic structures formation in thewake of an intense laser pulse, which was experimentally observed thanks to the use of laser-driven proton beams as a time and space–resolved probe of electromagnetic fields, contributedto the final version of the manuscript.20 G. Sarri, A. Macchi, C. A. Cecchetti, S. Kar, T. V. Liseykina, X. H. Yang, M. E. Dieckmann,J. Fuchs, M. Galimberti, L. A. Gizzi, R. Jung, I. Kourakis, J. Osterholz, F. Pegoraro, A. P. L.Robinson, L. Romagnani, O. Willi, & M. BorghesiDynamics of self-generated, large amplitude magnetic fields following high-intensity laser mat-ter interactionPhys. Rev. Lett. , 109, 205002 (2012)T. V. L. contributed to the modeling (designed the numerical framework and performed the 2DPIC simulations) and interpretation magnetic field generation experimentally observed thanksto the use of laser-driven proton beams as a time and space–resolved probe of electromagneticfields, contributed to the writing of the manuscript.31 RADIATION PRESSURE ACCELERATION OF IONS2. Radiation pressure acceleration of ionsThe list of publications:T. V. Liseikina and A. Macchi, Features of ion acceleration by circularly polarized laser pulses, Appl.Phys. Lett. 91, 171502 (2007)A. Macchi, A. Bigongiari, F. Ceccherini, F. Cornolti, T. V. Liseikina, M. Borghesi, S. Kar and L.Romagnani, Ion dynamics and coherent structure formation following laser pulse self-channeling,Plasma Phys. Contr. Fusion 49, B71 (2007)T. V. Liseikina, D. Prellino, F. Cornolti, A. Macchi, Ponderomotive Acceleration of Ions: CircularVersus Linear Polarization, IEEE Transaction of Plasma Science 36, 1866-1871 (2008)T. V. Liseykina, M. Borghesi, A. Macchi, S. Tuveri, Radiation pressure acceleration by ultraintenselaser pulses, Plasma Phys. Contr. Fusion 50, 124033 (2008)A. Macchi, T. V. Liseikina, S. Tuveri, S. Veghini, Theory and simulation of ion acceleration withcircularly polarized laser pulses, C. R. Physique 10, 207–215 (2009)A. Macchi, S. Veghini, T. V. Liseykina, F. Pegoraro, Radiation pressure acceleration of ultrathinfoils, New J. Phys. 12, 045013 (2010)M. Tamburini, A. Di Piazza, T. V. Liseykina, C. H. Keitel, Plasma-based generation and controlof a single few-cycle high-energy ultrahigh-intensity laser Pulse, Phys. Rev. Lett. 113, 025005(2014)Features of ion acceleration by circularly polarized laser pulsesT. V. LiseikinaaDipartimento di Fisica “E. Fermi,” Università di Pisa, Pisa I-56127, ItalyA. MacchibpolyLAB, CNR-INFM, and Dipartimento di Fisica “E. Fermi,” Università di Pisa, Pisa I-56127, ItalyReceived 26 May 2007; accepted 8 October 2007; published online 26 October 2007The characteristics of a MeV ion source driven by superintense, ultrashort laser pulses with circularpolarization are studied by means of particle-in-cell simulations. Predicted features include highefficiency, large ion density, low divergence, and the possibility of femtosecond duration. Acomparison with the case of linearly polarized pulses is made. © 2007 American Institute ofPhysics. DOI: 10.1063/1.2803318The short-duration, multi-MeV ion beams produced inthe interaction of high-intensity laser pulses with solid tar-gets have proven to be effective for applications such asproton radiography,1,2 diagnostic of highly transient electro-magnetic fields,2–4 isochoric heating of matter,5 isotopeproduction,6 and nuclear activation.7 Foreseen future appli-cations to medicine,8 nuclear fusion9 or particle physics10will require improvements in factors such as the conversionefficiency, peak ion energy, beam monochromaticity, andcollimation. Recent experiments performed with theseaims11–13 are based on sheath acceleration SA of ions on asurface layer at the rear side of the target.14 SA is driven bythe space-charge field of high-energy “fast” electrons ex-panding in vacuum. Numerical simulations have also ex-plored different regimes, such as “shock acceleration,”15“laser-piston,”16 “skin-layer ponderomotive acceleration,”17or acceleration by circularly polarized laser pulses.18–20 Inthese latter studies, as a common feature ion accelerationoccurs at the target front side and is in principle dominatedby the effect of the radiation pressure of the laser pulse.These regimes based on radiation pressure accelerationRPA might be the leading ones at ultrahigh intensities16 orbe most suitable for specific applications.17Here we report a numerical study on ion accelerationwith circularly polarized CP pulses with the aim to showthe peculiar features of the ion source high efficiency, largeion density, short duration, and good collimation which maybe advantageous for specific applications. A comparison withthe case of linearly polarized LP pulses is made to eviden-tiate the differences with CP and to provide a deeper under-standing of RPA mechanisms. This is possible because usingCP at normal incidence fast electron generation is almostsuppressed,18 thus related effects such as SA can be sepa-rated by those based on radiation pressure such as RPA.We compare two one-dimensional 1D particle-in-cellPIC simulations performed for LP and CP, respectively.Assuming a pulse wavelength L=1 m, the LP and CPsimulations have same duration L=26TL=86 fs, whereTL=L /c and intensity I=3.51020 W cm−2. To ensure thatI is the same, the peak field amplitude in the CP case,aL=11.3, is lower by a factor of 2 than in the LP caseaL=16. Here, aL is the dimensionless pulse amplitude givenby aL=0.85I218/, where I218 is the irradiance inunits of 1018 W cm−2 and =1 2 for LP CP. The plasmaparameters are the same for both simulations proton plasmaslab of 20 m thickness and electron density ne=10nc,where nc=1.11021 cm−3 is the cutoff density. The param-eters of the LP case are close to those of simulations reportedby Silva et al.15 to address “shock” acceleration. The tempo-ral and spatial resolutions are given by x=ct=L /400 and32 particles per cell are used. The high spatiotemporal reso-lution is necessary to ensure convergence of the results, sincevery sharp gradients are generated during the interaction,e.g., at the ion density spiking discussed below andelsewhere.18,21Figure 1 compares the ion density profiles and the phasespace of ions and electrons for CP and LP. For LP, stronglyrelativistic electrons with momenta pex up to 30mec aregenerated. The ion phase space shows at least three“groups:” ions accelerated by SA both at the front and therear side, with momenta pix up to 0.3mic, and ions accel-erated at the front surface propagating into the plasma withsimilar momentum values. For CP, electrons are relatively“cold,” since typical momenta are more than one order ofmagnitude lower than for LP. This is due to the fact that thenet force on electrons is small as the ponderomotive force ofthe laser pulse whose integral over the target depth gives thetotal radiation pressure and the electrostatic force producedby charge separation almost cancel each other.18 Due to theabsence of “fast” electrons, no significant SA ions are ob-aOn leave from Institute for Computational Technologies, SD-RAS,Novosibirsk, Russia.bElectronic address: macchi@df.unipi.itFIG. 1. Color online Snaphsots at t=140TL=467 fs of the ion density nitop and the x , px phase space projections of ions f i, middle and elec-trons fe, bottom from 1D PIC simulations in the CP left and LP rightcases. The laser pulse propagates from left to right. Length scales are nor-malized to L=1 m, the density to nc, and momenta to mc. Notice thedifferent scales on the momentum axes between the CP and LP cases.APPLIED PHYSICS LETTERS 91, 171502 20070003-6951/2007/9117/171502/3/$23.00 © 2007 American Institute of Physics91, 171502-1 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 139.30.84.198 On: Fri, 03 Jun 201614:28:3133 FEATURES OF ION ACCELERATION BY CP LASER PULSESserved, and most of the accelerated ions are located in abunch with longitudinal momentum px0.15mic.Figure 2 compares the absorption efficiency and the ionspectrum obtained in the CP and LP cases, respectively. Theabsorption into bunch ions is 13.7% for CP and is constantafter the laser pulse, confirming that all ions are acceleratedby RPA; absorption into electrons is negligible. For LP, ab-sorption into electrons is dominant during the interactionwith the laser pulse; later, energy transfer toward ions occursand the conversion efficiency into ion energy has a valuesimilar to CP, but including all the three ion groups observedin Fig. 1. The ion spectrum for CP is relatively narrow andpeaked around 10 MeV, while the LP spectrum is morethermal-like, with a broad maximum around 20 MeV. Aseries of simulations with parameters in the aL=1–15 andne=5–100 ranges confirm that these features are robust andindicate a scaling of both the peak ion velocity and the ab-sorption efficiency into ions proportional to aL.22While RPA is the only effective ion acceleration channelin the CP case, for LP its contribution overlaps with SA, andthe same analytical model proposed for the CP case18 may beused if the longitudinal force on ions is considered to be atemporal average. Thus, the forward accelerated ions at thefront surface observed for LP may be simply attributed toRPA rather than to shock acceleration15. However, thestrong absorption into electrons for LP reduces the total ra-diation pressure and so the RPA efficiency. An estimate ofthe relative contributions of SA and RPA is provided by ananalysis of particle energy versus position, showing that atthe time corresponding to Fig. 1 the energy belonging to ionslocated within the original plasma slab position 60 mx80 m is 76% of the total energy for LP with 10% and14% being the contributions of ions emitted from the frontand rear sides, respectively and almost 100% for CP.The reason why the RPA peak in the ion energy spec-trum is much more prominent for CP than for LP is attributedto the fact that the ponderomotive force tends to focus theion spatially at the end of the skin layer, creating a very sharpdensity peak.18,21 For LP, the strong electron pressure coun-teracts the piling up of the ions, leading to “explosion” of theproton bunch and to a broader energy spectrum; the maxi-mum ion energy is higher for LP than for CP, as a few of theRPA ions gain additional energy from the fast electrons. Thecomparison with the CP simulation shows that in this lattercase it is the relatively low electron temperature which al-lows for a narrow ion energy spectrum.The comparison of 1D, plane-wave simulations best en-lightens the different regimes of ion acceleration between LPand CP and the particular features of RPA versus SA. For arealistic laser pulse with a finite spot size, the differencesbetween LP and CP are somewhat weakened by the effects ofpulse focusing which introduces electric field componentsnormal to the target surface at the edges of the spot, leadingto electron heating. Figure 3 shows the effect on fractionalabsorption in 2D simulations with the same parameters of the1D case, and a tightly focused pulse with a Gaussian inten-sity distribution and a spot radius rL=2L. In the 2D case aLgives the peak amplitude at the center of the spot. The dif-ferences between LP and CP and between ion and electronabsorptions in the latter case are less dramatic as expected,but still substantial. The ion energy and the conversion effi-ciency into ions 11.3% observed in 2D simulations areclose to the values expected from the 1D analysis.The angular spread of ions depends on their energy E.The fE , distribution of ions in Fig. 3, where is theemission angle with respect to the target normal, shows thatthe most energetic the ions the most collimated they are: forinstance, ions having energy exceeding 0.01mic2 are foundwithin a cone with an aperture angle of about 10°. Widerspot sizes or smoother intensity profiles may yield a lowerdivergence.The narrow ion energy spectrum of the CP case is anecessary condition to obtain a dense ion bunch with veryshort duration, in addition to the requirement of a laser pulsewith duration of the order of the bunch acceleration time.18These particular features of ions accelerated using CP pulsesmay be useful for ultrafast, localized energy deposition inmatter, and are essential for a proposed concept of sources offusion neutrons with duration of a few femtoseconds.23The production of a single, ultrashort ion bunch can beobserved in the simulation of Fig. 4 for which I=5.51018 W cm−2, L=33 fs, and rL=4 m where, in addition,a linear density profile was used instead of a steplike one toaddress the effect of early plasma production by a prepulse inexperiments. We observe that ion acceleration initially oc-curs near the cutoff layer where ne=nc and produces at t30TL a narrow ion bunch with a density larger than 10nc.The divergence of the bunch ions is about 4°.To our knowledge no laser-plasma interaction experi-ments investigating RPA with CP have been reported so far.Our results suggests that this regime may lead to high effi-ciency and large numbers of ions which can be useful forapplications requiring fast deposition of energy in matter. Atthe same time, recent works19,20 suggest that the accelerationof very thin targets for sufficiently long times using CPpulses may lead to very high ion energies and monoenergeticspectra. Since no unaffordable problem seems to prevent theuse of high-intensity CP pulses at normal incidence for laser-FIG. 2. Color online Comparison of absorption efficiency into ions andelectrons vs time left and of ion energy spectra right from 1D PIC simu-lations of ion acceleration with LP or CP pulses for the same plasma param-eters and laser energy and duration.FIG. 3. Color online 2D simulation results. Left: comparison between CPand LP of absorption efficiency into ions and electrons vs time. Right:energy vs angle distribution of ions for the CP case. The pulse radius is rL=2 m. Other parameters are the same of the 1D runs of Figs. 1 and 2.171502-2 T. V. Liseikina and A. Macchi Appl. Phys. Lett. 91, 171502 2007 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 139.30.84.198 On: Fri, 03 Jun 201614:28:3134plasma interactions, the regime of RPA with CP pulses maybe investigated in present-day experiments allowing progressin the use of laser-accelerated ions for specific applications.This work has been supported by the Italian Ministeryfor University and Research MIUR via the PRIN project“Ultraintense laser-plasma interaction,” by CNR-INFM andCINECA Italy through the supercomputing initiative, andby JSCC Moscow, Russia.1J. A. Cobble, R. P. Johnson, T. E. Cowan, N. R.-L. Galloudec, and M.Allen, J. Appl. Phys. 92, 1775 2002.2A. J. Mackinnon, P. K. Patel, D. W. Price, D. Hicks, L. Romagnani, andM. Borghesi, Appl. Phys. Lett. 82, 3188 2003.3M. Borghesi, L. Romagnani, A. Schiavi, D. H. Campbell, M. G. Haines,O. Willi, A. J. MacKinnon, M. Galimberti, L. Gizzi, R. J. Clarke, and S.Hawkes, Appl. Phys. Lett. 82, 1529 2003.4C. K. Li, F. H. Seguin, J. A. Frenje, J. R. Rygg, R. D. Petrasso, R. P. J.Town, P. A. Amendt, S. P. Hatchett, O. L. Landen, A. J. MacKinnon, P. K.Patel, V. A. Smalyuk, T. C. Sangster, and J. P. Knauer, Phys. Rev. Lett.97, 135003 2006.5P. K. Patel, A. J. MacKinnon, M. H. Key, T. E. Cowan, M. E. Foord, M.Allen, D. F. Price, H. Ruhl, P. T. Springer, and R. Stephens, Phys. Rev.Lett. 91, 125004 2003.6K. Nemoto, A. Maksimchuk, S. Banerjee, K. Flippo, G. Mourou, D.Umstadter, and V. Y. Bychenkov, Appl. Phys. Lett. 78, 595 2001.7P. McKenna, K. E. D. Ledingham, T. McKanny, R. P. Singhal, I. Spencer,E. L. Clark, F. N. Beg, K. Krushelnick, M. S. Wei, J. Galy, J. Magill, R. J.Clarke, K. L. Lancaster, P. A. Norreys, K. Spohr, and R. Chapman, Appl.Phys. Lett. 83, 2763 2003.8V. Malka, S. Fritzler, E. Lefebvre, E. d’Humieres, R. Ferrand, G. Grillon,C. Albaret, S. Meyroneinc, J. P. Chambaret, A. Antonetti, and D. Hulin,Med. Phys. 31, 1587 2004.9S. Atzeni, M. Temporal, and J. J. Honrubia, Nucl. Fusion 42, L1 2002.10F. Terranova, S. V. Bulanov, J. L. Collier, H. Kiriyama, and F. Pegoraro,Nucl. Instrum. Methods Phys. Res. A 558, 430 2006.11B. M. Hegelich, B. J. Albright, J. Cobble, K. Flippo, S. Letring, M. Paffett,H. Ruhl, J. Schreiber, R. K. Schulze, and J. C. Fernandez, NatureLondon 439, 441 2006.12H. Schwoerer, S. Pfotenhauer, O. Jaeckel, K. U. Amthor, B. Liesfeld, W.Ziegler, R. Sauerbrey, K. W. D. Ledingham, and T. Esirkepov, NatureLondon 439, 445 2006.13T. Toncian, M. Borghesi, J. Fuchs, E. d’Humieres, P. Antici, P. Audebert,E. Brambrink, C. A. Cecchetti, A. Pipahl, L. Romagnani, and O. Willi,Science 312, 410 2006.14S. C. Wilks, A. B. Langdon, T. E. Cowan, M. Roth, M. Singh, S. Hatchett,M. H. Key, D. Pennington, A. MacKinnon, and R. A. Snavely, Phys.Plasmas 8, 542 2001.15L. O. Silva, M. Marti, J. R. Davies, R. A. Fonseca, C. Ren, F. S. Tsung,and W. B. Mori, Phys. Rev. Lett. 92, 015002 2004.16T. Esirkepov, M. Borghesi, S. V. Bulanov, G. Mourou, and T. Tajima,Phys. Rev. Lett. 92, 175003 2004.17J. Badziak, S. Jablonski, and S. Glowacz, Appl. Phys. Lett. 89, 0615042006.18A. Macchi, F. Cattani, T. V. Liseykina, and F. Cornolti, Phys. Rev. Lett.94, 165003 2005.19X. Zhang, B. Shen, X. Li, Z. Jin, and F. Wang, Phys. Plasmas 14, 0731012007.20A. P. L. Robinson, M. Zepf, S. Kar, R. G. Evans, and C. Bellei, e-printarXiv:0708.2040.21Y. Sentoku, T. E. Cowan, A. Kemp, and H. Ruhl, Phys. Plasmas 10, 20092003.22T. V. Liseikina, D. Prellino, F. Cornolti, and A. Macchi, Proceedings of theLaser and Plasma Accelerators Workshop 2007, Azores, Portugal, 2007IEEE Trans. Plasma Sci. to be published.23A. Macchi, Appl. Phys. B: Lasers Opt. 82, 337 2006.FIG. 4. Color online Snapshots of ion density from a 2D, CP simulationfor a linear density profile rising from 0 to 10nc over 2. Laser pulse pa-rameters are aL=2, L=10TL, and rL=4. The white dotted line indicates theinitial position of the ne=nc surface. The arrow indicates the location of theshort ion “bunch” at t=44TL.171502-3 T. V. Liseikina and A. Macchi Appl. Phys. Lett. 91, 171502 2007 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 139.30.84.198 On: Fri, 03 Jun 201614:28:3135 FEATURES OF ION ACCELERATION BY CP LASER PULSESIOP PUBLISHING PLASMA PHYSICS AND CONTROLLED FUSIONPlasma Phys. Control. Fusion 49 (2007) B71–B78 doi:10.1088/0741-3335/49/12B/S05Ion dynamics and coherent structure formationfollowing laser pulse self-channelingAndrea Macchi1,2, Alessandra Bigongiari2, Francesco Ceccherini2,Fulvio Cornolti2, Tatiana V Liseikina2, Marco Borghesi3,Satyabrata Kar3 and Lorenzo Romagnani31 polyLab, CNR-INFM, Pisa, Italy2 Dipartimento di Fisica ‘Enrico Fermi’, Università di Pisa, Largo Bruno Pontecorvo 3, I-56127Pisa, Italy3 School of Mathematics and Physics, the Queen’s University of Belfast, Belfast BT7 1NN, UKE-mail: macchi@df.unipi.itReceived 6 July 2007Published 14 November 2007Online at stacks.iop.org/PPCF/49/B71AbstractThe propagation of a superintense laser pulse in an underdense, inhomogeneousplasma has been studied numerically by two-dimensional particle-in-cellsimulations on a time scale extending up to several picoseconds. The effects ofthe ion dynamics following the charge-displacement self-channeling of the laserpulse have been addressed. Radial ion acceleration leads to the ‘breaking’ of theplasma channel walls, causing an inversion of the radial space-charge field andthe filamentation of the laser pulse. At later times a number of long-lived, quasi-periodic field structures are observed and their dynamics is characterized withhigh resolution. Inside the plasma channel, a pattern of electric and magneticfields resembling both soliton- and vortex-like structures is observed.(Some figures in this article are in colour only in the electronic version)1. IntroductionThe propagation of superintense laser pulses through low-density plasmas gives rise to a varietyof nonlinear electromagnetic phenomena [1,2]. As a general issue the response of the plasmais nonlinear due to both relativistic effects (hence the definition of ‘relativistic optics’ [3]) andthe intense ponderomotive force (i.e. the radiation pressure), which strongly modifies the localplasma density. Probably, the example of such dynamics which has been mostly investigated isthe self-focusing, channeling and filamentation of the laser pulse [4–10]. Another prominenteffect is the generation of coherent structures such as electromagnetic solitons or vortices.Numerical simulations (see e.g. [11–13]) show that such structures are generated during theinteraction with the laser pulse on an ultrafast (femtosecond) time scale, but they may leadto typical field structures which last for much longer times (e.g. ‘post-solitons’) [12], i.e.in the picosecond range, allowing for their experimental observation [14]. On such a scale0741-3335/07/SB0071+08$30.00 © 2007 IOP Publishing Ltd Printed in the UK B7137 ION DYNAMICS AND COHERENT STRUCTURE FORMATIONB72 A Macchi et althe temporal evolution of such field structures must be studied including the effects of themotion of the plasma ions. Stability and evolution of coherent structures on the ion timescale has been studied theoretically and numerically in several papers, for various regimes anddimensionalities [11–13, 15–20].In this paper we report a theoretical study of nonlinear effects during and after thepropagation of a superintense laser pulse in an underdense, longitudinally inhomogeneousplasma. The work was motivated by experiments on laser propagation in a low-density plasmawhere the dynamics of self-generated, slowly varying electromagnetic fields was investigatedusing the proton diagnostic technique [21]. In this paper we focus on the simulation resultsand on their theoretical interpretation, while a comparison with the experimental results willbe reported elsewhere [22, 23].2. Simulation set-upThe laser–plasma interaction simulations were performed using a particle-in-cell (PIC) codein 2D with Cartesian geometry. Reduction to 2D was dictated by the need to address relativelylong spatial and temporal scales, close to the experimental ones. Moreover (as will be clearfrom the discussion of the results) during the interaction sharp gradients in the field andcurrent patterns are generated. Thus, a reasonable resolution is mandatory to resolve suchdetails, pushing the memory requirements in 3D much beyond present-day supercomputingcapabilities. Among the set of 2D simulations that were performed for this study, the largestones employed a 7750 × 2400 grid, with spatial resolution x = y = λ/10 (where λ isthe laser pulse wavelength) and 16 particles per cell for both electrons and ions, requiring atotal of 5000 CPU hours on 100 processors to simulate more than 1500 laser periods of theinteraction. The code is fully parallelized and the simulations were performed at the CINECAsupercomputing facility in Bologna (Italy).In the following, lengths are given in units ofλ, times in units ofTL = λ/c = 2π/ω, electricand magnetic fields in units of E0 = meωc/e, and densities in units of nc = meω2/4πe2.For λ = 1 µm, E0 = 3.213 × 1010 V cm−1 = 107.1 MG and nc = 1.11 × 1021 cm−3. Thedimensionless parameter aL, giving the peak field amplitude of the laser pulse normalized to E0,is related to the laser intensity I and the wavelength by aL = 0.85(Iλ2/1018 W cm−2 µm2)1/2.In all the 2D simulations reported below, the plasma is inhomogeneous along the x axis,i.e. in the direction of propagation of the laser pulse. The electron density profile rises linearlyfrom zero value at x = 25λ to the peak value n0 = 0.1nc at x = 425λ, and then remainsuniform. The pulse duration τL was either 150 or 300 TL, corresponding to 0.5 and 1 ps,respectively, for λ = 1 µm.The laser pulse was S-polarized, i.e. the electric field of the laser pulse was in the zdirection perpendicular to the simulation plane. In the following we restrict the discussion tothe S-polarization case which has some advantages for the data analysis and visualization (forinstance, the space-charge field generated in the radial (y) direction during self-channeling isseparated by the electromagnetic field Ez, which is representative of the pulse evolution). Itis known, however, that at high intensity the details of nonlinear effects in pulse propagationdepend on the polarization leading to differences between the S- and P -polarization casesin 2D geometry and to asymmetry effects in 3D for what concerns self-focusing [24] andalso to differences in the type and stability of solitons and vortices [ [2, 3] and referencestherein]. A preliminary simulation performed for P -polarization showed slight, but nosubstantial differences for what concerns the early self-channeling evolution which we discussin section 3.1. The discussion of the effect of different polarizations on the coherent structuresgeneration and evolution is more involved and will be addressed in future work.38Ion dynamics and coherent structure formation B73Figure 1. Contours of the electric field of the laser pulse Ez and the ion density ni from a 2D PICsimulation, at t = 1000TL, showing the self-channeling and filamentation of the laser pulse and thegeneration of isolated solitary structures and of field patterns. The laser pulse propagates from theleft to the right. In the Ez frame, the contour levels in the leftmost region (50 < x < 250) has beenrescaled by a factor of 3 to show the presence of field structures with relatively low amplitudes.The laser pulse parameters are aL = 2.7, rL = 8λ and τL = 300TL.3. ResultsTo illustrate the variety of nonlinear effects observed in the simulation results, figure 1 showssnapshots at t = 103TL of the ion density (ni) and the electric field of the laser pulse (Ez) overnearly the whole length of the plasma, for a simulation with aL = 2.7 and τL = 300TL. Figure 1contains most of the prominent features we observed throughout the set of our simulations,which may be summarized as follows.In the low-density region, the laser pulse bores a single charge-displacement channel,which in the higher density region breaks up into three main channels and a few secondary,narrow filaments. In the following (see section 3.1) we trace back the appearance of the‘trifurcated’ channel to the effects of radial ion acceleration, which lead to the ‘breaking’ ofthe channel walls.Different types of electromagnetic structures are observed in regions of different density.In the lower density region (approximately between x = 100 and x = 150 in figure 1) a patternof fields with approximate axial symmetry is observed. A detailed analysis of the electric andmagnetic fields, including an estimate of their characteristic frequency from the simulation(see section 3.4), shows that this type of structures combines both features of low-frequencyelectromagnetic post-solitons or ‘cavitons’ and steady current vortices. In the higher densityregion a number of slowly evolving field structures, either appearing as ‘solitary’ structures ororganized into patterns, are observed both outside the main low-density channels and inside thelatter. There is some experimental indication of the growth of regularly spaced field structuresinto the main channel [23].3.1. Ion and electric field dynamics following self-channelingFor intensities up to aL 2, in the early stage of the interaction the laser pulse bores a regularcharge-displacement channel in the inhomogeneous region of the plasma, i.e. at densitiesne < 0.1nc. This is the case for the simulation of figure 2 (aL = 2, τL = 300TL, transversewidth rL = 4λ), which shows a snapshot of the ion density ni and the electric field componentsEz and Ey (results from this simulation are also reported in [22]). The laser pulse undergoesself-focusing as indicated both by the reduction of its transverse radius to 3λ and by theincrease in its amplitude by a factor 1.2.In the leading edge of the channel the transverse field Ey is in the outward directionfrom the axis, indicating that the channel is positively charged due to the radial expulsion ofelectrons. In the trailing part of the pulse, the radial profile of Ey changes qualitatively, as twoambipolar fronts appear on each side of the channel. On the inner side of the ambipolar fronts39 ION DYNAMICS AND COHERENT STRUCTURE FORMATIONB74 A Macchi et alFigure 2. Simulation results addressing electric field dynamics following self-channeling, showingthe transition in the radial field profile [22]. Left column: 2D PIC results. Top frame: ion density(ni) and electric field components (Ez and Ey ) at t = 600TL. Bottom frame: lineout of Ey (blue)and ni (red) along the y-axis at two different x-positions. Parameters are aL = 2, τL = 300TL,rL = 4λ. Right column: snapshots at various times of radial electric field Er (blue, thick line)and ion density ni (red, dashed–dotted line), and the phase space distributions of ions fi(r, pr )and electrons fe(r, pr ) from 1D simulations using a ponderomotive, electrostatic model [25].Parameters are aL = 2.7, ne/nc = 0.01, rL = 7.5λ, τL = 300TL.Ey now points in the inward direction, i.e. towards the axis. The onset of an ‘inversion’ in theradial field has been noticed in experimental investigations of channel dynamics [22].3.2. One-dimensional modeling and the electric field ‘echo’The dynamics leading to the evolution of the radial electric field can be studied in detail usinga one-dimensional, electrostatic PIC model where the laser pulse action is taken into accountonly via the ponderomotive force. The model assumes a non evolving radial profile of the laserpulse and cylindrical symmetry taking only the radial, cycle-averaged dynamics of electronand ions into account. Details about the model and its results are reported elsewhere [25].Here we focus on the most prominent features of electric field dynamics.Figure 2 shows snapshots of the radial electric field Er and the ion density ni at varioustimes, for a 1D simulation in the same regime of 2D electromagnetic runs. Initially, theponderomotive force Fp pushes electrons away from the axis, creating a back-holding space-charge field which is found to balance Fp almost exactly. At the end of the pulse, when Fp = 0,Er has almost vanished. However, Er appears back at a later time, with an ambipolar profilevery similar to that observed in the 2D simulations. This ‘echo’ effect originates from the iondynamics of ions which are accelerated by the electric force ZeEr = ZFp during the laserpulse. The spatial profile of Fp is such that the ions are focused towards a very narrow regionat the edge of the channel, producing a very sharp spike of the ion density and leading tohydrodynamical breaking as the fastest ions overturn the slowest ones. Looking at the profileof the ion density we observe that the latter may be said to ‘break’ in literal meaning, as asecondary density spike moving outwards is formed. The process is also accompanied bystrong heating of electrons near the breaking point, leading to the appearance of an ambipolarsheath field around the density spike. The negative field is strong enough to slow down andinvert the velocity of the slowest ions, which are directed back to the axis where they are foundto form a local density maximum at later times.3.3. Laser beam breakupA simple analytical model shows that the time required for the ions in the channel to reachthe ‘breaking’ point is proportional to the channel radius and inversely proportional to the40Ion dynamics and coherent structure formation B75Figure 3. Evolution of the laser field Ez at different times showing the breakup of the laser pulseinto three main beams. The two secondary beams propagating in the oblique direction originatefrom near the location of the ‘breaking’ of the channel walls. The laser pulse parameters areaL = 2.7, rL = 8λ and τL = 150TL.laser field amplitude [25]. For high intensities, the ‘breaking’ effect due to ion accelerationmay occur early during the laser pulse, i.e. when the electromagnetic energy density inside thechannel is very high, and cause a fast, strong variation of the density at the edge of the channel.In turn, this may affect the propagation of the laser pulse, similarly to what would happen ina wave guide where a sudden ‘leak’ in its walls occurs. A possible signature of this effect isthe appearance of two secondary beams, propagating in the oblique direction, and originatingnear the point where the breaking of the channel walls occurs, as can be observed in figure 3.From the ‘leaking waveguide’ picture we roughly estimate these secondary beams topropagate at an angle θ with respect to the axis given by tan θ ky/kx , where ky π/dis the transverse wavevector of the guided mode, d is the local channel diameter andkx √ω2/c2 − k2y . In this estimate the pulse in the channel is modelized as a TE modeof lowest order in a square guide. From the simulation result we get tan θ 0.065, whilewith d 7λ we obtain ky/kx (π/7λ)/(2π/λ) = 0.071.3.4. Slowly varying electromagnetic structuresAs already noted in figure 1 an impressive number of localized, slowly varying structures aregenerated in the interaction. In the denser plasma region, the several small-scale structureswhose most evident signature is a strong depression in the plasma density are likely to be rathersimilar to the so-called post-solitons [12, 14] having zero propagation velocity and slowlyexpanding due to ion acceleration driven by the internal radiation pressure. They may bedescribed as small cavities trapping electromagnetic radiation whose frequency is less than theplasma frequency of the surrounding plasma (hence they may be also appropriately named as‘electromagnetic cavitons’). We notice that we do not observe a drift of such structures towardsthe low-density region. This difference from the observations of [11] might be ascribed to thesmoother electron gradient in our case.The regular structures, forming an axially symmetrical row, observed in the low-densityregion near the plasma boundary (far left side in figure 1) have indeed features which are similarto both electromagnetic cavitons and magnetic vortices. This ‘dual’ nature can be observed infigure 4, which shows the components of the fields Ez and Bz perpendicular to the simulationplane as a contour plot and the components in the (x, y) plane as a vector plot. By analyzingthe frequency spectrum of the fields inside the density depression, we find that the fields Ez,Bx and By are oscillating at a frequency of approximately 0.1ω, lower than the local value41 ION DYNAMICS AND COHERENT STRUCTURE FORMATIONB76 A Macchi et alVector Field E (Ex , Ey) t=650100 125 150110130Vector Field B (Bx , By) t=650100 125 150110130100 125 150110130100 125 150110130-0.25-0.000.25-0.150.000.15Figure 4. (Anti-)symmetrical row of slowly varying structures in the low-density region of theplasma at t = 625TL. The left column shows the fields Ez (contour plot) and Bx + By (vector plot)oscillating at a frequency 0.1ω. The right column shows the quasi-static fields Bz (contour plot)and Ex + Ey (vector plot). The laser pulse parameters are aL = 2.7, rL = 8λ and τL = 150TL.of the plasma frequency (for unperturbed plasma) ωp 0.15ω. Qualitatively, the oscillatingfields are similar to those of the lowest TM resonant mode in a cylindrical cavity.The frequency analysis of Ex , Ey and Bz shows that these field components are quasi-static, their spectrum being peaked around zero frequency. The electric field components Exand Ey are in the radial direction with respect to the axis of the structure, as it is expected for acavity expanding under the action of the radiation pressure of the trapped radiation. The staticmagnetic field component Bz is associated with current rings flowing around the axis of thestructure.Apart from being associated with ‘post-soliton’-like structures, the fact that the magneticvortices form a symmetrical row and are localized near the boundary of the channel makesthem different from those observed in the wake of a much shorter laser pulse, for which thecreation of a low-density channel does not occur, and which seem to form an antisymmetricalrow [1,26]. It is nevertheless possible that the current filamentation instability discussed in [26]plays a role in vortex formation also in the present case. In the early stage we observe a strongelectron current in the main channel and two narrow return current sheets just outside thechannel boundaries; later, the current layers seem to bend locally forming vortices aroundmagnetic field maxima.The axial symmetry of these particular structures suggests that in ‘realistic’ 3D geometrythey may have a toroidal or ‘donut’ shape. To get an impression of such a 3D structureone should imagine the field patterns of the 2D simulations rotating around the x-axis. Thisparticular type of coherent structure would be characterized by azimuthal components of E(oscillating) and B (quasi-static) directed along the torus circumference, a solenoidal andoscillating magnetic field coiled up round the torus and by an electrostatic field componentperpendicular to the torus surface. The 3D soliton discussed in [13] has a toroidal magneticfield and a poloidal magnetic field; however, in our case we have no clear indication of thecharge oscillations inside the solitons observed in [13].Inside the main and secondary low-density channels generated in the denser region ofthe plasma, the growth of field patterns which are less regular than those of figure 4, but42Ion dynamics and coherent structure formation B77500 510 520 530 540 550105120105120500 510 520 530 540 550105120135105120135500 520 540105120135500 520 540105120135500 510 520 530 540 550105120135500 510 520 530 540 550105120135500 520 540105120135500 520 540500 510 520 530 540 550105120135500 510 520 530 540 550105120135500 520 540105120135500 520 540Figure 5. Details of the evolution of field structures in the denser region of the plasma, from thesimulation of figure 1. The laser pulse parameters are aL = 2.7, rL = 8λ and τL = 300TL.qualitatively similar, can be observed. Figure 5 gives details of their evolution. We observea tendency of this type of structures to grow inside the channels and to be correlated withrippling and bending of the channel walls. Theoretical work will be required to address thephysics of formation of such structure patterns.4. ConclusionsThe main results emerging from the series of 2D PIC simulations reported in this paper maybe summarized as follows. Ion acceleration due to the space-charge field in the channel drilledby the laser pulse leads to hydrodynamical breaking of the plasma profile at the channel walls.Two side effects of the ion-driven ‘breaking’ have been identified: a change in the radial profileof the electrostatic field (including a sort of ‘echo’ effect for pulses shorter than the breakingtime) and a breakup of ‘long’ laser pulses due to the sudden ‘leak’ generated in the channelwalls. The evolution of coherent, slowly varying field structures has been monitored in timeup to thousands of laser cycles, corresponding to several picoseconds in ‘real’ experiments.Patterns of multi-peak structures appear inside low-density channels, and the formation ofstructures having both oscillating and static field components with a hybrid soliton-vortexnature has been observed. These results, and the perspective of experimental investigations ofsuch field patterns, support the view of relativistic ‘laser plasmas’ as environments showing ahigh degree of self-organization and a wealth of coherent structures, which are thus of greatinterest for the physics of nonlinear systems.AcknowledgmentsThis work has been supported by the Royal Society (UK) via a Joint Project, by the Ministry ofUniversity and Research (Italy) via a PRIN project, and by CNR-INFM and CINECA (Italy)43 ION DYNAMICS AND COHERENT STRUCTURE FORMATIONB78 A Macchi et althrough the supercomputing initiative. Valuable discussions with S V Bulanov and F Pegoraroand the help of E Echkina in the analysis of some simulations are warmly acknowledged.References[1] Askaryan G A et al 1997 Magnetic interaction of ultrashort high-intensity laser pulses in plasmas Plasma Phys.Control. Fusion 39 A137[2] Bulanov S V et al 2001 Relativistic interaction of laser pulses with plasmas Rev. Plasma Phys. 22 227[3] Mourou G A, Tajima T and Bulanov S V 2006 Optics in the relativistic regime Rev. Mod. Phys. 78 309[4] Sun G Z, Ott E, Lee Y C and Guzdar P 1987 Self-focusing of short intense pulses in plasmas Phys. Fluids 30 526[5] Mori W B, Joshi C, Dawson J, Forslund D W and Kindel J M 1998 Evolution of self-focusing of intenseelectromagnetic waves in plasmas Phys. Rev. Lett. 60 1298[6] Borisov A B et al 1992 Relativistic and charge-displacement self-channeling of intense ultrashort laser pulsesin plasmas Phys. Rev. A 45 5830[7] Monot P et al 1995 Experimental demonstration of relativistic self-channeling of a multiterawatt laser pulse inan underdense plasma Phys. Rev. Lett. 74 2953[8] Borghesi M et al 1997 Relativistic channeling of a picosecond laser pulse in a near-critical preformed plasmaPhys. Rev. Lett. 78 879[9] Krushelnick K et al 1997 Plasma channel formation and guiding during high intensity short pulse laser plasmaexperiments Phys. Rev. Lett. 78 4047[10] Sarkisov G S et al 1997 Observation of the plasma channel dynamics and Coulomb explosion in the interactionof a high-intensity laser pulse with a He gas jet JETP Lett. 66 828[11] Sentoku Y et al 1999 Burst of superreflected laser light from inhomogeneous plasmas due to the generation ofrelativistic solitary waves Phys. Rev. Lett. 83 3434[12] Naumova N M et al 2001 Formation of electromagnetic postsolitons in plasmas Phys. Rev. Lett. 87 185004[13] Esirkepov T, Nishihara K, Bulanov S V and Pegoraro F 2002 Three-dimensional relativistic electromagneticsubcycle solitons Phys. Rev. Lett. 89 275002[14] Borghesi M et al 2002 Macroscopic evidence of soliton formation in multiterawatt laser-plasma interactionPhys. Rev. Lett. 88 135002[15] Farina D and Bulanov S V 2001 Relativistic electromagnetic solitons in the electron-ion plasma Phys. Rev. Lett.86 5289[16] Lontano M, Passoni M and Bulanov S V 2003 Relativistic electromagnetic solitons in a warm quasineutralelectron-ion plasma Phys. Plasmas 10 639[17] Lehmann G, Laedke E W and Spatschek K H 2006 Stability and evolution of one-dimensional relativistic solitonson the ion time scale Phys. Plasmas 13 092302[18] Weber S, Lontano M, Passoni M, Riconda C and Tikhonchuk V T 2005 Electromagnetic solitons produced bystimulated brillouin pulsations in plasmas Phys. Plasmas 12 112107[19] Riconda C, Weber S, Tikhonchuk V T, Adam J-C and Heron A 2006 Two-dimensional particle-in-cell simulationsof plasma cavitation and bursty brillouin backscattering for nonrelativistic laser intensities Phys. Plasmas13 083103[20] Baiwen Li, Ishiguro S, Skoric M M and Sato T 2007 Stimulated trapped electron acoustic wave scattering,electromagnetic soliton and ion vortices in intense laser interaction with subcritical plasmas Phys. Plasmas14 032101[21] Borghesi M et al 2001 Proton imaging: a diagnostic for inertial confinement fusion/fast ignitor studies PlasmaPhys. Control. Fusion 43 A267–76[22] Kar S et al 2007 Dynamics of charge-displacement channeling in intense laser-plasma interactions (preprintphysics/0701332)[23] Liseykina T V et al 2007 Generation and observation of coherent, long–lived structures in a laser–plasma channel(preprint physics/0702177)[24] Naumova N M et al 2001 Polarization, hosing and long time evolution of relativistic laser pulses Phys. Plasmas8 4149–55[25] Macchi A, Ceccherini F, Cornolti F, Kar S and Borghesi M 2007 Ponderomotive laser ion acceleration andelectric field dynamics following charge–displacement channeling (preprint physics/0701139)[26] Bulanov S V, Lontano M, Esirkepov T Zh, Pegoraro F and Pukhov A M 1996 Electron vortices produced byultraintense laser pulses Phys. Rev. Lett. 76 356244C. R. Physique 10 (2009) 207–215Laser acceleration of particles in plasmas / Accélération laser de particules dans les plasmasTheory and simulation of ion acceleration with circularly polarizedlaser pulsesAndrea Macchi a,b,∗, Tatiana V. Liseikina c,1, Sara Tuveri b, Silvia Veghini ba CNR/INFM/polyLAB, 56127 Pisa, Italyb Department of Physics “E. Fermi”, University of Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italyc Max Planck Institute for Nuclear Physics, 69029 Heidelberg, GermanyAvailable online 29 April 2009AbstractIon acceleration driven by the radiation pressure of circularly polarized pulses is investigated via analytical modeling andparticle-in-cell simulations. Both thick and thin targets, i.e. the “hole boring” and “light sail” regimes are considered. Parametricstudies in one spatial dimension are used to determine the optimal thickness of thin targets and to address the effects of preformedplasma profiles and laser pulse ellipticity in thick targets. Three-dimensional (3D) simulations show that “flat-top” radial profilesof the intensity are required to prevent early laser pulse breakthrough in thin targets. The 3D simulations are also used to addressthe issue of the conservation of the angular momentum of the laser pulse and its absorption in the plasma. To cite this article:A. Macchi et al., C. R. Physique 10 (2009).© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméThéorie et simulation de l’accélération des ions par impulsions laser à polarisation circulaire. L’accélération des ions par lapression de radiation des impulsions laser avec polarisation circulaire a été étudiée à l’aide de modèles analytiques et de simulations“particle-in-cell”. Les deux régimes de cibles épaisses et minces, c’est-à-dire de “hole boring” et “light sail” ont été considérés.Des études paramétriques dans une dimension spatiale ont été réalisées afin de déterminer l’épaisseur optimale des cibles minceset pour étudier l’effet des profils de densité plasma préformés et l’effet de l’ellipticité de l’impulsion dans cibles épaisses. Lessimulations tridimensionnelles (3D) montrent que des impulsions avec des profils radiaux plats en intensité sont nécessaires pourprévenir la pénétration de l’impulsion à travers la cible. Les simulations 3D ont aussi été utilisées pour étudier la conservation dumoment angulaire de l’impulsion laser et son absorption dans le plasma. Pour citer cet article : A. Macchi et al., C. R. Physique10 (2009).© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Keywords: Laser–plasma acceleration; Ion acceleration; Radiation pressure; Circular polarizationMots-clés : Accéleration laser–plasma ; Accélération des ions ; Pression de radiation ; Polarisation circulaire* Corresponding author at: Department of Physics “E. Fermi”, University of Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy.E-mail address: macchi@df.unipi.it (A. Macchi).URL: http://www.df.unipi.it/~macchi (A. Macchi).1 On leave from Institute of Computational Technologies, SD-RAS, Novosibirsk, Russia.1631-0705/$ – see front matter © 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crhy.2009.03.00245 ION ACCELERATION WITH CP LASER PULSES: THEORY AND SIMULATIONS208 A. Macchi et al. / C. R. Physique 10 (2009) 207–2151. IntroductionRadiation Pressure Acceleration (RPA) is a possible route to the acceleration of ions up to energies in the relativisticdomain (GeV/nucleon), alternative to the Target Normal Sheath Acceleration (TNSA) mechanism which explains ionacceleration from thin solid targets in present-day experiments (see [1] for a review). The interest in RPA was stimu-lated by Particle-In-Cell (PIC) simulations of Esirkepov et al. [2,3] showing that at intensities I > 5 × 1021 W cm−2RPA starts to dominate over TNSA, and that at I 1023 W cm−2 and using a thin foil target, an efficient generationof GeV ions and a linear scaling of ion energy vs. the pulse energy may be obtained. These characteristic features ofRPA, very appealing for foreseen applications of ion acceleration, may be qualitatively understood using the simplemodel of a perfectly reflecting mirror accelerated by a normally incident plane wave. Due to the Doppler effect, thefrequency ω of each incoming photon reflected by the mirror moving at the velocity V = βc in the laboratory frame isdownshifted to ω′ according to the relation ω′ = ω(1 − β)/(1 + β), so that almost all of the energy h̄ω of the photonis delivered to the target in the limit β → 1 and, since the number of photons is conserved for a perfect mirror, acomplete conversion of the wave energy into mechanical energy is obtained.Intensities exceeding 1023 W cm−2 are not available yet, but a dominance of RPA over TNSA may be alreadyobtained at much lower intensity if a laser pulse with circular polarization (CP), instead of linear polarization (LP), isused. In fact, in such conditions the acceleration of “fast” electrons at the laser–plasma interface is almost suppressed,ruling out TNSA which is driven from the space charge produced by energetic electrons escaping in vacuum. Thesuppression of fast electrons for CP can be understood by noting that models of electron acceleration at a sharpplasma surface ([4,5] and references therein) require the driving force to have an oscillating component along thedensity gradient. For normal incidence, such component is given by the J × B term at 2ω (with experimental evidencegiven by Ref. [6]) which however vanishes for CP (see Section 2 for a discussion based on a simple model).Radiation Pressure Acceleration using Circular Polarization (CP-RPA) was first studied by our group [7,8] byconsidering “thick” targets, such that during the laser pulse only a finite layer of the target at its front surface is ac-celerated, forming a dense bunch of ions (neutralized by “cold” electrons) entering the target. Dramatic differences inion acceleration were observed between CP and LP [7,8]. Later, three groups [9–12] studied the acceleration of “thin”foils, such that the whole target is accelerated. In this regime, the use of CP is particularly important to prevent the foilexpansion due to the “thermal” pressure of electrons, allowing the acceleration of the target as a single “rigid” objectand preserving the inherent monoenergeticity after the acceleration stage. In good accordance with the predictionsof the “mirror” model, it was found that sub-micrometric targets may be accelerated up to energies correspondingto GeV/nucleon using petawatt pulses with picosecond duration, which are feasible with current technology. Thisperspective has stimulated additional theoretical and numerical investigations [13–16].In this article, we review our analytical and simulation work [7,8,17,18] on both the “thick” and “thin” regimesof CP-RPA which are also named “hole boring” (HB) and “light sail” (LS) regimes, respectively. Most recent workfrom our group included a) a study of the “optimal” target thickness in the LS case, b) a preliminary evaluation of“preplasma” effects, and c) three-dimensional (3D) simulations of CP-RPA to address in particular the issue of theabsorption of the angular momentum of the laser pulse by the plasma.2. The role of the pulse polarization: a simple modelA simple and possibly pedagogical “minimal” model to show the role of the polarization in laser interaction withan overdense plasma at normal incidence may be described as follows. We assume a plane, elliptically polarized waveof frequency ω incident on an overdense plasma with a step-like profile of the electron density ne(x) = n0Θ(x), andωp > ω being ωp =√4πn0e2/me the plasma frequency. For sufficiently low intensity (neglecting relativistic andstrong charge separation effects) the vector potential inside the plasma has the formA(x, t) = A(0)√1 + ε2 e−x/dp (ŷ cosωt + εẑ cosωt) (1)where dp = c/√ω2p − ω2 and ε is the ellipticity (0 < ε < 1). The longitudinal force on electrons due to the v×B termis then obtained asFx = − e22mec2∂xA2 = F0e−2x/dp(1 + 1 − ε21 + ε2 cos 2ωt)(2)46A. Macchi et al. / C. R. Physique 10 (2009) 207–215 209where F0 = (e2A(0)2/2dpmec2). By solving the equations of the longitudinal motion of electrons for the secular and2ω components we obtain the electric field and the electron density perturbationEx = F0ee−2x/dp(1 + 1 − ε21 + ε2cos 2ωt1 − 4ω2/ω2p), δne = − 14πe∂xEx (3)The denominator in the oscillating term of Ex arises from the resonant excitation of plasma oscillations when2ω = ωp . For LP (ε = 0), the peak amplitude of the oscillating term is larger than the secular term, and as a con-sequence some electrons are dragged into the vacuum side, as can be evidenced by calculating the quantity Ne(t) =+∞∫0δne dx = F04πe2(1 + 1 − ε21 + ε2cos 2ωt1 − 4ω2/ω2p)(4)When Ne(t) > 0, electrons are piling up inside the plasma and a positive surface charge appears to ensure quasineu-trality. However, if the oscillating term becomes larger than the secular one (and this always happens if ε = 0, i.e. forLP), Ne(t) < 0 occurs implying that some electrons cross the x = 0 surface entering the vacuum side. In this casea regular, periodic solution does not exist anymore. This is a signature of the onset of self-intersection of electrontrajectories in vacuum leading to heating and appearance of fast electron “jets” twice per laser cycle. For CP (ε = 1),however, the oscillating terms vanish, the electric force on electrons balances the ponderomotive force (Fx −eEx = 0)and no electron heating occurs. The model suggests an ellipticity threshold εT = (ω2p/2ω2 − 1)−1/2 such that forε > εT we have Ne(t) > 0 at any t , and “vacuum heating” should be inhibited.These conclusions can be verified to hold also at higher intensity with numerical methods. For example, a PICsimulation with fixed ions and a CP, semi-infinite laser pulse with a sufficiently slow rising edge shows that theequilibrium profiles of Ex and δne are in excellent agreement with an exact analytical solution including relativisticand strong charge separation effects [19,20].3. Thick targets: density breaking and ion bunch generationThe electrostatic field Ex created by the electron displacement accelerates ions modifying the density profile.A self-consistent, time-dependent solution of the wave penetration is not available. In Ref. [7] a phenomenologicalmodel giving simple analytical solutions and accounting for the most prominent features observed in simulations forthe “thick” target or “hole boring” regime was obtained by taking an “ad hoc” spatial and temporal profile of theponderomotive force Fx and assuming the quasi-equilibrium condition for electrons to hold as the ions move. A briefdescription of the model is given in Fig. 1 and its caption.The equations of motion are valid until the time τb at which all the ions initially located in the layer of electroncompression (xd < x < xs in Fig. 1) reach the point x = xs ; here, the ion density becomes singular and the fastestions overcome the slowest ones so that hydrodynamical or “wave” breaking for the ion fluid occurs. At the time ti , themaximum ion velocity and the corresponding energy for a laser pulse of constant intensity are given byvimc= 2√ZAmempncneaL, Em = 12miv2im = 2Zmec2ncnea2L (5)where aL = 0.85(Iλ2/1018 W µm2/cm2)1/2 is the laser amplitude in dimensionless units. These formulas are validfor non-relativistic ion velocities; relativistic corrections are considered in Ref. [16].The details of the ion spectrum depend on the highly transient stage of wave breaking and may be studied mostlyvia simulations. Fig. 1b) shows three snapshots from a 1D PIC simulation, showing the generation of the electrostaticfield, the very strong spiking of the ion density at the instant of wave breaking and the formation of a short-duration,high-density “bunch” moving in the forward direction at velocity vim. The heating of electrons is significant onlyaround the “breaking” event and the electron energies are lower by nearly two order of magnitudes than the valueobserved for LP interaction at the same intensity [7,8]. A detailed study of the dynamics of wave breaking in a verysimilar context has been reported in Ref. [21].47 ION ACCELERATION WITH CP LASER PULSES: THEORY AND SIMULATIONS210 A. Macchi et al. / C. R. Physique 10 (2009) 207–215Fig. 1. a): model for RPA of ions at the front target surface [7,17]. The plots show the spatial profiles of Ex (red), ni (blue) and ne (green) atthree different times. The ponderomotive force Fx = eEx in the region where ne = 0. Due to conservation laws, model parameters are relatedby E0 = 4πen0xd , n0xs = np0(xs − xd ), and E0enp0(xs − xd )/2 2I/c where xs − xd c/2ωp . At the time t1, ions have not moved yetand electrons have been pushed creating the charge separation field; at t2, Ex accelerates ions and pile them up in the evanescence region, whileelectrons rearrange themselves to keep the force balance condition; at t3, the density peaks up to infinite values since all ions starting from thexd < x < xs get to the point x = xs at the same time, and “wave breaking” occurs. b): 1D PIC simulation of the interaction of a short CP laserpulse with an overdense, step-boundary plasma [17]. The figure shows ni and Ex (top row) and the (x,px ) phase space distribution for ions (fi ,middle row) and electrons (fe , bottom row) at different times in units of the laser period TL . The laser pulse has peak amplitude a = 2 and durationτ = 6TL (FWHM). The initial density is n0 = 5nc where nc is the cut-off density. The x coordinate is normalized to λ, the density to nc , theelectric field to meωc/e and the momenta to mec and mic, respectively.Some further insight into the dynamics of ion bunch formation is obtained by looking at simulations with ellipticalpolarization for different values of the ellipticity ε, since this allows to vary the relative importance of secular andoscillating components in the accelerating fields. They are also of interest to test the sensitivity of the RPA regime tovalues of ε < 1 as it may occur in experiments. The snapshots shown in Fig. 2 for simulations having the same plasmatarget and a laser pulse of fixed energy and duration but different values of ε show the tendency to the formation ofmultiple ion “bunches”. This effect may be explained by noting that, due to the oscillating component in Ex , ionsnow cross the evanescence point at different times corresponding to positive maxima of Ex . The phase space forε = 0.5 shows the “X”-type structure whose origin has been discussed in Ref. [21]. The electron heating is observedto increase as ε decreases from 1 (CP) to 0 (LP). Very recently a similar study has been reported in Ref. [22], showinga “threshold” value for ε which depends weakly on the laser intensity. This is in qualitative agreement with oursimple model (Section 2) where the intensity dependence may arise in the resonant denominator due to the relativisticcorrection to the plasma frequency ω2p → ω2p/√1 + a2.Parametric 1D runs were also performed to study the effect of a “preplasma”, i.e. a density profile smoother than astep-like one, which could be typically produced in experiments by a prepulse preceding the ultrashort, superintensepulse. Such study is also of interest because the scaling laws (5) predict the ion energy to be inversely proportional tothe density. Although these relations have been obtained for a step-like profile, they suggest that in a preplasma theinteraction occurs near the ne nc layer leading to higher ion energies. This effect was confirmed in the preliminarysimulations shown in Ref. [18], which show that ion bunch formation occurs also in a preplasma, suggesting thepossible use of prepulse control to achieve higher energies. The dependence of ion energy and conversion efficiency(defined as the ratio of the total energy of the ions over the energy of the laser pulse) on the density scalelengthL = nc/|∂xn0|n0=nc for a given laser pulse are shown in Fig. 2b).In general, because of the large value of ne/nc (∼102) for solid densities and the typical laser wavelength ∼1 µm,any possibility (e.g., by special target materials) to achieve a laser–plasma interaction at smaller values of nc/ne wouldbe important to obtain high ion energies in the “hole boring” regime. We note that, usually, 2D and in particular 3Dsimulations are performed for relatively small values of ne/nc because of computational limitations, since the needto resolve length scales smaller than c/ωp ∼ 1/√ne and density regions where ne < nc forces to use large numbersof small cells and large numbers of particles. At least, an estimate of the scaling of all relevant quantities such asion energy and conversion efficiency with ne is thus needed to extrapolate simulation results to feasible experimentalparameters.48A. Macchi et al. / C. R. Physique 10 (2009) 207–215 211Fig. 2. Results from parametric 1D simulations. a): snapshots of ni and fi(x,px) (in log10 scale) from simulations with different ellipticity ε.Common parameters to all runs are n0 = 10nc , a = 2, and τ = 12TL . b): interaction with preformed plasmas [18]. Conversion efficiency (red filleddiamonds, solid line) and peak energy (blue empty diamonds, dashed line) of ions as a function of the density scalelength L in preplasma and thelaser amplitude a. The density profile is described by a ∼(x − x0)4 function up to a peak density n0 = 16nc . The pulse duration τ = 9TL .4. Thin targets: optimal thickness and heating effectsThe analysis of the RPA dynamics for “thick” targets outlined above now allows us to define “thin” targets in thiscontext. In thick targets, after wave breaking the fastest ions overturn the slowest ones, penetrate into the overdenseplasma and are not accelerated anymore. To obtain high energies the laser pulse must be able to repeat the accelerationstage over the same ions. Thus, the target thickness must be close to the depth of the compression layer (the parameters = xs − xd in Fig. 1) in order for repeated or “cyclic” RPA to occur. If the solid target contains hydrogen ions,they will gain higher velocity in a first stage, but overturning other ions they will be screened by the laser pulseuntil heavier ions reach them. Thus, ions of different Z/A ratio will tend to ultimately be accelerated to the samevelocity, so that RPA of thin foils appears to be most suitable for the acceleration or ions heavier than protons, e.g.,carbon.Although due to the above considerations the target may not be considered as a “rigid” object, the acceleratingmirror or “light sail” model gives reasonable estimates for the target velocity V = βc as a function of the laser pulseenergy. The equations of motion ared(γβ)dt= 2I (t − X/c)ρdc21 − β1 + β R(ω′), dXdt= βc (6)where ρ and d are the mass density and thickness of the mirror and R is its reflectivity, which can be written as afunction of the pulse frequency in the rest frame ω′. An analytical solution is found for R = 1 and constant intensityγβ =[sinhψ(t) − 14 sinhψ(t)], ψ(t) = 13arcsinh(6I tρdc2+ 2)(7)Higher velocities and efficiency are expected for lighter targets, but as ρd decreases the target tends to become trans-parent to the laser light. One thus expect to find an “optimal” thickness dopt to exist for a given laser intensity. Fig. 3a)shows a set of 1D parametric simulations in a range of presently accessible intensities (1019–1021 W cm−2). It is foundthat dopt increases with pulse intensity (presumably due to induced relativistic transparency) and has typical values of≈10−2 µm. Such ultrathin targets are technologically feasible but will require the complete suppression of prepulsesin experiments.We obtained an analytical solution for the motion of the mirror even in the case of partial reflectivity by using forthe latter the model of a “delta-like” foil [23] and including the effects of self-induced transparency. The resultingenergy per nucleon as a function of the target thickness and the pulse fluence (energy per unit surface) is shown inFig. 3b). There is qualitative agreement with the numerical simulations although the model predicts lower ion energies.49 ION ACCELERATION WITH CP LASER PULSES: THEORY AND SIMULATIONS212 A. Macchi et al. / C. R. Physique 10 (2009) 207–215Fig. 3. a): energy conversion efficiency (red filled diamonds) and peak energy per nucleon (blue empty diamonds) as a function of the targetthickness d and pulse amplitude a from 1D PIC simulations [18]. Common parameters to all runs are τ = 9TL , ne = 250nc and Z/A = 1/2.b): analytical calculation of the energy per nucleon as a function of the areal density ned of the target and of the pulse fluence. c): energy spectraof ions and electrons from a simulation of the interaction of a 400 fs, 1.8 × 1020 W cm−2 pulse with a 0.056 µm thick carbon target (ne = 250nc),for two different times (in fs).A simple energy balance argument shows that to accelerate a solid target of “optimal” thickness ≈10−2 µm up tovelocities corresponding to energies per nucleon ≈1 GeV, a pulse energy per unit surface ≈108 J cm−2 is needed.The width of the energy spectrum is also an issue. The simulation results shown in Fig. 3c) show that a spectrum with5% spread is obtained at the end of the laser pulse. However, some broadening of the energy spectrum is observedalso after the acceleration stage. This seems to be related to a relatively abrupt increase of the absorption fraction intoelectrons near the end of the laser pulse, as the radiation pressure decreases. The resulting electron “temperature”,although orders of magnitude lower than the quiver energy, may be sufficient to cause some expansion of the thin foiland to broaden the ion spectrum.5. Multi-dimensional effectsThe above analysis is based on 1D modeling and simulations, which for example do not account for effects relatedto the finite transverse size of the laser pulse. As an obvious consequence, the intensity distribution implies that theradiation pressure on the plasma is inhomogeneous, leading to a broader ion energy spectrum, with less energetic ionsat the periphery of the laser spot. Moreover, a finite pulse waist of radius rL implies longitudinal field componentsE‖ ≈ (λ/rL)E⊥ at the edge of the spot, causing local heating of electrons. The bending of the plasma surface causedby hole boring or foil deformation also increases the electron heating because the local angle of incidence does notvanish anymore. However, 2D simulations [7,8,11,12] show that these effects do not cause a failure of the CP-RPA, andthe differences with the LP case are still evident [8]. It has been shown that pulses with a “flat-top” (e.g., supergaussian)[11] intensity profile allow to preserve a narrow ion spectrum and a very low divergence (a few degrees).The onset of surface instabilities, e.g. of density rippling, has also been studied by 2D simulations. A Rayleigh–Taylor-like instability driven by radiation pressure has been characterized in Ref. [24] for thin foils accelerated by LPpulses. However, the comparison of thick targets simulations for CP and LP show that the surface instability is weakerfor CP [17]. This suggests that additional effects besides RPA contribute to the dynamics of the instability.Since a CP wave carries a net angular momentum, conservation of the latter poses an additional constraint onthe interaction. This gives a specific motivation for 3D simulations of CP-RPA. The issue of angular momentumabsorption in the plasma has been studied in the past mostly for underdense plasmas and in close connection withthe Inverse Faraday Effect, i.e. the generation of axial magnetic fields, and apparently it has often been a subjectof controversy and misunderstanding (see e.g. [25,26] and references therein). In the present context it is noticeable50A. Macchi et al. / C. R. Physique 10 (2009) 207–215 213Fig. 4. Three-dimensional simulations of CP-RPA of a thin foil target. a): the ion density ni at t = 40TL for two different radial profiles of thelaser intensity, a supergaussian ∼exp(−r4/r4s ) or “flat-top” (left) and a Gaussian ∼exp(−r2/r2g ) (right) profile, with rs = 2.5λ and rg = 2.4λ.Common simulation parameters are n0 = 16nc , d = 0.4λ, a = 5 and τ = 10TL . b): the azimuthal ion current Ji,ϕ averaged over the transverseplane (y, z), t=130 fs versus longitudinal coordinate, showing the angular momentum absorption into ions [18]. The black and red lines correspondto an average over a circle with radius r = 4.5λ and 2.5λ, respectively, around the x-axis.that, according to the accelerating mirror model, absorption of angular momentum by the target may vanish even fornearly complete energy absorption. In fact, the photon “spin” is h̄ and independent from the frequency, while in thereflection of a photon from the mirror the direction of the spin is not reversed (as it happens for the momentum), so thatthe conservation of photon number in any frame implies that no angular momentum is absorbed by the target at all.Thus, the amount of angular momentum absorbed during CP-RPA gives an indication of non-adiabatic or dissipativeprocesses at play in the laser–plasma interaction.Fig. 4a) shows a 3D snapshot of the ion density for two simulations having identical parameters but different radialprofiles of the intensity, i.e. a Gaussian and a supergaussian “flat-top” profile. The thickness of the target was chosento be close to the “optimal” value inferred from the 1D parametric study. However, for the Gaussian pulse case, thelateral expansion of the central region of the target causes an early breakthrough of the laser pulse through the plasma,with a detrimental effect on the ion acceleration. The use of a “flat-top” pulse prevents the breakthrough and leads toan ion beam with very low divergence and energy values close to those inferred from the 1D analysis.The fraction AL of the angular momentum of the pulse transferred to the plasma was evaluated directly fromthe phase space distribution of the particles and compared to the energy absorption AE . It is found that AL AEshowing that a substantial part of the energy is absorbed via non-adiabatic processes (i.e. violating the conservationof photon number). It is noticeable that the density of axial angular momentum of the incident pulse is given byx = x(r) = − r2cω ∂rI (r), which peaks at the edge of the beam where the axial components of the oscillating electricfield have their maximum, i.e. where most of the electron heating occurs. Thus, non-adiabatic heating of electronsat the edge of the beam is likely to provide a channel for angular momentum absorption. However, the simulationsshow that eventually, after the end of the laser pulse, most of the absorbed angular momentum is given to ions. Theabsorption of angular momentum should lead to a torque on the ions and to the appearance of a steady azimuthalcurrent. Such a net current can be evidenced by an average over the transverse plane, as in Fig. 4b), as the actualdistribution is rather complicated. No regular magnetic field is found on the axis, i.e. there is no significant InverseFaraday Effect associated to CP-RPA.6. DiscussionThe “light sail” regime of CP-RPA is very attractive for the possibility to accelerate a large number of ions (pre-sumably of carbon or heavier elements) to GeV energies. The scheme requires ultrathin targets and, consequently,laser pulses with extremely high contrast [27]. Our simulations show that, in addition, flat-top intensity profiles maybe crucial to reach the highest energies as well as to achieve high collimation and a monoenergetic spectrum.The thick target or “hole boring” regime allows one to reach much lower energies with present-day intensities andsolid target densities. However, this regime requires less critical conditions than the “light sail”, and may be useful forapplications requiring high densities of ions with moderate energies (up to a few MeV). Moreover, our preliminary51 ION ACCELERATION WITH CP LASER PULSES: THEORY AND SIMULATIONS214 A. Macchi et al. / C. R. Physique 10 (2009) 207–215study of RPA in preformed plasma profiles suggest that the scaling with density allows to obtain energies higher thanthe values expected for solid densities and same pulse parameters. It might be also of interest to investigate RPA withCO2 pulses in gas jets, where the gas density can be tuned to be only slightly over nc and, in addition, high repetitionrates would be possible.Presently, no experimental results on CP interaction at normal incidence (or in conditions significantly close tosuch “optimal” conditions) have been reported yet in publications, but devoted experiments are planned or havebeen proposed in several facilities. For interactions with linearly polarized pulses, radiation pressure effects havebeen claimed to play a dominant role in a few experiments for different conditions [28–30] as well as in severalphenomena observed in simulations (e.g. “laser piston” [2], surface rippling [24] or “shock acceleration” [31]). Thestudy of circularly polarized interactions, besides its potential for ion acceleration, may be very useful for a detailedunderstanding of radiation pressure effects as these latter can be separated from effects due to fast electrons.AcknowledgementsWe are grateful to M. Borghesi, F. Cattani, F. Cornolti and D. Prellino for their previous contributions on the topicsof this paper and to many colleagues including D. Bauer, F. Ceccherini, T. Ceccotti, S. Kar, E. Lefebvre, F. Pegoraro,S. Propuzhenko, and M. Zepf, for useful discussions or suggestions. We thank I. Klonova for the French translationof the abstract. The 3D simulations were performed on the computing clusters at CINECA, Bologna (sponsored bythe CNR/INFM supercomputing initiative) and at MPI-K, Heidelberg. Part of the work was performed during a stayof two of the authors at Queen’s University, Belfast, supported by a Visiting Research Fellowship (A.M.) and byCOST-P14 (S.T.). 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Liseikina, Domenico Prellino, Fulvio Cornolti, and Andrea MacchiAbstract—Ponderomotive (or radiation pressure) acceleration isa route to the generation of high-energy ions in laser–solid inter-actions, which does not rely on the generation of fast electrons.It may be optimized by the use of circularly polarized (CP) laserpulses rather than linearly polarized (LP) ones, since for CP, fastelectron generation is strongly inhibited, quenching competingmechanisms such as target normal sheath acceleration. We presenta comparison of 1-D and 2-D particle-in-cell simulations for CPand LP, which shows the potential of CP pulses for ion accel-eration. The comparison also enables us to discriminate “pure”ponderomotive effects from those due to fast electrons, aidingthe understanding of basic mechanisms of ion acceleration andsurface rippling.Index Terms—Circular polarization (CP), ion acceleration,laser–plasma accelerators, ponderomotive force (PF), radiationpressure acceleration (RPA).I. INTRODUCTIONWHEN a laser pulse with intensity typically in the1017–1020 W/cm2 range impinges on a solid target,there is more than one possible route to energy transfer from thelaser pulse to ions. The laser–plasma interaction at the front sur-face of the target can generate large numbers of “fast” electronswith several MeV of energy, which attempt to escape outsidethe target, creating a very strong electric field. At the rear side ofthe target, the electric field efficiently accelerates ions initiallylocated on the surface, in particular, protons from hydrogencontaminants. This is the basis of the target normal sheathacceleration (TNSA) mechanism, leading to the productionManuscript received November 15, 2007. This work was supported in part bythe Italian Ministry for University and Research (MIUR) via the PRIN project“Ultraintense laser–plasma interaction,” by the Italian National ResearchCouncil–National Institute for the Physics of Matter (CNR–INFM) and theInteruniversity Consortium High Performance Systems (CINECA), Bologna,Italy, through the supercomputing initiative, and by the Joint SupercomputerCenter, Moscow, Russia.T. V. Liseikina is with the Dipartimento di Fisica “Enrico Fermi,” Universitádi Pisa, 56127 Pisa, Italy, with the Institute of Computational Technolo-gies, Siberian Division, Russian Academy of Sciences (SD RAS), 630090Novosibirsk, Russia, and also with the Max Planck Institute for NuclearPhysics, 69117 Heidelberg, Germany.D. Prellino was with the Dipartimento di Fisica “Enrico Fermi,” Universitádi Pisa, 56127 Pisa, Italy. He is now with the German Cancer Research Center,69120 Heidelberg, Germany.F. Cornolti is with the Dipartimento di Fisica “Enrico Fermi,” Universitá diPisa, 56127 Pisa, Italy.A. Macchi is with the polyLAB, CNR/INFM, 56127 Pisa, Italy and also withthe Dipartimento di Fisica “Enrico Fermi,” Universitá di Pisa, 56127 Pisa, Italy(e-mail: macchi@df.unipi.it).Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TPS.2008.927141of proton beams with remarkable properties and several pos-sible applications (see [1] for a review). Recent results havesuggested that appropriate target engineering can improve thebeam quality and control for both protons [2], [3] and heavierions [4].Another route to ion acceleration is based on the fundamen-tal concept of the radiation pressure exerted by the laser pulseon the target. In an elementary description, the target maybe represented by a mirror boosted by the total force F (2I/c)S, where I is the laser intensity, and S is the mirror sur-face. In a more detailed description, and within the assumptionof adiabatic and quasi-equilibrium dynamics, the local forceacting on ions is given by Zfp, where Z is the ion charge, andfp is the steady ponderomotive force (PF) acting on electrons.In general, fp = −mec2∇(1 + 〈a2〉)1/2, where a = eA/mec,A is the vector potential, and the brackets denote a cycleaverage. Several theoretical studies on ion acceleration at thefront surface are based on a ponderomotive picture where theions “directly” gain momentum from the laser pulse [5]–[7]. Aphenomenological description [6] shows that ion accelerationdriven by the PF occurs at the front side of the target as asequence of short ion bunches.While in most of present-day experiments ion accelerationfrom solid targets is dominated by TNSA, the interest in pon-deromotive or, more generally speaking, radiation pressure ac-celeration (RPA) arises from theoretical and simulation studiespredicting that RPA becomes dominant at intensities exceeding1023 W/cm2 [8] and leads to very efficient production of highlycollimated ions with energies approaching GeV values. Therequired intensities may become available in the next decadedue to the building of novel laser facilities such as the ExtremeLight Infrastructure.It is, however, possible to obtain a regime of RPA dominancewith present-day laser systems using circularly polarized (CP)pulses, instead of linearly polarized (LP) ones, at normal in-cidence. In fact, in this configuration, the generation of fastelectrons is almost suppressed [6], strongly quenching TNSAand other effects driven by energetic electrons. Simulations ofion acceleration with CP in a thick overdense plasma targetshow high conversion efficiency, strong collimation, and thepossibility of the ultrashort duration of the ion bunch [6], [9].The use of ultrathin targets may further increase the ion energyand may lead to a mononergetic spectrum [10]–[12].In this paper, we review our previous results on ion acceler-ation with CP, emphasizing the differences with the LP case.We also discuss more recent works related to issues such asanalytical modeling, scaling laws for ion energy and conversionefficiency, and rippling instabilities at the target surface.0093-3813/$25.00 © 2008 IEEE55 PONDEROMOTIVE ACCELERATION OF IONSLISEIKINA et al.: PONDEROMOTIVE ACCELERATION OF IONS: CIRCULAR VERSUS LINEAR POLARIZATION 1867Fig. 1. One-dimensional PIC simulation of the interaction of a short CP laserpulse with an overdense step-boundary plasma. The figure shows the (top row,solid line) ion density ni, (top row, dotted) electrostatic field Ex, and(x, px) phase-space distribution for (middle row) ions fi and (bottom row)electrons fe at different times in laser cycles. For a laser wavelength λL =1 µm, the laser and plasma parameters correspond to an irradiance Iλ2L =5.5× 1018 W · µm2/cm2, a pulse duration τL = 20 fs, and an initial electrondensity ne = 5.5× 1021 cm−3. The x-coordinate is normalized to λL =1 µm, the density to nc = 1.1× 1021 cm−3, and the momenta to mec andmic, respectively. The electric field is normalized to En = meωLc/e.II. THEORYThe physical basis for the difference between LP and CPwhen a laser pulse impinges at normal incidence on an over-dense plasma (density ne > nc, where nc = meω2L/4πe2 isthe cutoff density for an EM wave of frequency ωL) is theabsence of the oscillating component (at frequency 2ωL) inthe longitudinal J×B force for CP. In fact, fast electrongeneration at normal incidence is due to forced nonadiabaticoscillations of electron driven by the oscillating component,which is present only for LP. For CP, the J×B force has onlythe secular cycle-averaged component (i.e., the PF). Hence, thecomparison between LP and CP is useful to separate effects dueto the PF (i.e., due to radiation pressure) from those due to fastelectrons. A simple phenomenological model of ponderomotiveacceleration by ultrashort laser pulses was then formulated [6]on the basis of the dynamics observed in particle-in-cell (PIC)simulations.Fig. 1 shows three snapshots from the interaction of a shortCP pulse with an overdense plasma slab. In the early stage,we observe the generation of a strong electrostatic field due tothe pushing of electrons by the PF. It can be shown that thePF and the electrostatic force produced by charge separationalmost cancel each other, so that a quasi-equilibrium dynamicsfor electrons can be assumed and no substantial electron heatingoccurs, as can be observed in the phase-space distribution. Theelectrostatic field accelerates ions, leading to the formation ofa very sharp peak of the ion density at the end of the skinlayer, i.e., near the evanescence point of the EM field. Whenthe fastest ions (coming from the target surface) overturn theslowest ones, hydrodynamical breaking occurs (which is appar-ent in the ion phase-space plot) and the density peak “splits” inFig. 2. Simple model of ponderomotive acceleration. (a) Initial equilibriumconfiguration with immobile ions. (b) Ion acceleration leads to compression ofthe ion profile. (c) Density peaks up to infinite values since all ions get to thepoint x = d+ ls at the same time.two parts, creating a narrow bunch of ions propagating into thetarget. We also observe that longitudinal heating of electrons(up to the modest energies) occurs after the “breaking” ofthe ion fluid. It may be interesting to notice that a rathersimilar dynamics has been observed in simulations of radialponderomotive acceleration by a laser pulse propagating in anunderdense (ne < nc) plasma [13].The simple model accounting for the ion density spiking andsubsequent breaking and providing scaling laws for ion bunchgeneration is described as follows: A linear profile of Ex is as-sumed both in the depletion layer (Ex = E0x/d for 0 < x < d)and in the compression region (Ex = E0[1− (x− d)/ls] ford < x < d+ ls), and a self-consistent uniform electron den-sity np0 is assumed in this latter region [see Fig. 2(a)]. Theparameters E0, np0, d, and ls are related by the equationsE0 = 4πen0d (due to Poisson equation), n0(d+ ls) = np0ls(due to global charge conservation), and E0en0ls/2 2IL/c(due to the balance between the total radiation and electrostaticpressures). The “evanescence length” ls can be chosen as theonly free parameter. It can be estimated by a proper approxima-tion of the exact initial equilibrium profile of PF for immobileions, which can analytically be found [14], by the “ad hoc”profiles of Fig. 2. This approximation yields ls c/2ωp in thelimit d ls and shows a relatively weak dependence of ls onthe laser intensity and plasma density [15].Starting from this configuration, the equations of ion motionare solved in Lagrangian coordinates, keeping the assumptionof quasi-equilibrium between the PF and the electrostatic force.Since the electric field on a Lagrange particle is a constant in1-D, this implies that the PF behaves in a similar way. Solvingthe equations of motion, it is easily found that the ions pileup in the region between the laser–plasma interface and theevanescence point [Fig. 2(b)], all reaching the latter at the sametime τi. Here, the fastest ions overrun the slower ones, the den-sity becomes singular [Fig. 2(c)], and hydrodynamical breakingoccurs. After τi, the fluid description becomes inadequate, anda kinetic approach is needed. However, at least the fastest ionskeep the velocity acquired before τi; thus, their energy can beestimated using the fluid model. The maximum ion velocity vimand the breaking time τi [6] are given in the limit d ls byvimc= 2√ZAmempncneaL τi 12ωLaL√AZmpme. (1)Here, aL = 0.85(Iλ2L/1018 W · µm2/cm2)1/2 is the dimen-sionless laser amplitude. Noting that the number of accelerated561868 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 36, NO. 4, AUGUST 2008Fig. 3. Maximum ion velocity and energy conversion efficiency into ionsversus laser pulse amplitude and plasma density from a set of parametric 1-DPIC simulations. (Red squares) Electron density ne = 5.4nc. (Blue diamonds)ne = 10.3nc. The straight lines in the velocity plot correspond to (1) for vim.The simulations have been obtained for a pulse with a constant intensity startingfrom a configuration of an initial equilibrium with fixed ions.ions per unit surface is n0ls, the ion energy flux can beestimated as mi(vm/2)2n0ls/2, corresponding to the energyconversion into ions vm/c. If d is not small with respect tols, the model predicts a less favorable scaling of vm versus aL.The predicted scalings have been tested by a set of para-metric 1-D simulations. To separate a possible dependenceupon the pulse duration, an initial equilibrium with fixed ionsand a constant laser intensity are assumed in the simulations.To achieve the equilibrium configuration in the simulations,the laser pulse is switched off, adiabatically keeping the ionsimmobile; ions are released after the fields have reached theequilibrium profile (corresponding to the analytical solutionsof [14]. Results are shown in Fig. 3 for two values of theplasma densities. The aforementioned linear dependence of ionvelocity and conversion efficiency on the laser amplitude aL isobserved up to high amplitudes, beyond the range where d lsmay be assumed. These observed scalings (indeed promisingfor ponderomotive acceleration) suggest that a refinement of thesimple model is required for very high intensities.We conclude this section by noting that, since the secularcomponent is also present in the LP case, the aforementionedmodel may also be used to estimate the contribution of pondero-motive acceleration at the front side for LP if the longitudinalforce on ions is considered to be a temporal average overthe laser cycle. Features such as the sharp spiking of the iondensity (see, e.g., [16]) or the generation of dense bunches ofions with energies similar to what is predicted by the model(see next section) are also indeed observed in simulations withLP. However, the simultaneous presence of fast electrons affectsthe dynamics of ponderomotive acceleration, as, for example,the strong absorption into electrons reduces the total radiationpressure on the target, whereas the compression of ions into asingle density spike can be counteracted by the strong thermalpressure of electrons.III. CP VERSUS LPTo show the fundamental differences between the LP andCP cases, we compare simulations having the same targetparameters (i.e., the same electron density ne, charge-to-massratio Z/A, and target thickness L), as well as pulse intensity Iand duration τL, the only difference being the laser polarization.The chosen parameters are close to those of simulations byFig. 4. Comparison of 1-D simulation results for CP and LP. (Top frame)Snapshots of the ion density ni and the (x, px) phase-space projections ofions fi and electrons fe for (left column) CP and (right column) LP. The timet = 467 fs is measured from the instant at which the front of the laser pulse(propagating from left to right) reaches the target. Normalizations are as inFig. 1. (Bottom frame) Energy conversion efficiency into ions and electronsversus (left) time and (right) energy spectrum of ions. Simulation parametersare τL = 86 fs, I = 3.5× 1020 W/cm2, ne = 1022 cm−3, L = 20 µm, andZ/A = 1.Silva et al. [5], where (using LP) it is claimed that the ionacceleration due to the generation of a collisionless shock waveis dominant over TNSA. We compare the simulation resultsfor CP and LP both in 1-D and 2-D geometries. Details of thesimulation setup are reported in [9].A. 1-D SimulationsFig. 4 compares the ion density profiles and the phase-spaceprojection f(x, px) of ions and electrons for CP and LP. ForCP, at the chosen irradiance I = 3.5× 1020 W · µm2/cm2,a very few electrons with a longitudinal momentum largerthan 0.5mec are found. In contrast, for LP, the generationof strongly relativistic electrons with maximum longitudinalmomenta pex 30mec is observed. In the ion phase-space plot,for CP, we observe only one population of “fast” ions, withtypical momenta px 0.15mic. No ions accelerated by TNSAare found. For LP, ions accelerated by TNSA are observed bothat the front and rear sides of the target, with similar momentumvalues (up to 0.3mic). This “symmetric” feature of TNSAoccurs when the density gradient at the front side is as sharp asat the rear side and has recently been experimentally confirmedby prepulse-free measurements [17]. A third group of ions,accelerating at the front side and propagating inside the targetat the time shown, is observed and attributed to ponderomotiveacceleration.The conversion efficiency of laser pulse energy into ionenergy (Fig. 4, left bottom frame) for CP is close to 0.14,whereas the conversion into electron energy is negligible. No-tice that the ion energy remains constant after the laser pulseis over. In contrast, for LP, the conversion into electron energyis dominant at early times, and ions continue to acquire energy57 PONDEROMOTIVE ACCELERATION OF IONSLISEIKINA et al.: PONDEROMOTIVE ACCELERATION OF IONS: CIRCULAR VERSUS LINEAR POLARIZATION 1869Fig. 5. Comparison of 2-D simulation results for CP and LP. (Left) Energyconversion efficiency into ions and electrons versus time. (Right) Energy versusangle distribution of ions. The laser pulse profile is Gaussian with a waist radiusrL = 2 µm. The peak intensity on axis is equal to the intensity of the 1-D case.Other parameters are the same as in Fig. 4.at late times due to TNSA. The conversion efficiency for theponderomotively accelerated ions is, however, also relativelyhigh (around 0.1) for LP. Strong differences are also noticeablein the ion energy spectra (Fig. 4, right bottom frame), showingfor CP a peak around 10 MeV but a thermal-like spectrum forLP overlapping over a broad maximum around 20 MeV.B. 2-D SimulationsThe absence of oscillating forces driving electrons across thesharp laser–plasma interface for CP pulses is strictly true onlyfor a plane wave. A laser pulse focused to a spot size of a fewwavelengths has off-axis longitudinal electric field componentsoscillating at the laser frequency ωL, whose amplitude is onthe order of λL/rL times the peak field amplitude, where rLis the waist radius. However, 2-D simulations show that evenfor tightly focused pulses, electron heating indeed occurs atdistances rL from the axis and increases energy absorptioninto electrons, but weakly affects the features of ion accelera-tion with CP. This can be noticed in Fig. 5(a), which shows theenergy conversion into ions and electrons for 2-D simulationswith the same plasma parameters and the same pulse durationof the 1-D case (see Fig. 4) and a Gaussian transverse profilewith rL = 2λL. The intensity at the center of the spot in 2-Dsimulations is equal to the intensity in 1-D simulations. Theefficiency of energy conversion into ions is about 0.1 in the 2-DCP case. Fig. 5(b) shows the f(E,Θ) distribution of ions,where Θ is the emission angle with respect to the targetnormal. Ions with energy > 0.01mic2 have an angular spreadof 10◦.C. Surface RipplingThe 2-D simulations also show that the onset of density rip-pling at the target surface is affected by the pulse polarization.The comparison between CP and LP results in Fig. 6 shows thatat early times (t = 80 laser cycles), surface ripples are muchdeeper for LP than for CP, and that the density in the regionnear the surface evolves in a more “turbulent” state in the LPcase. For CP, density rippling is also evident, but the structuresare more regular and less deep than in the LP case.Rippling of the front surface was already observed in earlysimulations of high-intensity laser pulse interaction with over-Fig. 6. Contours of the ion density from 2-D simulation results for CP andLP at two different times, showing the growth of surface rippling at the laser-irradiated surface. Simulation parameters are the same as in Fig. 5.dense plasmas [18], [19]. Recently, this topic has been revisitedfor thin plasma foils accelerated in the radiation-pressure-dominated regime [20]. Explanations of the surface ripplinghave mostly been based on Rayleigh–Taylor-like instabilitiesdue to the strong acceleration of the target driven by the radia-tion pressure. To test such theoretical description and its scalingwith laser and plasma parameters, the case of CP interactionis most adequate as radiation pressure dominance holds at anyintensity. On the other hand, the difference observed betweenCP and LP suggests that additional effects are at play forLP. These effects might be due to fast electrons or stimulatedsurface instabilities [21].The surface rippling appears to affect ion acceleration at thefront surface. In the density plot for CP at t = 120 cycles inFig. 6, the curved front of ponderomotively accelerated ionsis clearly visible, which is consistent with the quite regularangular distribution shown in Fig. 5. No well-defined ion frontis observed for LP. In this latter case, we also notice thegeneration of density “holes” (e.g., at (x, y) (31, 25) fort = 120). Analysis of electromagnetic fields inside such holesshows that they behave as “optical microcavities,” trappingradiation at a frequency lower than the plasma frequency ofthe surrounding plasma. These structures are similar to the so-called “postsolitons” (see, e.g., [22]), which, however, havemostly been studied in underdense plasmas so far.581870 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 36, NO. 4, AUGUST 2008IV. DISCUSSION AND CONCLUSIONThe use of CP, instead of LP, at normal incidence stronglyaffects the dynamics of ion acceleration in laser interaction withoverdense plasmas. Due to the quenching of fast electron gener-ation, using CP, a regime of purely ponderomotive acceleration,i.e., of radiation pressure dominance, may already be obtainedat moderate intensities. Our theoretical and simulation studiesof “thick” targets showed that the ponderomotively acceleratedions may have properties such as high efficiency, large density,and good collimation, which may be interesting for specificapplications.The CP radiation-pressure-dominated regime should extendover a wide range of intensities above the typical values, wherecollective effects, instead of collisional ones, dominate thelaser–plasma interaction. This is usually the case at intensitiesabove 1018 W/cm2. The scaling of ion energy and conversionefficiency that have been inferred from parametric PIC simu-lations look promising for experiments at higher intensities. Itis possible that CP represents an optimal choice for ion accel-eration in the ultrahigh-intensity regimes, where ions becomerelativistic [8], [23]. However, our theoretical picture may notanymore be appropriate at such intensities. For example, as theions become faster while electrons acquire increased relativisticinertia in the laser fields, the two species tend to stick together,and it is not anymore justified to assume different temporalscales for their motion. In addition, as the threshold for inducedrelativistic transparency is reached for a given density, the laserpulse penetrates into the plasma, and the laser–plasma couplingcompletely changes.From the point of view of theoretical understanding, the useof CP in simulations is useful to separate purely ponderomotive(radiation pressure) effects from those due to fast electrons.For example, as we have shown in 2-D simulations, this mayhelp unfold mechanisms driving surface rippling of the plasma.It is interesting to notice that for LP, where ion accelerationdue to TNSA occurs, a significant fraction of ponderomotivelyaccelerated ions at the front surface are observed in simulations.One may then wonder why so far in experiments the observedgeneration of MeV ions (mostly protons) has successfullybeen explained by the effect of TNSA only, leaving weakor no evidence for ponderomotive acceleration. In addition tochecking if the interaction conditions and the diagnostics inthe experiments allowed the detection of such ions, a possibleexplanation is due to the fact that ions accelerated at the frontsurface in the forward direction have to cross the target atsolid density. Thus, collisional stopping may play an importantrole, particularly if the ponderomotively accelerated ions arenot protons from the surface but bulk ions at higher Z, aswould probably be the case for an interaction affected byprepulse effects. Thus, quite thin targets (on the order of 1 µm)would probably be required to observe ponderomotively accel-erated ions. Ultrathin targets (typically less than one tenth of amicrometer thick) may be accelerated as a whole by CP pulsesand, under proper conditions, allow one to reach very high ionenergies [10]–[12]. 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[Online]. Available:http://link.aip.org/link/?PHP/10/2009/159 PONDEROMOTIVE ACCELERATION OF IONSLISEIKINA et al.: PONDEROMOTIVE ACCELERATION OF IONS: CIRCULAR VERSUS LINEAR POLARIZATION 1871[17] T. Ceccotti, A. Levy, H. Popescu, F. Reau, P. D’Oliveira, P. Monot,J. P. Geindre, E. Lefebvre, and P. Martin, “Proton acceleration withhigh-intensity ultrahigh-contrast laser pulses,” Phys. Rev. Lett., vol. 99,no. 18, p. 185 002, 2007. [Online]. Available: http://link.aps.org/abstract/PRL/v99/e185002[18] S. C. Wilks, W. L. Kruer, M. Tabak, and A. B. Langdon,“Absorption ofultra-intense laser pulses,” Phys. Rev. Lett., vol. 69, no. 9, pp. 1383–1386,1992. [Online]. Available: http://link.aps.org/abstract/PRL/v69/p1383[19] S. C. Wilks and W. L. Kruer, “Absorption of ultrashort, ultra-intenselaser light by solids and overdense plasmas,” IEEE J. Quantum Electron.,vol. 33, no. 11, pp. 1954–1968, Nov. 1997. [Online]. Available: http://dx.doi.org/10.1109/3.641310[20] F. Pegoraro and S. V. 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Naumova, H. Ruhl, and V. A. Vshivkov, “Ion acceleration regimesin underdense plasmas,” IEEE Trans. Plasma Sci., vol. 28, no. 4,pp. 1177–1183, Aug. 2000. [Online]. Available: http://dx.doi.org/10.1109/27.893311[24] X. Q. Yan, C. Lin, Z. M. Sheng, Z. Y. Guo, B. C. Liu, Y. R. Lu, J. X. Fang,and J. E. Chen, “Generating high-current monoenergetic proton beams bya circularly polarized laser pulse in the phase-stable acceleration regime,”Phys. Rev. Lett., vol. 100, no. 13, p. 135 003, 2008. [Online]. Available:http://link.aps.org/abstract/PRL/v100/e135003Tatiana V. Liseikina was born in 1973. She receivedthe M.S. degree from Novosibirsk State University,Novosibirsk, Russia, in 1995 and the Ph.D. degreein physics and mathematics from the Institute ofTheoretical and Applied Mechanics, Siberian Di-vision, Russian Academy of Sciences (SD RAS),Novosibirsk, in 1998.She has been a Senior Researcher with the Insti-tute of Computational Technologies, SD RAS, since2004. Since 2007, she has been a Visiting Scientistwith the Dipartimento di Fisica “Enrico Fermi,” Uni-versitá di Pisa, Pisa, Italy. Since April 2008, she has been a Visiting Scientistwith the Max Planck Institute for Nuclear Physics, Heidelberg, Germany.Her current research interests are the theory and simulation of laser–plasmainteractions, as well as the development of algorithms and numerical codes forthe simulation of many-particle systems.Domenico Prellino was born in 1981. He receivedthe B.S. and M.S. degrees in physics from the Uni-versitá di Pisa, Pisa, Italy, in 2005 and 2007, re-spectively, with an M.S. thesis on “Ponderomotiveacceleration of ions via laser–plasma interaction.”He is currently working toward the Ph.D. degreeat the German Cancer Research Center, Heidelberg,Germany, focusing on theoretical and numericalmodeling of biological systems.Fulvio Cornolti was born in 1949. He received theM.S. degree from the Universitá di Pisa, Pisa, Italy,in 1974.He is currently an Associate Professor of plasmaphysics and nonlinear models in physics with theDipartimento di Fisica “Enrico Fermi,” Universitá diPisa. His main interests are plasma physics, nonlin-ear optics, and laser–matter interactions.Andrea Macchi was born in 1970. He received theM.S. degree from the Universitá di Pisa, Pisa, Italy,in 1995 and the Ph.D. degree in physics from theScuola Normale Superiore, Pisa, in 1999.He has been a Research Fellow with the ItalianNational Research Council–National Institute for thePhysics of Matter (CNR–INFM), Pisa, Italy, since2003 and has been affiliated with the polyLAB since2005. His current research interests are the theoryand simulation of laser–plasma interactions, partic-ularly ion acceleration and generation of coherentelectromagnetic structures.60IOP PUBLISHING PLASMA PHYSICS AND CONTROLLED FUSIONPlasma Phys. Control. Fusion 50 (2008) 124033 (9pp) doi:10.1088/0741-3335/50/12/124033Radiation pressure acceleration by ultraintense laserpulsesTatiana V Liseykina1,2, Marco Borghesi3, Andrea Macchi3,4,5 andSara Tuveri51 Max Planck Institute for Nuclear Physics, Heidelberg, Germany2 Institute of Computational Technologies SD RAS, Novosibirsk, Russia3 School of Mathematics and Physics, the Queen’s University of Belfast, Belfast, UK4 polyLab, CNR-INFM, Pisa, Italy5 Dipartimento di Fisica ‘Enrico Fermi’, Università di Pisa, Pisa, ItalyE-mail: Tatyana.Liseykina@mpi-hd.mpg.deReceived 2 June 2008, in final form 15 July 2008Published 5 November 2008Online at stacks.iop.org/PPCF/50/124033AbstractThe future applications of the short-duration, multi-MeV ion beams producedin the interaction of high-intensity laser pulses with solid targets willrequire improvements in the conversion efficiency, peak ion energy, beammonochromaticity and collimation. Regimes based on radiation pressureacceleration (RPA) might be the dominant ones at ultrahigh intensities and mostsuitable for specific applications. This regime may be reached with present-dayintensities using circularly polarized (CP) pulses thanks to the suppression offast electron generation, so that RPA dominates over sheath acceleration at anyintensity. We present a brief review of a previous work on RPA with CP pulsesand a few recent results. Parametric studies in one dimension were performedto identify the optimal thickness of foil targets for RPA and to study the effectof a short-scalelength preplasma. Three-dimensional simulations show theimportance of ‘flat-top’ radial intensity profiles to minimize the rarefactionof thin targets and address the issue of angular momentum conservation andabsorption.(Some figures in this article are in colour only in the electronic version)1. IntroductionIn 1962, Forward considered the possibility of interstellar travel by a rocket propelled by anEarth-based laser beam (see [1] and references therein). The concept is as simple as follows:the rocket’s engine and fuel are replaced by a light sail, i.e. a mirror, and the force exerted onthe sail due to the radiation pressure of the laser light boosts the rocket. In 1966, Marx [2]found that the efficiency of the system, i.e. the ratio between the mechanical energy of theobject accelerated by the laser beam and the energy contained in the laser beam itself, would0741-3335/08/124033+09$30.00 © 2008 IOP Publishing Ltd Printed in the UK 161 RADIATION PRESSURE ACCELERATION BY CP LASER PULSESPlasma Phys. Control. Fusion 50 (2008) 124033 T V Liseykina et alapproach unity as the velocity of the object approaches the speed of light. An heuristic(though incomplete) argument might be given in terms of light quanta, i.e. photons, althoughthe system can be described as entirely classical: let us consider a ‘perfect’ mirror irradiatedby a monochromatic light wave of frequency ω, which contains a certain number N of photonsand thus has a total energy Nh̄ω. The mirror reflects photons conserving their number in anyreference frame. If the mirror has an (instantaneous) velocity V = βc in the laboratory frame,the frequency of the reflected photons is ω′ = ω(1 − β)/(1 + β). Thus, since N is invariant,the energy of the reflected pulse tends to zero if V → c, so that a mirror moving at a speedclose to c absorbs almost all the energy of the incident pulse.Marx paper’s conclusions turned out to be right although its approach needed a criticalrevision, as can be found in the rigorous and pedagogical description of [3]. Accordingto the formulas in [3] it would take three years for a 10 TW laser to accelerate a 103 kgrocket to V = (4/5)c. Scaling this result to the typical parameters of superintense laserpulses and micro-targets, we obtain that about 5 × 1010 carbon ions might be acceleratedto the same speed in 1 ps by a 1 PW laser, which is within the capabilities of the presenttechnology. Note that the required acceleration length would be of the order of 100 µm,which is a suitable value for the Rayleigh length, so that laser pulse diffraction shouldnot be a strong limiting factor on the achievable energies. This makes the perspective ofradiation pressure acceleration (RPA) attractive for applications requiring large numbers ofrelativistic ions.Nearly all of the experiments reported in the last decade on the acceleration of ions (mainlyprotons) by superintense laser pulses (see e.g. [4, 5] and references therein) are not based onRPA but instead on the target normal sheath acceleration (TNSA) mechanism, in which ions areaccelerated by space-charge fields created by multi-MeV electrons escaping into vacuum. Thedominance of RPA over TNSA in thin solid targets irradiated at intensities higher than thoseof present-day experiments has been claimed by Esirkepov and coworkers [6, 7], on the basisof simulations showing a transition occurring at some intensity value above 1021 W cm−2,with a strong dominance leading to the so-called ‘piston’ regime over 1023 W cm−2. Suchan intensity may be available only in several years from now thanks to the development ofadvanced laser facilities. Experimentally, a preliminary indication of RPA effects in thin targetsat intensities approaching 1020 W cm−2 has been published recently [8] (some experimentalresults on ion acceleration at intensities 1018 W cm−2 were also interpreted in terms of purelyponderomotive, i.e. radiation pressure effects [9]).The question then arises whether it is possible to achieve an RPA-dominated regimealready at lower intensity; this corresponds in practice to quenching the generation of high-energy electrons which drive TNSA but do not contribute to RPA. This may be possible usingcircularly polarized (CP) laser pulses at normal incidence because the oscillating componentsof the Lorentz force in the direction perpendicular to the sharp density gradient vanish (for aplane wave) or are relatively small (for a finite laser spot size): as a consequence, the motion ofelectrons at the interaction surface is predominantly adiabatic and electron heating is stronglyreduced, while the space-charge field created to balance the local radiation pressure (i.e. theponderomotive force) accelerates ions.The strong differences between the cases of linearly polarized (LP) and CP pulses havebeen evidenced in some papers by our group [10, 11], mostly for the case of ‘thick’ targets,i.e. thicker than the skin layer. These studies showed that for CP the interaction acceleratesall the ions in the skin layer and the fastest ones produce a very dense ‘bunch’ with a narrowenergy spectrum, directed in the forward direction.Recently, the experimental availability of ultrathin targets (i.e. with thickness down to a fewnanometers) and high contrast laser pulses (see, e.g. [12–14]) has called for studies of CP-RPA262Plasma Phys. Control. Fusion 50 (2008) 124033 T V Liseykina et alwith such pulses and targets. The simulations performed independently by several groups[15–19] suggest that indeed the whole target may be accelerated, leading to efficient generationof large numbers of ions with monoenergetic spectra in the near-GeV range. Presently, noexperiment using CP pulses at normal incidence has been reported in publications yet, butseveral related proposals have been made, so that the CP-RPA concept is expected to beexplored soon.This paper reviews the main issues of CP-RPA and reports novel numerical results onparametric studies in one spatial dimension (1D), showing the role of the target thickness andthe case of RPA in short-scalelength preformed plasmas, as well as first results in fully 3Dgeometry where, in particular, the issue of angular momentum conservation can be addressed.2. Theory and earlier workFirstly we shall provide a brief description of the interaction in the case of a thick target. Theponderomotive force of the laser pulse causes the electrons to pile up in the skin layer untilit is balanced by the charge separation field that accelerates the ions. The ions produce asharp density spike at the end of the skin layer where hydrodynamical breaking occurs, withthe faster ions creating a dense bunch (with a narrow spectrum) that moves ballistically intothe plasma (a rather similar dynamics has been noted in the case of radial ponderomotiveacceleration of ions in an underdense plasma [20]). A detailed physical description and asimple model of RPA in thick targets (assuming a priori non-relativistic ions) have been givenin previous works [10, 21]. The model provides the following scaling, valid for sub-relativisticion velocities, for the maximum ion velocity (which almost corresponds to the bunch velocity)and the corresponding energy:vimc= 2√ZAmempncneaL, Em = mi2v2im = 2mec2Zncnea2L, (1)where nc = meω2/4πe2 = 1.1 × 1021 cm−3 is the cut-off density for the laserwavelength λ = 2πc/ω, ne is the background electron density, aL = eEL/meωc =0.85(Iλ2/1018 W cm−2) is the dimensionless amplitude of the laser pulse with electric fieldEL and intensity I and other symbols are standard. Actually, this result has been derived in thelimit of a relatively low intensity, but this scaling has been found to hold up to a much higherintensity in parametric 1D simulations [21].The ion bunch is formed in a time of the order of ∼c/ωpvim (where ωp is the plasmafrequency) after which it exits the skin layer accompanied by neutralizing electrons, and thelaser pulse may accelerate a new layer. If energies higher than the above estimate mustbe reached, it is necessary to repeat the acceleration stage on the same ions, i.e. the targetmust be thin enough in order to bunch and accelerate all ions via several cycles. Simulationresults on the acceleration of ultrathin targets have been interpreted with the model of theaccelerating mirror [17] (where the mirror is assumed to be a ‘rigid’ object, neglecting anyinternal dynamics).Although a few authors have proposed the RPA of a thin foil as a way to generate high-energy protons, this approach seems to be most interesting for the acceleration of higher-Zions. In fact, while it seems technologically unfeasible to have an ultrathin target made ofhydrogen only, in a target made of multiple species all the ions will be accelerated to thesame velocity, resulting in higher energies for the heavier species. If a lighter species (e.g.hydrogen) is present, these ions will be first accelerated overtaking the heavier ones. This willcause them to decouple from the laser pulse, which is screened by the heavier ion layer. The363 RADIATION PRESSURE ACCELERATION BY CP LASER PULSESPlasma Phys. Control. Fusion 50 (2008) 124033 T V Liseykina et alheavier ions will be accelerated until they reach the lighter ions, allowing the laser to reachthem and accelerate them further. In the end all species will have the same velocity. Due to thiseffect, for an ultrathin target of a single material (e.g. carbon) a monoenergetic ion spectrumis expected.The above picture of the acceleration dynamics should change when the ions finally reacha speed close to c, as they will no longer be separated from the electrons. The ‘laser-piston’regime investigated by Esirkepov et al [6] corresponds to conditions in which the ions arepromptly accelerated to relativistic velocities and stick to the electrons, which may not beassumed to be in a mechanical quasi-equilibrium anymore. In this paper we restrict our analysisto the regime of non-relativistic ions because near-term experiments on RPA are unlikely tohave the potential to accelerate ions up to strongly relativistic energies.The theoretical picture and the predicted scalings have been supported by 1D simulations.So far multi-dimensional effects have been addressed at most by 2D simulations for boththick [10, 11, 21] and thin [17–19] targets. In the thick target cases, the energy spectrumis basically determined by the convolution of the 1D scaling law with the intensity profileof the pulse. The angle of emission of ions is energy dependent, but a good collimationis already obtained for a Gaussian pulse profile [11]. For thin targets, the use of a flat-topprofile increases the monoenergeticity and collimation and quenches the heating of electrons,as expected [17, 18].The issue of target stability during RPA has been addressed in thin foil 2D simulationsfor CP [18] and also for linear polarization in the ultraintense regime [22], showing a bendinginstability which has been interpreted to be of the Rayleigh–Taylor type and hence can bedescribed in terms of the radiation pressure only. Simulations for thick targets, however, haveshown that surface instabilities are weaker for CP pulses than for the LP ones. The quality ofthe ion beam is expected to be lower for linear polarization [21].3. 1D simulations3.1. The role of the target thicknessWe performed a parametric study to determine the optimal values of target thickness d toobtain a higher efficiency and/or ion energy for given laser parameters. In order to be ableto cover a quite wide range of parameters and to simulate ‘realistic’ target densities we used1D simulations, which in the CP-RPA regime have so far proved to yield efficiency and ionenergy values close to those from 2D or 3D simulations for those cases where a comparison ispossible (i.e. for moderate density values). Results are shown in figure 1. The electron densityof the target and the pulse duration were kept constant for all runs and corresponded, for a laserwavelength λ = 0.8 µm, to ne = 4.3 × 1023 cm−3 and τL = 24 fs. The three values of thedimensionless amplitude that were studied (a =2.9, 9.2 and 29) corresponded to intensitiesI = 1.8 × 1019 W cm−2, 1.8 × 1020 W cm−2 and 1.8 × 1021 W cm−2, respectively.The values of d for which efficiency and ion energy have their maximum are close toeach other and, as expected, they correspond to ultrathin, sub-micrometer targets. The strongdecrease in efficiency and energy for smaller values of d may be explained with the onset ofrelativistically induced transparency in the thin foil when d λa(nc/ne) [23], so that the totalradiation pressure on the target decreases. This point will be further discussed below whenthree-dimensional effects are addressed (section 4).The energy per nucleon reported in figure 1 can be scaled to all species with Z/A = 1/2.For carbon (A = 12) the highest energy of 0.96 GeV is obtained for a = 29 and d = 0.025λ.Note that these are ‘peak’ energies which correspond to a distinct maximum in the ion464Plasma Phys. Control. Fusion 50 (2008) 124033 T V Liseykina et alFigure 1. Energy conversion efficiency into ions (red filled diamonds) and the ‘peak’ energy pernucleon (blue empty diamonds) as a function of the target thickness, investigated by parametric1D PIC simulations. The top, middle and bottom plots are for a pulse amplitude a = 29, 9.2 and2.9, respectively. In all the runs, the laser pulse had a duration of 9 cycles (FWHM), the electronsdensity was ne = 250nc and the charge-to-mass ratio was Z/A = 1/2.spectra. However, depending on the interaction parameters some tail of higher energy ionsappears. Moreover, the width of the ion energy peak also varies throughout the simulationsand does not remain constant in time, as some broadening is observed after the laser pulseis over. This broadening appears to be related to electron heating which occurs at the endof the acceleration stage, creating ‘warm’ electrons which are much less energetic than thoseproduced for LP interaction but may already drive the expansion of the thin plasma foil. Hence,monoenergeticity of ions appears to be a non-trivial issue already in 1D.In higher dimensionality it is known that the intensity distribution in the laser spot givesrise to an energy spread correlated with the direction of laser-accelerated ions [11], so thata ‘flat-top’ distribution, whenever feasible, may improve monoenergeticity as well as beamcollimation (see e.g. 2D simulations in [17]). Additional effects of the pulse profile are alsodiscussed in section 4.3.2. RPA in preformed plasmasThe use of ultrathin targets in experiments will require the use of systems with an extremely highcontrast ratio, otherwise the prepulse preceding the main interaction pulse will destroy the targetcompletely. Interaction experiments in such a regime appear to be presently possible [12, 14],thanks, for example, to the use of plasma mirrors to improve the contrast [13]. Such conditionsare optimal to test the CP-RPA of ultrathin targets, provided that the strategies implemented toimprove the pulse contrast are compatible with preserving the circular polarization of the pulse.It is also worth stressing that the need for normal pulse incidence might also be non-trivial tobe experimentally satisfied due to the danger of back-reflection from the overdense plasma.It is interesting in any case to consider the possibility of the interaction of the CP pulsewith a non-uniform preplasma, as this may be present in experiments where the contrast ratio565 RADIATION PRESSURE ACCELERATION BY CP LASER PULSESPlasma Phys. Control. Fusion 50 (2008) 124033 T V Liseykina et alFigure 2. Interaction with preformed plasmas. Left: snapshots of the profiles of E2T = E2y + E2z(dashed blue line), ni (thick black line) and ne (thin red line) soon after the formation of the‘fast’ ion bunch (evidenced by the arrow). The pulse intensity was a = 3 corresponding to1.2×1019 W cm−2 for λ = 1 µm, the density profile was rising with a ∼(x −x0)9 law up to a peakdensity n0 = 40nc. Right: conversion efficiency (red filled diamonds, solid line) and peak energy(blue empty diamonds, dashed line) of ions as a function of the density scalelength L, for threevalues of the laser amplitude a. In all the simulations the laser pulse had a duration of 9 cycles(FWHM) and the density profile was rising with a ∼(x − x0)4 law up to a peak density n0 = 16ncand then remained constant.is modest. Moreover, the expected scaling of the ion energy with the inverse of the plasmadensity suggests that, in a preformed plasma, a given laser pulse may produce a lower totalnumber of ions but with higher energy, as the interaction occurs with the layer at the cut-offdensity nc which is typically less than one hundredth of the solid density.We performed a set of parametric 1D simulations assuming initial density profiles ofthe power-law type (i.e. n0(x) ∼ (x − x0)k for x > x0) and different values of the densityscalelength at the cut-off layer, L = nc/|∂xn0|n0=nc . The snapshot of the ion density ni infigure 2 shows that ni undergoes spiking and ‘breaking’ and that a ‘fast’ bunch forms nearthe cut-off density layer, with features very similar to the case of a sharply rising density(no preplasma) [10]. The bunch density is several times nc. As a function of L, boththe maximum ion energy and the conversion efficiency have their maxima for a very shortscalelength L 0.5λ, as also shown in figure 2 where L = 0 corresponds to the case of nopreplasma, i.e. a step-like profile. When compared with the energy scaling (1), the observedion energy would correspond to a density value intermediate between nc and the peak density(16nc) of the profile. The decrease in energy and efficiency for larger values of L might berelated to the weaker coupling of the laser pulse to the cut-off layer; a relevant part of the pulseenergy is found to be absorbed in the underdense plasma, e.g. by excitation of plasma waves,causing the absorption degree into electrons to be higher than into ions and consequentlydecreasing the total radiation pressure. The stronger heating of electrons may account forthe broad energy spectrum that is observed for non-optimal values of L; figure 2 reports themaximum or cut-off energy, but several and broad ‘peaks’ may appear in the spectrum undersuch conditions, while the spectrum is narrow for the case of absolute maximum energy. Thesepreliminary results suggest that RPA may be strongly affected by prepulse effects. Howeverthis may allow one to achieve dense bunches of multi-MeV ions using ultrashort pulses withcontrolled contrast. For longer pulses (hundreds of femtoseconds), this approach may becomeineffective because of strong steepening effects during the rise of the pulse, decreasing the valueof Lc. The width of the target layer that remains undamaged by the prepulse may also play a666Plasma Phys. Control. Fusion 50 (2008) 124033 T V Liseykina et alFigure 3. (a) The distribution of ion the density at t = 75, 100 and 130 fs in the (xy) plane,(b) the distribution of the ion density at t = 130 fs (the laser pulse is over), (c) the integral of theion poloidal current∮Jϕ dy dz at t = 130 fs versus longitudinal coordinate, showing the angularmomentum absorption: black line for the region with radius r = 4.5 µm around the x-axis; redline for the r = 2.5 µm region around the x-axis.role because the ions may undergo relevant collisional losses while crossing the solid-densityregion (see, e.g. the discussion in [8]).4. 3D simulationsAs is always the case for computational plasma physics, 3D simulations would be requiredfor a ‘realistic’ description, but the limits of computing power force the restriction to a narrowset of ‘feasible’ parameters. This is the case for CP-RPA where, furthermore, the resolutionmust be high enough to resolve effects such as the strong spiking of the density observed in1D and 2D. Thus, only a few 3D runs could be performed and for plasma densities much lessthan solid-density values, though well above nc.The comparison with 1D and 2D results is also important because a CP pulse carries a netangular momentum whose conservation law appears as an additional constraint in 3D. Notethat, despite the strong absorption of pulse energy, no absorption of angular momentum isexpected, at least as long as the acceleration is adiabatic (as is assumed in the ‘perfect mirror’model). In fact, coming back to the heuristic argument of the introduction, if the number ofphotons is conserved and the reflected beam conserves helicity (which can be shown to hold),no angular momentum is left in the target because the ‘spin’ of any photon is h̄, independentof the frequency.In order to study these issues in detail, we performed several 3D simulations. In allthe runs discussed here the normally incident laser pulse was CP with a peak intensity of3.4 × 1019 W cm−2 and ∼60 fs duration and the target consisted of electrons and protons.Figure 3 presents the results of the interaction of a laser pulse with a ‘flat-top’ intensity profileof 6 µm width with a target of density ne = 16ncr = 1.7 × 1022 cm−3 and 0.3 µm thick.Figure 3(a) shows the projected 2D distributions of ion density at t = 75, 100 and 130 fs and(b) the 3D plot of the ion density when the laser pulse is over. The density of the ‘bunch’ isapproximately 0.7 of the initial density of the target, the peak energy of ions in this bunch is∼4 MeV the number of accelerated ions is ∼4×1010. When the laser pulse is over most of theabsorbed angular momentum (∼4%) is transferred to the ions (the energy absorption in thiscase was ∼7%). To prove that a torque on the plasma ions exists, we plot in figure 3(c) theintegral (over (y, z)) of the poloidal current Jϕ of the ions. We thus see that on average there767 RADIATION PRESSURE ACCELERATION BY CP LASER PULSESPlasma Phys. Control. Fusion 50 (2008) 124033 T V Liseykina et alFigure 4. The distribution of ion density at t = 75, 100 and 130 fs and the electromagnetic energyat t = 50 fs and t = 100 fs in the case of tightly focused (3 µm width) laser pulse interaction withan ultrathin foil.is a net ‘rotation’ of the ions, while the same plot for the electron current shows that the latteraverages over x almost to zero. The poloidal ion current is concentrated near the edge of thelaser spot where the angular momentum density has a maximum.Angular momentum absorption in laser–plasma interactions has been mostly studied so farin underdense plasmas as a problem closely related to the generation of a steady magnetic field(inverse Faraday effect). Haines [24] reported a short critical review of a previous work anddiscussed the effects leading to a torque on the plasma ions. The issue of angular momentumabsorption in overdense plasmas has received much less attention so far. In the present context,the observation of some degree of angular momentum absorption is a signature of non-adiabaticor ‘dissipative’ effects (which are an interesting issue in collisionless systems) not includedin the ‘perfect mirror’ model of RPA. They may be related to the onset of hydrodynamicalbreaking during the acceleration process [10], violating the adiabaticity condition.In figure 4 the results of the interaction of a tightly focused (3 µm width) Gaussian laserpulse with a target of density ne = 9ncr = 1 × 1024 cm−3 and thickness of 0.4 µm are shown.In this case the pulse focusing was tight enough to contribute dramatically to the inducedtransparency of the target.In both cases presented here the density and the width of the targets were chosen in away to ensure their opacity on the basis of the 1D analysis. However, 3D effects decrease thetransparency threshold because the foil tends to expand in the perpendicular direction. For thetightly focused Gaussian laser pulse this effect is very pronounced so that the foil becomestransparent even if initially it was opaque. Since the use of a target with densities not very farfrom the transparency threshold is more suitable to achieve an efficient acceleration rate, theshape of the laser pulse becomes a critical issue and the use of ‘flat-top’ laser pulses, wheneverpossible, may help.AcknowledgmentsThis work was supported by CNR-INFM and CINECA (Italy) through the super-computinginitiative and by CNR via a RSTL project. Some of the simulations were performed on theLinux Cluster of MPI-K, Heidelberg. A part of the work was performed during a stay of two ofthe authors at Queen’s University, Belfast, UK, supported by a Visiting Research Fellowship(AM) and by COST-P14 (ST). TVL also acknowledges the support from RFBR (via 08-02-08244 grant). The authors are grateful to D Bauer, F Cornolti and F Pegoraro for a criticalreading and comments.868T h e o p e n – a c c e s s j o u r n a l f o r p h y s i c sNew Journal of PhysicsRadiation pressure acceleration of ultrathin foilsAndrea Macchi1,2,6, Silvia Veghini1, Tatyana V Liseykina3,4 andFrancesco Pegoraro1,51 Department of Physics ‘E. Fermi’, Largo B Pontecorvo 3, 56127 Pisa, Italy2 CNR, Istituto Nazionale di Ottica (INO), Pisa, Italy3 Max Planck Institute for Nuclear Physics, Heidelberg, Germany4 Institute of Computational Technologies, SD-RAS, Novosibirsk, Russia5 Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia(CNISM), unità di ricerca di Pisa, ItalyE-mail: macchi@df.unipi.itNew Journal of Physics 12 (2010) 045013 (18pp)Received 13 October 2009Published 30 April 2010Online at http://www.njp.org/doi:10.1088/1367-2630/12/4/045013Abstract. The acceleration of sub-wavelength, solid-density plasma foilsby the ultraintense radiation pressure of circularly polarized laser pulses isinvestigated analytically and with simulations. An improved ‘Light Sail’ oraccelerating mirror model, accounting for nonlinear self-induced transparencyeffects, is used for estimating the optimal thickness for acceleration. The modelpredictions are in good agreement with one-dimensional simulations. Theselatter are analyzed in detail to unfold the dynamics and self-organization ofelectrons and ions during the acceleration. Two-dimensional simulations arealso performed to address the effects of target bending and of laser intensityinhomogeneity.6 Author to whom any correspondence should be addressed.New Journal of Physics 12 (2010) 0450131367-2630/10/045013+18$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft69 RADIATION PRESSURE ACCELERATION OF ULTRATHIN FOILS2Contents1. Introduction 22. Thin-foil modeling 32.1. Review of the LS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2. Nonlinear reflectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3. Improved LS formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4. Optimal thickness for ion acceleration . . . . . . . . . . . . . . . . . . . . . . 73. Electron and ion dynamics in one dimension 94. 2D simulations 135. Conclusions 15Acknowledgments 16References 161. IntroductionPresent-day laser systems may deliver at their focal spot intensities up to 1022 W cm−2,and possibly even higher in the near future. The radiation pressure on a reflecting objectcorresponding to such intensities would exceed terabar values, driving strong compressionand acceleration of matter. A few experiments have already given some evidence of strongradiation pressure (or ponderomotive) effects in various regimes [1]–[3]. The possibility todevelop laser-driven sources of high-energy ions based on radiation pressure acceleration (RPA)is of particular interest. Theoretical studies [4]–[6] have shown that in the interaction withultrathin (sub-micrometric) foils at intensities exceeding 1023 W cm−2, the dominant mechanismof ion acceleration may be RPA instead of target normal sheath acceleration (TNSA), whichhas been intensively investigated experimentally (see [7] for a review). A regime of radiationpressure dominance may also exist at lower intensities, if circular polarized (CP) laser pulsesat perpendicular incidence on the target are used. In such conditions, TNSA may be ruled outby the strong quenching of high-energy electron generation for CP pulses [8]. The regime ofRPA with CP pulses has been recently investigated in many theoretical papers for differentlaser and target conditions, including thick targets (‘hole boring’ regime) [8]–[13], structured orcomposite targets [14]–[18], and ultrathin foils [19]–[29]. The effects of elliptical polarizationwere also studied [30, 31]. First experimental results were communicated recently [32]–[34].In this paper, we focus on ultrathin foil targets. Although experimentally challenging, sucha regime appears to be feasible due to the present possibility of both manufacturing targets asthin as a few nanometers, e.g. diamond-like carbon (DLC) foils, and avoiding early disruptionof such targets by prepulse effects thanks to the use of techniques such as the plasma mirror toobtain extremely ‘clean’ pulses [35]. Early theoretical studies both for linearly polarized pulsesin the radiation pressure-dominated regime [4] and for CP pulses at lower intensities [19]–[22]suggested that an ultrathin foil may be accelerated as a whole, leading to a monoenergeticspectrum of ions. In addition, the acceleration process is highly efficient and the energy pernucleon scales favorably with the laser pulse energy. These features are in agreement with thepredictions of the simplest model of thin-foil acceleration, which assumes the thin foil targetto be a perfectly reflecting, undeformable, plane mirror boosted by the radiation pressure of anelectromagnetic (EM) plane wave (corresponding to the laser pulse) at perpendicular incidence.New Journal of Physics 12 (2010) 045013 (http://www.njp.org/)703This regime of RPA has also been named ‘Light Sail’ (LS), and in the following we refer to the‘accelerating mirror’ model as the LS model.This paper is organized as follows. In section 2, we present an analytical model thatimproves the basic LS model by including the effect of the nonlinear self-induced transparency(SIT) of the foil at ‘relativistically’ strong laser intensities. It is found that SIT effects determinean ‘optimal’ value of the foil thickness for which, given a laser pulse, the foil velocity ishighest. The predictions of the model are found to be in good agreement with one-dimensional(1D) particle-in-cell (PIC) simulations for what concerns the energy of the spectral peak of‘monoenergetic’ ions and the conversion efficiency.In section 3, the dynamics of thin-foil RPA is analyzed in 1D PIC simulations. It is foundthat the distribution of electrons and ions undergoes a dynamic self-organization to maintain anequilibrium condition for electrons where the electrostatic and ponderomotive forces balanceeach other. It is found that in general the spectral peak contains just a fraction of the total ions,the actual value depending on the target and pulse parameters.In the final section 4, the acceleration of foils with thicknesses close to the ‘optimal value’is further investigated with two-dimensional (2D) simulations, addressing the effects of targetbending and of laser intensity inhomogeneity.2. Thin-foil modeling2.1. Review of the LS modelThe LS model takes into account an EM plane wave of frequency ω and intensity I = I (t),perpendicularly incident on a foil of mass density ρ, thickness ` and reflectivity R = R(ω). Inthe rest frame of the foil, neglecting dissipative effects, the radiation pressure is given byPrad = 2RIc. (1)Note that equation (1) is different from the known expression for the case of incidence on asemi-infinite mediumPrad = (1 + R)Ic, (2)which does not vanish if R = 0 because the transmitted wave does not leave the target anddeliver all its momentum there.By a Lorentz transformation of the force on the foil and of the intensity and frequency ofthe EM wave, the following equations of motion for the foil in the laboratory frame are obtained:ddt(βγ ) = γ 3dβdt=2I (t − X/c)ρ`c2R(ω′)1 − β1 + β, (3)dXdt= βc, (4)where β = V/c is the velocity of the foil in units of the speed of light c, ω′ = ω√(1 − β)/(1 + β)is the EM wave (laser) frequency in the rest frame and γ = (1 − β2)−1/2.New Journal of Physics 12 (2010) 045013 (http://www.njp.org/)71 RADIATION PRESSURE ACCELERATION OF ULTRATHIN FOILS4For R = 1, i.e. taking the foil as a perfectly reflecting mirror, the model yields a simplesolution for β = β(w), where w = t − X/c is the retarded time, as a function of the pulse fluenceF (the pulse energy per unit surface) [36, 37]:β(w) =[1 + E(w)]2 − 1[1 + E(w)]2 + 1, (5)E(w) =2F(w)ρ`c2, F(w) =∫ w0I (w′) dw′. (6)This expression can also be retrieved from the more general derivation given in section 2.3. Thecorresponding energy per nucleon isE(w) = mpc2[γ (w) − 1] = mpc2 E2(w)2[E(w) + 1]. (7)The instantaneous efficiency η, i.e. the ratio between the mechanical energy delivered to the foiland the incident pulse energy, is also a simple function of β [36, 37] or, equivalently, of E :η =2β(w)1 + β(w)= 1 −1[E(w) + 1]2. (8)Thus, η → 1 when β(w) → 1, i.e. when E→ ∞.The final velocity of the foil βf and the corresponding energy per nucleon En are obtainedtrivially by rewriting equations (5)–(7) as a function of the total energy, i.e. by taking E = E(∞).In practical units,E ' 2.2F108 J cm−2(ρ1 g cm−3)−1 (`10 nm)−1. (9)It is also useful to rewrite E as a function of the relevant dimensionless parameters: theamplitude of the pulse a0 =√I/mec3nc (with nc = meω2/4πe2 the cutoff density), the surfacedensity ζ = π(ne/nc)(`/λ) (with λ = 2πc/ω the laser wavelength) and the duration of thelaser pulse τ in units of the laser period T = 2π/ω = λ/c. By estimating the total fluenceF ' I (T τ) = mec3ncT (a20τ), we obtain for EE '2mec2nca20τλm ini`c3= 2πZAmempa20τζ. (10)Figure 1 shows βf and En as a function of a20τ and ζ (Z/A = 1/2 has been assumed). Toconvert from/to practical units, for a laser wavelength λ = 0.8 µm and a fully ionized DLCtarget (ρ = 2.2 g cm−3, ne/nc = 384), we find a20τ = 1.73F/(104 J cm−2) and ζ = 1.5`/(1 nm).2.2. Nonlinear reflectivityTo include the effects or partial reflectivity (R < 1) and SIT into the LS model, we first lookfor a suitable analytical expression of R. We use the model of a delta-like ‘thin foil’ [38], i.e.a plasma slab located at x = 0 with electron density ne(x) = n0`δ(x). The model gives thefollowing expression for R in the rest frame of the foil:R =11 + ζ−202, (11)New Journal of Physics 12 (2010) 045013 (http://www.njp.org/)725Figure 1. Analytical results from the LS model. The final velocity of the foilβf and the corresponding energy per nucleon En are given as a function of thedimensionless pulse fluence a20τ and for the values of surface density of the foilζ = 1 (black line), 3.16 (green), 10 (blue), 31.6 (orange) and 100 (red). Thevalues on the upper horizontal axis give the fluence in J cm−2 correspondingto a20τ for λ = 0.8 µm. For a DLC target, ζ = 2.6`/(1 nm). The solid linescorrespond to equation (5) i.e. to the case of a ‘perfect’ mirror with reflectivityR = 1. The dotted lines correspond to equation (17) where the effects of pulsetransmission through the foil have been taken into account (for ζ > 10, solid anddotted lines cannot be distinguished).0 =1 + a20 − ζ22+√(1 + a20 − ζ 22)2+ ζ 2, (12)which is very well approximated byR '{1/(1 + ζ−2) (a0 <√1 + ζ 2),ζ 2/a20 (a0 >√1 + ζ 2).(13)Note that 0 is the relativistic factor for electrons in the thin foil (calculated self-consistently withthe EM fields). It is thus apparent that the SIT is due to the relativistically increased inertia ofelectrons, as me is replaced by me0 in their equation of motion. The expression for a0 <√1 + ζ 2corresponds to that obtained in the linear regime, i.e. for sub-relativistic intensities.Equations (11) and (13) for R are both plotted in figure 2, showing that the transitionfrom the linear to the nonlinear regime is quite sharp and that equation (13) provides anexcellent approximation. We may thus define a threshold condition for relativistic SIT asa0 =√1 + ζ 2 ' ζ when ζ 1, i.e. in most cases of interest. Combining equations (1) and (13),the total radiation pressure Prad on the target becomes independent of a0 for a0 > ζ . Thus, a0 ' ζNew Journal of Physics 12 (2010) 045013 (http://www.njp.org/)73 RADIATION PRESSURE ACCELERATION OF ULTRATHIN FOILS6Figure 2. Nonlinear reflectivity R as a function of incident wave amplitudea0 and for several values of the ‘optical thickness’ ζ . Thick and dashedlines correspond to the ‘exact’ formula (11) and to its approximation (13),respectively.gives the condition of maximum radiation pressure on the (immobile) foil, and one would expectthe RPA of the foil to saturate for a0 > ζ . Actually, simulations show that in this regime theinteraction is rather dominated by the effect of expulsion of electrons from the foil, leading toa transition from RPA to a Coulomb explosion [28]. Hence, the LS model is appropriate fora0 < ζ .2.3. Improved LS formulaAccording to equation (11), the reflectivity R is independent of a0 in the a0 <√1 + ζ 2 range,and for ζ 1, corresponding to most cases of experimental relevance, R ' 1 is a very goodapproximation. Nevertheless, thin-foil manufacturing technology seems able even to approachvalues of ζ not much larger than unity; hence it is of some interest to consider how the LSformulae are modified in the a0 > ζ & 1 regime.Using R ' (1 + ζ−2)−1 and switching to the retarded time w, equation (3) becomesdβdw=2ρ`c2I (w)γ −3(1 + β)−1(1 + ζ−21 − β1 + β)(14)(note that dw = (1 − β)dt). By switching to the variable b(w) =√(1 − β)/(1 + β), we obtainb−2(w)[1 + b2(w)ζ−2] dbdw= −2ρ`c2I (w), (15)which can be integrated with the initial condition b(0) = 1, corresponding to β(0) = 0, to yieldb−1(w) − 1 − ζ−2b(w) + ζ−2 =2ρ`c2∫ w0I (w′) dw′ = E(w). (16)New Journal of Physics 12 (2010) 045013 (http://www.njp.org/)7470 2 4 6 8 1000.20.40.60.81.0Ωtβ.Figure 3. Foil velocity as a function of t defined in (18) for ζ = 1 (green lowercurve) and for ζ 2 = ∞, i.e. a perfectly reflecting foil (blue upper curve).By solving (16) for b(w) and rewriting the result for β, we obtainβ(w) =[1 + E(w) − ζ−2]2(1 +√1 + 4ζ−2[1 + E(w) − ζ−2]−2)+ 2ζ−2 − 2[1 + E(w) − ζ−2]2(1 +√1 + 4ζ−2[1 + E(w) − ζ−2]−2)+ 2ζ−2 + 2. (17)In the limit of ζ → ∞, i.e. R → 1, one recovers equation (5). As shown in figure 1, thedifference between (5) and (17) (and between the corresponding energies) is important forζ ∼ 1.For a constant intensity I , i.e. for a ‘flat-top’ temporal profile of the laser pulse, the fluencecan be written as a function of time asE(t) =2I tρ`c2= 2πZAmempa20ζtT≡ t, (18)where T is the period of the laser radiation. In this case, explicit analytical expressions for β(t)and X (t) can be given as a function of ζ and t [39]. The result for β(t) is shown in figure 3for the two cases ζ = 1 and ζ = ∞. The latter corresponds to the known solution for a perfectlyreflecting mirror [4, 21], which is formally identical to the solution of the longitudinal motionof a charge during Thomson scattering from a plane wave [40].2.4. Optimal thickness for ion accelerationWe now use the LS model with SIT effects included to provide an estimate for the value of foilsurface density ζmax (corresponding to thickness `max for a given target material), which, for agiven laser pulse, leads to the highest foil velocity. In the regime of sub-relativistic intensity, ζmaxmay be obtained by differentiating (17) with respect to ζ at fixed pulse fluence. However, forvery high intensities, ζmax is ultimately determined by SIT effects; thus the complete nonlinearexpression for R (11) must be used. In the LS equation of motion (3), the expression for R(ω′)New Journal of Physics 12 (2010) 045013 (http://www.njp.org/)75 RADIATION PRESSURE ACCELERATION OF ULTRATHIN FOILS8now readsR(ω′) = R(β) =11 + ζ−202(β)(1 − β)/(1 + β), (19)0 =1 + a20 − ζ2(1 + β)/(1 − β)2+√√√√(1 + a20 − ζ 2 1+β1−β2)2+ ζ 21 + β1 − β. (20)By using the variable b(w) as in equation (15), we now obtain[b−2(w) + 02(b)ζ−2]dbdw= −2ρ`c2I (w), (21)which can be integrated for a constant intensity I (w) = I :b−1(w) − 1 − ζ−2∫ b10(b′) db′ = E(w). (22)Using the implicit equation (22), b(w) and β(w) can be obtained numerically. Bydifferentiating (22) with respect to ζ at a fixed acceleration time τ (which should approximatelycorrespond to the duration of the laser pulse), it can be shown [39] that ζmax lies in the interval√(τ2π)2+ 1 + a20 −τ2π< ζmax <√1 + a20, (23)where has been defined in (18). Equation (23) shows that for τ 1 the optimal thicknessis ζmax '√1 + a20 ' a0, corresponding to the maximum radiation pressure on the immobile thinfoil as found in section 2.2. As τ & 1 increases, e.g. for increasing pulse duration (keepingother parameters fixed), ζmax may become smaller than a0. This effect may be ascribed to theincrease of the reflectivity and the decrease of the intensity in the rest frame of the foil as itis accelerated to high values of β. Since −1 ' 6 × 102 T(ζ/a20), this effect is relevant onlyfor long (hundreds of periods) pulses. We may thus conclude that the condition of ‘optimal’thickness, given by our improved LS model with SIT effects included for a pulse of constantamplitude a0, is given byζmax ' a0. (24)In figure 4, the energy per nucleon En and the efficiency η obtained from the numericalsolution of (22) for β(∞) are compared with the results of the 1D PIC simulations that werepreviously shown in figure 1 of [24]. In these simulations, η is calculated as the ratio between thetotal energy of the ions and the energy of the laser pulse. Both En and η are shown as a functionof the target thickness ` and for three different values of the peak pulse amplitude a0. For allsimulations, ne = 250nc, Z/A = 1/2 and the pulse has a full-width at half-maximum (FWHM)duration of 9 periods with a sin2 envelope for the field. A fairly good agreement between themodel and simulations is found. Part of the discrepancy may be ascribed to the effect of thepulse envelope; for simulations with a flat-top envelope (i.e. a nearly constant amplitude), theobserved ‘optimal’ thickness is in closer agreement with equation (24).Several other authors give for the optimal thickness the same scaling of equation (24) butoften with different numerical coefficients. Yan et al [23] give the following range,ζπ(1 + R)1/2∼ a0 . 2ζ, (25)New Journal of Physics 12 (2010) 045013 (http://www.njp.org/)769501005100 0.01 0.02 0.03 0.04 0.05 0.0600.510.20.40.10.20 0.01 0.02 0.03 0.04 0.05 0.0600.010.020.03E(MeV)η./λ /λa0 = 29 a0 = 29a0 = 9.2 a0 = 9.2a0 = 2.9 a0 = 2.9(a) (b)Figure 4. Energy per nucleon (a) and efficiency of energy conversion into ions(b) obtained as a function of target thickness ` and pulse amplitude a0 for a laserpulse of duration τ = 9 cycles and an electron density ne = 250nc. The solidline gives the values corresponding to the numerical solution of (22) for β. Thediamonds give the values obtained from parametric 1D PIC simulations.using an argument of pressure balance but not accounting for SIT effects, since equation (2)is used for Prad. A critical comment by Ji et al [41] about the a0 . 2ζ condition appearsto originate from a missing 2 factor [42]. Tripathi et al [26] give a condition identical toequation (24), accounting for the ‘relativistic’ modification of the plasma refractive index butusing a calculation for a semi-infinite plasma profile. Finally, Esirkepov et al [5] observein multi-parametric simulations a condition of the form ζmax ' 3π + b√I , where b is anappropriate constant factor, and notice its similarity with the SIT condition of a thin foil;however, the simulations are performed for linearly polarized pulses and for parameters notcorresponding to the radiation pressure-dominated regime. An additional source of differencemay be indeed the 2D (instead of 1D) nature of the simulations, as we also find ζmax to be largerthan indicated by equation (24) in 2D simulations (see section 4 below).3. Electron and ion dynamics in one dimensionThe comparison with 1D PIC simulation results shown in figure 4 shows that the simple LSmodel of section 2 is remarkably successful in predicting ion energy as a function of pulseenergy and foil thickness. Moreover, for a0 . ζmax the further simplifying assumption of a totallyreflecting foil (R = 1) is appropriate. The dynamics underlying the RPA of the thin foil is,however, more complex than what the simple ‘mirror’ model might suggest. Unfolding suchdynamics with the help of simulations helps to explain why some of the observed features seemto be in contrast with the model assumptions. Such features have been discussed in [28]. Inthe following, we start by recalling the results we obtained in [28] and go a step further in theanalysis.The LS model assumes the foil as a ‘rigid’ object, so that all the ions and electrons movecoherently at the same velocity V = βc of the foil. However, the 1D simulations show that thespectral peak, centered at the energy predicted by the LS model, contains only a fraction of theNew Journal of Physics 12 (2010) 045013 (http://www.njp.org/)77 RADIATION PRESSURE ACCELERATION OF ULTRATHIN FOILS10(a) (b)Figure 5. Cartoon showing how two ion populations are formed. The schematicprofiles of the ion density ni (green), electron density ne (blue) and electrostaticfield Ex (red) are shown at two different times. Part (a) corresponds to the earlystage of the interaction where ions have not moved yet, so that their densityequals the initial value n0, and electrons have piled up under the action of theponderomotive force fp in the d < x < ` or ‘Sail’ (S) region. The 0 < x < dregion of electron depletion is named as the ‘Tail’ (T) region. Part (b) showsthe stage of ion acceleration. Ions in the T region are accelerated by their ownspace-charge field as in a Coulomb explosion, so that the ion density decreasesin this region. Ions in the S region (x > X (t)) are bunched and accelerated byEx , which in this region equals fp/e, and move with the electrons at velocity V .total number of the ions. Those ‘monoenergetic’ ions originate from a layer at the rear surfaceof the foil. The rest of the ions form an exponential-like ‘pedestal’ in the spectrum.We explain the formation of the two ion populations with the help of the cartoon in figure 5.As the laser pulse impinges on the overdense plasma, the electrons are pushed by the steadyponderomotive force fp, whose integral over the whole target depth equals the radiation pressure.Thus, electrons pile up in the region d < x < ` of figure 5, which we call the ‘Sail’ region. Theelectron displacement generates the space-charge field Ex ' fp/e to balance the ponderomotiveaction. (If R ' 1, almost no electrons are pushed out of the foil, because the EM field and theponderomotive force vanish at the rear surface of the foil; the electron compression does notchange R in 1D since the product ne` is constant.) Ions in the Sail are accelerated and bunchedby Ex [8, 28] and move together with electrons. The Sail is thus negatively charged during theacceleration because it contains all of the electrons in the foil but just a fraction F ' 1 − d/`of the ions. An estimate for the condition of pressure equilibrium gives F ' 1 − a0/ζ , showingthat F can be significantly smaller than unity as a0 approaches ζ .The analysis of 1D simulations confirms the above-described scenario. Typical snapshotsof density and field profiles are shown in figure 6(a) for a simulation with a0 = 12, τ = 9,ne/nc = 100 and ` = 0.05, so that ζ = 15.7. Both a fine spatiotemporal resolution (1x = c1t =λ/2000) and a high number of particles per cell were used to resolve accurately the dynamics,and in particular the density variation that involves both the strong spiking in the Sail region andthe rarefaction in the Tail region.The Sail moves with a well-defined velocity V , so that ions in the Sail are monoenergetic.The velocity V satisfies the LS equation (3), where, according to the comparison withNew Journal of Physics 12 (2010) 045013 (http://www.njp.org/)7811(a)(b)Figure 6. Snapshots from a 1D PIC simulation of the interaction of a laserpulse with a thin plasma slab. Ion density ni (green), electron density ne (blue),longitudinal electric field Ex (red, dashed) and pulse field amplitude EL =√E2y + E2z (red, dotted) are shown. Plots in row (a) correspond to the early stageof the interaction when electrons pile up at the rear side of the foil and a chargedlayer at the rear side is accelerated. Plots in row (b) correspond to later timeswhen the laser pulse decouples from the foil and excess electrons are acceleratedbackward leaving the charged layer. Note that the scales of the x and ne,i axeshave been changed in (b) with respect to (a) for better visibility. The target leftboundary is at x = 0 where the pulse impinges at t = 0. Times are normalized tothe laser period T , fields to E0 = meωc/e and densities to nc = meω2/4πe2. Thelaser pulse has peak amplitude a0 = 12 and a ‘flat-top’ profile with a durationof 9T (FWHM) and 1T rise and fall times, the foil thickness is ` = 0.05λ, theelectron density ne = 100nc and Z/A = 1/2.simulations, the total thickness ` of the foil, i.e. the total mass, must be used. This may actuallysound surprising because the Sail’s mass is ' F times the total mass and the total pressure onions is lower than Prad. In fact, in 1D the equilibrium condition for electrons is∫(−e)ne Ex dx =∫ne fp dx = Prad, (26)New Journal of Physics 12 (2010) 045013 (http://www.njp.org/)79 RADIATION PRESSURE ACCELERATION OF ULTRATHIN FOILS12Figure 7. Red solid line: temporal evolution of the peak of the electrostatic fieldEx from the same simulation of figure 6, Em(t) = Ex(x = X (t)). Blue solidline: velocity of the peak position V (t) = dX/dt . Purple dashed line: functionEd(t) = Em(0)(1 − V/c)/(1 + V/c).while the total electrostatic pressure Pes on ions is given byPes =∫Zeni Ex dx (27)(the ponderomotive force on ions is smaller than fp by a factor ∼ me/m i and is thus negligible).Only in the case of local charge neutrality Zni = ne, would we obtain Pes = Prad. However, in theSail, Zni < ne, so that Pes < Prad. The calculation of Pes from the approximate initial profiles ofni and Ex (figure 5(a)) in equilibrium conditions shows that Pes = F P rad [28]; thus the equationof motion for the Sail can be written asddt(βγ ) =Pesρ`Fc2=Prad Fρ`Fc2=Pradρ`c2, (28)which is equivalent to equation (3).In the 0 < x < d region, which we call the ‘Tail’ region, complete electron depletionoccurs. In the Tail, ions are accelerated by their own space-charge field, as in a Coulombexplosion, resulting in an ion density that decreases with time and in a broad energy spectrum.Actually, not all the ions in the Tail remain at all times behind the Sail front, i.e. in thex < X (t) region. This is related to maintaining a mechanical quasi-equilibrium for electronsduring the acceleration. As the Sail is accelerated toward higher velocities, the radiation pressuredecreases by the factor (1 − β)/(1 + β). Thus, the field Ex must decrease by the same factor tokeep the electrons in equilibrium. This is evident in figure 7, where we plot the peak fieldEm(t) = max[Ex(x, t)] = Ex(x = X (t)) and show that Em(t) ' Em(0)(1 − β(t))/(1 + β(t)), sothat the pressure equilibrium is maintained. In 1D, the value of Ex at a point x ′ is proportionalto the amount of charge contained in the x < x ′ region; thus, there must be a positive currentthrough the x = X (t) surface during the acceleration stage. By computing the total number ofions in the x > X (t) region, we find that there is a flow of ions in the forward direction at theNew Journal of Physics 12 (2010) 045013 (http://www.njp.org/)8013Figure 8. Space–time contours of ni = ni(x, t) and ne = ne(x, t) from thesame simulation of figure 6, in logarithmic scale. Labels ‘b’ and ‘f’ indicatetwo electron bunches leaving the Sail in the backward and forward direction,respectively, at t ' 11.x = X (t) surface. These ions are accelerated by the space-charge field in the Tail region andoverturn the position of the Sail front.The number of electrons in the Sail remains constant up to near the end of the laser pulse,i.e. when the laser amplitude at the foil surface drops down. At a certain time, Prad does notbalance Pes anymore and thus electrons no longer remain in mechanical equilibrium. As aconsequence, a bunch of electrons is accelerated backward by Ex , as shown in figure 6(b).Some other electrons leave the Sail at later times and also in the forward direction, acquiringa larger velocity than the Sail. This dynamics is best visualized by plotting the space–timecontours of ni(x, t) and ne(x, t) in logarithmic scale, as in figure 8. The detachment of excesselectrons eventually leaves the Sail neutral. At the same time, the loss of mechanical equilibriumcauses heating of the electrons, driving in turn the expansion of the Sail (which can be noticedin figure 8) and the broadening of the ion spectrum.Despite the formation of two ion populations, of which the ‘monoenergetic’ one may beless numerous, the fraction of laser pulse energy converted into kinetic energy of the ions is stillin good agreement with the LS efficiency η given by equation (8), which in turn depends onlyon the Sail velocity. Since work has to be done by the laser pulse to push the negatively chargedSail against the electrostatic field Ex generated by the ions in the Tail, part of the energy is storedas electrostatic energy [28] and eventually converted into kinetic energy of ions in the Tail.In a recent similar study, Eliasson et al [27] also noticed the formation of two ionpopulations. In their work, ions in the Sail are referred to as ‘trapped’ in an effective potential inthe Sail frame. Using an argument of force balance (instead of pressure balance), they estimatethe fraction of ions initially trapped as F ' 1 − (a20/ζ2)/2.4. 2D simulationsThe analysis of the two preceding sections is based on a 1D approach, which may be accurateonly under certain conditions, i.e. when the intensity distribution of the laser pulse in the focalplane is flat-top and the foil displacement is smaller than the focal spot radius. To check thepredictions of the 1D model, we performed 2D simulations for laser and target parameters closeNew Journal of Physics 12 (2010) 045013 (http://www.njp.org/)81 RADIATION PRESSURE ACCELERATION OF ULTRATHIN FOILS14Figure 9. 2D simulation results. Snapshots of ne, ni and electric field componentsEx and Ez are shown at three different times. The laser pulse has eight cycleduration and eight wavelength diameter (both FWHM) and peak amplitudea0 = 20. The plasma foil has initial density ne = 50nc and thickness ` = 0.2λ.Normalizations are as in figure 6.to the optimal thickness condition (24). In 2D geometry, several effects of possible relevance aretaken into account, including e.g. laser intensity distribution, bending of the target, increasedheating of electrons due to longitudinal components of the electric field out of the laser axis andonset of surface rippling instabilities. A few groups also reported 3D simulations of thin-foilRPA [4, 14, 24]. These simulations are generally very demanding numerically, also because, asshown by the 1D analysis, high values of the spatial resolution and of the number of particlesper cell are needed to resolve the huge density variations in the plasma. With respect to the2D case, in 3D geometry the additional physical constraint of the conservation of the angularmomentum delivered by the circularly polarized laser pulse is taken into account. A theoreticalanalysis [24] shows that angular momentum absorption in the plasma may occur only viadissipative effects; thus, in general it does not provide the generation of an axial magnetic field,according to the mechanism hypothesized in [26].Figure 9 shows snapshots of ion and electron densities and of Ex and Ez field componentsfrom 2D simulations. The laser pulse has a peak amplitude a0 = 20 and a fourth-ordersupergaussian (i.e. an exp(−u4) function) profile both in the longitudinal (x) and in thetransverse (y) directions, with eight cycle duration and eight wavelength width (both FWHM).The foil target has initial electron density ne = 50nc and thickness ` = 0.2λ so that ζ = 31.4.At the intermediate time t = 18, the formation of the low-density Tail of ions can benoticed in the contour plot of ni. The electron density distribution is more complicated dueto 2D effects. As the foil is accelerated by the laser pulse, owing to target deformation the laserpulse incidence is effectively oblique at the edges of the laser spot. Moreover, as soon as the foildisplacement is a few times λ, the light reflected from one edge is incident back on the oppositeNew Journal of Physics 12 (2010) 045013 (http://www.njp.org/)8215Figure 10. Ion spectra (in energy per nucleon) from the simulations of figure 9(a0 = 20) and figure 11 (a0 = 30). The dotted line gives the spectrum where onlyions inside a 10◦ cone around the axis are included. Both spectra correspond tothe latest time in figures 9 and 11.side, and a standing wave-like pattern is observed (t = 18 T). These effects enhance electronheating, driving the formation of a low-density electron cloud around the target and leading tobroadening of the ion spectrum due to thermal expansion of the foil. In turn, absorption intoelectrons and oblique incidence reduce the radiation pressure at the edges of the focal spot, andmay thus enhance the effect of the inhomogeneous distribution of the laser intensity. A similardynamics is also noticed for even ‘flatter’ intensity profiles (e.g. eighth-order supergaussian).As a likely consequence of stronger electron heating in 2D, the ion spectrum shown infigure 10 is quite broad, even when only ions moving into a 10◦ cone around the axis areconsidered. According to these results, to obtain a monoenergetic spectrum is a challengingissue. In two recent papers, Chen et al investigated strategies to improve monoenergeticity basedon tailoring either the laser intensity distribution [12] or the thickness of the target [14].It also appears that 2D effects make the optimal thickness condition (24) more stringent,so that a lower intensity is required. Figure 11 shows snapshots from simulations with the sameparameters as in figure 9, but with the pulse amplitude a0 = 30, close to the value of ζ = 31.4.We observe a strong transmission of the laser pulse, causing a strong turbulence in the electrondensity. The strong transmission appears to be correlated with a strong rarefaction of the electrondensity out of the axis (around y = ±2λ as in figure 11), leaving a denser bunch of ions alongthe axis. The corresponding spectrum shows that the RPA peak is barely distinguishable fromthe exponential-like background.5. ConclusionsWe reported an analytical and simulation study of RPA of ultrathin foils. The effects of thenonlinear SIT of the foil were addressed and incorporated into the 1D ‘LS’ model for foilacceleration. The model predictions for ion energy and conversion efficiency as a function oftarget thickness were in fairly good agreement with the result of 1D simulations. The latterwere further analyzed to unfold the dynamics and self-organization of foil acceleration. 2Dsimulations for parameters close to the ‘optimal’ thickness value given by the 1D theory showedNew Journal of Physics 12 (2010) 045013 (http://www.njp.org/)83 RADIATION PRESSURE ACCELERATION OF ULTRATHIN FOILS16Figure 11. Same as figure 9, but for a pulse amplitude a0 = 30.that deformation of the target causes strong heating of electrons, leading to a broad ion spectrumand giving a lower threshold for the penetration of the laser pulse through the foil.AcknowledgmentsSupport from CNR via an RSTL project and use of supercomputing facilities at CINECA(Bologna, Italy) sponsored by the CNR/INFM supercomputing initiative are acknowledged.References[1] Badziak J, Glowacz S, Jablonski S, Parys P, Wolowski J, Hora H, Krása J, Láska L and Rohlena K 2004Production of ultrahigh ion current densities at skin-layer subrelativistic laser-plasma interaction PlasmaPhys. Control. 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Beams 11031301[23] Yan X Q et al 2008 Generating high-current monoenergetic proton beams by a circularly polarized laser pulsein the phase-stable acceleration regime Phys. Rev. Lett. 100 135003[24] Liseykina T V, Borghesi M, Macchi A and Tuveri S 2008 Radiation pressure acceleration by ultraintense laserpulses Plasma Phys. Control. Fusion 50 124033[25] Qiao B, Zepf M, Borghesi M and Geissler M 2009 Stable GeV ion-beam acceleration from thin foils bycircularly polarized laser pulses Phys. Rev. Lett. 102 145002[26] Tripathi V K, Liu C S, Shao X, Eliasson B and Sagdeev R Z 2009 Laser acceleration of monoenergeticprotons in a self-organized double layer from thin foil Plasma Phys. Control. Fusion 51 024014[27] Eliasson B, Liu C S, Shao X, Sagdeev R Z and Shukla P K 2009 Laser acceleration of monoenergetic protonsvia a double layer emerging from an ultra-thin foil New J. Phys. 11 073006[28] Macchi A, Veghini S and Pegoraro F 2009 ‘Light Sail’ acceleration reexamined Phys. Rev. Lett. 103 085003[29] Yan X Q, Wu H C, Sheng Z M, Chen J E and Meyer–ter–Vehn J 2009 Self-organizing GeV, nanocoulomb,collimated proton beam from laser foil interaction at 7 × 1021 W cm−2 Phys. Rev. Lett. 103 135001[30] Rykovanov S G, Schreiber J, Meyer–ter–Vehn J, Bellei C, Henig A, Wu H C and Geissler M 2008 Ionacceleration with ultra-thin foils using elliptically polarized laser pulses New J. Phys. 10 113005[31] Macchi A, Liseikina T V, Tuveri S and Veghini S 2009 Theory and simulation of ion acceleration withcircularly polarized laser pulses C. R. Phys. 10 207–15[32] Borghesi M 2009 Talk at the Coulomb09 Workshop Ion Acceleration with High Power Lasers: Physics andApplications (Senigallia, Italy, June 15–18, 2009)New Journal of Physics 12 (2010) 045013 (http://www.njp.org/)85 RADIATION PRESSURE ACCELERATION OF ULTRATHIN FOILSPlasma-Based Generation and Control of a Single Few-CycleHigh-Energy Ultrahigh-Intensity Laser PulseM. Tamburini,1,* A. Di Piazza,1 T. V. Liseykina,2 and C. H. Keitel11Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany2Institut für Physik, Universität Rostock, D-18051 Rostock, Germany(Received 3 August 2012; published 10 July 2014)A laser-boosted relativistic solid-density paraboloidal foil is known to efficiently reflect and focus acounterpropagating laser pulse. Here we show that in the case of an ultrarelativistic counterpropagatingpulse, a high-energy and ultrahigh-intensity reflected pulse can be more effectively generated by arelatively slow and heavy foil than by a fast and light one. This counterintuitive result is explained with thelarger reflectivity of a heavy foil, which compensates for its lower relativistic Doppler factor. Moreover,since the counterpropagating pulse is ultrarelativistic, the foil is abruptly dispersed and only the first fewcycles of the counterpropagating pulse are reflected. Our multidimensional particle-in-cell simulationsshow that even few-cycle counterpropagating laser pulses can be further shortened (both temporally and inthe number of laser cycles) with pulse amplification. A single few-cycle, multipetawatt laser pulse withseveral joules of energy and with a peak intensity exceeding 1023 W=cm2 can be generated alreadyemploying next-generation high-power laser systems. In addition, the carrier-envelope phase of thegenerated few-cycle pulse can be tuned provided that the carrier-envelope phase of the initial counter-propagating pulse is controlled.DOI: 10.1103/PhysRevLett.113.025005 PACS numbers: 52.59.Ye, 42.65.Re, 52.38.−r, 52.65.RrAwide range of novel studies in nonlinear optics as wellas the major new regimes of extreme field physics requirelaser pulses which simultaneously exhibit the followingthree key features: few-cycle duration, high-energy, andultrahigh intensity. Already in nonrelativistic atomic phys-ics, it has been demonstrated that quantum processes can becontrolled by manipulating the pulse shape of few-cyclelaser pulses [1]. In order to achieve the same goal also in theultrarelativistic regime and in the realm of nonlinear QED,few-cycle laser pulses with tunable carrier-envelope phase(CEP) are required with peak intensities largely exceeding1020 W=cm2 [2–4]. At such high intensities, for example,the nonlinear Compton emission spectrum is expected toshow pronounced pulse-shape effects [5,6].Although next-generation 10-PW optical laser systemsare expected to generate laser pulses with 150–300 J energyand 15–30 fs duration [4,7] [full width at half maximum(FWHM) of the pulse intensity], the limited bandwidthrenders the generation of few-cycle pulses with multijouleenergy very challenging [8,9]. Indeed, the only laser systemaiming at 1-PW power and few-cycle duration is thePetawatt Field Synthesizer [10]. Several methods forfurther shortening and amplifying laser pulses have beenproposed, e.g., Raman [11,12] and Brillouin [13,14] back-scattering, interaction with plasma waves [15,16] andionization induced self-compression effects [17,18].However, none of the pulses generated employing theabove-mentioned methods simultaneously exhibit few-cycle duration, multijoule energy, and ultrarelativisticintensity. In fact, the initial intensity is bounded torelatively moderate values and the generated pulses aretransversely and temporally modulated, which might pre-vent their subsequent focusing to ultrarelativistic inten-sities. In addition, the CEP control, which is crucial formany applications, has not been demonstrated in any of theabove-mentioned methods.In this Letter, we put forward the concept of a laser-boosted solid-density paraboloidal relativistic “mirror,” inter-acting with a superintense counterpropagating laser pulse, togenerate a CEP tunable few-cycle pulse with multijouleenergy and peak intensity exceeding 1023 W=cm2. Contraryto intuition, it is found that a heavy and therefore relativelyslow “mirror” should be employed to maximize the intensityand the energy of the reflected pulse, since its largerreflectivity compensates for the lower velocity. Further-more, the short duration of the reflected pulse is achievedby employing a superintense incident pulse, which abruptlydisperses the plasma mirror after only the first few cycles.Multidimensional particle-in-cell (PIC) simulations indicateboth the feasibility of the presented setup by employingnext-generation multi-PW laser systems and a considerableshortening with amplification even for already few-cyclelaser pulses.In the proposed setup, a “driver” pulse with frequency ωand (average) intensity Id accelerates a “mirror” to rela-tivistic velocities along the positive x direction and a“reflected” pulse is generated in the collision of the mirrorwith a counterpropagating “source” pulse, also with fre-quency ω and with intensity Is. Here and below, thesubscript s (d) and the upper (lower) sign refer to thePRL 113, 025005 (2014) P HY S I CA L R EV I EW LE T T ER Sweek ending11 JULY 20140031-9007=14=113(2)=025005(5) 025005-1 © 2014 American Physical Society87 PLASMA-BASED GENERATION AND CONTROL OF ULTRASHORT PULSESsource (driver) pulse counterpropagating (copropagating)with respect to the mirror, and T ¼ 2π=ω (λ ¼ cT) is thelaser period (wavelength). Our aim here is to determine theconditions for maximizing both the intensity and the energyof the reflected pulse. In order to develop an analyticalmodel, for the thin foil we employ the Dirac-δ densityprofile nðxÞ ¼ nelδðxÞ [19,20], where ne and l are the foildensity and thickness, respectively. If the foil moves withvelocity vx ¼ βc > 0, its reflectivity is given by Rs=d ¼ζ2s=d=ðζ2s=d þ Γ2s=dÞ [20], where Γ2s=d ¼ f1þ a2s=d − ζ2s=d þ½ð1þ a2s=d − ζ2s=dÞ2 þ 4ζ2s=d1=2g=2 and ζs=d ≡ ζ0=D.Here we have introduced the normalized (average) fieldamplitude a2s=d ≡ Is=d=I with I ≡m2eω2c3=4πe2, theDoppler factors D ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1 βÞ=ð1∓βÞp ⋛ 1, and the sur-face density ζ0 ≡ πnel=ncλ, with nc ≡meω2=4πe2 beingthe critical density. Notice that for a linearly polarized (LP)pulse the peak intensity Î is approximately twice theintensity I, whereas they coincide for a circularly polarized(CP) pulse. If both the source and the driver pulse fields areultrarelativistic (as=d ≫ 1), the reflectivity can be approxi-mated as [20,21] Rs=d ≈ 1 if ζs=d > as=d and Rs=d ≈ζ2s=d=a2s=d if ζs=d < as=d, which presents the reflectivitywith accuracy better than 2% for as=d > 50. Hence, thecondition ζs=d > as=d has to be fulfilled to secureRs=d ≈ 1.In our model the foil is initially at rest and it isaccelerated along the positive x direction by the driverpulse. In order to determine the value of the Doppler factorafter the acceleration phase Dþ0 , we assume that ζ0 > adand thus Rd ≈ 1. The velocity of a foil accelerated by theradiation pressure [22] of the driver pulse can be calculatedanalytically by employing the “light sail” equation for aperfectly reflecting mirror [20,21,23] and the result for Dþ0is Dþ0 ¼ 1þ Ed=ζ0, where Ed ¼ 2πZmeRa2dðwÞdw=Ampis the “effective” energy of the driver pulse. Here Z (A) isthe ion atomic number (weight) and a2dðwÞ ¼ IdðwÞ=Iis the field amplitude as a function of the foil phasew ¼ ½t=T − xðtÞ=λ.Since the foil undergoes a recoil due to the radiationpressure of the source pulse, the Doppler factor Dþ of thefoil at the maximum of the source pulse intensity is smallerthanDþ0 . On this respect it is convenient to employ a sharp-rising, high-contrast source pulse, as those generated withthe plasma mirror technique [24,25]. By proceeding as forthe calculation of Dþ0 , we obtainDþ ¼ Dþ01þDþ0 Es=ζ0¼ ζ0ðζ0 þ EdÞζ20 þ Esðζ0 þ EdÞ; ð1Þwhere Es ¼ 2πZmeRa2sðwÞdw=Amp which, for a sharp-rising pulse, is the part of the source pulse energy before thesource pulse intensity reaches its maximum (see below fordetails). Since we seek Rs ≈ 1, we require ζs > as, whichprovides the constraint ζ0 > ζ0;m withζ0;m ¼ as½1 − ϵþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1 − ϵÞð1 − ϵþ 4Ed=asÞp=2; ð2Þwhere ϵ≡ Es=as accounts for the effect of the recoil. Inorder to maximize the energy and the intensity of thereflected pulse atRs ≈ 1 for fixed driver and source pulses,we have to maximize the Doppler factorDþ as a function ofζ0 with the condition ζ0 > ζ0;m. From Eq. (1),Dþðζ0Þ has amaximum at ζ0 ¼ EdffiffiffiffiffiEsp=ð ffiffiffiffiffiEdp−ffiffiffiffiffiEsp Þ and monotoni-cally decreases for ζ0 > ζ0. Assuming sufficiently smallrecoil [ϵ < 1=2 and Ed > Esð1 − ϵÞ2=ð1 − 2ϵÞ2], thenζ0;m > ζ0 and the maximum Dþðζ0Þ compatible with ζ0 >ζ0;m is at ζ0;m, and it is Dþm ¼ ζ0;m=as.Note that, for a flat foil and fixed driver and sourcepulses, both the maximum intensity Ir ¼ Dþ4RsIs andenergy Er ≈ IrSΔts=Dþ2 of the reflected pulse areachieved at the minimum ζ0 such that ζs > as, i.e., atζ0;m. Here S is the surface area of the focal spot and Δts isthe source pulse duration. In fact, for ζs < as the reflec-tivity isRs ≈ ζ2s=a2s thus Ir ¼ Dþ2ζ20I and Er ≈ ζ20ISΔts,which are monotonically increasing functions of ζ0. Thefact that there exists an optimal value of the surface densityhas a simple physical interpretation: for fixed driver andsource pulses, if ζ0 is too large, the foil slows down and theDoppler factor is small. If ζ0 becomes too small, the velocityof the foil increases and the reflectivity rapidly decreasesbecause ζs tends to vanish. Moreover, at ζ0;m the reflectedpulse energy Dþ2m IsSΔts is a monotonically increasingfunction of Is. If ϵ < 1=3 and Ed < asð1 − ϵÞ2ð1 − 3ϵÞ=4ϵ2, i.e., if the effect of the recoil is sufficiently small, themaximum reflected pulse intensity Dþ4m Is is also a mono-tonically increasing function of Is. For fixed source pulse,the above conditions account for the slowdown of the foildue to the recoil, which becomes increasingly important forincreasing foil velocity [see Eq. (1)]. In a three-dimensionalgeometry, a paraboloidal mirror can focus the sourcepulse to its diffraction limit. Since the laser wavelength isDoppler reduced in the rest frame of the foil, the reflectedpulse can be focused down to λ2=Dþ2 and the intensity atthe focus is Ir;f¼Dþ6RsIsS=λ2. If ζ0 > Ed, ϵ < 1=4 andEd<2asð1−ϵÞ2ð1−4ϵÞ=ð1þ2ϵÞ2, the maximum of theintensity at the focus Ir;f is achieved at ζ0;m and it is anincreasing function Is. In other cases, the maximum of Ir;fcan be a decreasing function of Is or the maximum of Ir;fcan be achieved at Rs < 1. However, in these cases ahigher intensity at the focus is achieved at the expense of alower reflected pulse power Pr ¼ Dþ4RsIsS and energyEr ≈Dþ2RsIsSΔts.For simplicity, we first consider a driver and source pulsewith one-cycle sin2-function rise and fall, and with a five-cycle constant plateau. Figure 1 reports the maximumreflected pulse amplitudeffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDþ4RsIs=Ipas a function of ζ0for ad ¼ 130 and for as ¼ 130; 100; 80. In each case thereflected pulse amplitude initially increases for increasingPRL 113, 025005 (2014) P HY S I CA L R EV I EW LE T T ER Sweek ending11 JULY 2014025005-288ζ0, reaches its maximum at Rs ≈ 1, and then decreases asthe Doppler factor decreases. The three triangles in Fig. 1are centered at (ζ0;m,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDþ4m Is=Ip) and their positioncoincides with the maximum of the reflected pulse ampli-tude, confirming our analytical estimates. Since in all casesEd < asð1 − ϵÞ2ð1 − 3ϵÞ=4ϵ2, the maximum reflectedamplitude rises for increasing Is (see Fig. 1). The resultsof one-dimensional PIC simulations with CP driver andsource pulses are also reported in Fig. 1 (colored circles),the foil being a slab of fully ionized carbon withne ¼ 400nc. The spatial resolution is λ=4000 and thenumber of particles per cell per species 1000. Our PICsimulation results agree with the model predictions atζ0 > ζ0;m, i.e., at Rs ≈ 1.In a multidimensional geometry, the onset of transverseRayleigh-Taylor-like (RT) [22] instabilities renders the foil“porous” to the source pulse. RT instabilities in the radiationpressure acceleration regime have been investigated analyti-cally [26,27], numerically [26–28] and experimentally [29].In particular, in Refs. [26,27] it was shown that in the linearapproximation the RT instability grows as exp½ΦdðwÞ withΦdðwÞ ¼Rw0 2π½Zmea2dðuÞλ=Ampζ0λRT1=2du where λRT isthe wavelength of the perturbation. Our simulations indicatethat in order to effectively reflect the source pulse, ΦdðwÞ ≲5.7 for λRT ≈ λ [29], which can be fulfilled by increasing thevalue of ζ0 > ζ0;m.In our two-dimensional PIC simulations both the driverand the source pulse have a sin2-function temporal fieldprofile with 15.5 fs duration (FWHM of the intensity),Gaussian transverse profile and wavelength λ ¼ 800 nm.The driver (source) pulse is CP (LP with the electric fieldalong the y axis) with intensity Id ≈ 3.4 × 1022 W=cm2(Is ≈ 5.6 × 1022 W=cm2) and spot radius σd ¼ 3.8λ(σs ¼ 1.2λ), corresponding to a power Pd ≈ 9.9 PW(Ps ≈ 1.6 PW). These parameters are envisaged at theAPOLLON laser system [4,7,30]. The foil consists of fullyionized carbon with electron density ne ¼ 400nc and it isinitially shaped transversely with a thickness distributionl¼max½l1;l0expð−y2=2σ2fÞ, with l1¼0.02λ, l0¼0.20λ,σf ¼ 2.6λ and localized at x ¼ 5λ. Note that the propertiesof such carbon foils can be engineered with high precisionnowadays [31,32]. It has been shown that Gaussian pulsesand shaped foils can be employed to generate collimatedion beams [33,34]. Here we propose to use shaped foils togenerate paraboloidal relativistic mirrors. Indeed, for σd >ffiffiffi2pσf the acceleration factor a2dðyÞ=ζ0ðyÞ [33] is larger inthe outer part of the foil, which therefore takes a focusingprofile for the source pulse. Since for many applicationsslow focusing and defocusing are desirable, we have setσd ≈ffiffiffi2pσf so the relativistic mirror is nearly flat beforeinteracting with the source pulse (see Fig. 2). The size ofthe computational box is 20λðxÞ × 20λðyÞ, the correspond-ing grid is 20 000ðxÞ × 8000ðyÞ and 900 particles per cellfor each species are used.Figure 2 displays the evolution of the square root of theenergy density u ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðE2 þB2Þ=2pand of the electrondensity distribution ne. The driver (source) pulse reachesthe edge of the foil at t ≈ 0 (10T). An accurate synchro-nization between two laser pulses can be achieved, e.g., bygenerating the two pulses from the same seed pulse beforethe amplification stage. Although instabilities have devel-oped (Φd ≈ 4.7 with our parameters) and density fluctua-tions are clearly visible before the source pulse impinges onthe foil, the foil remains sufficiently compact to reflect thefirst part of the source pulse (see Fig. 2 at t ≤ 16T and theSupplemental Material [35] for a movie of the laser-foilinteraction). As the source pulse amplitude at the foilFIG. 1 (color online). The maximum amplitude of the reflectedpulseffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDþ4RsIs=Ipas a function of ζ0 for ad ¼ 130 and as ¼130 (solid red line), as ¼ 100 (dashed blue line) and as ¼ 80(dotted black line). See the text for further details.FIG. 2 (color online). Snapshots of u ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðE2 þ B2Þ=2p(first row) and ne (second row) in normalized units. See the textfor details.PRL 113, 025005 (2014) P HY S I CA L R EV I EW LE T T ER Sweek ending11 JULY 2014025005-389 PLASMA-BASED GENERATION AND CONTROL OF ULTRASHORT PULSESposition increases, the source pulse “digs through” thelower-density regions and abruptly disperses the foil, whichbecomes transparent to the remaining part of the pulse (seeFig. 2 from t ¼ 16T to t ¼ 18T). Finally, at t ¼ 22T asingle few-cycle reflected pulse separated from the foilremnants is observed. The peak intensity and peak power ofthe reflected pulse are: Îr ≈ 2.3 × 1023 W=cm2 (for thesource pulse Îs ≈ 9.6 × 1022 W=cm2), and P̂r ≈ 2.2 PW,with 5.8 fs duration and 6.8 J energy [see Fig. 3(c)].Figure 3(a) displays the y component of the electric field ofthe reflected pulse along the central axis for the case of zero(solid black line) and π=2 (dotted red line) CEP of thesource pulse showing that the reflected pulse inherits theCEP of the source pulse. Inclusion of radiation reaction(RR) effects, according to Refs. [36,37], does not signifi-cantly alter the reflected pulse [see Fig. 3(b)]. Our explan-ation is that when the reflected pulse is generated, the foildensity is still high and the fields inside the foil are muchsmaller than in vacuum [36]. Moreover, we ensured that theprobability of electron-positron pair production remainsnegligible. The influence of a randomly distributed pre-plasma on the front surface of the foil is also considered inFig. 3(b) (dashed red line). The preplasma thicknesscorresponds to 10% of the foil thickness and its averagedensity is ne=2. The presence of the preplasma reduces thepeak intensity, peak power and energy of the reflected pulseto Îr ≈ 1.8 × 1023 W=cm2, P̂r ≈ 2.0 PW and 5.8 J, respec-tively. This can be explained by the increased electronheating due to the enhanced penetration of the driver pulseinto the preplasma. The modulus of the Fourier transformof the y component of the electric field along the centralaxis jEr;yðkxÞj, where kx denotes the wave number andk≡ 2π=λ, is reported in Fig. 3(d) (solid black line) showingthat the reflected pulse is chirped and peaked at λr ≈ 593 nm.For comparison, the spectra of two Gaussian pulses with thesame wavelength and with two (dotted red line) and three(dashed blue line) cycles FWHM of the field profile arealso reported [see Fig. 3(d)].In order to account for the slowly rising profile of thesource pulse and estimate the wavelength λr and peakintensity Îr of the reflected pulse, we approximate the sin2-function field profile with a linearly rising profile b0w=N.Here N is the number of cycles before the source pulsemaximum and b0 ¼ 3as=2ffiffiffi2pso the source pulse and itslinear profile approximation have the same duration andenergy before their maximum. Assuming Rs ≈ 1, themaximum reflected intensity is achieved at min½N; ŵ withŵ ¼ ½4AmpN2ζ0=15πZmeDþ0 a2s 1=3. For a slowly-risingprofile ŵ ≤ N, thus ÊsðŵÞ ¼ ζ0=5Dþ0 which does notdepend on the source pulse parameters. Hence, fromEq. (1) we get D̂þðŵÞ ¼ 5Dþ0 =6. By inserting our numeri-cal parameters we obtain: λr ≈ 656 nm and Îr ≈ 1.2 ×1023 W=cm2 for the linearly rising profile, and λr ≈593 nm and Îr ≈ 1.4 × 1023 W=cm2 for the more realisticsin2-function profile. While λr is in good agreement withthe two-dimensional simulation results, Îr is underesti-mated because, by definition, the one-dimensional modeldoes not include focusing effects. Indeed, our simulationsshow that increasing the ratio σd=σf by reducing σf from2.6λ to 2.4λ improves the focusing and further enhances Îrfrom 2.3 × 1023 W=cm2 to 2.8 × 1023 W=cm2. In addition,higher intensities are expected in a fully three-dimensionalgeometry, where, in contrast to two-dimensional simula-tions, the source pulse is focused also along the z axis.We also mention that increasing Ps=d by doubling σ2s=d andσ2f with the other parameters as reported above enhances P̂rto 3 PW but reduces the intensity enhancement Îr=Îs from2.4 to 1.8 because the pulse focusing decreases.Finally, we stress that even a few-cycle source pulse can befurther shortened and amplified. Indeed, by employing al0 ¼ 0.17λ, σf ¼ 2.1λ shaped foil and a driver (source)pulse with 15.5 fs (5.8 fs) duration, Id ≈ 5.1 × 1022 W=cm2(Is ≈ 5 × 1022 W=cm2) intensity and σd ¼ 3.1λ (σs ¼ 1λ)radius [corresponding to a driver (source) power Pd ≈9.9 PW (Ps ≈ 1 PW)], a single 1.5 cycles (2.1 fs duration),2 J energy, P̂r ≈ 1.8 PW and Îr ≈ 1.4 × 1023 W=cm2reflected pulse is generated (Îs ≈ 7 × 1022 W=cm2). Mor-eover, in contrast to the previous case of a relatively longsource pulse, a 2.7 fs duration, 1.3 J energy 1 PW peak powerand 4.7 × 1022 W=cm2 peak intensity transmitted pulse isalso generated (see the movies in the Supplemental Material[35]). Similar parameters for the driver and source pulses areenvisaged at the Extreme Light Infrastructure [4,38].We acknowledge useful discussions with B. M.Hegelich, N. Kumar, A. Macchi and G. Sarri. We thankA. Macchi for providing his one-dimensional PIC code.Some PIC simulations were performed using the comput-ing resources granted by the Research Center Jülich underthe Project No. HRO01.FIG. 3 (color online). (a) Er;y along the central axis for zero(solid black line) and π=2 (dotted red line) CEP of the sourcepulse. (b) Er;y with RR effects (solid black line) and with apreplasma on the front surface of the foil (dashed red line).(c) Power contained in a spot with 1λ radius centered on the axis.(d) jEr;yðkxÞj (solid black line) and the corresponding quantity fora Gaussian pulse with two (dotted red line) and three (dashed blueline) cycles FWHM of the field profile. The inset shows a zoomof the main peak region.PRL 113, 025005 (2014) P HY S I CA L R EV I EW LE T T ER Sweek ending11 JULY 2014025005-490*matteo.tamburini@mpi-hd.mpg.de[1] F. Krausz and M. Ivanov, Rev. Mod. Phys. 81, 163(2009).[2] S. Meuren and A. Di Piazza, Phys. Rev. Lett. 107, 260401(2011).[3] A. I. Titov, H. Takabe, B. Kämpfer, and A. Hosaka, Phys.Rev. Lett. 108, 240406 (2012).[4] A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H.Keitel, Rev. Mod. Phys. 84, 1177 (2012).[5] F. Mackenroth and A. Di Piazza, Phys. Rev. A 83, 032106(2011).[6] M. Boca, V. Dinu, and V. Florescu, Phys. Rev. A 86, 013414(2012).[7] A. V. Korzhimanov, A. 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Reau, P. d’Oliveira, P. Audebert,R. Marjoribanks, and P. Martin, Nat. Phys. 3, 424 (2007).[25] C. Rödel, M. Heyer, M. Behmke, M. Kübel, O. Jäckel, W.Ziegler, D. Ehrt, M. Kaluza, and G. Paulus, Appl. Phys. B103, 295 (2011).[26] F. Pegoraro and S. V. Bulanov, Phys. Rev. Lett. 99, 065002(2007).[27] S. V. Bulanov, T. Z. Esirkepov, F. Pegoraro, and M.Borghesi, C.R. Phys. 10, 216 (2009).[28] M. Chen, N. Kumar, A. Pukhov, and T.-P. Yu, Phys. Plasmas18, 073106 (2011).[29] C. A. J. Palmer et al., Phys. Rev. Lett. 108, 225002 (2012).[30] G. Chériaux, F. Giambruno, A. Fréneaux, F. Leconte, L. P.Ramirez, P. Georges, F. Druon, D. N. Papadopoulos, A.Pellegrina, C. Le Blanc, I. Doyen, L. Legat, J. M. Boudenne,G. Mennerat, P. Audebert, G. Mourou, F. Mathieu, and J. P.Chambaret, AIP Conf. Proc. 1462, 78 (2012).[31] A. V. Krasheninnikov and F. Banhart, Nat. Mater. 6, 723(2007).[32] W. van Dorp, C. Hagen, P. Crozier, B. van Someren, andP. Kruit, Microelectron. Eng. 83, 1468 (2006).[33] M. Chen, A. Pukhov, T. P. Yu, and Z. M. Sheng, Phys. Rev.Lett. 103, 024801 (2009).[34] M. Chen, T.-P. Yu, A. Pukhov, and Z.-M. Sheng, New J.Phys. 12, 045004 (2010).[35] See the Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.113.025005 for amovie of the laser-foil interaction with the parameterscorresponding to those of Fig. 2 (Movie_long_source_pul-se.gif) and with the parameters reported in the last paragraphof the Letter (Movie_few-cycle_source_pulse.gif).[36] M. Tamburini, F. Pegoraro, A. Di Piazza, C. H. Keitel, andA. Macchi, New J. Phys. 12, 123005 (2010).[37] M. Tamburini, T. V. Liseykina, F. Pegoraro, and A. Macchi,Phys. Rev. E 85, 016407 (2012).[38] “Extreme Light Infrastructure,” http://www.eli-laser.eu/(2014).PRL 113, 025005 (2014) P HY S I CA L R EV I EW LE T T ER Sweek ending11 JULY 2014025005-591 PLASMA-BASED GENERATION AND CONTROL OF ULTRASHORT PULSES93 HOT ELECTRON GENERATION AND BRUNEL’S MODEL3. Hot electron generation and Brunel’smodelThe list of publications:P. Mulser, S.M. Weng, T. Liseykina, ”Analysis of the Brunel model and resulting hot electron spectra,Phys. Plasmas 19, 043301 (2012)T. Liseykina, P. Mulser, M. Murakami, Collisionless absorption, hot electron generation, and en-ergy scaling in intense laser-target interaction, Phys. Plasmas 22, 033302 (2015)Analysis of the Brunel model and resulting hot electron spectraP. Mulser,1,a) S. M. Weng,1,b) and Tatyana Liseykina21Theoretical Quantum Electronics (TQE), Technische Universität Darmstadt, D-64289 Darmstadt, Germany2Institut für Physik, Universität Rostock, Universitätsplatz 3, D-18051 Rostock, Germany(Received 9 December 2011; accepted 3 March 2012; published online 11 April 2012)Among the various attempts to model collisionless absorption of intense and superintense ultrashortlaser pulses, the so-called Brunel mechanism plays an eminent role. A detailed analysis revealsessential aspects of collisionless absorption: Splitting of the electron energy spectrum into two groupsunder p-polarization, prompt generation of fast electrons during one laser cycle or a fraction of it,insensitivity of absorption with respect to target density well above nc, robustness, simplicity, andlogical coherence. Such positive aspects contrast with a non-Maxwellian tail of the hot electrons, toolow energy cut off, excessively high fraction of fast electrons, and inefficient absorption at moderateangles of single beam incidence and intensities. Brunel’s pioneering idea has been the recognition ofthe role of the space charges induced by the electron motion perpendicular to the target surface thatmake irreversibility possible. By setting the electrostatic fields inside the overdense target equal tozero, anharmonic resonance and mixing of layers leading to Maxwellianization are excluded. Towhat extent the real electron spectra and their scaling on laser intensity are the product of theinterplay between Brunel’s mechanism and anharmonic resonance is still an open question. VC 2012American Institute of Physics. [http://dx.doi.org/10.1063/1.3696034]I. INTRODUCTIONDespite nearly three decades of research on superintenselaser-target interaction collisionless absorption is not yetunderstood. Simple estimates show that from intensities ofI ¼ 1016 W/cm2 on collisional heating becomes less and lessimportant and the excellent degree of absorption at relativis-tic and subrelativistic intensities, demonstrated by experi-ments and simulations, has to rely on collisionlessprocesses.1 Various mechanisms and models have been pro-posed: Anomalous skin effect,2,3 vacuum heating,4,5 wavebreaking,6 Landau damping,7 diffusive heating,8 stochasticelectron acceleration in fluctuating fields,9 j B heating,10surface plasmon excitation,11,12 and “laser dephasingheating.”13 The models are either based on simulations only(j B heating, “laser dephasing”), or they are in disagree-ment with basic properties observed in experiments and sim-ulations. The anomalous skin effect at high intensities yieldsan absorption of a few percentage only. In addition, theassumption of regular or diffuse electron reflection at thetarget-vacuum interface is severely modified by electrontrapping in front of the target as well as in its interior.14Absorption by excitation of surface plasmons requires corru-gated targets.12 In flat targets,11 coupling is very delicate andunlikely to occur. For a discussion in more detail, the readermay consult Refs. 1, 14, and Sec. 8.3.2 in Ref. 15.One of the prominent features of collisionless absorptionin intense laser beam-dense matter interaction is the genera-tion of fast electrons. Energetic electrons in the MeV regionare observed in all computer simulations of particle in cell(PIC), Vlasov and molecular dynamics type, and appear inall corresponding experiments as well. As several applica-tions are connected with this phenomenon, like efficient gen-eration of intense electron and ion beams by noveltechniques, fast ignition of inertial fusion pellets, new radia-tion sources in the extreme ultraviolet (XUV) domain, cancertherapy, the spectral composition of the fast electrons, theirmaximum energy, and the question of possible tailoring ofthe spectra is in the focus of common interest. Nevertheless,no systematic spectral investigations on the experimental sitehitherto exist and, on the other hand, the theoretical under-standing of the electron spectra is far from being satisfactory.Even worse, no general agreement exists on how the fastelectrons are created in the laser field.In long laser pulses of ps duration and subrelativisticintensities, several collisionless processes may be responsi-ble for absorption and fast electron generation as soon as ararefaction density profile has formed, to mention first reso-nance absorption at critical electron density16 and subsequentelectron trapping in the electron plasma wake (Landaudamping). Collisionless absorption becomes particularlycomplex owing to crater formation at late times and the evo-lution of instabilities, Secs. 2.3 and 6.2 in Ref. 15. The studyof collisionless absorption in the general 3-dimensional (3D)case with no restrictions on symmetry is the domain ofsimulations and experiments. Both need interpretation; simu-lations do not reveal the underlying physics either. Neverthe-less, the situation is not hopeless. Basic understanding meansprediction of the degree of absorption, of the origin of fastelectrons and their spectrum, their percentage and scaling ofthe “temperature” on laser intensity as well as the meanenergy of the slow electrons. What we have to aim at is simi-lar to atomic physics once when understanding the hydrogenspectrum led to understanding the complex spectrum of iron.In plane target geometry, there is the hope to arrive at such ahydrogen atom analogy, i.e., at a simple picture of physicsa)Electronic mail: Peter.Mulser@physik.tu-darmstadt.de.b)Present address: Institute of Laser Engineering, Osaka University, Osaka565-0871, Japan.1070-664X/2012/19(4)/043301/8/$30.00 VC 2012 American Institute of Physics19, 043301-1PHYSICS OF PLASMAS 19, 043301 (2012) Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 139.30.84.198 On: Fri, 03 Jun2016 14:12:0395 ANALYSIS OF THE BRUNEL MODELgoverning collisionless absorption of intense pulses (a) whentheir length is of sub 100 fs duration and the ions may beconsidered as fixed or (b) in the (sub)ps domain when theirintensity is relativistic. In both cases, the electron dynamicsof the critical region may be limited to 1D of a sharp edgedplasma, owing to the absence of a rarefaction wave (a) orbecause of the extreme profile steepening and almost ab-sence of subcritical plasma, Sec. 5.3.1 in Ref. 15 (b). Fortu-nately, there exist 1D single beam experiments showingabsorption up to 78% under oblique incidence and 1D PICsingle beam simulations reproducing them.17 Important hintsfor a simple model may come from simulations. In detail,simulations show that the fast electrons are generated (1) firstbefore the slow electrons; (2) nearly instantaneously, i.e., thesingle electron during one laser cycle; and (3) efficiently,regardless of how overdense the target is. In contrast, anabsorption mechanism of diffusive type, e.g., electron-ioncollisions and collisions of the electrons with the fluctuatingelectromagnetic field according to the fluctuation-dissipationtheorem, is slow and extends over a high number of cycles.Hence, fast electron generation is not of diffusive type.In 1987, Brunel proposed an absorption model foroblique incidence of a p-polarized sub-relativistic laser beamonto a step-like highly overdense flat target, ne nc; neelectron density, nc ¼ 0mex2=e2 critical density, me elec-tron mass, x laser frequency, and e electron charge.18 As weshall see in Sec. II, this model is capable of explaining sev-eral features of collisionless absorption in a much more con-vincing way than others mentioned above. In the stronglyrelativistic domain, the zero vector potential mechanism per-haps may assume a similar role.19 Indeed, until recently Bru-nel’s model has been the only successful attempt so far toexplain efficient collisionless absorption on physical groundsunder a minimum of assumptions and simplest geometry.For normal incidence, j B heating is generally invoked.10However, the reader must be aware that no explanation wasgiven for it by the authors, absorption was found by simula-tions only. Nevertheless, it is a pioneering paper, similar toBrunel’s, because the authors recognized that electronmotion strictly parallel to the laser field would define a per-fectly reversible process. Irreversibility comes from themotion perpendicular to the target surface induced by themagnetic field B and its discontinuity there, in analogy toBrunel’s model for oblique incidence. Brunel’s model occu-pies a special place. Therefore, it is indicated to subject it toa thorough analysis, never given before, the author included,with particular attention to its predictions on fast electrongeneration. It seems to be believed by a majority of research-ers that the fast electrons are “Brunel electrons.” In this con-text here, an electron is defined fast somewhat arbitrarilywhen its energy exceeds the mean oscillation energy it hasgot in the laser beam in vacuum.The analysis of Brunel’s model will be done in Sec. II.By comparing with measurements, it will be shown to whatextent this model is able to make valid predictions and bywhich simple additions partial improvement in absorptioncould be achieved. Subsequently, we shall compare with PICsimulations and discover also serious shortcomings and weshall indicate the main reason for them. The purpose of theanalysis is to make a further step towards understanding thephysics of collisionless absorption and the splitting of theelectron spectrum into a low energetic main part and a hottail. Incidentally, the detailed study of the plasma layer dy-namics in Brunel’s model will reveal already a qualitativeunderstanding of the existence of two spectral groups.Contrary to the general believe that all electrons are pushedback to the target by the laser field after reversion, half ofthem (the slow ones) return against the laser field. Finally, itwill result that the belief of the fast electrons being of Brunelorigin is of very limited validity.II. BRUNEL’S MODEL IN DETAILA highly overdense, e.g., solid target of constant elec-tron density ne ¼ n0 and zero temperature filling the halfspace x 0 is given. From the vacuum, the laser field com-ponent EðsÞ ¼ E0 sins is normally incident onto its surface,s ¼ xt. From Poisson’s law follows that the skin depth attime sl isx0l ¼ 0E0sinslen0; (1)see Fig. 1. With the amplitude of the oscillation velocityv0 ¼ eE0=ðmexÞ in x direction, the normalized velocity wl ¼v=v0 of the layer number l is given bywlðsÞ ¼ ðcoss cosslÞ þ ðs slÞsinsl; s sl: (2)Correspondingly, the normalized trajectory sl ¼ xlx=v0results asslðsÞ ¼ ðsins sinslÞ ðs slÞcossl þ 12ðs slÞ2sinslþ x0lxv0; s sl: (3)The number of particles per unit cross section that move dur-ing one cycle follows from Eq. (1) as N0 ¼ n0x0 ¼ 0E0=e.FIG. 1. Configuration of Brunel’s absorption model. A p-polarized planelaser wave EL (intensity IL) impinges under angle a and is partially reflectedas REL under a. At time instant sl ¼ xtl, the resulting x component E0ðslÞpenetrates up to the position x0l given by Eq. (1). At the instant s, its magni-tude there is E0ðsins sinslÞ for jsinsj > jsinslj and zero for jsinsj jsinslj.043301-2 Mulser, Weng, and Liseykina Phys. Plasmas 19, 043301 (2012) Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 139.30.84.198 On: Fri, 03 Jun2016 14:12:0396A. The genuine modelOwing to the smallness of x0l in a high density target incomparison to the oscillation amplitude v0=x, it is set zero.When the lth particle is driven back to the target surface atthe instant s ¼ sf , it enters the region x > 0 that is assumedto be field-free. Thus, it has gained the kinetic energymev20w2l ðsf Þ=2 from the laser field. For Eqs. (2) and (3) tohold the orbits are not allowed to cross. A sufficient condi-tion for this is that@sl@sl¼ 12ðs slÞ2cossl (4)does not change sign. In the interval 0 sl p=2, it issatisfied.In Fig. 2, representative orbits starting in the first interval0 sl 2p are shown. All of the N0 electrons are pulled outinto the vacuum during the first quarter time period 0 sl 0 and s ¼ p=2, it follows wl ¼ sinþ cos¼ 2=2 < 0. In contrast, sl ¼ p=2 yields wl ¼ 3=3! > 0. Inthe second half period p < sl < 2p, the driving laser field isreversed and neutralized by the (infinitely) strong space chargefield. Thus, when considering only particles with reentry timess 2p, no crossing of orbits happens. Figure 2, upper plot,shows that not all particles return during one period of s. Thefront layer drifts away into the vacuum at the speed v0 becauseof missing free fall term. The closer a layer is positioned to thefront, the longer is its reentry time. This gives rise to someorbit crossing for reentry times larger than one cycle.In Fig. 3, the number of particles returning up to time sin units of N0 is plotted. It shows that N0=2 particles reenterin the interval p=2 < s p but only a small fraction (2.2%)comes back after one period 2p.The velocity wf at which a particle enters the target isshown as a function of sl in Fig. 4. The maximum reacheswf ¼ 2:13. High-speed electrons in the neighborhood of themaximum of wf return late. Particles lifted into vacuum atsl < 13 ¼ 13p=180 come back only after two periods andshow strongly oscillating reentry velocities.Of particular interest is the absorbed energy as a func-tion of s. The dashed curve in Fig. 5 shows the average reen-try energy E in units of the mean oscillation energy mev20=4in the interval ½0; s]. If normalization is done by the numberof all particles N0 that have been moved during one lasercycle, the solid line is obtained. The results do not differmuch from each other because the electrons N0=2 reenteringduring the second quarter period contribute energetically byless than 1%. If summation is extended to all particles N0pulled out into the vacuum during one period, the result isthe dotted line. All N0 electrons contribute to absorption.However, the target is heated directly by the laser only bythose coming back during the duration of the laser pulse.Crossing of trajectories does not occur in the intervalp < s 5p=2. In order to obtain a measure for target heat-ing, we extend the integration over a whole cycle from p to3p, although intersection of orbits occurs for s > 5p=2, inva-lidating in principle Eqs. (2) and (3) (Brunel18 does the inte-gration from p=2 to 5p=2). However, according to Fig. 3,only 0.5% of electron orbits show crossing. This is a toosmall number to influence the equations of motion (2) and(3) by their space charge. Together with Fig. 5 follows thatthe energetic contribution from the extension remains belowFIG. 2. Single layer displacement sðsÞ. The individual layer starts from posi-tion x0l ’ 0 at the time sl, indicated in degrees on the ordinate at the rightð10; 20; ::::; 90() p=2; 100; :::::Þ. Only displacements xðsÞ 0 are physi-cally real (a). In the region x>0, the electric field E0ðsÞ is zero at all times s (b).FIG. 3. Number of particles per unit area NðsÞ returned to the target in theinterval ½0; s.FIG. 4. Reentry velocity component wf normal to the target of a layerstarted at sl.043301-3 Mulser, Weng, and Liseykina Phys. Plasmas 19, 043301 (2012) Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 139.30.84.198 On: Fri, 03 Jun2016 14:12:0397 ANALYSIS OF THE BRUNEL MODEL1% and that the strongly oscillating fraction in Fig. 4 contrib-utes even considerably less. From Fig. 5 for the averageenergy ghmev20=4 gained per electron heating the target in theinterval ½p; 3p, one reads gh ¼ 1:74 (Brunel: gh ¼ 1:57).The corresponding factor gt of all electrons contributing toabsorption is gt ¼ 1:87. One could think that the run awayelectrons could accumulate over several cycles and disturbthe dynamics Eqs. (2) and (3). This may not happen because,in three dimensions, the space charge cloud rarefies rapidlyand may also be quickly neutralized by a return currentwithin the target.Let us come now to the two most interesting questions,namely to the energy spectrum of the electrons and to theabsorption coefficient in Brunel’s model. The fastest electronof wf ¼ 2:13 finds back to the target at s ’ 2p correspondingto the cut off energy Ef ¼ ð2w2f ¼ 9:07Þ mev20=4. Thespectral distribution f(E) normalized to unity is presented inFig. 6. Its peculiarity is the peak of f at the cut off energyexceeding the flat minimum by a factor of 7.5. If only nonintersecting orbits are considered, i.e., s 5p=2, almost nochange is observed; in particular, cut off energy and the peakthere in f(E) are preserved.As the only free parameter is 0, this spectrum is self-similar and universal. Equating the energy gain by the reen-tering electrons per period 2p to the energy supplied to thetarget,14N0ghmev20 ¼ AIcos a2px; (5)a angle of incidence, the absorption coefficient A isA ¼ 12ghv0cE20E2L12p cos a; (6)EL laser field amplitude. With the help of the reflection coeffi-cient R, the field component E0 follows from Fig. 1 asE0 ¼ ð1þffiffiffiRp ÞELsina. Introducing the electron oscillationamplitude in vacuum vos ¼ eEL=mex, v0 ¼ vosð1þffiffiffiRp Þsina,the heating and total absorption coefficients (indexes h, t) readA ¼ 14pgh;tvoscð1þffiffiffiRpÞ3 sin3acosa ðIk2Þ1=2;gh4p¼ 0:138; gt4p¼ 0:149: (7)In general, the connection between A and R may be com-plex. In the simplest case, R ¼ 1A may be set and Eq. (7)may be solved for vos=c ¼ 0:3 as a function of a. The resultis presented in Fig. 7 for gt. For a¼86, total absorption isreached and no solution exists beyond. At relativistic inten-sities, an effective Lorentz factor c has to be introduced. Asa rule, for a60, Eq. (7) yields only modest absorption inp-polarization. Even setting vos ¼ c (weakly relativistic case)and R¼ 1 produces collisionless absorption not exceedingA ¼ 120sin3a=cosa% at moderate angles of incidence(a 25). On the other hand, for a ¼ 82A exceeds unity atvos=c ¼ 1. In the subrelativistic intensity range, the absorbedenergy scales like ðIk2Þ3=2.Equation (7) can directly be compared with measure-ments and PIC simulations of the angular absorption depend-ence in p polarization for I ¼ 2 1016 W/cm2 Ti:Sa,17 seeTable I, AC in the 1st row. They are taken with a few cycleTi:Sa laser and correspond to the conditions of Brunel’smodel, i.e., no prepulse, no hydrodynamic motion. The lastvalues (80) refer to I ¼ 5 1016 W/cm2. The Brunel valuesAB in the 2nd row are calculated from Eq. (7) under the con-dition Aþ R ¼ 1.In the 3rd row under AE, older measurements at I ¼3 1016 W/cm2 twice the harmonic of the fundamentalTi:Sa frequency are reported for additional comparison.20 InFIG. 5. Normalized average reentry energy as a function of reentry time s.Bold curve: average over all particles pulled into vacuum during one lasercycle; and dashed curve: average over NðsÞ from Fig. 3.FIG. 6. Electron spectra f(E) and FðEÞ ¼ Ð f ðE0ÞdE0.FIG. 7. Absorption coefficient A as a function of angle of incidence a forvos=c ¼ 0:3 under condition R ¼ 1A; vos=c electron oscillation amplitudein vacuum normalized to light speed c.043301-4 Mulser, Weng, and Liseykina Phys. Plasmas 19, 043301 (2012) Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 139.30.84.198 On: Fri, 03 Jun2016 14:12:0398this latter case, the plasma density scale length L in units ofthe fundamental laser wavelength L=k ¼ 5% 6% is esti-mated. The table shows good agreement between the twoexperiments, in addition AC is in agreement with PIC simu-lations, see Ref. 17. However, the predictions on absorptionAB from Brunel’s genuine model differ drastically. Onlynear grazing incidence realistic values are obtained for AB;for all angles below the AB values are far too low, i.e., themodel works inefficiently in this intensity range. From Eq.(7), the scaling AB I1=2 is deduced. Owing to the universalcharacter of the spectrum in Fig. 6, except slight modifica-tions due to R, this is also the scaling of the fast electrons.The scaling extracted from Ref. 17 is AB I0:1260:02. Thescaling of the average energy of the fast electrons (“electrontemperature Te”) in Brunel’s model results proportional tothe intensity I, whereas in the experiments, simulations, andtheory, the state of the art is Te I1=2 to I1=3 with tendencyto a 1/4 exponent in the strong relativistic domain, see Refs.21–26. Thus, the Brunel mechanism predicts very differentscalings for the total absorption coefficient and the absorbedtotal energy; these quantities in turn are proportional to thenumber and the energy of the hot electrons in Brunel’smodel.There is a natural subdivision of the electron spectruminto a low energy and a high energy group, separated by asmooth transition zone. Where does it originate from? Witha view on Figs. 2–6 and bearing in mind what has been statedon particle dynamics in the combined space charge and laserfields, in particular on reentry, the decay of the electron ener-gies into a slow and into a fast group has to be attributed tothe following characteristics of Brunel’s model. The particlesreturning to the target from vacuum during the secondquarter cycle ðp=2; pÞ are those closer to the target. Theyfeel a weakly screened space charge field and a weakenedlaser field. As a consequence, they are not pushed far outinto to vacuum and return already when the two fields areopposed to each other. This results into a low energy gain forhalf the total number N0 of particles lifted into the vacuum(see Figs. 3, 5, and 9). We give them the name “free fallingparticles.” For the N0 particles reentering the target in theinterval ðp; 2pÞ, the situation is reversed; they are effectivelyaccelerated back by the two fields in phase.B. Extended Brunel modelIn the sharp-edged plasma, the energy gained by the sin-gle electron and absorbed in the field-free region is given bythe sum of the contributions from wl normal and wy parallelto the target rather than by wl alone, thus12mew2f ¼12meðw2l þ w2yÞ; wy ¼eE0ð1ffiffiffiRp Þcosasinslmexv0:(8)If not stated differently, in the following two pictures, theangle of incidence is chosen a¼45, and R¼ 0 is set fordirect comparison with Figs. 4–6. The significant contribu-tion to wf by the extension is seen in Fig. 8 from the boldcurve. There is a significant change for sl < p=12, as closerinspection shows, but on the same ground as in Fig. 4, itdoes not contribute noticeably to the average reentry energy.However, the extension has a strong impact on it arounds ¼ p=6, compare the dotted and dashed graphs. As a conse-quence, the absorbed energy per electron is increased consid-erably, gh ¼ 4:04 and gt ¼ 4:27 now, see the change of scalealmost by a factor of 2.5 in Fig. 9 compared to Fig. 5, bothunder R¼ 0. Only its shape is basically preserved owing towf reducing to zero for sl approaching p=2.Again, the most interesting aspect is the energy distribu-tion function f(E) and its integral F(E). The number of slowelectrons (E < 1Þ is decreased and there is a significantincrease of hot electrons over the entire spectral range with apronounced maximum around E¼ 2, Fig. 10. Needless tomention that it is far from Maxwellian. The location of thelocal maximum of f(E) is very sensitive to the reflectioncoefficient R. The fraction of the fast electrons has increasedfrom 34% to 71%.TABLE I. Comparison of the absorption coefficients as functions of theincident angle from two experiments (AC;AE) and from Brunel’s model(AB;ABE). The meaning of AC;AB;AE, and ABE is given in the text. ABand ABE obey the conditionAþ R ¼ 1.a 20 30 40 50 60 70 80AC 0.12 0.17 0.20 0.30 0.40 0.60 0.77AB 0.003 0.012 0.028 0.054 0.098 0.171 0.334AE 0.11 0.12 0.24 0.38 0.50 0.49 —ABE 0.006 0.020 0.046 0.089 0.155 0.262 0.476FIG. 9. Normalized average reentry energy from the extended Brunelmodel, corresponds to that from the Brunel model in Fig. 5.FIG. 8. Extended Brunel model, corresponds to Fig. 4 with motion wy paral-lel to the target surface included. Dashed: genuine Brunel model,wf ¼ ðw2yf þ w2xf Þ1=2.043301-5 Mulser, Weng, and Liseykina Phys. Plasmas 19, 043301 (2012) Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 139.30.84.198 On: Fri, 03 Jun2016 14:12:0399 ANALYSIS OF THE BRUNEL MODELIn Fig. 7, A is calculated self consistently fulfilling thecondition Aþ R ¼ 1. The same should be done in theextended model for wf from Eq. (8). This is much moreinvolving because now gh depends on angle of incidence aand on R. For vos=c ¼ 0:3, the numerical result is shown inFig. 11. The difference to the genuine Brunel model is 45% ata ¼ 60, see Fig. 11, bold curve, with dashed curve takenfrom Fig. 7. It shifts into the right direction, but not enough atmoderate intensities, as, in particular, the 4th row of Table Iillustrates with ABE self consistently determined for the pa-rameters of AB of the 2nd row for intensities vos=c. 0:1. Theextension introduced here is not as successful one would havehoped with a view on the drastically increased values ofgh and gt. It is a consequence of the minus sign in Eq. (8). If,as a test, the R values are chosen from Ref. 17, the experimen-tal values from there are reproduced; however, this is incon-sistent. The situation with the spectrum of the fast electronsdeteriorates in the extended model owing to the presence ofthe local maximum at normalized E¼ 2.III. THE PROBLEM OF THE FAST ELECTRONSPECTRA AND THE MISSING PHYSICS IN BRUNEL’SMODELAmong the various attempts to arrive at a quantitativedescription of the angular dependence of absorption, there isthe one dimensional hydrocode MULTI-fs (Ref. 27) that canhandle linear resonance absorption. It has been widely usedin the past and has still a non ignorable group of supporters.In a first trial, it was also applied to the experiment in Ref.17 with the result that it reproduced the experimental valuesAC as well as the PIC simulations could do or even slightlybetter. Nevertheless for that experiment, it had to be dis-carded because as a test showed almost all absorption,locally given by jE cycle-averaged, took place in a pre-plasma in front of the target of density by orders of magni-tude lower than that of the target. This is unphysical and isdue to the artificial viscosity introduced in the code to pre-vent hydrodynamic breaking of flow, in the experimentunder consideration occuring two orders below the intensityused. It could be shown in general that collisionless absorp-tion was reproduced when introducing a collision frequencyratio =x > 2 artificially. A realistic estimate for the criticalregion led to ratios =x not exceeding 102. Hence, whenusing MULTI-fs, it must be separately tested that absorptionoccurs in the right place and is not produced by the artificialviscosity term. Such caution is in order when reexaminingany papers on absorption of ultrashort laser pulses where“good agreement” is found of MULTI with experiments.So far, there are several arguments in favor of theextended Brunel model: (1) simplicity and coherence, (2)prompt generation of the fast electrons within a fraction of alaser cycle, (3) increased absorption values for nearly relativ-istic intensities at angles from which on the electric forceprevails on the Lorentz force (“j B heating”), and (4)absorption efficiency not depending on the target density assoon as it is much higher than the critical density. At thesame time, there are severe cons against the model: (1)absorption exceeding 100% at grazing incidence, (2) wrongabsorption scaling on intensity, (3) wrong prediction of thefraction of fast electrons, and (4) energy spectrum far fromMaxwellian with a cutoff that is too low. In this respect, theextension presented here on physical grounds makes the sit-uation even considerably worse; compare Figs. 6 and 10.In the geometry of the Brunel model, the PIC simulationsunambiguously show that the hot electron spectrum is Maxwel-lian, fhðEÞ expðbEÞ; b ¼ const, and that its fraction is afew percent only. Examples of f(E) and F(E) from a 1D simula-tion for Nd pulses of constant intensity Ik2 ¼ 1:37 1018a2W/cm2lm2, a ¼ eEL=mecx ¼ 0:3; 1:0; 3:0; and10 at obliqueincidence of a ¼ 45 after 40 fs are presented in Fig. 12.The normalization of f(E) is done on all electrons having beenexposed once to the laser light before escaping into thetarget and having got energies superior to E¼ Eos;Eos ¼ mev20=4; ¼ 0;104;103;102. The resulting percent-age of the fast electrons is seen from FðE¼ EosÞ. It is nearly 2orders of magnitude lower than in Brunel’s model. It clearlyshows that the bulk of electrons is heated up to lowtemperature.The question arises: What part of physics is missing inBrunel’s model? To give a first answer, Eqs. (2) and (3) maybe solved numerically by decomposing the overdense targetinto a number of layers and determining the driving field inthe evanescent region from Poisson’s law. This is an easilysolvable system having all advantages of dynamics ofFIG. 10. Electron spectra f(E) and FðEÞ ¼ Ð f ðE0ÞdE0 from the extendedBrunel model under the condition Aþ R ¼ 1 (solid line). Dashed line fromFig. 5 for comparison.FIG. 11. Absorption coefficient A from the extended Brunel model forvos=c ¼ 0:3 (solid line) and from Fig. 7 (dashed).043301-6 Mulser, Weng, and Liseykina Phys. Plasmas 19, 043301 (2012) Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 139.30.84.198 On: Fri, 03 Jun2016 14:12:03100Hamiltonian character. The answer has been given in Ref. 28.The system shows a chaotic behavior which manifests itself ina continuous crossing of layers and stochastic order of reenter-ing, see their order given there. Closer inspection then showsthat single layers happen to resonate anharmonically, therebyundergoing a phase shift by p or a fraction of it and movingagainst the bulk. This is a very efficient process of wavebreaking, introduced first in Ref. 28 (“resonant breaking offlow”), and generator of a high level of fluctuations. This factshifts the question of missing physics in Brunel’s model to thesearch for the origin of the exclusion of resonance. In fact, asshown by our sufficient condition Eq. (4) for non-crossing oforbits, the model is mathematically correct as long as eachparticle undergoes one oscillation only, in contrast to what thenumerical Hamiltonian treatment shows. When an electron isdisplaced by the weak evanescent laser field, it undergoessmall forced oscillations that later may become resonant whenentering the nonlinear regime. Thus, by assuming zero electro-static potential inside the target (x > 0), all electrons undergoone single oscillation only with the consequence that reso-nance and chaos are excluded automatically in Brunel’smodel, with the severe impact above on the hot electron spec-trum. From test runs with the potential inside the target setequal zero in the Hamiltonian model, the absorption decreasedby more than an order of magnitude. Such a boundary condi-tion is equivalent to the assumption of an instantaneous onsetof the neutralizing return current. In reality, the building up ofthe return current takes time according to its inertia whichinvolves many layers in contrast to the few layers driven inthe evanescent laser front.Going one step further by calculating correctly the laserfield in each layer with the help of Fresnel formulas for thinlayers (thickness much less than the local wavelength) andincluding the Lorentz force term in Eqs. (2) and (3), one is ledat additional insight on when a layer reenters the target and onthe spatial distribution in vacuum where this disruption-likemotion originates from Ref. 29. We show the resulting elec-tron spectrum in Fig. 13: There is the bulk of cold electrons,similar to the slope of the main fraction of cold Brunel elec-trons in Fig. 6(a), however continued to the right now by a fastMaxwellian like tail extending up to 17Eos. Recirculation ofelectron layers enabling even higher cut offs is excluded in themodel. The percentage of the particles in the tail is below10%. Although the Hamiltonian multilayer model is physicallyricher and more complete in the spectrum of Fig. 13 remnantsof Brunel’s original spectrum are still present, like the concaveshape of the hot electron distribution and an attenuated maxi-mum shifted now from normalized E¼ 9.3 to E¼ 11.IV. SUMMARY AND DISCUSSIONAn analysis of the Brunel model has revealed essentialaspects of collisionless absorption: Prompt generation of fastelectrons during one laser cycle or a fraction of it underp-polarization, insensitivity of absorption with respect to tar-get density well above nc, robustness, and simplicity. Finally,the characteristic splitting of the electron energy spectruminto the two groups of slow and of fast electrons, observed inall experiments and simulations, PIC and Vlasov, comesabout in the most natural way. Such positive aspects contrastwith a non-Maxwellian tail of the hot electrons, too low cutoff, unrealistic fraction of fast electrons, scaling of meanelectron energy with intensity drastically different from theexperiment by Cerchez et al.17 under Brunel conditions, aswell as from relevant findings in the literature so far,21–26and inefficient absorption below very oblique incidence.These are clear indications of some essential physics missingin Brunel’s model. It is difficult to see which effect has beenoverlooked in this simple mechanism that the consequencesare such a strong scaling of the absorption coefficient AB I1=2 on intensity and, in particular, the astonishing depend-ence of the energy absorbed per cycle I3=2. In tendency, itlooks like energy production from vacuum.Brunel’s pioneering idea has been the recognition of thespace charges induced by the electron motion perpendicular toFIG. 12. 1D PIC simulation of collisionless absorption of constant inten-sities a¼ 0.3, 1.0, 3.0, and 10 under 45 p-polarized incidence. (a) Spectralenergy distribution f(E) of the electrons; vertical line marks the position of Eequalizing the mean electron oscillation energy. (b) F(E) total number ofelectrons having energies greater than Eos and less or equal E; ¼ 102.The number of fast electrons amounts to a few percentage.FIG. 13. Electron energy spectrum f(E) from Hamiltonian dynamics of alayered target under the action of the Lorentz force from a p-polarized laserbeam of intensity I ¼ 1017 Wcm–2 Nd under 45 incidence. The local fieldsare calculated from Fresnel’s formulas. Anharmonic resonance allows formixing of layers. Energy cutoff occurs at E ¼ 17 Eos and Eos ¼ mev20=4.043301-7 Mulser, Weng, and Liseykina Phys. Plasmas 19, 043301 (2012) Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 139.30.84.198 On: Fri, 03 Jun2016 14:12:03101 ANALYSIS OF THE BRUNEL MODELthe target surface that make irreversibility possible. However,one essential root of failure evidently is setting the electrostaticfields inside the overdense target equal to zero. It implies theneglect of any inertia and resistivity of the cold return currentas well as of excitation of plasmons by the jets of the hot elec-trons and their mutual energy transfer. It must be kept presentthat in the skin layer, the total density of electron currentsstreaming into the vacuum and coming back undergo strongrarefaction compared to the situation in the target interior, i.e.,the return current undergoes strong acceleration, accomplishedby an intense electrostatic field. Even more significant seemsto be the following effect. When the dynamics of the Brunellayers is studied properly as in Refs. 28 and 29, they undergoseveral oscillations before escaping into the target interior. InBrunel’s model, they are suppressed with the consequence ofinhibition of anharmonic resonance and, in concomitance, ofmixing of layers (“resonant breaking of flow”). Mixing pro-duces stochastic motion, longer life time in front of the target,the tendency towards Maxwellization and fast tail formation ofthe energetic electrons and a natural reduction of the reentrycurrents. We think that exactly here there is space for improve-ments. Hence, it remains to be investigated next to whichextent the Brunel effect and anharmonic resonance competeand mix up to generate the correct electron spectra seen in sim-ulations and experiments and to produce higher absorption atmoderate angles of incidence in p-polarization. Furthermore,the Brunel mechanism and anharmonic resonance have a dif-ferent impact on the symmetry of fast electron flows into thetarget and towards the irradiating laser. Brunel is extremelyunidirectional into the target, the anharmonic resonance mech-anism tends, like PIC simulations, to a more symmetric distri-bution of electron flows into the target and into the vacuum. Inthe latter case, target normal sheath acceleration (TNSA)yields similar maximum ion energies in front and behind thesolid target of suitable thickness.30Finally, one additional consideration on viscosity andMULTI may be in order. Perhaps stimulated by the fluid-liketreatment of j B heating in its analytical part in Ref. 10, onecould be tempted to think of absorption being connected some-how with the nonlinear motion and bunching of the electronfluid, here assumed as collisionless. On a fluid scale, this canonly be modeled by introducing an artificial viscosity or by ananalogous technique. However, whatever the nonlinear motionof an ideal fluid is it does not lead to any absorption as long theflow represents a continuous mapping in space, i.e., as longbreaking and/or splitting into beamlets for example is absent.V. CONCLUSIONCollisionless absorption in overdense plasma cannot betreated in the framework of ideal fluid dynamics.ACKNOWLEDGMENTSThe support of this work by the Alexander von Hum-boldt Foundation is gratefully acknowledged.1J. R. Davies, Plasma Phys. Controlled Fusion 51, 014006 (2009).2T.-Y. B. Yang, W. L. Kruer, R. M. More, and A. B. Langdon, Phys. Plas-mas 2, 3146 (1995).3W. Rozmus and V. T. 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Mulser, Phys. Plasmas 14, 023301 (2007).15P. Mulser and D. Bauer, High-Power Laser-Matter Interaction (Springer,Heidelberg, 2010).16A. D. Piliya, Sov. Phys. Tech. Phys. 11, 609 (1966); H.-J. Kull, Phys. Flu-ids 26, 1881 (1983).17M. Cerchez, R. Jung, J. Osterholz, T. Toncian, O. Willi, P. Mulser, and H.Ruhl, Phys. Rev. Lett. 100, 245001 (2008).18F. Brunel, Phys. Rev. Lett. 59, 52 (1987).19T. Baeva, S. Gordienko, A. P. L. Robinson, and P. A. Norreys, Phys.Plasmas 18, 056702 (2011).20K. Eidmann, R. Rix, T. Schlegel, and K. Witte, Europhys. Lett. 55, 334(2001).21F. N. Beg, A. R. Bell, A. E. Dangor, C. N. Danson, A. P. Fews, M. E. Glin-sky, B. A. Hammel, P. Lee, P. A. Norreys, and M. Tatarakis, Phys.Plasmas 4, 447 (1997).22C. D. Chen, J. A. King, M. H. Key, K. U. Akli, F. N. Beg, H. Chen, R. R.Freeman, A. Link, A. J. Mackinnon, A. G. MacPhee, P. K. Patel, M. Por-kolab, R. B. Stephens, and L. D. Van Woerkom, Rev. Sci. Instrum. 79,10E305 (2008).23A. G. MacPhee, K. U. Akli, F. N. Beg, C. D. Chen, H. Chen, R. Clarke, D.S. Hey, R. R. Freeman, A. J. Kemp, M. H. Key, J. A. King, S. Le Pape, A.Link, T. Y. Ma, H. Nakamura, D. T. Offermann, V. M. Ovchinnikov, P. K.Patel, T. W. Phillips, R. B. Stephens, R. Town, Y. Y. Tsui, M. S. Wei, L.D. Van Woerkom, and A. J. Mackinnon, Rev. Sci. Instrum. 79, 10F302(2008).24S. C. Wilks, W. L. Kruer, M. Tabak, and A. B. Langdon, Phys. Rev. Lett.69, 1383 (1992).25M. G. Haines, M. S. Wei, F. N. Beg, and R. B. Stephens, Phys. Rev. Lett.102, 045008 (2009).26T. Kluge, T. Cowan, A. Debus, U. Schramm, K. Zeil, and M. Bussmann,Phys. Rev. Lett. 107, 205003 (2011).27K. Eidmann, J. Meyer-ter-Vehn, Th. Schlegel, and S. Hüller, Phys. Rev. E62, 1202 (2000).28P. Mulser, D. Bauer, and H. Ruhl, Phys. Rev. Lett 101, 225002(2008).29P. Mulser and D. Bauer, AIP Conf. Proc. 1209, 75 (2010).30T. Ceccotti et al., Phys. Rev. Lett. 99, 18502 (2007).043301-8 Mulser, Weng, and Liseykina Phys. Plasmas 19, 043301 (2012) Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 139.30.84.198 On: Fri, 03 Jun2016 14:12:03102Collisionless absorption, hot electron generation, and energy scalingin intense laser-target interactionT. Liseykina,1,2,a) P. Mulser,3 and M. Murakami41Institut f€ur Physik, Universit€at Rostock, Universit€atsplatz 3, 18051 Rostock, Germany2Institute of Computational Technologies SD RAS, Acad. Lavrentjev Ave. 6, 630090 Novosibirsk, Russia3Theoretical Quantum Electronics, Technische Universit€at Darmstadt, 64289 Darmstadt, Germany4Institute of Laser Engineering, Osaka University, Osaka 565-0871, Japan(Received 23 December 2014; accepted 26 February 2015; published online 12 March 2015)Among the various attempts to understand collisionless absorption of intense and superintenseultrashort laser pulses, a whole variety of models and hypotheses has been invented to describe thelaser beam target interaction. In terms of basic physics, collisionless absorption is understood nowas the interplay of the oscillating laser field with the space charge field produced by it in theplasma. A first approach to this idea is realized in Brunel’s model the essence of which consists inthe formation of an oscillating charge cloud in the vacuum in front of the target, thereforefrequently addressed by the vague term “vacuum heating.” The investigation of statisticalensembles of orbits shows that the absorption process is localized at the ion-vacuum interface andin the skin layer: Single electrons enter into resonance with the laser field thereby undergoing aphase shift which causes orbit crossing and braking of Brunel’s laminar flow. This anharmonicresonance acts like an attractor for the electrons and leads to the formation of a Maxwellian tail inthe electron energy spectrum. Most remarkable results of our investigations are the Brunellike spectral hot electron distribution at the relativistic threshold, the minimum of absorptionat Ik2 ffi ð0:3 1:2Þ 1021 Wcm2lm2 in the plasma target with the electron density ofnek2 1023cm3lm2, the drastic reduction of the number of hot electrons in this domain and theirreappearance in the highly relativistic domain, and strong coupling, beyond expectation, of the fastelectron jets with the return current through Cherenkov emission of plasmons. The hot electronenergy scaling shows a strong dependence on intensity in the moderately relativistic domainIk2 ffi ð1018 1020ÞWcm2lm2, a scaling in vague accordance with current published estimates inthe range Ik2 ffi ð0:14 3:5Þ 1021 Wcm2lm2, and again a distinct power increase beyondI ¼ 3:5 1021 Wcm2lm2. The low energy electrons penetrate normally to the target surface, theenergetic electrons propagate in laser beam direction. VC 2015 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4914837]I. INTRODUCTIONIntense and superintense laser beam interaction withdense matter is characterized by one prominent phenomenon,that is, the generation of superthermal high energy electronsand ions. It leads to the spontaneous question as to the effectsthat generate them in the absorption process of intense mono-chromatic light beams. Latest when the kinetic temperaturereaches 103Z2 eV in the plasma, Z ion charge, collisionalabsorption is ineffective and other effects of non-collisionalnature have to become active in order to ensure absorption.The best known non-collisional candidate so far was reso-nance absorption at oblique laser incidence.1,2 It consists inthe direct conversion of laser light into an electron plasmawave resonantly excited at the critical electron density wherethe laser frequency x equals the plasma frequency xp. Highintensity laser pulses in the intensity domain I¼ 1016–1022Wcm2 with good contrast ratio are so fast rising that there isno time to form a preplasma in front of an irradiated solidsample that could couple to a resonantly excited plasmawave. Therefore, the search begun for new collisionlessabsorption processes. The first successful proposal was theso-called jB heating.3 The authors could show by particle-in-cell (PIC) simulations that at normal incidence the Lorentzforce induces non-resonant electron oscillations at 2x normalto the target surface which lead to appreciable absorption, tar-get heating, and production of superthermal electrons at anydensity above critical. However, no attempt was made toexplain how the observed absorption, i.e., irreversibility,comes into play. Not long after, a remarkable step forwardwas made by Brunel4 in understanding high-power collision-less absorption. He could show after introducing a few modi-fications that the resonance absorption concept could beadapted to steep highly overdense plasma profiles and signifi-cant absorption could be achieved under oblique incidencedespite total absence of plasma resonance at x¼xp (not-so-resonant, resonant absorption4) and no possibility for aplasma wave to propagate into a shallow preplasma in frontof the target. Instead, now the energy imparted to the elec-trons is transported into the target and deposited there. Underthe assumptions of cold (i) infinitely dense plasma (ii) withdiscontinuous interface to vacuum (iii) Brunel could formu-late the laser-matter interaction dynamics in the vacuum infront of the target in terms of three ordinary differentiala)Electronic mail: tatyana.tiseykina@uni-rostock.de1070-664X/2015/22(3)/033302/11/$30.00 VC 2015 AIP Publishing LLC22, 033302-1PHYSICS OF PLASMAS 22, 033302 (2015)103 COLLISION ABSORPTION, HOT ELECTRON GENERATION AND ENERGY SCALINGequations. Perhaps, for this reason, Brunel’s mechanism ofthe electrons pulled out into the vacuum and then pushedback into the field-free target interior is frequently identifiedwith the term “vacuum heating,” an expression coined laterfor the particles forming a kind of thermal cloud at thevacuum-ion interface in PIC simulations.5To be more specific, the reason why the term vacuumheating played an ominous role in the past and partially stilldoes presently is because, often invoked as the leading colli-sionless absorption mechanism, it has never been definedproperly. To introduce some rating in this respect, it seemsthat two groups of authors can be distinguished. By the con-cept of vacuum heating, the first group addresses the elec-trons in front of the sharp-edged target that circulate in thevacuum and do not cross the interface during one lasercycle.5,6 Identification of vacuum heating with Brunel’smechanism is made by the second group7,8 to contrast withanomalous skin layer absorption.9–11 By the latter, all motionis strictly confined to the target inside. Sometimes, vacuumheating is interpreted as a consequence of unspecified wavebreaking.12,13 Meanwhile, it has been clarified that Brunel/vacuum heating prevails distinctly on skin layer absorptionand that vacuum heating in the restricted sense, i.e., the con-tribution to absorption of the electrons not entering the targetwith the periodicity of the laser, is almost insignificant.14Only recently, a detailed analysis and discussion ofBrunels’s model has been given in Ref. 15. With a view onthe means of the present paper, a compact summary of theresults may be of interest. The laser field component perpen-dicular to the target is assumed to have the structureEðtÞ ¼ E0 sin xt. It generates electron jets of periodicitys ¼ 2p=x. All of them are ejected during the first quarterperiod, and all, except 2.2%, return to the target during thesecond half period (p, 2p). During the remaining 3/4 period,no further electron ejection is possible as a consequence ofpartial screening by the outer layers and driver field inver-sion. Contrary to a common believe that all electrons in onejet are pushed back by the inverted field, only half of them,lifted in the interval (0, p/4), are in phase with the driver, theother half experience a weakened driver due to screeningand fall back to the target, attracted by the immobile ions,before the laser field has changed direction. This leads quitenaturally to a classification into energetic and less energeticelectrons. If not specified differently, throughout the paper,we define, somehow arbitrarily, those electrons as hot whosereturn energy exceeds the quiver energy Eos of the free elec-tron. Accordingly, 34% of the Brunel electrons are hot andcarry 82% of the energy in the single jet. In contrast to PICsimulations, the Brunel spectrum is non-Maxwellian with apronounced maximum at E ¼ 9:1Eos, followed by a sharpcut off (see Fig. 1). Absorption A ¼ Iabs=I is considerablylower than measured at intermediate angles of incidence16but reaches unity at 86 of incidence. A and the absorbedenergy scale like I1=2 and I3=2, respectively. Evidently, this isthe price Brunel pays for oversimplification. We want tostress that in Brunel’s model, crossing of layers among eachother during the laser action is excluded, except a few frontlayers whose contribution to absorption is negligible. Inother words, the electron flow dynamics is laminar, no wavebreaking or, more appropriate in the context, no breakingof flow occurs. Brunel’s model offered, within limits, thefirst physical explanation of jB heating at 2x. And yet,Brunel’s model does not give a direct physical feeling for theabsorption process. The numerous wrong, at least inexactinterpretations of it that are still around are a direct indicator(example: “all electrons are pushed back by the laser field”contrasts with the true “free fall” of half of them).From an early sional (1D) PIC simulation, the energyscaling Ehot I1=2 has been extracted for the hot electrons;17it has been re-“confirmed” by independent simulations18 andapparently by experiments.19,20 However, the scaling seemsto be questionable for its too strong dependence on intensity;it contrasts with other experiments21 and more sophisticatedanalysis.22 In turn, corrections to the latter have been givenrecently on the basis of a relativistic kinematic model.23Analogous scaling laws have been proposed by numerousother authors.16,24–26 Nevertheless, there is no convergencetowards a definite scaling.27 In order to achieve furtherprogress, the investigation has to start from a discussion ofcollisionless absorption, a flow analysis of the absorbedenergy into the various plasma components, a definition ofthe hot electron component, and completed by analyticalmodeling in combination with concomitant simulations. Inwhat follows, we present our considerations on the degree ofunderstanding collisionless laser beam absorption, the processof fast electron generation, and their interaction with the lowelectron energy component in order to arrive at more firmlyvalidated scaling relations in forthcoming work. The analysiswill enable us also to get insight into shortcomings of theexisting models for intense laser-dense matter interaction.II. COLLISIONLESS ABSORPTION BASICALLYUNDERSTOODLet us consider the phenomenon of collisionless absorp-tion of high-power laser beams from a more fundamentalFIG. 1. (a) Electron spectrum f(E) from Brunel’s nonrelativistic model;4 (b)FðEÞ ¼ Ð f ðE0ÞdE0. Energy E in units of mean oscillation energy in vacuum;energy cut off is at E ¼ 9:1Eos.033302-2 Liseykina, Mulser, and Murakami Phys. Plasmas 22, 033302 (2015)104point of view. Under quasi-steady state conditions,Poynting’s theorem averaged over one laser cycle reduces torS ¼ jE: (1)The energy flux density is the Poynting vector S ¼ e0c2E B;it relates to the laser intensity by I ¼ S. With ne the electrondensity and v the mean electron flow velocity, the current den-sity is j ¼ enev. Equation (1) describes all kinds of absorp-tion, collisional, noncollisional, classical, or quantized; in thelatter case, the current density and the electric field E are to besubstituted by their operators acting on the corresponding statevector jwi. In the intense laser field, despite the high particledensities involved, the classical picture is an excellent approxi-mation. If the laser field evolves in time as E sin xt, thecurrent density follows as j cosðxtþ /Þ andrS ¼ jE cos xtþ /ð Þsin xt ¼ 12sin /: (2)Dephasing between driver field and current determines thedegree of absorption. Thus, collisionless absorptionreduces to the problem of finding out which effects lead toa finite phase shift in j. In collisional absorption, it is thefriction originating from the collisions between electronsand ions,jE ¼ e0x2px2 þ 2 jEj2 > 0; (3) collision frequency. At ¼ 0, the collisional phase shiftvanishes and any finite / can only be of dynamic origin. Upto I¼ 5 1020–1021 Wcm2, this dynamic origin is found inthe space charge induced by rv 6¼ 0. The space charge gen-erates an electrostatic field that, superposed to the laser field,determines the electron motion and leads to the desired finitephase shift /. This can be seen most immediately with a con-stant electric field E0. It yields per electronjE ¼ 2pe2E02mex> 0: (4)It is interesting to note, and it can be formally shown; how-ever, it is also physically evidents that jðELaser þ EsÞ¼ jELaser; all work is done by the driver field; the spacecharge field Es is inert; it provides for the phase shift only.Some authors may attribute absorption to the Brunel likeabrupt reduction of the laser wave amplitude in the skinlayer. Due to this asymmetry, the energy gained by an elec-tron in the vacuum cannot be given back anymore to thewave when entering the evanescent region. However, forthis picture to work, an electrostatic field component isneeded too; transverse and longitudinal components cannotbe isolated from each other. In the standard resonanceabsorption, at the critical density, it is the space chargefield of the electron plasma wave that provides for colli-sionless absorption up to 49% through a phase shift/ 6¼ 0.2 On the fundamental level of Eqs. (2) and (4), colli-sionless absorption of superintense ultrashort laser pulsesmay be classified as fully understood now for I< 1021Wcm2 for optical wavelengths.All kinds of difficulties and complications arise whenthe degree of absorption has to be quantified. This step canonly be done by introducing appropriate models. Numerousattempts into this direction have been undertaken with theintention to explain (i) the origin of the hot and the warmelectron components, (ii) when and how they are created,during one laser period by direct resonant and nonresonantacceleration, or by stochastic processes over several lasercycles, and (iii) where is absorption localized, in vacuum orin the skin layer. Correspondingly, the existing absorptionmodels may be characterized as statistic or as dynamic.Examples of the first class are vacuum heating in the re-stricted sense,28,29 wave breaking,12,13 and skin layer absorp-tion, e.g., Ref. 9, linear and nonlinear Landau damping.30,31Candidates of the second class are, first of all, sharp edgeabsorption,4 longitudinal32 and transverse33 ponderomotiveheating, “zero vector potential mechanism,”18 relativistic ki-nematic model,23 and anharmonic resonance.34 Let our PICsimulations decide on questions (i)–(iii) and to what degreestatistics is involved in the dynamics induced by the laser indense targets.Now, after three decades of intense studies on superin-tense laser-matter interaction, one would expect that suchbasic questions (i)–(iii) as labeled above should have found afinal answer. The numerous models presented on performedexperiments tell the opposite and show that no convergencehas been reached yet. It is instructive to have a look at over100 experimental and theoretical results on the absorptiondegree collected up to 2009 in Ref. 35, (Fig. 1 in Ref. 35), inthe irradiance regime from Ik2 ¼ 1018 Wcm2lm2, to 1021Wcm2lm2. The absorption degrees range from 5% to 95%,and yet, at the constant irradiance of 6 1018 Wcm2lm2,absorption between 35% and 85% is reported. It drasticallyreflects the difficulties encountered in performing unambigu-ous experiments with all essential parameters well defined. Itis this situation that justifies still basic 1D simulations inorder to learn more about which are the essential parametersdefining the underlying physical processes. For example, agood portion of difficulties and uncertainties arising in thecontext of hot electron scaling have their origin in differentunderstanding of when an electron is “hot.”III. LOCALIZATION OF ABSORPTION AND ORIGIN OFFAST ELECTRONSWe consider always linear polarized (in y-direction)laser pulses impinging under 45 angle of incidence ontostrongly overdense fully ionized cold hydrogen targets withinitial electron density ne0 such that ne0k2 ’ 1023 cm3lm2.The interaction of the laser beam with the target is studiedby 1D relativistic PIC simulations using the boost tech-nique.36 If not specified differently, throughout the text, wecall all electrons with energies E exceeding their mean oscil-lation energy Eos “hot” or “fast,”E > Eos ¼ mec2 1þ a2=2 1=2 1h i; a ¼ ejÂjmec;Ik2 ¼ 1:37 1018a2 W=cm2lm2;Â maximum vector potential amplitude.033302-3 Liseykina, Mulser, and Murakami Phys. Plasmas 22, 033302 (2015)105 COLLISION ABSORPTION, HOT ELECTRON GENERATION AND ENERGY SCALINGA. Brunel model and vacuum heatingThe energy spectra f(E) of the hot electrons assume thetypical shape of Fig. 2, here, for an I sin4 laser pulse offull width 50 cycles and a¼ 0.3, 1, 3, 10 after 30, 35, 40, and50 laser cycles. The straight lines of lnf ðEÞ starting from theenergy E between 2Eos=3 for a¼ 10 and from E ¼ Eos fora 3 are a clear signature of a Maxwellian type distributionfunction, f ðEÞ expðE=kBTeÞ, kB Boltzmann constant (forexample, the genuine nonrelativistic Maxwellian containsthe degeneracy factorffiffiffiEpto be subtracted from lnf ðEÞ toyield a straight line, see related argument in Sec. V). Fromthe slope, an electron “temperature” Te is determined, withthe significance of kBTe the mean energy if the straight linesare extrapolated down to E¼ 0. In the figure, Te is indicatedin units of MeV. For a¼ 10, one notices already cooling byenergy transfer to the cold electrons and beginning plasmaexpansion during laser irradiation. Similar hot electronspectra have been reported by other authors, for example, inRef. 37.The interaction of intense laser beams with dense targetsis very complex and rich of peculiar facets. On the otherhand, consequences of basic effects, like collisionless interac-tion under non-harmonic resonance, are not clarified as theyshould. Here, we report on phenomena which we believe willsurvive in 2D and 3D also. Let us tentatively identify vacuumheating with the energy gained by all Brunel-like electrons(Brunel electrons). It is the sum of energies gained during theexcursion into vacuum. This energy fraction is identified as“vacuum heating” and compared with the energy absorbed byall electrons. The target thickness is chosen such that no par-ticles are reflected from the target backside and falsify thestatistics. The laser beam intensity rises during one lasercycle to its full intensity, is subsequently held constant for 30cycles, and then sinks to zero during another full cycle. InFig. 3, the energies of all Brunel particles and of all particlesthat have crossed the skin layer at a depth of half a vacuumwavelength are plotted at their crossing time for a¼ 1 anda¼ 60. The salient feature is their jet like structure predictedby the Brunel model. At the low intensity (a¼ 1), there is aclear distinction between the Brunel electrons and all elec-trons having undergone heating. The increase of the energymaxima from 6:5Eos to 8Eos and the more diffuse energy pro-files of the jets at half wavelength in depth is a clear indica-tion that some heating is localized in the skin layer, incontrast to the Brunel mechanism. At high intensity (a¼ 60),the jets assume a pronounced double structure due to theincreased vB heating operating at 2x, but the patterns ofthe two groups appear equally diffuse. The increase in energyof the fastest electrons is almost no longer visible (increaseby 0:5Eos). For a> 1, the fraction of Brunel electrons resultsFIG. 2. Electron energy spectra lnf ðEÞ at (a) 30, (b) 35, (c) 40, and (d) 45 cycles after the beginning of the interaction. A sin4 laser pulse of peak amplitudea¼ 0.3, 1, 3, 10 and full width of 50 cycles impinges under 45 onto a hydrogen target with electron density ne0 such that ne0k2 ¼ 9 1022cm3lm2. Thepulses are identical in all four frames. Target thickness varies from 40 to 60 k. Vertical dashed lines mark the mean oscillation energies. The hot electrons fol-low a Maxwellian distribution. The maximum mean energies kBTe for a 1 are by the factors 1.04, 1.09, 1.7 higher than the associated Eos. kBTe increases dur-ing the evolution of laser pulse for a¼ 3; for a¼ 10, it decreases. Power scaling kBTe Ia; a 0 not detected.FIG. 3. Energy spectra of Brunel jets (upper pictures) and jets at depth k/2as function of time (units in laser cycles) for a¼ 1 and a¼ 60. Double struc-ture is due to vB acceleration. Strong reduction of Emax=Eos with increas-ing a is noticeable.033302-4 Liseykina, Mulser, and Murakami Phys. Plasmas 22, 033302 (2015)106always higher than the fraction of electrons moving inwardand crossing the boundary at k/2 in the skin layer. The reasonfor the difference is to seek in the accumulation of Brunelelectrons in the skin layer with increasing laser cycle number,randomized there and repeatedly crossing the target-vacuuminterface before disappearing in the depth of the target. It isinstructive to analyze the spectral distribution function f(E) ofthe Brunel electrons and all electrons just when crossing posi-tions x¼ 0.5k, k, 1.5k, and 2k for the laser intensities corre-sponding to a¼ 1, 7, 30, and 60, see Fig. 4. Surprisingenough, at low intensity (a¼ 1) and, to a minor degree, alsoat a¼ 7, the Brunel electrons from the PIC simulationsresemble much Brunel’s analytical spectrum from Fig. 1: Thesharp cut offs and the adjacent maxima of f(E) are repro-duced; their positions, however, lie at much lower energies.The maximum of f(E) is still clearly visible for a¼ 15 (not inthe figure), this time at E ¼ Eos, but the sharp cut off changesinto a transition extending over 0:4 Eos. The formation of aMaxwellian tail in the fast electron spectrum occurs in theskin layer and even deeper inside the target. From a ’ 20, onno difference in the spectra from Brunel and total electronscan be observed, they are all “thermalized.” At a¼ 30, thespectra extend up to 1:4 Eos; at a¼ 60, the maximum energyis shifted to E ¼ 3:7 Eos. This is in agreement with the de-pendence of the fast electron number on laser intensity, seeSec. IV A. From Fig. 3, we conclude that at moderate inten-sities (a 15), the skin layer contributes sensitively to theproduction of the most energetic electrons, either by laser-space charge resonance and/or by stochastic Brunel electron-plasmon interactions.B. Localization and mechanism of heatingPlasma density and velocity distributions as functions oftime are the natural outcome in standard PIC simulations.Additional insight in the heating mechanism is obtainedfrom the orbits x(t) of randomly chosen electrons. We haveanalyzed numerous such computer runs each with 200 trajec-tories stochastically selected from (i) all particles heated bythe laser and (ii) from the set of the hot electrons only. InFig. 5, their time histories are depicted for the intensitiesa¼ 7 (left) and a¼ 60 (right). The salient features character-istic of the two groups are the following:(1) Heating of the energetic electrons is well localized at thevacuum-target interface and takes place during one lasercycle or a fraction of it. This excludes stochastic heatingof the hot electrons. Only a low fraction of them get thehigh energy in the skin layer without ever emerging intovacuum.(2) Contrary to the standard assumption, the “slow” returncurrent is highly irregular as a consequence of the inter-action of the jets from Fig. 3 with the background. It isclearly recognized in Fig. 5 that irregular flow sets in justwith the arrival of the first jets and it becomes the moreirregular the more jets it is exposed to. The jets areaccompanied by strong localized electrostatic fields thatforce electrons from the return current to reverse theirdirection towards the back of the target or, if they suc-ceed to cross the charge cloud of an incoming jet, theyare heavily accelerated towards the target front to inter-act further with the laser field. In short words, the lami-nar flow of the return current is heavily perturbed by theCherenkov emission of the plasmons excited by the jets.The stochastic interaction, both, return electron accelera-tion and deceleration, has been observed in test particlemodels in the past.14(3) The plasma flow in the skin layer breaks (like “wavebreaking”), i.e., the orbits cross each other, in contrast toBrunel’s laminar model of infinite target density.(4) Excursion into vacuum (vacuum heating) of the energeticelectrons decreases continuously with a increasing toreach a minimum at around a¼ 30 and then to increaseFIG. 4. Energy distributions f(E) of Brunel electrons and of electrons cross-ing positions x¼ 0.5k, 1k, 1.5k, and 2k at intensities a¼ 1, 7, 30, and 60.The distributions are taken after 37 laser cycles when all electrons havereturned to position x¼ 2k. The non-Maxwellian structure of the Brunelelectrons is preserved up to a¼ 15. For higher intensities, there is almost nodifference between electrons heated in the vacuum and additional heating inthe skin layer region.FIG. 5. Arbitrary selection of orbits xi(t), i¼ 1–100, for a¼ 7 (a) and a¼ 60(c). The lower (b) and (d) show the same number of stochastically chosenorbits from the hot electrons only with E Eos. Their analysis shows accel-eration, i.e., heating almost during one laser cycle or a fraction of it in thelaser field at the target front and their strong interaction with the returncurrent.033302-5 Liseykina, Mulser, and Murakami Phys. Plasmas 22, 033302 (2015)107 COLLISION ABSORPTION, HOT ELECTRON GENERATION AND ENERGY SCALINGagain. Owing to the significance of effects (1)–(3) forlocalization and understanding hot electron generation,and understanding collisionless laser beam absorption ingeneral, we analyze further the acceleration process. InBrunel’s model, the density of the target is assumed infi-nite. Consequently, all electrons start from the same posi-tion and no crossing of orbits occurs; the particle flowinto vacuum and back to the target is laminar. DespiteBrunel’s oversimplification, his model explains basicproperties of the collisionless interaction: formation ofsteady state jets (Fig. 3), two groups of electron energies(energetic electrons co-moving with the laser field, slowfree fall electrons), majority of fast electrons stemmingfrom the excursion into vacuum, and dominant fractionof laser energy delivered to hot electrons. If therefore“vacuum heating” is identified with Brunel’s mechanism,it acquires a precise meaning. However, there is the miss-ing link to the physics of acceleration in this simplemodel. Not to forget that in Brunel’s model, all heatedelectrons are lifted into vacuum only during the first quar-ter laser period. The reality with skin layer included isdifferent: The phase for lifting is stochastic, as expectedfrom broken flow; period doubling, tripling, quadrupling,etc., of electron oscillations occurs in the skin layer (seeFig. 5); acceleration to high energies is a resonanceeffect. To see this, we must concentrate once more indetail on single orbits selected statistically. In Fig. 6, thetime history of four particles starting from different depthin the target together with the electric field (white traces)they “see” during their motion is depicted for a¼ 1, i.e.,the orbits x(t) and the momenta px/mec normal to thetarget. Resonant interaction in the first two pictures isclearly recognized by the abrupt changes in x(t) and px(t).Out of resonance, the phase difference between field andmomentum is p/2, see px(t) and laser field (white line) inthe first two pictures. The transition to resonance, i.e.,field and momentum antiparallel, occurs during halfa cycle or less in the kink of x(t), seen best by zoomingFig. 6. The essential point of this resonance is its anhar-monic character. In contrast to the harmonic oscillator inthe oscillator with anharmonic potential resonance is anattractor: Given an excitation by the periodic laser, abovea certain threshold, transition to resonance is unavoidable.The reason for this behavior is as follows. The harmonicpotential is the only one in which the degree of excitationdoes not change its periodicity, and therefore, it either isdriven in or out of resonance. The average stochasticallyperturbed space charge potential of the plasma is flatterthan harmonic, and so, depending on the excitation level,its eigenfrequency changes continuously from the highlevel xp at low excitation down below the laser frequencyx. At the crossing point, resonance occurs. It has twoconsequences: (i) Driver, when in phase with the electrondisplacement, transforms it into a runaway particle ingeneral;34 (ii) the resonant phase switch forces the elec-tron to move against the bulk, the plasma flow breaks.Breaking of flow or wave breaking, respectively, ofteninvoked as acceleration or absorption mechanism12,13 isnever their origin, it is their consequence.We conclude that at I< (5 1020–1021) Wcm2, themajority of energetic electrons is produced by resonant inter-action of the laser field with the longitudinal space chargefield over a fraction of one laser cycle in the vacuum as wellas in the skin layer. However, there is also indirect accelera-tion of stochastic nature of the target background, evidencedby the last two pictures in Fig. 6 with electrons heated sto-chastically by the plasmons emanating from the jets. Therapidly oscillating stochastic field of the plasmons increaseswith the number of jets produced; its influence on stochasticacceleration of electrons is evident in the last two pictures ofFig. 6.IV. FAST ELECTRONS AND ENERGY PARTITIONAs seen in Sec. III, there are several thermalizing mech-anism: breaking of flow, skin layer noise, and Cherenkovplasmons from jets. As a consequence, one would expectthat such effects dominate the low energy component of theelectrons and that this should propagate mainly normally tothe target. The more energetic an electron is, the more it feelsthe Lorentz force in vB direction forcing its motion intolaser beam direction, 45 in this paper. For the single freeelectron starting from rest in a traveling plane wave, themaximum energy gain DE and the lateral angular spreadtan a of the velocity component vk in propagation directionto the velocity component in the E field direction vE are38DE ¼ 12a2mec2; tan a ¼ vEvk ¼2a; (5)thus confirming this tendency. In Fig. 7, the momenta py par-allel to the target vs px along the target normal of the heatedelectrons are depicted for a¼ 1, 7, 15, 60, and 100. TheFIG. 6. Anharmonic electron resonance and stochastic interaction.34 Regularshadow structure: laser field; black trajectory: orbit x(t) (left) and momentumpx(t) (right), white traces: electromagnetic/electrostatic field at the particle’sposition. Primary interaction is by resonance between transverse and longi-tudinal field during a fraction of laser cycle. Particles in the last two picturesexperience stochastic acceleration by plasmons only.033302-6 Liseykina, Mulser, and Murakami Phys. Plasmas 22, 033302 (2015)108corresponding distributions of the momenta px normal to thetarget over the space coordinate are shown in Fig. 8. Fromthe pictures, it is not directly seen that the majority ofslow electrons move in the direction of the target normal;however, as expected from (5), with increasing energy theelectrons follow, indeed, the direction of the laser beam. Inaddition, at a¼ 60 and 100, an appreciable percentage isaccelerated into specular direction. The reduction of theabsolute number of hot electrons with increasing intensity,their almost vanishing at a¼ 15, and their impressive reap-pearance towards a¼ 100 is particularly striking. This effectwill have direct impact on every attempt to formulate scalinglaws for the “hot electron” production. We have countedtheir number as a function of intensity; the result is reportedin Table I. Drop and increase with intensity are beyondexpectation.The formation of spatial spikes within groups of ener-gies and laser intensities is depicted in Fig. 8. It has to beseen as complementary to the spike distribution in time inFig. 3. At low laser intensity, only the fastest electrons formjets in space as long as they are “young.” As they travelfurther into the target, they become increasingly diffuse as aconsequence of their interaction with the Cherenkov plas-mons. The electrons of varying velocity undergo mixing inphase space, see uniform background in Figs. 3 and 8; thespikes only are accompanied by strong elecrostatic fields.Their damping by friction is given through the collisionfrequency coll ¼ 2ðe4ne0=0m2ev3ecÞlnK. With the Lorentzfactor, c¼ 1, ve¼ c, and ne0¼ 1023 cm 3, this results incoll ¼ 5 1010lnK s1; hence, collisional damping of spikesis unimportant. Anomalous interaction of the laser heatedelectrons with the background has been studied recently.39All electrons after having entered the distinctly relativisticregime show a neat spiky structure because they all fly atlight speed and behave much stiffer now against their con-comitant space charge field. The inclination of groups ofspikes with respect to the normal to the abscissa at subrela-tivistic speeds is self-explaining. We note also that the excur-sion of the slow electrons into vacuum (px negative) reduceswith increasing a.A. Partition of the absorbed energySometimes, it is claimed (at least in the past) that allelectrons are “hot” in intense laser-solid target interaction.This rises the question on the percentage of the hot electronswith respect to number, to average energy, or to average fluxdensity. Here, we must stress that a percentage in particlenumber cannot be given, neither in the experiment nor in thesimulation for the simple reason that the fraction dependsvery sensitively on the total number of particles involved:Where to put the lower threshold? Should the shock heatedportion of the target be counted also, or is it reasonable torestrict counting on those electrons that have “seen” the laserat least once? However, the situation is totally different withFIG. 7. Direction of heated electrons: Momenta py/mec vs px/mec for a¼ 1,7, 15, 60, and 100 (left picture a¼ 100: with radiation damping, right: with-out) at the end of the standard laser pulse. Electron energies: E < Eos=3within inner black circle, E 2 ½1=3; 2=3 Eos within dashed circle, E 2½2=3; 1 Eos within bold circle, E > Eos (hot electrons) outside. Circles in thelast pictures are very small; therefore, see the two insets). The color of theparticles in this pictures indicates their number according to the color bar.Low energy electrons within the inner circle (majority in number) penetratethe target normally; energetic electrons follow the laser beam direction(dashed black lines), in a¼ 60, 100 also along the reflected laser beam.FIG. 8. Energy distribution of heated electrons: Momentum px/mec vs targetnormal x for a¼ 1, 7, 15, 60, and 100 (left: with radiation damping, right:without) at the end of the standard laser pulse. The color of the particlesindicates their number according to the color bar of Fig. 7. The spiky struc-ture is a rough indicator of relativistic jets.TABLE I. Number of hot electrons per unit area (arbitrary units) in depend-ence of a for ne0 ¼ 100nc ¼ 100mex20=e2.a 1 3 7 15 30 60Nhot 7819 7991 17 464 147 265 19 273033302-7 Liseykina, Mulser, and Murakami Phys. Plasmas 22, 033302 (2015)109 COLLISION ABSORPTION, HOT ELECTRON GENERATION AND ENERGY SCALINGrespect to energy fractions because no ambiguity arises onthat. In Table II, we present such an absorbed energy parti-tion as a function of laser intensity (parameter a I1=2) fortwo overdense targets, ne0 ¼ 100nc and ne0 ¼ 200nc (fora¼ 15 also ne0 ¼ 400nc): overall fraction of absorption Iabs;percentage of energy which is found in the electrons; totalfraction of energy absorbed by the hot electrons, E Eosand by hot þ medium hot electrons of E > Eos=2 (warm e);energy fraction transmitted to the ions; energy fraction foundin the electrostatic space charge field (fields). A first view onthe table tells that the main absorption is accomplished bythe energetic electrons (see 5th and 6th column). The absorp-tion by the ions (protons in the table) remains modest fora 15, however, Cherenkov plasmons (fields) assume a nonnegligible portion of laser energy, more than we predicted.The increase in ion energy beyond a¼ 15 is due to thedeeper penetration of the laser as a consequence of the reces-sion of the electrons by the radiation pressure and hence,increased energy coupling to the ions.The overall absorption drops continuously with increas-ing laser intensity. In contrast to the runaway energy E in(5), the free quiver energy at fixed oscillation center isEos ¼ mc2½ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ a2=2p 1 I1=2. However, this reductionis counterbalanced by the relativistic increase of the criticaldensity, ncr cnc. As the speed of the moderately hot elec-trons approaches c, the absorption into energetic electrons,which is the major portion, should not change; the drop musthave a different, nonrelativistic origin. Although the scalingof Eos and ncr may be oversimplified (see Ref. 40 for ncr scal-ing), it is correct in its tendency. Our current explanationattributes the very pronounced reduction of absorption to thelimiting effect of the electrostatic field on the oscillation am-plitude of the single electron: With increasing intensity I, theelectrons are pushed more and more inward by the radiationpressure. The electron oscillating in the neighborhood of thevacuum-ion interface oscillates in a narrow anharmonicpotential, the half width of which towards the target interioris a small fraction of the wavelength (profile steepening). Asimilar reduction of absorption has been reported for normalincidence, with an explanation that agrees qualitatively withours.41 Latest, beyond I ’ 1021 Wcm2, absorption by thefastest electrons increases again. They are runaway elec-trons. The electrostatic potential is strong but finite. Thephase at which the electrons enter the laser beam is stochas-tic. Within them, there will some of them happen to be inresonance with the field and subject to the Doppler shiftx0 ¼ cðx kvÞ; (6)with c Lorentz factor, k wave vector. If such an electron ismoving inward from the vacuum, it sees the incident wave ata Doppler downshifted low frequency and is acceleratedover a longer distance, whereas the reflected wave is seen ata highly upshifted frequency and represents merely a highfrequency disturbance. For an electron moving outwardtowards the vacuum, the accelerating field is that of thereflected wave. The proof of this acceleration mechanism isbased on the study of single particle motions, and is directlyconfirmed by the appearance of energetic electrons flowinginto vacuum in the reflected wave direction in the picture fora¼ 60 and a¼ 100 of Fig. 7. To give a numerical exampleof electron displacement lengthening Dx/k in a plane TN:SAlaser wave (k¼ 800 nm) during a forth cycle Du ¼ p=2:Dx=k ¼ 3 at I¼ 1021 Wcm2 and Dx=k ¼ 30 at 1022Wcm2. For comparison, at I¼ 1018 Wcm2, this shift is0.03 only. Beyond I¼ 1022 Wcm2 radiation, reaction on theelectron motion has to be taken into account.43–45 A sum-mary of absorption into all plasma channels (electrons, ions,plasmons), its fraction into electrons, the decrease of absorp-tion towards a pronounced minimum close to zero at a ’15–20, and its rise beyond is presented in Fig. 9.V. ON SCALING LAWS OF THE “HOT ELECTRONS”From intensity scaling, the experimentalist and theoreti-cian expect analytical formulas of the shape of the electronspectrum as a function of the laser intensity. As such, a goalTABLE II. Partition of the incident laser energy: fraction of absorbed inten-sity Iabs transmitted to the electrons, the hot and warm electrons, and theions and the plasmons (fields) at the end of the standard laser pulse.Energy partitiona0 ne0/nc A All e Hot e Warm e Ions Fields0.3 100 0.377 0.25 0.213 0.217 0.009 0.118200 0.313 0.24 0.161 0.167 0.007 0.0660.5 100 0.43 0.28 0.228 0.233 0.01 0.1381 100 0.358 0.238 0.2 0.211 0.008 0.112200 0.354 0.24 0.199 0.206 0.0077 0.1063 100 0.18 0.122 0.08 0.092 0.003 0.055200 0.18 0.123 0.08 0.092 0.003 0.0545 100 0.2 0.136 0.088 0.102 0.0033 0.061200 0.2 0.136 0.089 0.103 0.0026 0.0617 100 0.19 0.131 0.073 0.089 0.0033 0.056200 0.19 0.132 0.076 0.093 0.0025 0.05515 100 0.067 0.036 0.0002 0.003 0.01 0.021200 0.051 0.028 0.0005 0.003 0.007 0.016400 0.064 0.039 0.0022 0.008 0.0057 0.01930 100 0.105 0.0525 0.0002 0.0018 0.0206 0.032200 0.045 0.0168 0 0.00012 0.0168 0.01160 100 0.23 0.126 0.01 0.027 0.033 0.071200 0.092 0.031 0.00006 0.0003 0.034 0.027FIG. 9. Total absorption (triangles) of a 30 cycles standard laser pulse (see text)and the absorption by electrons (blue diamonds) is given as a function of a. Thereduction is due to oscillation inhibition by the induced electrostatic field; itsrise beyond is mainly a consequence of entrainment (runaway electrons).033302-8 Liseykina, Mulser, and Murakami Phys. Plasmas 22, 033302 (2015)110seems to be beyond reach; at present, the high power lasercommunity has limited its focus on the energetic electrons.There, the generation of a Maxwellian tail is one of the char-acteristics of high power interaction. It is also the most inter-esting part of the spectrum because, as seen from Table II, itcontains the main part of the absorbed energy and, last butnot least, it is relevant to applications for collective ionacceleration, radiation sources, medical applications, andothers. It is aimed at how the number of energetic electrons,the degree of absorption, and the mean energy scale withintensity. On the basis of present knowledge, scaling of thefirst two quantities is not feasible. Regarding the meanenergy, or the hot temperature kBThot, respectively, despitethe frequent attempts in experiment and theory, no conver-gence has been achieved so far at all. In the light of our fore-going analysis, there is not much surprise about.The frequently invoked ponderomotive scaling (Wilks’scaling)17 of Thot I1=2 is based on the idea that eachlaser cycle energetic electrons with energy average in therange of about Eos ¼ mec2ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ a2=2p 1Þ are generated.Subsequently, this scaling has been recognized as too strongand, in first place guided by experiments,21 has beenreplaced by the milder power law46 Thot I0:340:4 in the in-tensity range 1018–1021 Wcm2. It is intended as to beapplied to the unidirectional Maxwellian electrons reachingthe analyzer at time t ¼ 1. The search for the right meanenergy scaling is equivalent to the search for the process ofabsorption. In the absence of the stochastic element inherentin collisions, it is important to understand whether aMaxwellian tail is one of the signatures of the interactionand, if it is, why.To arrive at a Maxwellian distribution in an ensemble, itis sufficient to know that, given a certain amount of particlesnhot containing the amount of energy Ehot, all possible statesin the relevant phase space are equally likely and that theHamiltonian is given by the sumH ¼Xnhoti¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2ec4 þ c2p2iq; jHj ¼ Ehot; (7)which expresses the property that the single electrons areuncorrelated. If the relevant phase space is fðp; qÞg, as—forexample—in statistical thermodynamics, the resulting distri-bution is the Maxwellian momentum distribution f ðEÞ ffiffiffiEp expðE=kBThotÞ, in disagreement with Fig. 2.However, as outlined in the foregoing chapters, fast electrongeneration is by anharmonic resonance. This has the impor-tant properties: (i) resonance is an attractor for all electronsabove a certain oscillation energy, in contrast to harmonicresonance; the always present crossing of trajectories is aclear indicator of it. (ii) All nhot electrons have the samechance to resonate anharmonically regardless of their phasewith respect to the laser driver. This makes it very likely thatthe relevant phase space is the energy acquired at resonancerather than the momentum. Then, from (i) and (ii) followsthat collisionless absorption is accompanied by aMaxwellian tail of energetic electrons (consequence ofanharmonic attractor) and the spectrum scales likef ðEÞ expðE=kBThotÞ, without a degeneracy factorffiffiffiEp.This is what we also deduce from Fig. 2.Let us first examine Fig. 2 for pulses I sin4 in the in-tensity range 1018–1020 Wcm2. The uncertainties on themean energy (slope of log scale) in pictures (a) (30 cycles)and (c) (40 cycles) are considerable; nevertheless, we canconclude with certainty that neither Wilks’ original pondero-motive scaling17 (I0.5) nor its improved version is met tosome extent. They are far too weak. However, the analysisshows that the assumption kBThot ¼ j Eos with the con-stant j not far from unity works. This means that at theserelatively low intensities (from a ’ 3 to a¼ 10), the scalingiskBThot ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ a2=2p 1; (8)in agreement with Ref. 42. From a¼ 12 on, Eos is wellapproximated by mec2a I1=2. In Fig. 10, we extended thesearch for scaling from a¼ 10 up to a¼ 60; the latter isalready in the runaway regime of absorption. Satisfactoryagreement with Ref. 42 is obtained for I from 1020 to 1021Wcm2. Beyond the change in the absorption mechanismand the stiffer, coupling to the ions is noticeable in theincrease of slope relative to Ref. 42.VI. SUMMARY AND CONCLUSIONThe focus of the present paper is on the physics of colli-sionless absorption of intense laser beams in dense targets inthe intensity domain I¼ 1018–1022 Wcm2 for optical wave-lengths, on the variation of the spectral composition of theenergetic electrons with intensity and on their scaling with thelatter. Most remarkable results are the Brunel like spectral hotelectron distribution at the relativistic threshold, the minimumof absorption at a ffi 15 30, the drastic reduction of thenumber of hot electrons in this domain and their reappearancebeyond, the strong coupling with the return current beyondexpectation, a strong hot electron scaling in a ffi 1 10, and aFIG. 10. Hot and warm electron energy scaling hEei with laser intensityI ¼ 1017 5 1021 Wcm2, standard pulse. Stars: Ee Eos (hot electrons);diamonds: Ee 0:5Eos (warm electrons). In contrast to Fig. 2, here, hEei isthe average taken over the single energies Ee. Solid line: scaling after,42kBThot ¼ mec2½ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ðIk2Þ=ð2:74 1018Þq 1 . Dashed line: scaling after,21kBThot ¼ mec2½Ik2=ð1:37 1018Þ 0:34.033302-9 Liseykina, Mulser, and Murakami Phys. Plasmas 22, 033302 (2015)111 COLLISION ABSORPTION, HOT ELECTRON GENERATION AND ENERGY SCALINGscaling in vague accordance with current published estimatesin a ffi 10 50 and a strong increase beyond.On a fundamental level understanding, collisionlessabsorption is equivalent to the search for the non-orthogonality of induced current density to the laser field.The answer is found in the interplay of the laser field withthe space charge field induced by it. The idealized model ofBrunel works already on this basis. It is capable of explain-ing important effects at the relativistic intensity thresholdand below, like the generation of two groups of electrons, ahot and a cold component. The non-Maxwellian spectrumpredicted by the model is found in our simulations at the rel-ativistic threshold and below; with the increase in a, it iswashed out. By following test orbits, we are able to localizeabsorption at the vacuum-target interface and skin layerfor all intensities below the radiation reaction limit atI ffi 1022 Wcm2 in linear polarization, in agreement withBrunel for non relativistic intensities. If, therefore, theominous “vacuum heating” is invoked as responsible forabsorption, this is correct if it is identified with Brunel’smechanism. What it does not explain, and Brunel does noteither, is the underlying physics, i.e., the phase shift and, inconcomitance, orbits crossing.An explanation in terms of physics has to show that (i)such a breaking of flow is not by accidence and (ii) a hotMaxwellian tail in the spectrum is a natural outcome fromstrong drivers. Anharmonic resonance is currently the bestmodel explaining both aspects. It rules stochastic heatingand “weave breaking” out automatically. Anharmonic reso-nance constitutes an attractor (fix point). This kind of reso-nance always happens in presence of a sufficiently strongdriver at any laser frequency and any target density, incontrast to harmonic resonance, which is bound to x¼xp.When crossing resonance, the momentum of an electronundergoes a phase shift by p or a fraction of it with respectto the bulk of the plasma. Wave breaking, here more appro-priately called breaking of flow owing to profile steepeningon lengths of a small fraction of a laser wavelength, is a con-sequence of absorption and energetic electron generation,not its origin.From I ffi 5 1021 Wcm2, anharmonic resonance isstrengthened by the generation of runaway electrons due totrapping in both, the incident and the reflected laser wave.The reduction up to nearly disappearance of hot electronsE Eos is attributed to oscillation inhibition by the pondero-motive space charge field. In the whole intensity domainconsidered, the major fraction of laser energy is deposited inthe hot and moderately hot electrons. The next significantportion goes into Cherenkov plasmons excited by the peri-odic plasma jets. From the analysis of the test trajectories,their coupling to the neutralizing return current is apparent.We consider it as an important aspect when modeling anom-alous transport of heat and fast electrons in compressedmatter.Finally, we reexamined the hot electron scaling in 1D,perhaps the most controversially discussed subject in the per-tinent literature. Acceptable coincidence with the leadingapproximations is only found in the intensity range1020–1021. The deviations below a ffi 10 are to be attributedmainly to the imprecise proportionality between Eos andI a2. In the runaway absorption regime, the governingscaling law is still to be discovered. The main reason for thecurrent misunderstandings and disagreements have to beattributed to the poor knowledge of the electron energy spec-trum f(E). What is missing most at present in the experimentand in the theory is a clear definition of what means “hot”and “cold” electrons. In order not to fall into this deficiency,we define electrons as “hot” and “moderately hot” if theirenergy is higher than Eos and ð0:5 2=3ÞEos. It is approxi-mately the range of the Maxwellian tail. The commonly usedterminology “Maxwellian” is misleading because it refers tothe electron velocities v or momenta with a distributionlaw df ðEÞ ¼ v2 expðE=kBThotÞdv ffiffiffiEpexpðE=kBThotÞdE.This differs from our findings (and, implicitly, others) of aBoltzmann distribution df ðEÞ ¼ expðE=kBThotÞdE for therelevant restricted phase space of total energies E ¼PEi.ACKNOWLEDGMENTSThis work was in part supported by the DFG within theSFB 652. PIC simulations were performed using thecomputing resources granted by the John von Neumann-Institute for Computing (Research Center J€ulich) under theProject No. HRO01. 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Pegoraro, and A. Macchi, Phys. Rev. E85, 016407 (2012); M. Tamburini, F. Pegoraro, A. Di Piazza, C. H. Keitel,and A. Macchi, New J. Phys. 12, 123005 (2010).45L. L. Ji, A. Pukhov, I. Yu. Kostyukov, B. F. Shen, and K. Akli, Phys. Rev.Lett. 112, 145003 (2014).46S. C. Wilks and W. L. Kruer, IEEE J. Quantum Electron. 33, 1954(1997).033302-11 Liseykina, Mulser, and Murakami Phys. Plasmas 22, 033302 (2015)113 COLLISION ABSORPTION, HOT ELECTRON GENERATION AND ENERGY SCALING115 RADIATION FRICTION EFFECT IN LASER-PLASMA INTERACTIONS4. Radiation friction effect in laser-plasmainteractionsThe list of publications:M. Tamburini, F. Pegoraro, A. Di Piazza, C. H. Keitel, T. V. Liseykina, A. Macchi, Radiation reac-tion effects on electron nonlinear dynamics and ion acceleration in laser–solid interaction, NuclearInstruments & Methods in Physics Research A 653, 181–185 (2011)M. Tamburini, T. V. Liseykina, F. Pegoraro, A. Macchi, Radiation-pressure-dominant acceleration:Polarization and radiation reaction effects and energy increase in three-dimensional simulations,Phys. Rev. E 85, 916407 (2012)T. Liseykina, S. Popruzhenko, A. Macchi, Inverse Faraday effect driven by radiation friction, New J.Phys. 18 072001 (2016)S. Popruzhenko, T. Liseykina, A. Macchi, Efficiency of radiation friction losses in laser-driven ”holeboring” of dense targets, New J. Phys. 21, 033009 (2019)Radiation reaction effects on electron nonlinear dynamics and ion accelerationin laser–solid interactionM. Tamburini a,, F. Pegoraro a, A. Di Piazza b, C.H. Keitel b, T.V. Liseykina c,d, A. Macchi a,ea Dipartimento di Fisica ‘‘E. Fermi’’, Universita di Pisa, Largo Bruno Pontecorvo 3, I-56127 Pisa, Italyb Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germanyc Institute of Computer Technologies, SD-RAS, Novosibirsk, Russiad Institute of Physics, University of Rostock, Germanye Istituto Nazionale di Ottica, CNR, Research unit ‘‘A. Gozzini’’, Pisa, Italya r t i c l e i n f oAvailable online 16 December 2010Keywords:Radiation ReactionIon accelerationLaser–plasma interactionRadiation pressurea b s t r a c tRadiation Reaction (RR) effects in the interaction of an ultra-intense laser pulse with a thin plasma foil areinvestigated analytically and by two-dimensional (2D3P) Particle-In-Cell (PIC) simulations. It is foundthat the radiation reaction force leads to a significant electron cooling and to an increased spatialbunching of both electrons and ions. A fully relativistic kinetic equation including RR effects is discussedand it is shown that RR leads to a contraction of the available phase space volume. The results of our PICsimulations are in qualitative agreement with the predictions of the kinetic theory.& 2010 Elsevier B.V. All rights reserved.1. IntroductionCurrent laser systems may deliver intensities up to 1022 W cm2[1] and intensities up to 1026 W cm2 are expected at the ExtremeLight Infrastructure (ELI). In such ultrahigh-intensity regime and fortypical laser wavelength l 0:8 mm the motion of electrons in thelaser field is ultra-relativistic and Radiation Reaction (RR) effectsmay become important. The RR force describes the back-action ofthe radiation emitted by an accelerated electron on the electronitself and accounts for the loss of the electron energy and momen-tum due to the emission of such radiation. Apart from the need ofincluding RR effects in the dynamics of laser–plasma interactions inthe ultra-relativistic regime, the latter also offers for the first timethe opportunity to detect RR effects experimentally [2,3].In this paper we present an approach to a kinetic description oflaser–plasma interactions where RR effects are included via theLandau–Lifshitz (LL) force [4]. Some properties of the kineticequation with RR are discussed and in particular it is proved thatthe RR force leads to a contraction of the phase space volume. Then,Particle-In-Cell (PIC) simulations are used to study RR effects on theacceleration of a thin plasma foil in the regime of Radiation Pressuredominance [5]. Numerical simulations [5] suggested that RadiationPressure Acceleration (RPA) becomes the dominant mechanism ofion acceleration at intensities exceeding 1023 W cm2. Such RPAregime is attractive because of the foreseen high efficiency, thequasi-monoenergetic features expected in the ion energy spectrumand the possibility to achieve a potentially ‘‘unlimited’’ accelera-tion [6]. Previous PIC simulations [7] showed signatures of RReffects at intensities exceeding 5 1022 W cm2 and increasingnonlinearly with the laser intensity. More recent simulationsstudies of RPA both for thick targets [8,9] and ultrathin targets[10] suggested that the inclusion of the RR force cools the electronsand may improve the quality of the ion spectrum.Our approach to the inclusion of RR effects in a PIC code hasbeen discussed in detail in Ref. [11] where one-dimensional (1D)simulations of RPA have been also reported. In the present paper wereport both additional 1D simulations and first two-dimensional(2D) simulations using parameters similar to those of Ref. [12]where, in particular, the impact of a Rayleigh–Taylor-like instabil-ity on a thin foil acceleration was studied.In classical electrodynamics, the effect of RR can be included bymeans of the LL force [4]:FR ¼4p3rel g @@tþv = Eþv@@tþv = B Eþv Bð Þ Bþ v Eð ÞE½ þg2 Eþv Bð Þ2 v Eð Þ2h ivð1Þwhere v is the electron velocity, g is the relativistic factor, re e2=mc2 2:8 109 mm is the classical electron radius, l¼ 2pc=o is thelaser wavelength and we use dimensionless quantities as in the PICcode: time, space and momentum are normalized in units of o1,co1 and mc, respectively. Consequently, EM fields are normalized inunits of moc=jej and densities in units of the critical density nc ¼mo2=4pe2.Contents lists available at ScienceDirectjournal homepage: www.elsevier.com/locate/nimaNuclear Instruments and Methods inPhysics Research A0168-9002/$ - see front matter & 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.nima.2010.12.056 Corresponding author.E-mail addresses: matteo.tamburini@gmail.com,tamburini@df.unipi.it (M. Tamburini).Nuclear Instruments and Methods in Physics Research A 653 (2011) 181–185117 RF EFFECTS ON ELECTRON DYNAMICS AND ION ACCELERATIONThe LL approach holds in the classical framework and quantumeffects are neglected. As pointed out in Ref. [11], the first term of theLL force Eq. (1) i.e. the one containing the derivatives of the electricand magnetic fields, should be neglected because its effect issmaller than quantum effects such as the spin force. However, inEq. (2) we show the effect of each term of the LL force Eq. (1) on therate of change of the phase space volume.2. The kinetic equation with Radiation ReactionIn this section, a fully relativistic kinetic equation that includesthe RR effects is discussed. We show a few basic properties of thekinetic equation pointing out the peculiarities of the RR force whosemain new feature is that it does not conserve the phase space volume.Generalized kinetic equations for non-conservative forces andin particular for the RR force have been known since late 60s for theLorentz–Abraham–Dirac (LAD) equation [13,14] and late 70s forthe LL equation [15]. Recently, the generalized kinetic equationwith the LL force included has been used to study the RR effects onthermal electrons in a magnetically confined plasma [16] and todevelop a set of closed fluid equations with RR [17–19]. In thispaper, we give the kinetic equation in a non-manifestly covariantform, see Refs. [15,16] for the kinetic equation in a manifestlyLorentz-covariant form.The relativistic distribution function f ¼ f ðq,p,tÞ evolves accord-ing to the collisionless transport equation:@f@tþ=q ðf vÞþ=p ðf FÞ ¼ 0 ð2Þwhere q are the spatial coordinates, v¼ p=g is the three-dimen-sional velocity, g¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1þp2pis the relativistic factor and F¼ FLþFRis the mean force due to external and collective fields(FL ðEþv BÞ is the Lorentz force and FR is given in Eq. (1)).Physically, Eq. (2) implies the conservation of the number ofparticles.The new key feature compared to the usual Vlasov equation isthat for the RR force FR we have =p FRa0. Using Lagrangiancoordinates q(t), p(t), Eq. (2) can be recast in the equivalent form:dlnfdt¼=p F: ð3ÞAccording to Eq. (3), =p F provides the percentage of variation ofthe distribution function f within the characteristic time scale o1.Integrating Eq. (3) along its characteristics, we find that thedistribution function f remains positive as required.Introducing the entropy density in the phase spacesðq,p,tÞ¼ f ðq,p,tÞ ln f ðq,p,tÞ, from Eq. (2) we get the equationfor the evolution of the entropy density:@s@tþ=q ðsvÞþ=p ðsFÞ ¼ f =p F: ð4ÞIntegrating Eq. (4) in the phase space, we get the rate of variation ofthe total entropy S(t)¼Rd3 q d3 p s(q, p, t):dSðtÞdt¼Zd3qd3pfrp F: ð5ÞThe Lorentz force FL ðEþv BÞ gives =p FL ¼ 0 identically thus=p F¼=p FR. Moreover, the distribution function f ðq,p,tÞ isalways non-negative f Z0 thus the sign of dS=dt is given by =p FR solely.From the LL force Eq. (1) we get [20]=p FR ¼4p3rel =q Ev =q B@E@t þ2E2þB2g" #þ4g½ðv EÞ2þðv BÞ22v ðE BÞ): ð6ÞIn a plasma, the kinetic equation is coupled with the Maxwell equationsfor the self-consistent fields:=q E¼rrc¼1ncXj ¼ e,iZjZd3pf jðq,p,tÞ ð7Þ=q B@E@t¼jjc¼1nccXj ¼ e,iZjZd3pvfjðq,p,tÞ ð8Þwhererc jejnc , jc jejncc,Rd3q d3pf jðq,p,tÞ ¼Nj is the total numberof particles for each species (j¼e electrons, j¼ i ions) and Zj is the chargeof the particle species in units of jej (for electrons Ze¼1). For a plasma,Eq. (6) can be recast as=p FR ¼4p3rel rrcv jjc þ2E2þB2g" #(þ4g½ðv EÞ2þðv BÞ22v ðE BÞ): ð9ÞThe terms of Eq. (9) proportional to the charge density r and to thecurrent density j come from the first term of the LL force Eq. (1) i.e. theterm containing the derivatives of the fields. In general, these terms cangive either a positive or negative contribution to =p FR. The secondterm of Eq. (9) i.e. the term proportional to ðE2þB2Þ has always anegative sign, its effect decreases with increasing electron energy and itis typically negligible. The third term of Eq. (9) comes from the stronglyanisotropic ‘‘friction’’ term of the LL force i.e. the term proportional tog2in Eq. (1) (see Ref. [11] for a detailed discussion of this term) anddominates in the ultra-relativistic limit gb1.It is possible to prove [20] the following statement: for any vsuch that jvjA ½0,1½ thenðv EÞ2þðv BÞ22v ðE BÞh iþE2þB22g2" #Z0 ð10Þtherefore according with Eqs. (5, 9), the terms of the LL force Eq. (1)that do not depend on the derivatives of the fields always lead to acontraction of the available phase space volume dS/dt r 0 . In a fewspecial cases, the effect of the terms of the LL force Eq. (1) thatdepend on the derivatives of the fields (i.e. the terms proportionalto r and j in Eq. (9)) might lead to an expansion of the phase spacevolume. Anyway, their effect should be negligible compared toquantum effects as discussed in Ref. [11].We show explicitly the contraction of the phase space in thespecial case of a small bunch of electrons interacting with a planewave where collective fields are assumed to be negligible com-pared with the plane wave fields. Assuming an initial distributionf ¼ gðqÞd3ðpp0Þ, from Eqs. (5, 9) we havedSðtÞdt¼4p3rel Zd3qgðqÞ 2E2þB2gðp0Þ" #þ4gðp0Þ(½ðv0 EÞ2þðv0 BÞ22v0 ðE BÞ)ð11Þwhere v0 ¼ p0=gðp0Þ. If the electron bunch counter propagates withthe plane wave (½v0 ðE BÞo0) or propagates in the transversedirection (½v0 ðE BÞ ¼ 0), from Eq. (11) it is clear that RR leadsto a contraction of the phase space. In particular, in the case ofcounter-propagation (using jEj ¼ jBj, E B¼ 0) we have =p FR ¼ð4pre=3lÞ4E2½2gðp0Þjv0jð1þjv0jÞþ1=gðp0Þ. On the other hand, ifthe bunch propagates in the same direction of the plane wave(v0 parallel to E B), then the contribution of the friction term(proportional to g in Eq. (9)) becomes comparable with thecontribution of the second term (proportional to (E2 + B2) inEq. (9)) and we have =p FR ¼ð4pre=3lÞ½4E2=ð1þjv0jÞ2g3ðp0ÞM. Tamburini et al. / Nuclear Instruments and Methods in Physics Research A 653 (2011) 181–185182118which still leads to a contraction of the phase space but with a rateg4 smaller than the case of counter-propagation. This reinforces theevidence of the strongly anisotropic features of the LL forceEq. (1) (see Ref. [11] for further details).The physical interpretation of the above properties is that the RRforce acts as a cooling mechanism for the system: part of the energyand momentum are radiated away and the spread in bothmomentum and coordinate space may be reduced. This generalprediction is confirmed by our PIC simulations (see Section 3)where we found that RR effects lead to both an increased bunchingin space and to a noticeable cooling of hot electrons.Finally, it is worthwhile mentioning that Eq. (2) is more generalthan the Vlasov equation but the PIC approach is still valid i.e. thePIC approach provides a solution for Eq. (2) and it not limited to theVlasov equation [20].3. PIC simulationsSuitable approximations to the LL force and our approach to itsinclusion in a PIC code are described in Ref. [11]. The numericalapproach is based on the widely used Boris particle pusher and itcan be implemented in codes of any dimensionality. Inclusion of RReffects via this method in PIC simulations leads to only a 10%increase in CPU time, which may be essential to perform large-scalesimulations with limited computing power.3.1. 1D simulationsWe first report one-dimensional (1D3P) PIC simulations with laserand plasma parameters similar to Ref. [5]. Previous 1D simulations inthis regime have been reported in Ref. [11] where a detailedcomparison with other work is also made. In the present paper wereview the basic observations in the 1D case and we include results atintensities higher than those investigated in Ref. [11].The target is a plasma foil of protons with uniform initial densityn0¼100nc and thickness ‘¼ 1l where l¼ 0:8 mm is the laserwavelength and T ¼ l=c 2:67 fs is the laser period. In thesesimulations, the laser pulse front reaches the edge of the plasmafoil at t¼0, the profile of the laser field amplitude has a ‘‘trape-zoidal’’ shape in time with one cycle, sin2- function rise and fall anda five cycles constant plateau. We considered three intensitiesI¼ 2:33 1023,5:5 1023, and 1024 W cm2 for both Circular (CP)and Linear (LP) polarization of the laser pulse.In the CP case, we found that RR effects on the ion spectrum arenegligible even at intensities of I¼ 1024 W cm2 as shown in Fig. 1.For CP, electrons pile up and the numerical density grows exceed-ing thousand of times the critical density nc. The laser pulse doesnot penetrate deeply into the target (i.e. the effective skin depth is avery small fraction of the foil thickness) and electrons move in afield much weaker than the vacuum field.In Ref. [5] it was expected that RR effects in the radiation-pressure dominated acceleration of the thin foil would have beenweak because in this regime the whole foil becomes quicklyrelativistic, hence in the foil frame the laser wavelength lu increasesand the typical strength of the RR parameter re=l [see Eq. (1)]decreases. The present case of acceleration with CP pulses appearsto confirm this picture. The weakness of RR effects may also beexplained on the basis of the LL equation for an electron movinginto a plane wave [21]. As electrons move in the forward directioncoherently with the foil (while rotating in the transverse plane inthe CP field) and the amplitude of the reflected wave is weak whenthe foil is strongly relativistic, the situation is similar to an electronco-propagating with the plane wave at a velocity close to c, forwhich the LL force almost vanishes [11]. The relativistic motion ofthe foil also prevents the onset of Self-Induced Transparency byincreasing the optical thickness parameter z¼ pn0‘=ncl in the foilframe (see Ref. [22] and references therein). For smaller targetthickness, breakthrough of the laser pulse occurs and RR effects aregreatly enhanced also for CP [11].It may be worth noticing that, at the highest intensity con-sidered I¼ 1024 W cm2, in principle one would expect the classi-cal approach to RR to break down due to the onset of quantumelectrodynamics (QED) effects, as discussed in Ref. [11]. However, itcan be shown by a direct analysis of the simulation data that thethreshold condition for QED effect is not violated in the CP case.For linear polarization (LP), differently from the CP case, we foundthat RR effects are important leading to a reduction of the maximumachievable ion energy and to some narrowing of the width of the ionspectrum as shown in Fig. 2. This different dynamics for LP iscorrelated with the strong longitudinal oscillatory motion driven bythe oscillating component of the jB force which is suppressedin the CP case. This allows a deeper penetration of the laser pulse intothe foil with a significant fraction of electrons on the front surfacemoving in a strong electromagnetic field of the same order ofvacuum fields [11]. The relative reduction in the ion energy when RRis included is close to the percentage of the laser pulse energy whichis lost as high-energy radiation escaping from the plasma.Fig. 1. Ion energy spectra at t¼66 T with (top) and without (bottom) RR for CircularPolarization. The laser intensity I is 2:33 1023 W cm2 (yellow), 5:5 1023 W cm2(blue), 1024 W cm2 (red) and the target thickness is ‘¼ 1l. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)Fig. 2. Ion energy spectra at t¼14 T with (top) and without (bottom) RR for LinearPolarization. The laser intensity I is 2:33 1023 W cm2 (yellow), 5:5 1023 W cm2(blue), 1024 W cm2 (red) and the target thickness is ‘¼ 1l. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)M. Tamburini et al. / Nuclear Instruments and Methods in Physics Research A 653 (2011) 181–185 183119 RF EFFECTS ON ELECTRON DYNAMICS AND ION ACCELERATIONThe results for LP (Fig. 2) are shown for the same intensity valuesof the CP case (Fig. 1) for a direct comparison. However, at least forthe highest intensity case, the LP results must be taken with somecaution as the condition for the validity of a classical approachtends to be significantly violated. In such regime, an analysis basedon quantum RR effects might be necessary [23,24].3.2. 2D simulationsWe report preliminary two-dimensional (2D3P) PIC simulationswith laser and plasma parameters similar to Ref. [12]. To the best ofour knowledge, this is the first paper reporting results of two-dimensional PIC simulations with RR effects included.The target is a plasma slab of fully ionized deuterium (Z/A¼1/2)of width 40l, density n0¼169nc and thickness ‘¼ 0:5l. The sizeof the computational box is 95l 40l with a spatial resolutionDx¼Dy¼ l=80 and 625 quasi-particles per cell correspondingto a total of 8107 quasi-particles for each species. The laserpulse is s-polarized with the electric field along the z-axis. Itsnormalized amplitude is a0¼320 corresponding to an intensityI¼ 1:4 1023 W cm2 with a wavelength l¼ 1:0 mm and periodT ¼ l=c 3:3 fs. The pulse has a Gaussian transverse profile of width20l FWHM and a sin2 longitudinal profile of length 40l FWHM. Inthese simulations, the front of the laser pulse reaches the foil at t¼0.Comparing the results of our simulations with and without RR(see Fig. 3, we report the results at t¼70 T) it is apparent that RRleads to both an increased electron and ion bunching and to a strongcooling of electrons. These results are qualitatively consistent withour expectations from the kinetic theory that we have discussed inSection 2 and in particular with the prediction of a contraction ofthe electrons available phase space volume.A qualitative understanding of these results can be achieved byrecalling that the RR force Eq. (1) is mainly a strongly anisotropic andnon-linear friction-like force that reaches its maximum for electronsthat counter-propagate with the laser pulse [11]. The backward motionof electrons is thus impeded by RR, more electrons and consequentlyions are pushed forward leading to an enhanced clumping thatimproves the efficiency of the RPA mechanism. In fact, the ionspectrum with RR shows a region between about 300 and 600 MeVwith a significant increase in the number of ions compared to the casewithout RR (Fig. 3). This picture is confirmed by both the enhancementof the longitudinal electric field Ex and the formation of denser bunchesin the ion density compared to the case without RR (see Fig. 3).However, for linear polarization, hot electrons are always generated bythe oscillating component of the jB force. The generation of hotelectrons provides a competing acceleration mechanism to RPA andultimately leads to the generation of the fraction of ions with thehighest energy. The noticeable suppression of the jB heatingmechanism due to the RR force therefore leads to a lower maximumcut-off energy both in the electron and in the ion spectrum (see Fig. 3).These preliminary results for two-dimensional simulationswith RR effects included suggest that, in the LP case, the trendsfound in one-dimensional simulations hold qualitatively even forhigher dimensions. More detailed studies and quantitative com-parisons between one-dimensional and two-dimensional PICsimulations are left for forthcoming publications.4. ConclusionsWe summarize our results as follows. Radiation Reaction effectson the electron dynamics in the interaction of an ultra-intense laserpulse with a thin plasma foil were studied analytically and by one-dimensional and two-dimensional PIC simulations. The details ofthe numerical implementation of the RR force in our PIC code weredescribed in Ref. [11].In one-dimensional simulations, we checked RR effects for threedifferent intensities: I¼ 2:33 1023, 5:5 1023 and 1024 W cm2comparing the results for Circular and Linear Polarization of thelaser pulse. For CP, we found that RR effects are not relevant even atintensity of I¼ 1024 W cm2 whenever the laser pulse does notbreak through the foil. In contrast, for LP we found that RR effectsare important reducing the ion energy significantly.In two-dimensional simulations, we found that RR reduces thejB heating mechanism leading to a lower maximum cut-off energyboth in the electron and in the ion spectrum. Moreover, RR increasesthe spatial bunching of both electrons and ions which are collectedinto denser clumps compared to the case without RR. This might leadto a somewhat beneficial effect with a longer and more efficientradiation pressure acceleration phase whose signature would be anion energy spectrum peaking at an intermediate energy.Fig. 3. Plots of the 2D PIC simulations at t¼70 T. The laser pulse is s-polarized with anintensity I¼ 1:4 1023 W cm2 and the target thickness is ‘¼ 0:5l. From top tobottom, ion ni and electron ne density distributions with (left column) and without(right column) RR, longitudinal Ex (first row) and transverse Ez (second row) electricfield, ion and electron spectrum with (red) and without (blue) RR. (For interpretationof the references to color in this figure legend, the reader is referred to the web versionof this article.)M. Tamburini et al. / Nuclear Instruments and Methods in Physics Research A 653 (2011) 181–185184120A generalized relativistic kinetic equation including RR effectshas been discussed and we have shown that RR leads to acontraction of the available phase space volume. This predictionis in qualitative agreement with the results of our PIC simulationswhere we observed both an increased spatial bunching and asignificant electron cooling as discussed above.AcknowledgmentWe acknowledge the CINECA award under the ISCRA initiative(project ‘‘TOFUSEX’’), for the availability of high performancecomputing resources and support.References[1] V. Yanovsky, V. Chvykov, G. Kalinchenko, P. Rousseau, T. Planchon, T. Matsuoka,A. Maksimchuk, J. Nees, G. Cheriaux, G. Mourou, K. Krushelnick, Opt. Express 16(2008) 2109.[2] C.H. Keitel, C. Szymanowski, P.L. Knight, A. Maquet, J. Phys. B 31 (1998) L75.[3] A. Di Piazza, K.Z. Hatsagortsyan, C.H. Keitel, Phys. Rev. Lett. 102 (2009) 254802.[4] L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields, second ed., Elsevier,Oxford, 1975, p. 76.[5] T. Esirkepov, M. Borghesi, S.V. Bulanov, G. Mourou, T. Tajima, Phys. Rev. Lett. 92(2004) 175003.[6] S.V. Bulanov, E.Yu. Echkina, T.Zh. Esirkepov, I.N. Inovenkov, M. Kando,F. Pegoraro, G. Korn, Phys. Rev. Lett. 104 (2010) 135003.[7] A. Zhidkov, J. Koga, A. Sasaki, M. Uesaka, Phys. Rev Lett. 88 (2002) 185002.[8] N. Naumova, T. Schlegel, V.T. Tikhonchuk, C. Labaune, I.V. Sokolov, G. Mourou,Phys. Rev. Lett. 102 (2009) 025002.[9] T. Schlegel, N. Naumova, V.T. Tikhonchuk, C. Labaune, I.V. Sokolov, G. Mourou,Phys. Plasmas 16 (2009) 083103.[10] M. Chen, A. Pukhov, T.P. Yu, Z.M. Sheng, Plasma Phys. Control Fusion 53 (2011)014004, arXiv:0909.5144v1, doi:10.1088/0741-3335/53/1/014004.[11] M. Tamburini, F. Pegoraro, A. Di Piazza, C.H. Keitel, A. Macchi, New J.Phys. 12 (in press) 123005, arXiv:1008.1685v2, doi:10.1088/1367-2630/12/12/123005.[12] F. Pegoraro, S.V. Bulanov, Phys. Rev. Lett. 99 (2007) 065002.[13] R. Hakim, A. Mangeney, J. Math. Phys. 9 (1968) 116.[14] R. Hakim, A. Mangeney, Phys. Fluids 14 (1971) 2751.[15] L.S. Kuz’menkov, Dokl. Akad. Nauk. Arm. SSR 241 (1978) 322;L.S. Kuz’menkov, Sov. Phys. Dokl. 23 (1978) 469.[16] R.D. Hazeltine, S.M. Mahajan, Phys. Rev. E 70 (2004) 046407.[17] V.I. Berezhiani, R.D. Hazeltine, S.M. Mahajan, Phys. Rev. E 69 (2004) 056406.[18] R.D. Hazeltine, S.M. Mahajan, Phys. Rev. E 70 (2004) 036404.[19] V.I. Berezhiani, S.M. Mahajan, Z. Yoshida, Phys. Rev. E 78 (2008) 066403.[20] M. Tamburini, Ph.D. Thesis, in preparation.[21] A. Di Piazza, Lett. Math. Phys. 83 (2008) 305.[22] A. Macchi, S. Veghini, T.V. Liseykina, F. Pegoraro, New J. Phys. (2010)045013.[23] A. Di Piazza, K.Z. Hatsagortsyan, C.H. Keitel, Phys. Rev. Lett. 105 (2010) 220403in press, arXiv:1007.4914v1, doi:10.1103/105.220403.[24] I.V. Sokolov, J.A. Nees, V.P. Yanovsky, N.M. Naumova, G.A. Mourou, Phys. Rev. E81 (2010) 036412.M. Tamburini et al. / Nuclear Instruments and Methods in Physics Research A 653 (2011) 181–185 185121 RF EFFECTS ON ELECTRON DYNAMICS AND ION ACCELERATIONPHYSICAL REVIEW E 85, 016407 (2012)Radiation-pressure-dominant acceleration: Polarization and radiation reaction effects andenergy increase in three-dimensional simulationsM. Tamburini,1,2,* T. V. Liseykina,3 F. Pegoraro,2,1 and A. Macchi1,21Istituto Nazionale di Ottica, CNR, research unit “A. Gozzini,” Pisa, Italy2Dipartimento di Fisica “E. Fermi,” Università di Pisa, Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy3Institut für Physik, Universität Rostock, D-18051 Rostock, Germany(Received 11 August 2011; published 30 January 2012)Polarization and radiation reaction (RR) effects in the interaction of a superintense laser pulse (I >1023 W cm−2) with a thin plasma foil are investigated with three dimensional particle-in-cell (PIC) simulations.For a linearly polarized laser pulse, strong anisotropies such as the formation of two high-energy clumps in theplane perpendicular to the propagation direction and significant radiation reactions effects are observed. On thecontrary, neither anisotropies nor significant radiation reaction effects are observed using circularly polarized laserpulses, for which the maximum ion energy exceeds the value obtained in simulations of lower dimensionality.The dynamical bending of the initially flat plasma foil leads to the self-formation of a quasiparabolic shell thatfocuses the impinging laser pulse strongly increasing its energy and momentum densities.DOI: 10.1103/PhysRevE.85.016407 PACS number(s): 52.38.Kd, 52.65.RrI. INTRODUCTIONThe radiation pressure generated by ultraintense laserpulses may drive strong acceleration of dense matter, asexperimentally shown in various regimes [1]. Thus, radiationpressure may be an effective mechanism for the generation ofhigh-energy ions, especially in the regime of extremely highintensities and relativistic ion energies as foreseen with the ELIproject. In the case of solid-density thin foil targets, pioneeringparticle-in-cell (PIC) simulations showed that at intensitiesexceeding 1023 W cm−2 and for linear polarization of the laserpulse radiation pressure dominates the acceleration yieldinglinear scaling with the laser pulse intensity, high efficiency andquasimonoenergetic features in the ion energy spectrum [2].More recent two-dimensional (2D) simulations for a smalldisk target suggested a potentially “unlimited” energy gainfor the fraction of ions that get phase locked with the laserpulse [3].The above-mentioned studies showed that the radiationpressure dominant acceleration (RPDA) regime is very ap-pealing as a route to the generation of relativistic ions, butthey leave several theoretical issues open. First, transverse in-stabilities [4] and multidimensional effects may play a crucialrole as shown by 2D simulations [3]. Second, the use of circularpolarization (CP) instead of linear polarization (LP) quenchesthe generation of high-energy electrons [5], allowing radiationpressure to dominate even at intensities below 1023 W cm−2and leading to efficient acceleration of ultrathin foils [6]; ithas not been shown yet whether the use of CP is advantageousalso at ultrahigh intensities (I > 1023 W cm−2), i.e., whenthe radiation pressure generated by the laser pulse becomesthe dominant mechanism of acceleration both for CP and LP.Finally, it has been shown by one-dimensional (1D) simu-lations that radiation reaction (RR) effects may significantlyaffect the dynamics of radiation pressure acceleration both for*tamburini@df.unipi.itthick [7] and thin targets [8,9], and also depend strongly on thepolarization [8]. All of these phenomena may be affected by thedimensionality of the problem, and a fully three-dimensional(3D) approach is ultimately needed because, e.g., in 2Dsimulations and for LP the laser-plasma coupling is differentfor S and P polarization (i.e., for the electric field of thelaser pulse either perpendicular or parallel to the simulationplane, respectively) and the constraint of the conservationof angular momentum carried by CP pulses holds in 3Donly.In this paper, we address the role of polarization and RReffects in the RPDA regime using fully 3D PIC simulations.To our knowledge, these are the first 3D simulations of ionacceleration in the RPDA regime with RR effects included andamong the largest and most accurate 3D simulations reportedso far. Our results show that even in the RPDA regime CP leadsto higher ion energies and better collimation than LP, for whichan anisotropic ion distribution is observed. It is also found thatthe bending of the foil leads to a self-generated parabolic shellthat focuses the impinging pulse down to an almost λ3 volumeand that the energy density at the focus largely exceeds theinitial peak energy density. Compared to 2D simulations withanalogous parameters, the pulse focusing effect is remarkablyenhanced and the cut-off energy of ions is increased. Radiationreaction effects on the ion spectrum are found to be negligiblefor CP but quite relevant for LP where they increase the energycutoff.II. MODELING AND SIMULATION SET-UPOur approach is based on the numerical solution of kineticequations for the phase-space distributions of electrons andions, where RR is included in the motion of electrons via theLandau-Lifshitz (LL) force [10]. Details of our RR modelingand numerical implementation in a PIC code are given inRefs. [8,11]. The effective equation of motion for electrons,after neglecting terms that are negligible in the classical limit016407-11539-3755/2012/85(1)/016407(5) ©2012 American Physical Society123 POLARIZATION AND RADIATION FRICTION EFFECTS IN THE RPATAMBURINI, LISEYKINA, PEGORARO, AND MACCHI PHYSICAL REVIEW E 85, 016407 (2012)[8], isdpdt= −e(E + vc× B)+ fR,fR = 2r2c3{− γ 2[(E + vc× B)2−(vc· E)2]vc+[(E + vc× B)× B +(vc· E)E]}, (1)where rc = e2/mec2. The RR force contribution is importantfor ultrarelativistic electrons and it is usually dominated by thefirst term, while the second term ensures the on-shell condition[8]. Notice that the dominant term has almost the same formalso in different approaches to RR modeling [7,9]. For a planewave propagating along the x axis, the RR force is maximumor zero for counterpropagating (vx/c → −1) or copropagating(vx/c → +1) electrons, respectively.In order to clarify the new qualitative features due to RReffects, we recall that the phase-space volume element Jevolves according to dJ/dt = J∇p · fR . It has been shown[11] that ∇p · fR 0 and, therefore, the RR force leads to acontraction of the available phase-space volume. The physicalinterpretation of this property is that the RR force acts as acooling mechanism for the system accounting for the emissionof high-energy photons. These photons are assumed to escapefrom the plasma freely, carrying away energy and entropy [11].We present a total of four 3D simulations each with the samephysical and numerical parameters but different polarization,with and without RR effects. In these simulations, the laserfield amplitude has a sin2-function longitudinal profile with9λ full width at half-maximum (FWHM) (where λ = 0.8 μmis the laser wavelength) while the transverse radial profile isGaussian with 10λ FWHM and the laser pulse front reachesthe edge of the plasma foil at t = 0. The peak intensity atthe focus is I = 1.7 × 1023 W cm−2, which corresponds toa normalized amplitude a0 = 280 for LP and a0 = 198 forCP. The target is a plasma foil of electrons and protons withuniform initial density n0 = 64nc (where nc = πmec2/e2λ2is the critical density), thickness = 1λ, and initially lo-cated in the region 10λ x 11λ. The density n0 1.1 ×1023 cm−3 is slightly lower than that of solid targets butthe areal density n0 has fully realistic values. Moreover,laser pulses of ultrahigh contrast are now available [[12], andreferences therein] to avoid early plasma formation effects bythe prepulse, thus, a thin plasma with steplike profile is not anunrealistic assumption.The simulation grid is 1320 × 896 × 896 and the spatialstep is λ/44 for each direction. The time step is T/100 whereT = λ/c = 2.67 fs is the laser period. We use 216 particlesper cell for each species and the total number of particles is1.526 × 1010. The runs were performed using 1024 processorseach one equipped with 1.7 GB of memory of the IBM-SP6cluster at the CINECA supercomputing facility in Bologna,Italy.III. RESULTS AND DISCUSSIONFigure 1 shows the ion and the electron 3D spatialdistributions [13] at t = 20T for the LP case without (a)and with (b) RR and for the CP case without (c) and with(d) RR. The color (grayscale) corresponds to the range inkinetic energy. For CP, the ion spatial distribution follows thespatial intensity profile of the initial laser pulse, has rotationalsymmetry around the central axis, and a distribution in energymonotonically decreasing with increasing radial distance. Themost energetic ions are located near the axis. The number ofions having energy E 1100 MeV and E 800 MeV are2.3 × 1010 and 9.4 × 1010, respectively. The electron spatialdistribution has a helicoidal shape with step λ; Figs. 1(c),FIG. 1. (Color online) Spatial distributions of ions (upper row) and electrons (lower row) at t = 20 T and in the region (|y|,|z|) 5.7λ, forLP without (a) and with (b) RR and for CP without (c) and with (d) RR. Ions and electrons are divided into seven populations according to theirkinetic energy, with the color bar (grayscale) reporting the lower bound of the energy interval. In the LP case [(a) and (b)], the polarization isalong the y axis.016407-2124RADIATION-PRESSURE-DOMINANT ACCELERATION: . . . PHYSICAL REVIEW E 85, 016407 (2012)FIG. 2. (Color online) (x,y) and (x,z) sections of the 3D simulations of the laser pulse-foil interaction [(a1)–(a3), (b1)–(b3), (c1)–(c3), and(d1)–(d3)] and 2D simulations for CP and without RR with the same parameters as the 3D simulations [(e) and (f)], all at t = 20T . Each framereports the color (grayscale) contours of√E2 + B2 (normalized units) in the xy plane at z = 0 [(a1)–(a3) and (c1)–(a3)], in the xz plane at y = 0[(b1)–(b3) and (d1)–(d3)] and in the simulation plane [(e) and (f)] in the 2D case. Line contours of the ion and electron densities are superimposedin the upper and lower frames, respectively. The CP case with RR is almost identical to the CP case without RR and it is not reported.and 1(d). Radiation reaction effects play a minor role for CP,affecting only a small fraction of ultrarelativistic electrons[mostly removing fast electrons behind the foil with almostno influence on the ion distribution as seen by comparingFigs. 1(c) and 1(d)].The (x,y) and (x,z) sections of the total electromagneticenergy density and of the ion and electron densities for the CPcase in Fig. 2 [(a3)–(b3) and (c3)–(d3)] evince a self-generatedparabolic shell wrapping the laser pulse and focusing it up tonearly a λ3 volume, so both the energy and the momentumdensities at the focus reach values more than 8 times their peakvalue in the initial laser pulse. This effect is much weaker in 2Dsimulations with the same parameters as shown in Figs. 2(e)–2(f). Along the axis, the peak value and width of the ion densityprofile are 10nc and 0.5λ, showing a strong rarefaction dueto the transverse expansion, potentially leading to enhancedacceleration as described in Ref. [3].For LP, the peak ion energy is lower than for CP, the iondistribution is anisotropic, and RR effects are much stronger.The most energetic ions (800–1100 MeV) are grouped into twooff-axis clumps lengthened and aligned along the polarizationdirection, and their number is increased in the case with RR asseen by the comparison of Figs. 1(a) and 1(b) and also in Figs. 2[(a1),(a2) and (b1),(b2)] where sections of the ion density inthe (x,y) and (x,z) planes are shown. The contours of theelectromagnetic (EM) energy density in Figs. 2 [(a1),(a2) and(b1),(b2)] show that near the axis most of the laser pulse hasbeen transmitted through the target. The increased bunchingand higher density observed in the case with RR may be relatedto the higher ion energies since the local increase of the densityand, therefore, of the reflectivity leads to a longer and moreefficient RPDA phase. This is consistent with observing inFigs. 2(b1) and 2(b2) that the EM energy density is higherbehind the two high-density clumps, which correspond to themost energetic ions and are similar to the ion lobes observedin Ref. [14] at lower intensity and in a regime of strong pulsepenetration through the foil. The pulse focusing effect by theself-generated parabola is present also in the LP case althoughweaker than in the CP case and presumably reduced as thelaser pulse breaks through the parabolic shell.The differences between CP and LP in the accelerationdynamics are well explained, for planar geometry and non-relativistic ions, by the absence of the oscillating componentof the J × B force for CP [5], which maximizes the effectof the radiation pressure strongly suppressing the electronheating. The absence of the oscillating component of theJ × B force also accounts for the very different RR effects.For CP, a steady push of the foil and weak pulse penetrationare observed and most of the electrons move coherently withthe foil and in the same direction as the laser pulse so the RRforce becomes very small in accordance with Eq. (1) sincethe electrons effectively copropagate with the laser pulse (seealso Refs. [2,8]). For LP, the J × B-driven oscillations allowelectrons to collide with the counterpropagating laser pulsetwice per cycle producing temporal maxima in the RR forcein agreement with Eq. (1). Our present results indicate that CPleads to more efficient acceleration, producing higher energyand collimated ion beams, and making RR effects negligiblealso in the 3D case, accounting for target bending and pulsefocusing effects, and for relativistic ions.Large-scale 3D PIC simulations are limited by the size andavailability of computational resources both in the number ofruns that may be performed and in the achievable numericalresolution. This last issue may raise doubts on the accuracyof 3D results. To gain confidence on this side, as wellas to compare the 3D results to those obtained in lowerdimensionality, we performed 2D simulations both withnumerical parameters similar to those of 3D runs and withhigher resolution. The effect of increasing resolution andparticle number on the ion spectra in 2D simulations is shownin Fig. 3 where 2D results are reported for the three differentpolarization cases (CP, LP-S, and LP-P ) and compared withthe 3D results for both LP and CP. The spectra are normalizedto unity both in the 2D and 3D cases. In the CP case bothnumerical and RR effects on the spectrum are smaller while inthe P -polarization case these effects are larger. Changing thespatial resolution from λ/44 to λ/80 and increasing the numberof particles-per-cell for each species from 256 to 625 shifts theenergy cutoff by ∼2% in the CP case and by ∼15% (∼20%) inthe P -polarization case without RR (with RR). The stronger016407-3125 POLARIZATION AND RADIATION FRICTION EFFECTS IN THE RPATAMBURINI, LISEYKINA, PEGORARO, AND MACCHI PHYSICAL REVIEW E 85, 016407 (2012)FIG. 3. (Color online) Ion spectra from 2D [(a)–(c)] and 3D [(d)–(e)] simulations with same physical parameters, all at t = 20T . The2D spectra are reported for circular [CP, panel (a)] and linear (LP) “S”[panel (b)] and “P ” [panel (c)] polarization cases. In each plot, theblue (dark gray) and red (light gray) curves correspond to simulationswithout and with radiation reaction (RR) effects, respectively. In theupper plots of panels (a)–(c) the numerical resolution (number ofparticles per cell and of points per wavelength) is similar to those ofthe 3D simulations in panels (d) and (e), while in the lower plots theresults for higher resolution are shown.effect of the inclusion of RR for the higher resolution case maybe explained by noticing that RR mostly affects the highestenergy electrons [8], which are located in the high-energy tailof the distribution function that needs a very large numberof particles to be resolved properly. Nevertheless, the limitedresolution does not qualitatively affect prominent features inion spectra, such as the higher ion energy for CP and therelevance of RR effects for LP only, leading for this latter caseto an higher energy of ions with respect to the case of no RRas observed in 1D simulations [8]. As a novel feature of 2Dsimulations, P polarization leads to much stronger RR effectsthan S polarization. In fact, for P polarization the electric fieldcan drag a large fraction of electrons out in vacuum and towardthe laser pulse as the plasma foil begins to bend, enhancingthe RR effect.For CP, the maximum ion energy is higher in the 3D case(1600 MeV) than in the 2D case (1400 MeV). In turn,the latter value is higher than what found in 1D, plane-wavesimulations, for which we find a broad spectral peak whichat t = 20 extends up to 1100 MeV and is centered arounda value of 870 MeV. The latter value corresponds to theenergy ELS = (γ − 1)mpc2, where γ = 1/√1 − β2 and β areobtained from the “light sail” model [[15], and referencestherein] by numerically integrating the 1D equations of motionfor the foildγβdt= 2I (t − X/c)n0mpc2(1 − β1 + β),dXdt= βc. (2)From the 1D modeling we also evaluate a final ion energy forthe spectral peak of 1700 MeV that is reached at t 90.The 3D simulations could not be extended up to the end of theacceleration stage (estimated up to ∼90T in 1D simulations)but, since the efficiency of RPDA increases with the foilvelocity, the energy gain is expected to be even larger at thelater times (provided that 3D effects do not cause an early stopof the acceleration). Hence, the comparison at t = 20 shows anoverall increase of the ion energy in the 3D case with respectto 1D and 2D cases. Part of the energy enhancement can beattributed to the reduction of the areal density n0 due tothe transverse expansion, as was noticed in 2D simulationssupporting the model of “unlimited” acceleration [3]. Anadditional contribution may come from the above-describedfocusing of the laser pulse by the deformed foil, which isstronger in 3D geometry. This latter effect was absent in thesimulations of Ref. [3] because a target with a radius smallerthan the pulse waist was considered.IV. CONCLUSIONSIn conclusion, with three-dimensional particle-in-cell simu-lations of ultraintense laser interaction with solid-density foilswe showed that circular polarization improves ion accelerationalso in the radiation pressure dominant regime, confirming andextending previous results obtained for lower intensity and/orlower dimensionality. In detail, circular polarization leads tothe highest ion energies, to a symmetrical and collimateddistribution, and to negligible effects of radiation reaction.In addition, the maximum energy of ions in 3D is largerthan observed in corresponding 1D and 2D simulations. Thisenhancement is attributed both to the density decrease in thetarget, as noticed in the “unlimited acceleration” model [3], andto the strong focusing of the laser pulse by the parabolicallydeformed foil. In the linear polarization case, lower maximumenergies are achieved, the most energetic ions are groupedanisotropically into two off-axis clumps and radiation reactioneffects significantly affect the energy spectrum. We expectthese findings to be of relevance for the design of futureexperiments on laser acceleration of ions up to relativistic(GeV) energy.ACKNOWLEDGMENTSWork sponsored by the Italian Ministry of Universityand Research via the FIRB project “SULDIS.” We acknowl-edge the CINECA Grant No. HP10A25JKT-2010 for theavailability of high-performance computing resources.[1] S. Kar et al., Phys. Rev. Lett. 100, 225004 (2008); K. U. Akliet al., ibid. 100, 165002 (2008); A. Henig et al., ibid. 103, 245003(2009); C. A. J. Palmer et al., ibid. 106, 014801 (2011).[2] T. Esirkepov, M. Borghesi, S. V. Bulanov, G. Mourou, andT. Tajima, Phys. Rev. Lett. 92, 175003 (2004).[3] S. V. Bulanov, E. Y. Echkina, T. Z. Esirkepov, I. N. Inovenkov,M. Kando, F. Pegoraro, and G. Korn, Phys. Rev. Lett. 104,135003 (2010)[4] F. Pegoraro and S. V. Bulanov, Phys. Rev. Lett. 99, 065002(2007).016407-4126RADIATION-PRESSURE-DOMINANT ACCELERATION: . . . PHYSICAL REVIEW E 85, 016407 (2012)[5] A. Macchi, F. Cattani, T. V. Liseykina, and F. Cornolti, Phys.Rev. Lett. 94, 165003 (2005).[6] X. Zhang, B. Shen, X. Li, Z. Jin, and F. Wang, Phys. Plasmas14, 073101 (2007); A. P. L. Robinson, M. Zepf, S. Kar,R. G. Evans, and C. Bellei, New J. Phys. 10, 013021 (2008);O. Klimo, J. Psikal, J. Limpouch, and V. T. Tikhonchuk, Phys.Rev. ST Accel. Beams 11, 031301 (2008).[7] N. Naumova, T. Schlegel, V. T. Tikhonchuk, C. Labaune, I. V.Sokolov, and G. Mourou, Phys. Rev. Lett. 102, 025002 (2009).[8] M. Tamburini, F. Pegoraro, A. D. Piazza, C. H. Keitel, andA. Macchi, New J. Phys. 12, 123005 (2010).[9] M. Chen, A. Pukhov, T.-P. Yu, and Z.-M. Sheng, Plasma Phys.Contr. Fusion 53, 014004 (2011).[10] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields,2nd ed. (Elsevier, Oxford, 1975).[11] M. Tamburini, F. Pegoraro, A. D. Piazza, C. Keitel, T. Liseykina,and A. Macchi, Nucl. Inst. Methods A 653, 181 (2011).[12] C. Thaury et al., Nat. Phys. 3, 424 (2007); C. Rödel, M. Heyer,M. Behmke, M. Kübel, O. Jäckel, W. Ziegler, D. Ehrt,M. Kaluza, and G. Paulus, Appl. Phys. B: Lasers Opt. 103, 295(2011).[13] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevE.85.016407 for high-quality pictures of thethree-dimensional electron and ion spatial distributions reportedin Fig. 1.[14] L. Yin, B. J. Albright, K. J. Bowers, D. Jung, J. C.Fernández, and B. M. Hegelich, Phys. Rev. Lett. 107, 045003(2011).[15] A. Macchi, S. Veghini, T. V. Liseykina, and F. Pegoraro, New J.Phys. 12, 045013 (2010).016407-5127 POLARIZATION AND RADIATION FRICTION EFFECTS IN THE RPANew J. Phys. 18 (2016) 072001 doi:10.1088/1367-2630/18/7/072001FAST TRACK COMMUNICATIONInverse Faraday effect driven by radiation frictionTVLiseykina1,5, S VPopruzhenko2 andAMacchi3,41 Institute of Physics, University of Rostock, D-18051 Rostock, Germany2 National ResearchNuclearUniversity,MoscowEngineering Physics Institute, Kashirskoe Shosse 31, 115409,Moscow, Russia3 CNR,National Institute ofOptics (INO), AdrianoGozzini research unit, Pisa, Italy4 Enrico FermiDepartment of Physics, University of Pisa, largo Bruno Pontecorvo 3, I-56127 Pisa, Italy5 On leave from Institute of Computational Technologies, SDRAS, 630090Novosibirsk, Russia.E-mail: andrea.macchi@ino.itKeywords: radiation friction, inverse Faraday effect, laser–plasma interactions,magneticfield generation in plasmas, high field physicsAbstractAcollective,macroscopic signature to detect radiation friction in laser–plasma experiments isproposed. In the interaction of superintense circularly polarized laser pulses with high density targets,the effective dissipation due to radiative losses allows the absorption of electromagnetic angularmomentum,which in turn leads to the generation of a quasistatic axialmagnetic field. This peculiar‘inverse Faraday effect’ is investigated by analyticalmodeling and three-dimensional simulations,showing thatmulti-gigagaussmagnetic fieldsmay be generated at laser intensities> -10 W cm23 2.1. IntroductionThe development of ultrashort pulse lasers with petawatt power has opened new perspectives for the study ofhighfield physics and ultra-relativistic plasmas [1, 2]. In this context, the longstanding problemof radiationfriction (RF) or radiation reaction has attracted new interest. RF arises from the back-action on the electron ofthe electromagnetic (EM)field generated by the electron itself and plays a dominant role in the dynamics ofultra-relativistic electrons in strong fields. A considerable amount of work has been devoted both to revisitingthe RF theory [3, 4] and to its implementation in laser–plasma simulations [5–9], as well as to the study ofradiation-dominated plasmas in high energy astrophysics, see e.g. [10–12].While RF is still an openmatter both for classical and quantum electrodynamics [2], RFmodels have notbeen discriminated experimentally yet. This circumstance led to several proposals of devoted experimentsproviding clear signatures of RF, e.g. in nonlinear Thomson scattering [13–18], Compton scattering [19],modification of Raman spectra [20], electron acceleration in vacuum [21–23], radiative trapping [24–26] or γ-ray emission fromplasma targets [27, 28].Most of these studies are based on single particle effects, andRFsignatures are found inmodifications of observables such as emission patterns and spectra whenRF is includedin themodeling. Detecting suchmodificationsmay require substantial improvements in reducing typicaluncertainties in laser–plasma experiments. At very high intensities, RF lossesmay affect the collective dynamics,e.g. bymodifying the spectra of accelerated ions in the radiation pressure dominated regime [29, 30, andreferences therein] or the dynamics ofmagnetic field generation by thefilamentation instability in laser-generated colliding pair plasmas [31]. However, also in this case themodifications are quantitative, rather thanqualitative, and relativelymodest so that itmay be difficult to discriminate RF effects.Instead, in this paper we identify a collective,macroscopic effect induced byRF, namely the generation ofmulti-gigagauss, quasi-steady, axialmagnetic fields in the interaction of a circularly polarized (CP) laser pulsewith a dense plasma. This is a peculiar formof the inverse Faraday effect (IFE) [32–35] andmay bemoreaccessible experimentally than single-particle effects. In fact, the IFE has been previously studied in differentregimes of laser–plasma interactions [36–42, and references therein]. By using three-dimensional (3D) particle-in-cell (PIC) simulations, wefind that at laser intensities foreseeable with next generation facilities producingmulti-petawatt [43] or even exawatt pulses [44, 45], themagnetic field created by the RF-driven IFE in denseOPEN ACCESSRECEIVED4May 2016REVISED14 June 2016ACCEPTED FOR PUBLICATION21 June 2016PUBLISHED11 July 2016Original content from thisworkmay be used underthe terms of the CreativeCommonsAttribution 3.0licence.Any further distribution ofthis workmustmaintainattribution to theauthor(s) and the title ofthework, journal citationandDOI.© 2016 IOPPublishing Ltd andDeutsche PhysikalischeGesellschaft129 INVERSE FARADAY EFFECT DRIVEN BY RADIATION FRICTIONplasma targets reachesmulti-gigagauss values with a direction dependent on the laser polarization, whichconfirms its origin from the ‘photon spin’. Themagnetic field is slowly varying on times longer than the pulseduration andmay be detected via optical polarimetry techniques [46–50]. This would provide an unambiguoussignature of the dominance of RF effects, since the axialmagnetic field disappears in the absence of RF. The effectmight also be exploited to create stronglymagnetized laboratory plasmas in so far unexplored regimes (seee.g. [51]).2. Role andmodeling of radiative lossesThe IFE is due to absorption of EMangularmomentum6, which in general is not proportional to energyabsorption. As an example of direct relevance to the present work, let us consider amirror boosted by theradiation pressure of a CP (with positive helicity, for definiteness) laser pulse. From a quantumpoint of view, thelaser pulse of frequencyω propagating along x̂ corresponds toN incident photonswith total energy wN andangularmomentum ˆN x. If themirror is perfect,N is conserved in any frame. If themirrormoves along x̂, thereflected photons are red-shifted leading to EMenergy conversion intomechanical energy (up to 100% if themirror velocity~c) but there is no spin flip for the reflected photons, hence no absorption of angularmomentum.However, if the electrons in themirror emit high-frequency photons, a greater number of incidentlow-frequency photonsmust be absorbedwith their angularmomentum. From a classical point of view,absorption of angularmomentum requires some dissipationmechanism [42]which, in our example, implies anon-vanishing absorption in the rest frame of themirror.In the case here investigated, effective dissipation is provided by theRF force whichmakes the electrondynamics consistent with the radiative losses. In order to demonstrate IFE induced by RF, we consider a regimeof ultra-high laser intensity > -I 10 W cmL 23 2 and thick plasma targets (i.e. with thicknessmuch greater thanthe evanescence length of the laserfield)where the radiative energy loss is a significant fraction of the laser energyas shown by simulationswith RF included [29, 30, 54–56].We use a simplemodel to account for such losses andprovide a scaling lawwith the laser intensity. The power radiated by an electronmovingwith velocity vx along thepropagation axis of a CPpulse of amplitude w= º( )E m c e a B aL e 0 0 0 (withω the laser frequency) isw g= - ( )⎜ ⎟⎛⎝⎞⎠Pe acvc231 . 1xrad2 2 202 2The classical RF force on an electron is defined in order that thework done per unit time equals Prad. In oursimulations the Landau–Lifshitz (LL) expression for the RF force has been used [57]. Consistently withequation (1) the LL force in a planewave vanishes for =v cx , and has amaximum for = -v cx . The spectrumofthe emitted radiation peaks at frequencies w g wrad 3 , with γ the relativistic factor of electronswhich can beestimated as g wa0 . At such frequencies the radiation from the plasma is incoherent (see also the discussion insection 3), thus the total radiated power byN comoving electronswill be NPrad. For thin targets accelerated bytheCP laser pulse (‘light sail’ regime), all electronsmovewith the foil at v cx  , and there is no high-frequencyoscillation driven by the ´v B force. Thus the radiation is strongly suppressed by the factor -( )v c1 1x 2  ,as observed in simulations [7, 58]. In contrast, RF losses becomemuch larger for thick targets [29, 30, 54] (‘holeboring’ regime) because the acceleration of the plasma surface has a pulsed nature [55, 59, 60]with a densebunch of electrons being periodically dragged towards the incident laser pulse, i.e. in a counterpropagatingconfiguration ( ℓ( )c 1s , andwe consider arange of parameters such that >a n n 10 c 0 .We thus obtain » ( )n n a n n2 1p0 0 0 c 0 1 2  , in agreementwith adetailed theory of nonlinear wave propagation (see e.g. [62]) and numerical simulations. Thus, bywritingw p= ( )I m c a r4L e 2 02 c (where =r e m cc 2 e 2)we obtain lN a rx 0 c (where l p w= c2 is the laserwavelength), independently on the initial density7.The total radiated intensity is =I P Nxrad rad . In order to compare the radiated energy with the laser pulseenergywe take into account that the radiation is emitted as bursts corresponding to the periodic return ofelectrons towards the laser, i.e. for a fraction t t t+t ( )f e e i of the interaction stage where te is the timeinterval duringwhich the electronsmove backwards. Analysis of laser piston oscillations in [55] suggests thatt te i sowe take tf 1 2 for our rough estimate. Since -( ) v c1 1x 2 for returning electrons, we obtain forthe fraction of radiated energy to the laser pulse energyhplg ( )r a43. 3radc02If the energy of electrons ismainly due to themotion in the laser field, then g +( )a a1 02 1 2 0  for a 10 and h µ arad 03. For l m= 0.8 m, h ~ 1rad for ~a 4000 , corresponding to ~ ´ -I 7 10 W cmL 23 2. This order-of-magnitude estimate implies that for such intensities a significant part of the laser energy is lost as radiation,strongly affecting the interaction dynamics. Amore precise estimate would require to account both for theenergy depletion of the laser and for the trajectorymodification of the electrons due to the RF force.3. Simulation resultsA3Dapproach is essential tomodel the phenomena of angularmomentum absorption andmagnetic fieldgeneration, thuswe rely onmassively parallel PIC simulations inwhichRF is implemented following theapproach described in [7], based on the LL equation (see [9] for a benchmarkwith other approaches).Weremark that the inclusion of the radiation loss as a dissipative process via the RF force requires the followingassumptions: (i) the dominant frequencies in the escaping radiation aremuch higher than the highest frequencythat can be resolved on the numerical grid, (ii) the radiation at such frequencies is incoherent, (iii) the plasma istransparent to such frequencies. Since, as also stated above, for radiation in the field of a planeCPwave ofultrarelativistic intensity the radiation spectrumpeaks at frequencies of the order of w w» a ,rad 03 all the aboveassumptions arewell-satisfied in our conditions.The laser pulse is initialized in away that at thewaist plane x=0 (coincident with the target boundary) thenormalized amplitude of the vector potential = e m ca A e 2 would beFigure 1.Cartoon showing the electron dynamics during the ‘hole boring’ stage. Frame (a) shows the approximated profiles of thedensities of ions (ni) and electrons (ne) and of the electrostatic field (Ex) at the early time (t 0 )when ions have notmoved yet andelectrons from the depletion region ( < 1023Wcm−2)with high-density, thick plasmatargets, simulations show significant radiation friction losses, in contrast to thin targets for whichsuch losses are negligible.We present an analytical calculation, based on classical radiation frictionmodeling, of the conversion efficiency of the laser energy into incoherent radiation in the casewhen acircularly polarized pulse interacts with a thick plasma slab of overcritical initial density. By accountingfor three effects including the influence of radiation losses on the single electron trajectory, the global‘hole boring’motion of the laser-plasma interaction region under the action of radiation pressure, andthe inhomogeneity of the laserfield in both longitudinal and transverse direction, wefind a goodagreementwith the results of three-dimensional particle-in-cell simulations. Overall, the collectiveeffects greatly reduce radiation losses with respect to electrons driven by the same laser pulse invacuum,which also shift the reliability of classical calculations up to higher intensities.1. IntroductionThe continuous progress of laser techniquesmaking higher and higher electromagnetic (EM) intensitiesaccessible for experiments has stimulated the growth of research areas such as relativistic dynamics andnonlinear optics in classical plasmas [1] and quantum electrodynamics in extremely strong fields [2, 3].Radiation friction (RF) is a problemof central interest in both the abovementioned areas. In the classicalcontext, amodification of theNewton–Lorentz equation ofmotion for an electron by adding a new force term,named the RF force (RFF) or radiation reaction force, is necessary tomake the electron dynamics self-consistentwith the emission of radiation. Although the correct formof theRFF has been the subject of intense debate fordecades and until recently [4, 5], it now appears that in the classical limit the Landau–Lifshitz (LL) expression [6]gives a correct and consistent description [7, 8]. The LL expression of the RFF has become the basis of classicalsimulations of superintense laser-plasma interaction [9, 10]where RF losses (corresponding to the escape ofhigh-frequency, incoherent radiation from the plasma) are important enough to affect the plasma dynamics.When the frequency of the emitted radiation becomes sufficiently high that the energy andmomentumofsingle photons are not negligible with respect to those of the radiating electron, a quantum electrodynamics(QED) description becomes necessary. However, a correct and effective description of ‘quantumRF’ is an openissue. Thefirst two experiments claiming for evidence of quantumRF signatures in nonlinear Thomsonscattering of superintense laser pulses by ultrarelativistic electrons [11, 12] came to somewhat differentconclusions aboutwhichmodel better described the experimental results (see [13] for a discussion). Notice thatthese experiments involved laser-plasma physics in the generation via wakefield acceleration of a dense, shortduration bunch of relativistic electrons in order to increase the luminosity in the gamma-ray region; however,OPEN ACCESSRECEIVED30August 2018REVISED18December 2018ACCEPTED FOR PUBLICATION23 January 2019PUBLISHED15March 2019Original content from thisworkmay be used underthe terms of the CreativeCommonsAttribution 3.0licence.Any further distribution ofthis workmustmaintainattribution to theauthor(s) and the title ofthework, journal citationandDOI.© 2019TheAuthor(s). Published by IOPPublishing Ltd on behalf of the Institute of Physics andDeutsche PhysikalischeGesellschaft137 EFFICIENCY OF RADIATION FRICTION LOSSESthe dynamics of the laser-bunch interactionwas of single particle nature. The geometry of these experiments wasdesigned tomaximize RF losses in order tomake quantum signatures appearing at relatively low intensity. In thisregime, such signatures aremostly a reduction of RF losses with respect to the classical calculation because of thespectral cut-off which appears when the emitted photon energy approaches the photon energy. These effects canbe reproduced by a semiclassicalmodeling, similar towhat found in a different class of experiments involvinghigh energy electron scattering in crystals [14, 15].An alternative approach to investigate RF in the laboratory is to search for regimeswhere collective effects inthe laser-plasma interaction boost radiation losses, so that RF signaturesmay become strong and unambiguous.Several simulationworks have shownhighly efficient radiation losses (a few tens per cent of the laser pulseenergy) in the interaction of circularly polarized (CP) pulses with dense thick targets [16–22]. This is in sharpcontrast with thin targets accelerated byCP pulses in the so-called ‘light sail’ (LS) regime, for which the radiationlosses are veryweak [9, 23, 24]. Strong differences betweenCP and linear polarization (LP)were also evidenced[9, 23]. Hence, the collective laser-plasma dynamics can play a crucial role in determining the amount of RFlosses.In our previous work [21]wehavemade a first attempt of a classicalmodel to estimate the conversionefficiency ηrad of the laser energy into incoherent radiation in the case when a strongCP pulse interacts with athick plasma of overcritical initial density. In turn, the efficient absorption of CP light causes a strong transfer ofangularmomentum to the target, with the generation of ultrahighmagnetic fields (inverse Faraday effect)withstrength achieving several Giga-Gauss which can provide amacroscopic signature of RF [21].In [21] the scaling of ηrad with the laser intensity agreed reasonably with the results of three-dimensional(3D) particle-in-cell (PIC) simulations of the laser-plasma interaction, up to intensities approaching1024Wcm−2. Beyond this limit, however, themodel predicts unphysical values of ηrad>1 because neither themodification of the radiating electron trajectories due toRF nor the depletion of the laser pulse are taken intoaccount. In addition, andmore in general, at those intensities the classical description becomes questionable andquantum effects are expected to become relevant.The aimof this paper is to provide an accurate estimate of ηrad for CPfields via analyticalmodeling assumingthat the classical RF regime is retained. First, we use the solution by Zeldovich [25] to take self-consistently intoaccount the effect of RF losses on the electron trajectory. Then, we show that the amount of RF losses isconsiderably affected by the averagemotion of the plasma surface, the finite evanescence length of the EMfieldin the plasma, and the radially inhomogeneous distribution of the laser intensity. By accounting for these effects,analytical estimates in good agreement with the results of 3D simulations are obtained.We also provide anestimate for the value of the quantumparameter and show that, in the present context, the electron dynamicscan still bewell describedwithin the classical RFF approach.2. Review of previousmodeling and its limitationsIn the regime of interest here, an ultraintense laser pulse of frequencyω and dimensionless field amplitudew=a eE m cL e0 (withEL the electric field amplitude) interacts with a strongly overdense (electron densityw p=n n m e4e c e 2 2, the cut-off density) plasma target which remains opaque to the laser light. The radiationpressure of the laser light is high enough to produce ‘hole boring ’ (HB) in the target, i.e. the plasma surface isdriven at an average velocityr=X+ XX = = ( )vcIcZn mAn ma1, , 1L c ee pHB3 02where p= =I cE m c n a4L L e c2 302 is the laser intensity. Equation (1) can be obtained by balancing themass andmomentumflows at the surface [26] and is valid for total reflection of the laser light in the frame co-moving withthe surface, i.e. in the absence of dissipative effects. If a fraction η of the laser intensity is dissipated, for exampledue to RF losses, equation (1)may bemodified by replacing ILwith h-( )I 1 2L . In the case of our simulationsthis would lead atmost to a;5%decrease in vHB at the highest intensity considered (a0=800).In order for the interaction to remain in theHB regime during thewhole duration of the laser pulse, thetargetmust be ‘thick’ enough that vHBτL<D, where τL is the laser pulse duration andD is the target thickness.In the opposite ‘thin’ target limit vHBτL?D, the target can be accelerated as awhole and enter the ‘LS’ regime[27, 28], where the scaling of the velocity vLS with intensity becomesmuch faster than (1). Thus, the same laserpulse parametersmay enable to reach velocities vLS;c in an ultrathin target while yielding vHB to be a fractionof c in a thick target. In particular, for the parameters of calculations presented below vHB≈(0.3÷0.6)c. Thedifferent acceleration regimemay therefore explain the huge difference in the radiation efficiency between thickand thin targets. In fact, assuming that the electrons radiate in the field of a plane EMCPwave propagating alongx, the radiated power is [29]2New J. Phys. 21 (2019) 033009 SVPopruzhenko et al138w g= -⎜ ⎟⎛⎝⎞⎠ ( )Pe acvc231 , 2xrad2 2 202 2where g = - v c1 1 2 2 and vx is the velocity component parallel to thewavevector. Assuming thatmost ofthe radiating electrons co-movewith the target ions, the factor -( )v c1 x 2 leads to strong suppression ofradiation emission for thin targetsmoving at vLS;c, while the suppression ismuch less severe for thick targetsas far as vHB is significantly smaller than c.In the thick target case, the laserfield penetrates into the skin layer where the electrons pile up under theaction of the radiation pressure. The areal density of electrons in the skin layer can be estimated as [21, 30]l ( )N ar, 3x00where =r e mc0 2 2 is the classical electron radius. For a0?1, by estimating γ;a0 we obtain the radiatedpower per unit surface as = µI N P axrad rad 05, which implies aµa03 scaling for the radiation loss efficiency, ingood agreement with the simulation results. For vx=0, and assuming that the duration of the uncoherent high-energy emission is the same as the laser pulse, the conversion efficiency defined as a ratio of the energy emitted byradiating electronsUrad to that of the laser pulseUL is thus given byh x= = = ( )UUIIa , 4radradLradL03where the parameterxpl= ( )r4350is introduced, andλ is the laser wavelength.In [21] it was suggested that for thick targets an enhancement of radiation lossesmay originate from the non-steady dynamics ofHB acceleration [30]. In particular, ion acceleration by the space-charge field causes a pulsed‘collapse’ of the electron density with the excess electrons returning towards the laser with negative velocityvx<0, enhancing theRF losses by a sequence of radiation bursts.However, it is not straightforward to provideanalytical estimates for either the rate of the bursts or the value of vx for the returning electrons. In particular,estimating vxwould require tofind themotion of the returning electrons in an inhomogeneous electric fieldwiththe RFF included. For an order-of-magnitude estimate, we simply assumed - ( )v c1 1x 2 and the number ofthe returning electrons to be ;Nx [21] (i.e.most of the electrons in the skin layer to collapse). This leads again toan expression like (4) for the conversion efficiency, apart from a reduction factor< 1 accounting for the fact thatthe returning electrons radiate only for a fraction of the interaction time.Apparently, the h ~ arad 03 scaling fairly agrees with the results of 3D simulationswhich give h ~ arad 03.2 up tointensities a0;500, but the absolute value predictions of (4) aremuch higher than those observed in thesimulations (seefigure 1 below). This is not surprising since obviously (4) becomes invalidwhen approaching acritical value of the laser field amplitudeFigure 1.Conversion efficiency into radiation ηrad as a function of the dimensionless laser amplitude a0. The dashed line shows theperturbative result (4) [21]. Results of the selfconsistent calculation are shown by a thick black line (equation (16)with vx=0), thinblack line (equation (16)with vx=vHB from (1)) and thick red line (equation (32)with vx=vHB). Results of 3DPIC simulations areshown by empty black diamonds and filled red circles for circular and linear polarization of the laser pulse, respectively. The dashedred line is a~a03.2fit to the PIC results for circular polarization.3New J. Phys. 21 (2019) 033009 SVPopruzhenko et al139 EFFICIENCY OF RADIATION FRICTION LOSSESx= »- ( )a 400, 6cr 1 3where ηrad;1, which is unphysical. In the simulation [21], ηrad≈0.08 for a0;400. The quantitativedisagreementmakes also not possible, on the basis of the predicted scaling only, to understandwhether theradiation ismostly due to electrons either remaining in the skin layer or returning towards the laser.The very limited nature of the estimate (4) for the conversion efficiency is due to several underlyingshortcomings, such as the neglect of self-consistent RF effects on the electronmotion, the absence of amoreprecise estimate of vx, and the inhomogeneity of the laser field in both the longitudinal and transverse directions.In the followingwe show that accounting for these effects, even if still in an approximatedway, leads to aconsiderably smaller growth of the conversion efficiency at high intensities than that given by (4) and thereforesubstantially improves the agreement with the simulations.3. Self-consistent electronmotionThemodelfirst introduced by Zeldovich [25] describes a stationary electronmotion in the field of a strongCPplanewavewith RF effects included self-consistently. Since RF allows absorption ofmomentum from the planewave, a drag force is exerted on the electron along the direction of wave propagation (x for definiteness). Thus, inorder to obtain a stationary solution an electric field Ed along x is introduced in themodel balancing theradiation drag. The complete EMfields are thus given byj j j j= = -( ) ( ) ( ) ( ) ( )t x E E E t x E EE B, , cos , sin , , 0, sin , cos . 7d L L L LIn the stationary regime an electronmoves along a circle in the ( )y z, plane and drifts along the x axis with aconstant velocity:j q j q j w= - - - = -( ) ( ( ) ( )) ( )t v v v t kxv , sin , cos , . 8x 0 0The phase shift θ is generated by theRFF.Neglecting the latter in the equations ofmotion gives θ=0. For ultra-relativistic particles, the RFF is given by [6]= -( ) ( )PcFv v, 9radrad2with the radiation powerw g= -⎜ ⎟⎛⎝⎞⎠( ) ( )Pe vcvcv231 10xrad2 202 432which differs from (2) by the replacement ga02 2 reflecting the fact that the circularmotion of the electron isnowdetermined jointly by the Lorentz and the RFFs. In the stationary regime the total force (with the centrifugalcomponent included) vanishes. Projecting this condition on the axes of cylindrical coordinates and assumingthat the value of the longitudinal electric field Ed is knownwe obtain three equationswhich determine the valuesof γ, θ and vx:q g- - =( ) ( )eE eE vcP vvcsin , 0, 11dLxx0rad 2q g- + =⎜ ⎟⎛⎝⎞⎠ ( ) ( )eEvcP vvc1 sin , 0, 12Lxxrad02g w q= ( )m v eE cos . 13L0In principle the systemof equations (11)–(13)might be applied to study themotion of electrons in the space-chargefield created by the ponderomotive force action, see examples e.g. in [31]where an approximate analyticdescription for the case of standingwaveswas developed.However, such space-charge field is highlyinhomogeneous, whichwould alreadymake an analytical estimate difficult. In addition, in the case underinvestigation the electron density is high enough for screening effects to be non-negligible: considering as anexample the contribution of returning electrons, as those located exactly at the plasma-vacuumboundary returntowards the incoming laser, the space-charge field is partially canceled so that the electrons filling in inner layerswill experience a lower force. A complete description of this scenariowould require to resolve the electronplasma dynamics with RFF included.Since our primary aim is to relate the radiation losses to an average value of vx determined by the laser-plasma dynamics, we take vx as a parameter in the system, and following Zeldovich [25]we solve equations (11)–(13) in the reference framemovingwith the instant velocity vx of the radiating electron. In the following, we usethe notations g¢, ¢v0, x¢, etc for valuesmeasured in this reference frame. Setting ¢ =v 0x and taking into accountthat g¢ 1, onemay safely put ¢ »v c0 . Eliminating the angle q¢ from equations (12), (13)we obtain an equationdetermining g x¢ ¢( )a, 0 (note that a0 is relativistically invariant) [25]:4New J. Phys. 21 (2019) 033009 SVPopruzhenko et al140g x g¢ + ¢ ¢ =( ) ( )a1 . 142 2 6 02For low intensities, ¢a a0 cr it gives g¢ = a0, as was used in [21]. In the opposite limit, ¢a a0 cr, the gamma-factor growsmuch slowerwith a0:gx¢ =¢⎛⎝⎜⎞⎠⎟ ( )a. 1501 4Equation (15), previously obtained in [32], corresponds to the limit inwhich the oscillation energy of theelectron in the EMfield equals the energy radiated per cycle. Remarkably, this single particle result corresponds,in ourmodel where collective effects enter via (3) for the number of radiating electrons, to a total conversion ofthe laser pulse energy into radiation from the target. In fact it follows from equations (14), (10) and (3) thath xgxg= ¢ ¢ =-+¢ ( )av cv c a11, 16xxrad4040so that ηrad→1 for ¢a a0 cr. Note that ξ is determined by the laserwavelengthmeasured in the laboratory frame,and the factor 1/2 is added to take into account that only half of the electrons radiate efficiently [21]. In section 5,we refine the calculation, so that the empirical factor is no longer needed.We compare predictions of ourmodel to the results of 3DPIC simulations (see [21] for the numerical set-updetails)which describe the interaction of a laser pulse with a plasma of thicknessD>10λwhereλ=0.8 μmcorresponding to a Ti:Sapphire laser and initial density = = ´ -n n90 1.55 10 cm .c0 23 3 The charge-tomassratio for ionswas takenZ/A=1/2. The supergaussian laser pulse is introduced via the time-dependentboundary condition at the plasma surface, x=0, as described in [21]w w= = + - -( ) ( ( ) ( )) ( )( ) ( )r x t a t ta y z, 0, cos sin e , 17r r ct r0 L0 4 4with = +r y z2 2 , r0=3.8λ, rL=3.0λ and duration (full-width-half-maximumof the intensity profile) 14.6fs. In our PIC calculationswe varied the laser amplitude in the interval = ¸a 300 7500 which corresponds tothe peak intensities ¸ ´( )3.8 23.7 1023 Wcm−2 and the total pulse energy (1.08–6.71) kJ. The numerical boxhad a [30×25×25]λ3 size, with 40 grid cells perλ in each direction and 125 particles per cell for each species.The simulationswere performed on ¸5000 10 000 cores of the JURECAClusterModule atNIC (Jülich,Germany).As is seen onfigure 1, the values of ηrad obtained from (16) and shown by thick black line for vx=0qualitatively reproduce the behavior of conversion efficiency extracted from the PIC simulation (shown bydiamonds) in thewhole interval of a0, although the absolute values appear considerably overestimated. Belowweidentify the sources of these differences and improve themodel by accounting for the respective effects. For thesake of comparison, three values of ηrad are also shown for a LP pulse at all other parameters identical to those oftheCP simulation. As it is seen, the conversion efficiency is considerably higher in the LP case and also reaches itssaturation at lower intensities. This is in line with previous observations [9, 23] and can be traced back to effect ofthemagnetic (v×B) force driving longitudinal electron oscillations duringwhich  -v cx .4. Effects of the longitudinal velocityAn analysis of the 3Ddistribution functions of the radiation power density ( )x r v, , x (calculated as = - ·n v Fe rad) and of the electron and ion density ne, i(x, r, vx) extracted from the PIC simulation shows thatmost of the emitted radiation comes from electrons having velocities vx>0, and located close to the recedingfront of the ion density. This is illustrated for the a0=500 case infigure 2where space–time plots in the (x, t)plane are shown for the radiation power and the particle densities at r=1λ, where the former has its radialmaximum. The density frontsmove in the forward directionwith average velocity;0.41c, in fair agreementwith the value =v c0.47HB given by equation (1). Small oscillations in the front position are visible incorrespondence of the generation of plasma bunches in the forward direction, as discussed in [30]. The powerdensity plot shows thatmost of the emission originates close to theHB front. Emission due to returningelectronswith velocity;−c is visible after =t T11 L, but its contribution to the total emitted power is small,presumably because of the low density in the returning jets (as seen on the ne(x, t) plot).As clearly seen from the plot of the x-integrated radiation power shown on the upper panel, spikes ofradiation occur in correspondance of the generation of plasma bunches. Such spikesmay be explained by theenhanced penetration of the laserfield into the plasma at these time instants. Since the spikes remain close to theHB front, no strongmodification of vx is correlatedwith them.Consistently with these observations, we assumethat on the average the radiating electronsmovewith velocity vx=vHB given by (1). In this way, we obtain aresult shownonfigure 1 by a thin black line. The account of the longitudinalmotion improves the agreement,5New J. Phys. 21 (2019) 033009 SVPopruzhenko et al141 EFFICIENCY OF RADIATION FRICTION LOSSESalthough analytically calculated values still exceed the PIC results by approximately 5 times at a0=400 and3 times at a0=750.5. Effects offield inhomogeneityFinally, we account for the attenuation of the laser field in the plasma and the dependence of the laser intensityon time and its radial distribution in the focal spot. The laser field amplitude a0 is not constant within theevanescence lengthℓs, but dropping down, leading to a considerable decrease of the ‘efficient’ value of a0entering equations (15) and (16). Figure 3 based on theHBmodel of [30] sketches the electron and the iondensity distributions along the propagation direction at the initial stage of the interactionwhen the electrons arepushed forward by light pressure, while the ions still remain immobile and homogeneously distributed insidethe plasma layer. Taking the electron density for x>d in the form= + - - -( ) ( ) ( )ℓ( )n x n n n e , 18e p x d0 0 0 swe replace a step distribution employed in [21] by a decaying exponent. Assuming that n np0 0 with np0 beingthemaximal density of electrons and n0 is the initial density equal to that of ions, we obtain for the electric fieldinside the layerp= - - -( ) ( ) ( )ℓ( )E x e n n l4 e 19p s x d0 0 swith themaximal valuep pº = = - »ℓ ℓ( ) ( ) ( )E E x d e n n en4 4 20d p s p s0 0 0achieved at the electron surface. Taking into account that l= ¢ℓn N a rp s x0 0 0 (3)we obtain for themaximallongitudinal fieldFigure 2. Space–time plots of the radiation power density ( )x t, (top, logarithmic scale, arbitrary units), ion density ni(x, t) (middle)and electron density ne(x, t) (bottom) all evaluated at r=1λ distance from the axis. Awhite curve on the upper panel shows theradiation power integrated over x.6New J. Phys. 21 (2019) 033009 SVPopruzhenko et al142x» ¢ ( )E E a3 , 21d cl 0where = = = ´E e r m c e 1.81 10cl 02 2 4 3 18 V cm−1 is the criticalfield of classical electrodynamics which is1/α=ħc/e2≈ 137 times greater than that ofQED= ( )E m ce. 22cr2 3Note that forλ;1 μm, Ed;Ecr at a0;(400ξ)−1≈1.6×105, according to (21), so that in this caseEd;EL;Ecr.Within the same approximation the local equilibrium condition for the electrons inside the layer requiresthat the laser field amplitude drops accordingly, = - - ℓ( ) ( ( ) )a x a x dexp s0 . Then the global equilibriumcondition for thewhole layer readsòh p-¢= »¥ℓ( ) ( ) ( ) ( )Ice n x E x x e n2 d 2 . 23Lde p srad202 2Herewe take into account that the intensity of reflected radiation in the reference frame co-movingwith theelectrons is (1−ηrad) ¢IL. The arial radiation power (intensity) isòwx¢ = ¢ = =¢¢¥ ℓ( ) ( ) ( ) ( )I P x v n x x e a ncf a, 0 d23, , 24dx ep srad2 20400where the power ¢ =( )P x v, 0x is given by (10) in the reference frame co-movingwith the electrons andg x¢ ¢( )a, 0 is expressed from equation (14), so thatòx g x=( ) ( ¯) ¯ ( )f a a a a,1, d . 25a5 04This gives the following equation for the conversion efficiencyh h x x=¢¢= - ¢ ¢( ) ( )IIa f a2 1 2 , , 26Lradradrad 030with an approximate solutionh x x» ¢ ¢( ) ( )a f a2 , , 27rad 03 0which employs the fact that ηrad/2=1 up to very high values of a0; in particular, at a0=750whichwas at thelimit of our numerical calculation, ηrad/2≈0.11. In the limiting cases of weak and strong fields the integral in(25) can be solved analytically givingh x»  ( )a a a25, 28rad 030 crandh   ¥ ( )a0.78, . 29rad 0Figure 3.Distribution of electron (red line) and ion (blue line) charge densities calculated using theHBmodel for the initial stage ofthe charge separation, before the ions startedmoving under the action of the longitudinal electric field (green curve).7New J. Phys. 21 (2019) 033009 SVPopruzhenko et al143 EFFICIENCY OF RADIATION FRICTION LOSSESThe latter number is obtained directly from (26), as the approximation ηrad/2=1 is no longer valid in thislimit. As is clearly seen from (28) in the limit of low intensities (which in practicemeans the laser field amplitudeup to a0;400), thefield attenuation inside the plasma layer leads to further suppression of the radiation lossesby a factor;0.4.Remarkably, when the attenuation of the laser field in the plasma is accounted for along (26), no extra 1/2factor introduced in [21] is required to fulfill the requirement nrad⩽ 1.Note that the above results remain ratherrobust with respect to a particularmodel for the electron density and field distribution in the emitting layer. Asour simulations show,while the distribution of the electron density follows qualitatively that offigure 3, the onefor ions appears by farmore complicated.However, the only feature of the ion density distributionwe practicallyuse for the analyticmodeling is that there is a significant number of ions to the left from the sharp electrondensity profile. These ions, independently of the spatial shape of their distribution, create a quasistatic field Edwhich equilibrates the laser light pressure. The electron density profile can also be chosen in different forms, andthat given by equation (18) is not unique. The only essential point is that both the electron density and the laserfield amplitude drop downon the lengthℓswhich considerably reduces the effective value of a0 particularly inthe low-field regime, a0<acr. The value of ℓs itself is also not of crucial significance as it enters the equations inthe formof = ℓN nx p s0 . Comparing the values of the areal densityNx calculated from (3) and extracted from thesimulationwe found a reasonably good agreement: for a0=400 the simulation and equation (3) give= ¸ ´ -( )N 1.3 1.5 10 cmxsim 19 2, and = ´ -N 1.1 10 cmxmodel 19 2 correspondingly; for a0=500 thesenumbers are = ¸ ´ -( )N 1.4 1.7 10 cmxsim 19 2, and = ´ -N 1.3 10 cmxmodel 19 2.A similar suppression effect emerges due to the laser amplitude dependence on the transverse coordinateand time. Assuming that the dimensionless laser amplitude in the focal waist possess axial symmetry=( ) ( ) ( )a r t a g r r ct r, , 30L0 0and integrating the radiation power over the transverse coordinate and timewe obtain that the functionx¢( )f a, 0 in (27) is replaced by the factoròòxr t x r t r tr t r t¢ =¢( ) ( ) ( ( ))( ) ( )S ag f a gg,, , , d d, d d, 310502where ρ=(r/r0)2 and τ=ct/rL. Finallyh x x= ¢ ¢( ) ( )a S a2 , 32rad 03 0apparently leading to additional suppression of the convergence efficiency. For the supergaussian pulse (17)usedin the PIC simulationsò òx p x¢ = G - - ¢( ) ( ) ( ( )) ( ) ( )S ayy yy yy yf a y,12 1 4dlndln, . 33y0 1 4 0111 102 242 13 4 0 21In the strongfield limit ~( )f a a1 3 so that the integrands in (31) are proportional one to another, leaving thelimit (29) unchanged. Instead, in theweak-field limit f≈1/5, which gives for (33) »( )S a a 2 50 cr 3 4 7 4,and consequently h x» ¢a0.20rad 03. The resulting dependence ηrad(a0) calculated for a supergaussian pulse (17)along (32) and (33) is shown onfigure 1 by a solid red line and demonstrates an impressive improvement of (16):in the interval of intensities a0=400÷ 800 the calculated values do not deviate from the PIC result bymorethan 20%.Residual discrepanciesmay largely be ascribed to the fact that (1) tends to overestimate the actualrecession velocity since complete reflection is assumed.Notice that radiation losses also contribute to decreasethe reflectivityR=1−ηrad and hence reduce the recession velocity, which is principlemay create a positivefeedback for the enhancement of radiation emission.However, since ηrad is quite smaller than unity, these effectsappear not to play a significant role.6. Extension of the classical regime of interaction towards higher intensitiesAlthough the radiation losses appear high compared to those in the ‘LS’ regime, their significant relativesuppression caused by the RFF leads to a specific freezing of the electron lateralmotion, so that the relativistic γ-factor growsmuch slower (15) than in the perturbative domain a0=acr where theRFF is negligible. This inturn shifts the border between the classical and the quantum regime of interaction to considerably higherintensities. The significance ofQED effects is determined by the value of the relativistically invariant quantumparameterc = - mn n( ) ( )em c F p , 343 428New J. Phys. 21 (2019) 033009 SVPopruzhenko et al144where mnF is the EMfield tensor and p ν is the four-momentum vector. The value of (34) can be easily expressedvia the parameters in the reference framemovingwith =v vx HB whereg j q j q» ¢ - ¢ - - ¢m ( ( ) ( )) ( )p mc 1, 0, sin , cos , 35(see equation (8)). Calculating the tensor Fμν for thefields (7) and taking into account equations (11)–(13), weobtain for (34)cax g= ¢ ¢( ) ( )32. 362In theweak field regime a0=acr this gives for vHB=c a quantumparameter c a x= ( ) a3 2 02, so thatχ;0.1already at a0≈200. Recent work has shown that quantumquenching of radiation lossesmay be alreadysignificant at suchmodest values ofχ [24].However, in our particular conditions, due to (a)RF induced suppression in the growth of g¢ (15) and (b)reduction of x¢with increasing of vHB a further increase in the laser intensity results in a very slow growth ofχstarting from a0;acr. In the strong field limit,  ¥a0 theHB velocity approaches the speed of light, and theparameterx xx¢ =-+ ( )v cv c a A11 2, 37HBHB 01 4where =A Zn m An mc e e p (see equation (1)). For parameters of our simulationA≈3×10−6. This results inthe asymptotic value of the quantumparametercax»¥⎛⎝⎜⎞⎠⎟ ( )aA32 2. 3802 1 4In the range = ¸a 200 8000 the value of c increases from0.092 to 0.436, and even for an ‘extreme’ amplitudeof a0=2000we obtainχ≈0.636, showing that the onset of a full radiation-dominated regime is prevented. Inaddition, with regards to the interaction geometry investigated in our case, these estimates neglect the screeningof the laserfield in the ‘skin’ layer (section 5) fromwhichmost of the radiation is generated. This allows us topredict that, for the specific interaction geometry of CPpulses and thick overdense targets, QED effects will bestrongly quenched compared to the case when the laser pulse and electron bunch counter-propagate or at leastthe longitudinal electron velocity vx;0 in the laboratory frame.7. ConclusionsIn conclusion, we have presented a self-consistent analyticmodel for the interaction of superintense CP laserpulses with thick plasma in theHB regime. The inclusion of the RFF along the lines of Zeldovichʼs work [25]allowed calculating the conversion efficiency of the laser energy into high frequency radiation in thewide rangeof intensities. After accounting the effects of (a) the globalHBmotion of the plasma and (b) of the laserfieldinhomogeneity in space and time, our result demonstrated a good quantitative agreement with the outcome ofthe PIC simulation.Note that despite of its analytic simplicity themodel is robust with respect to assumptions onthe particular shape of electron and ion density distributions in the radiating layer. The effect of the RFF, incombinationwith the factors (a) and (b), results in amuch slower (compared to predictionsmade in [21])increase of the conversion efficiencywith the laser intensity, so that η≈0.25 at IL=3×1024W cm−2.Consequently, the quantumparameter also grows only slowlywith increasing of the laser intensity, c ~ a01 4,whichmay lead to quantum effects not to dominate even at the highest intensities we considered. Thispredictionmay be tested by simulationswithQED effects included.AcknowledgmentsAuthors acknowledge fruitful discussions with SVBulanov, AMFedotov, EGGelfer, GKorn, VTTikhonchuck,and SWeber. SVP acknowledges support of theMEPhIAcademic Excellence Project (ContractNo. 02.a03.21.0005) and of the Russian Foundation for Basic Research throughGrantNo. 16-02-00963a. Thedevelopment of numerical algorithmswas supported byRussian Science Foundation throughGrantNo.16-11-10028.Numerical simulationswere performed using the computing resources granted by the John vonNeumann-Institut für Computing (ResearchCenter Jülich) under the projectHRO04.9New J. Phys. 21 (2019) 033009 SVPopruzhenko et al145 EFFICIENCY OF RADIATION FRICTION LOSSESORCID iDsSVPopruzhenko https://orcid.org/0000-0001-6676-7974TVLiseykina https://orcid.org/0000-0002-5070-3543AMacchi https://orcid.org/0000-0002-1835-2544References[1] MourouGA, TajimaT andBulanov SV 2006Optics in the relativistic regimeRev.Mod. Phys. 78 309–71[2] Di Piazza A,Müller C,HatsagortsyanKZ andKeitel CH2012 Extremely high-intensity laser interactions with fundamental quantumsystemsRev.Mod. Phys. 84 1177–228[3] NarozhnyN and Fedotov A 2015 Extreme light physicsContemp. Phys. 56 249–68[4] Sokolov I V 2009Renormalization of the Lorentz–Abraham–Dirac equation for radiation reaction force in classical electrodynamicsJ. 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Phys. 12 045013[29] Landau LD and Lifshitz EM1975TheClassical Theory of Fields 2nd edn (Amsterdam: Elsevier) ch78[30] Macchi A, Cattani F, Liseykina TV andCornolti F 2005 Laser acceleration of ion bunches at the front surface of overdense plasmasPhys. Rev. Lett. 94 165003[31] Kostyukov I Y andNerush EN2016Production and dynamics of positrons in ultrahigh intensity laser-foil interactions Phys. Plasmas23 093119[32] Bulanov S S, Esirkepov TZ, ThomasAGR, Koga JK andBulanov SV 2010 Schwinger limit attainability with extreme power lasersPhys. Rev. Lett. 105 22040710New J. Phys. 21 (2019) 033009 SVPopruzhenko et al146147 HE-DROPLETS IN INTENSE LASER FIELD5. He-Droplets in intense laser fieldThe list of publications:T.V. Liseykina, S. Pirner, D. Bauer, Relativistic Attosecond Electron Bunches from Laser-IlluminatedDroplets, Phys. Rev. Lett. 104, 092002 (2010)T.V. Liseykina, D. Bauer Plasma-Formation Dynamics in Intense Laser-Droplet Interaction, Phys.Rev. Lett. 110, 145003 (2013)P. Sperling, T. Liseykina, D. Bauer, R. Redmer, Time-resolved Thomson scattering on high-intensitylaser-produced hot dense helium plasmas, New J. Phys. 15, 025041 (2013)Relativistic Attosecond Electron Bunches from Laser-Illuminated DropletsT.V. Liseykina,1,2 S. Pirner,1 and D. Bauer1,31Max-Planck-Institut für Kernphysik, Postfach 103980, 69029 Heidelberg, Germany2Institute for Computational Technologies, SD-RAS, Novosibirsk, Russia3Institut für Physik, Universität Rostock, 18051 Rostock, Germany(Received 5 June 2009; published 4 March 2010)The generation of relativistic attosecond electron bunches is observed in three-dimensional, relativisticparticle-in-cell simulations of the interaction of intense laser light with droplets. The electron bunches areemitted under certain angles which depend on the ratios of droplet radius to wavelength and plasmafrequency to laser frequency. The mechanism behind the multi-MeVattosecond electron bunch generationis investigated using Mie theory. It is shown that the angular distribution and the high electron energies aredue to a parameter-sensitive, time-dependent local field enhancement at the droplet surface.DOI: 10.1103/PhysRevLett.104.095002 PACS numbers: 52.38.Kd, 42.25.Fx, 52.27.Ny, 52.65.RrThe understanding of the laser energy conversion intofast electrons and ions is of utmost importance for thedesign of efficient ‘‘tabletop’’ particle accelerators [1–3].The laser-plasma-based acceleration schemes discussed sofar in the pertinent literature may be divided into twogroups according to whether the plasma is underdense,i.e., the plasma frequency !p is smaller than the laserfrequency !, or vice versa. Wakefield accelerators andthe so-called ‘‘bubble regime’’ (see, e.g., [4–6]) fall intothe former category while the interaction of intense laserpulses with solid surfaces or thin foils (see, e.g., [7–11])belongs to the latter, overdense regime. Of particular in-terest are finite-size targets where fast particles cannotescape into the field-free bulk material but yet the densityof the accelerated particles may be sizable. In recent yearsthe nonrelativistic interaction of intense laser light withsmall, subwavelength-size clusters has been thoroughlyinvestigated [12]. In particular, the efficient absorption oflaser energy, leading to high charge states and thus tointense line emission as well as to energetic electronsand ions was studied. In such small clusters of radii R< , where is the laser wavelength and is the skindepth of the cluster plasma, the effect of the cluster on thepropagation of the laser pulse need not be taken intoaccount. Although the plasma, which is created via fieldionization on a subcycle time scale, is overdense, screeningof the cluster interior only occurs due to polarization butnot due to a skin effect. Technically speaking, the dipoleapproximation can be applied to the nonrelativisticallyintense laser field, Eðr; tÞ ’ EðtÞ. As a consequence, theelectron dynamics mainly occurs in the laser polarizationdirection while the v B force in laser propagation direc-tion and the influence of the scattered electromagnetic fieldon the particle dynamics can be safely neglected.The other extreme of intense laser-matter interaction isconstituted by targets of sizes much larger than a wave-length, e.g., a laser beam impinging on a solid surface. Theabsorption mechanisms in this case have also been exten-sively investigated [13]. It is well known, for instance, thatfor perpendicular incidence electrons are accelerated inlaser propagation direction (i.e., into the bulk material).In this Letter we will focus on electron acceleration inthe regime where R and are of the same order of magni-tude (i.e., rather droplets than clusters) and the laser inten-sity is relativistic, i.e., the ponderomotive energy Upexceeds the electron rest energy mc2. By changing thedroplet radius from R (small-cluster limit) to R (solid-surface limit) the emission angle of electrons isexpected to decrease from ¼ =2 to 0 with respect to thepropagation direction k̂. Moreover, the emission is ex-pected to occur in the ðÊ; k̂Þ plane for a linearly polarizedlaser field of amplitude Ê.For a sphere in a plane electromagnetic wave all anglesof incidence occur simultaneously. It is known from laser-plasma interaction studies that, depending on the plasmascale length and the laser intensity, there exists an optimalangle of incidence for the absorption of laser energy [14]:the steeper the plasma gradient, the more this optimal angleis pushed towards =2, i.e., grazing incidence. For suffi-ciently large scale lengths, resonance absorption (see, e.g.,[1]) can occur most efficiently under a certain optimalangle. Indeed, this effect, resulting in the electron emissionopposite to the incoming laser pulse, has been observed inexperiments with droplets and two-dimensional modelsimulations [15].For R> the self-consistent electromagnetic fieldneeds to be calculated. Numerically we do this by meansof three-dimensional, electromagnetic, relativistic particle-in-cell (PIC) simulations using a code which originatedfrom the study in [16]. In such PIC simulations the mean-field particle dynamics, the deformation of the target, andthe corresponding modification of the field are automati-cally taken into account. Analytically, the self-consistentelectromagnetic field around and inside a laser-illuminateddroplet of given dielectric permittivity " can be calculatedusing the corresponding solutions to Maxwell’s equationsPRL 104, 095002 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending5 MARCH 20100031-9007=10=104(9)=095002(4) 095002-1 2010 The American Physical Society149 ATTOSECOND ELECTRON BUNCHES FROM LASER-ILLUMINATED DROPLETSput forward by Clebsch, Lorenz, Mie, Debye, and others.The electromagnetic scattering by dielectric or metallicspheres is commonly called ‘‘Mie theory’’ [17]. Of course,the self-consistent particle dynamics far from equilibriumin very intense laser fields cannot be captured by a dielec-tric constant and thus is not included in Mie theory while itis included in the PIC simulations. However, we will showthat Mie theory is nevertheless capable of explaining theangular distributions observed in the PIC simulations andthe high electron energies exceeding the ponderomotiveenergy significantly.We start by presenting typical results from PIC simula-tions of the interaction of an intense plane wave laser pulsewith a preionized, cold He droplet. A spatial resolution ofx ¼ y ¼ z ¼ =100 and 64 particles per cell for bothelectrons and ions were used. Absorbing boundary condi-tions for the fields and for the particles were employed inpropagation direction, periodic ones for the other direc-tions. A linearly (in x direction) polarized 16-cyclesin2-laser pulse enters the numerical box through theboundary z ¼ 0 and propagates into the z > 0 half-space.The dimensionless vector potential amplitude a ¼jeÂ=mcj ¼ jeÊ=m!cj of the laser pulse ( ¼ 800 nm)was a ¼ 1, corresponding to a laser intensity I ’ 21018 W=cm2. The initial density of the He droplet wasne0 ¼ 22nc, with nc ¼ 1:8 1021 cm3. Figure 1 showsa snapshot of the electron isocontour surfaces correspond-ing to 1% of the initial electron droplet density. Electronbunches emitted each half-cycle under plus or minus acertain angle (with respect to the propagation axis z) inforward direction are clearly visible. The bunches aremainly confined to the plane of incidence [i.e., the ðx; zÞplane].The electron energy and density for the same He dropletin a four-times more intense laser pulse is presented inFig. 2. Figure 2(a) shows that the bunches consist ofelectrons with energies up to ’ 6 MeV. The ponderomo-tive energy Up¼mc2ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þa2=2p 1Þ is only ’0:37 MeV.As the spatial width of the bunches is much smaller than awavelength, the temporal bunch structure is already in theattosecond domain. Similar bunches have been observed inPIC simulations of the interaction of ultraintense laserpulses with solid surfaces under grazing incidence orwith plasma channels [18]. Such bunches have potentialapplications in attosecond electron diffraction experimentsor the generation of coherent short-wavelength radiationvia Thomson scattering.In the following we will explain the multi-MeVelectronenergies and their emission angles using Mie theory. Thelatter gives the analytical electromagnetic field configura-tion around and inside the (unperturbed) droplet in terms ofseries expansions over Legendre and Bessel functions [17].In the case of interest to us, the droplet is conducting (" ¼1!2p=!2), the skin depth ’ c=!p R, and the elec-tric field on the droplet surface is perpendicular to it andquickly decays inside the droplet. Electrons at the dropletsurface can be pulled out of the droplet or pushed inside,depending on the time they appear at the surface [7].Figure 3 shows the absolute value of the radial electricfield for an !2p=!2 ¼ 22 times overdense droplet at thedroplet surface versus angle and time for R ¼ =4 andR ¼ =2 according Mie theory. Note that in our Miecalculations the incident plane wave field is of the formEinc;xðz; tÞ ¼ Ê cosðkz!tÞ, i.e., has a constant ampli-tude. Figure 3 shows that Mie theory predicts certain timesand angles at which the electric field at the surface islargest. Moreover, these electric field maxima may exceedthe field amplitude Ê [by a factor of 3 in Fig. 3(a) and 2.3 inFig. 3(b)]. For the bigger droplet the optimal angle issmaller, which means that the maximum electric fieldoccurs more in propagation direction ¼ 0. Secondarylocal maxima are visible at ’ 0:45.The two panels on the right-hand side of Fig. 3 show theelectric vector field in the y ¼ 0 plane according to Mietheory at the ‘‘optimal’’ times when the electric field at thesurface is highest and points inwards (i.e., it pulls electronsoutwards). In Fig. 3(a), for instance, t ’ 0:275 cycles andthe largest field occurs under the angle ¼ 0:27. It isclear that under the same angle but x < 0 the electric fieldFIG. 1 (color online). Electron isocontour surfaces (1% of ne0)of a R ¼ =4 He droplet in a laser pulse of intensity 21018 W=cm2 at t ¼ 10 cycles.FIG. 2 (color online). Kinetic electron energy (a) anddensity (b) contour plots (in the y ¼ 0 plane) of the He dropletof Fig. 1 in a laser pulse of intensity 8 1018 W=cm2 at t ¼8 cycles. In the bunch ejected around the pulse maximum, themaximum electron energy is ’ 6 MeV, and 2 105 electronshave an energy >3 MeV.PRL 104, 095002 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending5 MARCH 2010095002-2150has the same absolute value but points outward. As aconsequence, electrons entering from inside the dropletinto such a field configuration under this optimal anglewill be efficiently accelerated outwards (inwards for x <0). Half a laser cycle later the situation reverses andelectrons at x < 0 will be accelerated outwards (inwardsfor x > 0). This explains the observation of alternatingelectron acceleration into two directions each laser cycle.Figure 4 shows angle-resolved ion and electron energyspectra obtained from the PIC simulations. The spectrawere taken at time t ¼ 9 cycles when the fast electronsare still inside the simulation box. The ions are hardly setinto motion at such an early time. However, the angulardistribution of the ions in Figs. 4(a) and 4(c) has alreadyimprinted on them the anisotropy due to the field distribu-tion at the droplet surface. In fact, the emission angles aremore easily inferred from the ion distributions than fromthe electron distributions in Figs. 4(b) and 4(d) because theelectrons change their direction as they move away fromthe droplet. This is the why in Figs. 4(b) and 4(d) theenergetic electrons are aligned stronger in the forwarddirection than the slow electrons. The emission anglesinferred from the PIC results in Fig. 4 are ¼ 0:27 forthe R ¼ =4 droplet and ¼ 0:14 for the R ¼ =2droplet with a secondary maximum at 0:4. These resultsare in excellent agreement with theMie results presented inFig. 3. We have performed a systematic study, comparingthe emission angles predicted by Mie theory with thosefrom the PIC calculations (a ¼ 2) for droplet radii between=8 and =2, and in all cases found very good agreement.The unexpectedly high electron energies can be ex-plained by the field enhancements at the droplet surface.As shown in Fig. 3(a) the field amplitude at the surface ofthe R ¼ =4 droplet is 3 times the incident field amplitude.We investigated the electron acceleration mechanism byfollowing trajectories of typical PIC electrons ending up ina bunch. We found that for, e.g., the parameters of Fig. 2the enhanced, radial field at the droplet surface acceleratesthe bunch electrons over a tenth of a wavelength already upto 90% c. Their subsequent relativistic dynamics is mainlydetermined by the incoming laser pulse. In particular, theelectron bunches remain well-confined in propagation di-rection as long as they are in the laser pulse. The overdensedroplet in essence serves as an electron injector with theenhanced surface field as an efficient preaccelerator. Wehave also performed test-particle calculations where weplaced electrons with zero initial velocity on the dropletsurface at various phases with respect to the analytical,time-dependent Mie field configuration. The maximumtθ/π(laser cycles)tθ/π(a)(b)x/ λz/λx/ λz/ λFIG. 3 (color). Absolute value of the radial electric field for a22 times overdense droplet at the droplet surface versus angle and time, as predicted by Mie theory for R ¼ =4 (a) andR ¼ =2 (b) in the y ¼ 0 plane. The color indicates the electricfield in units of the incident field strength Ê. The plots to theright show the electric vector field in the y ¼ 0 plane at the‘‘optimal’’ times (indicated by white, vertical lines in the leftplots) when the electric field at the droplet surface is highestand pointing inwards, i.e., t ¼ 0:275 cycles (a) and t ¼0:675 cycles (b), respectively.FIG. 4 (color online). Angle-resolved ion and electron kineticenergy spectra for the R ¼ =4 droplet [(a) ions and(b) electrons] and the R ¼ =2 droplet [(c) ions and(d) electrons] after t ¼ 9 cycles in a laser pulse of intensity 21018 W=cm2. The emission angle [indicated in (c)] wasdetermined as ¼ arctanðpx=pzÞ with p the momentum. Thecolor coding is proportional to the logarithm of the particlenumber.PRL 104, 095002 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending5 MARCH 2010095002-3151 ATTOSECOND ELECTRON BUNCHES FROM LASER-ILLUMINATED DROPLETSkinetic energies acquired by such test electrons are in goodagreement with the maximum energies observed in the PICsimulations.In the limit of small plasma droplets kR ¼ 2R= 1the maximum radial field at the droplet surface predictedby Mie theory isEðmaxÞr ¼ 3Ê !2p=!2 1!2p=!2 3 : (1)For !2p !2 one obtains EðmaxÞr =Ê ¼ 3, i.e., a threefoldfield enhancement, as observed in Fig. 3(a) (although kR ¼=2> 1 in this case). For, e.g., !p ¼ 2!, the predictedfield enhancement according (1) is EðmaxÞr =Ê ¼ 9, so thatone could argue that even more energetic electron bunchescan be produced for lower-density droplets. However, onlyfor a 1 do the PIC calculations reproduce the predic-tions of Mie theory as far as the field enhancements at thedroplet surface are concerned. This is because for relativ-istic laser intensities an only few-times overdense dropletquickly dissolves during the rising edge of the laser pulse.Before concluding we show the simulation result for thedroplet of Fig. 2 in an ultraintense laser field of intensity8 1020 W=cm2. At the plotting time (t ¼ 8 cycles, i.e.,at the maximum of the pulse) the highest electron energyobserved is ’50 MeV. However, if one follows the elec-tron bunch indicated by an arrow in Fig. 5(b), in time onefinds a final energy of ’130 MeV, i.e., 20 times the pon-deromotive energy. From Fig. 5(b) one infers that almostall electrons are removed from the droplet. Nevertheless,the angle under which the electron bunches are emitted iswell described by Mie theory. However, the energeticelectrons are not only located inside the electron bunchesanymore at such high intensities and for such small drop-lets, as is clearly visible in Fig. 5(a). In order to keep thewell-localized electron bunch structure with increasinglaser intensity, the droplet size (or density) should beincreased too.In summary, we showed that multi-MeV attosecondelectron bunches are produced when intense laser fieldsinteract with overdense droplets of diameters comparableto the laser wavelength. The attosecond electron bunchesare emitted each half-laser cycle under plus or minus acertain angle in the polarization plane. The preferred elec-tron emission angles and the high kinetic energies arise dueto local field enhancements at the droplet surface that canbe calculated using Mie theory. Relativistic attosecondelectron bunches may be used for the generation of short-wavelength radiation, time-resolved structural imaging, orplasma diagnostics.T. V. L. acknowledges support from the RFBR GrantsNo. 09-02-12201 and No. 09-02-01103. S. P. is supportedby the Studienstiftung des deutschen Volkes. The numeri-cal simulations were performed on the MPI-K LinuxCluster (Heidelberg).[1] Paul Gibbon, Short Pulse Laser Interactions with Matter:An Introduction (Imperial College Press, London, 2005).[2] Gerard A. Mourou, Toshiki Tajima, and Sergei V.Bulanov, Rev. Mod. Phys. 78, 309 (2006).[3] Yousef I. Salamin et al., Phys. Rep. 427, 41 (2006).[4] W. P. Leemans et al., Nature Phys. 2, 696 (2006).[5] S. P. D. Mangles et al., Nature (London) 431, 535 (2004);C. G. R. Geddes et al., ibid. 431, 538 (2004); J. Faureet al., ibid. 431, 541 (2004).[6] A. Pukhov and J. Meyer-ter-Vehn, Appl. Phys. B 74, 355(2002).[7] F. Brandl et al., Phys. Rev. Lett. 102, 195001 (2009).[8] S. Kar et al., Phys. Rev. Lett. 100, 225004 (2008).[9] J. Fuchs et al., Phys. Rev. Lett. 99, 015002 (2007).[10] J. S. Green et al., Phys. Rev. Lett. 100, 015003 (2008).[11] Andrea Macchi et al., Phys. Rev. Lett. 94, 165003 (2005);F. Pegoraro and S. V. Bulanov, Phys. Rev. Lett. 99, 065002(2007).[12] U. Saalmann, Ch. Siedschlag, and J.M. Rost, J. Phys. B 39R39 (2006); Th. Fennel et al., arXiv:0904.2706 [Rev.Mod. Phys. (to be published)].[13] F. Brunel, Phys. Rev. Lett. 59, 52 (1987); P. Gibbon andA. R. Bell, Phys. Rev. Lett. 68, 1535 (1992); P. Gibbon,Phys. Rev. Lett. 73, 664 (1994); S. C. Wilks et al., Phys.Rev. Lett. 69, 1383 (1992); W. L. Kruer and K. Estabrook,Phys. Fluids 28, 430 (1985); T.-Y. Bryan Yang et al., Phys.Plasmas 2, 3146 (1995); M. Cerchez et al., Phys. Rev.Lett. 100, 245001 (2008); P. Mulser, D. Bauer, and H.Ruhl, Phys. Rev. Lett. 101, 225002 (2008).[14] H. Ruhl and P. Mulser, Phys. Lett. A 205, 388 (1995).[15] X. Y. Peng et al., Phys. Rev. E 69, 026414 (2004); JunZheng et al., Phys. Plasmas 12, 113105 (2005); J. PlasmaPhys. 72, 1253 (2006).[16] V. A. Vshivkov et al., Phys. Plasmas 5, 2727 (1998).[17] Max Born and Emil Wolf, Principles of Optics(Cambridge University Press, Cambridge, England, 2003).[18] N. Naumova et al., Phys. Rev. Lett. 93, 195003 (2004).FIG. 5 (color online). Kinetic electron energy (a) and density(b), as in Fig. 2 but for a ¼ 20 (8 1020 W=cm2). The arrowin (b) indicates an electron bunch which, after the pulse, acquiredan energy of ’130 MeV.PRL 104, 095002 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending5 MARCH 2010095002-4152Plasma-Formation Dynamics in Intense Laser-Droplet InteractionT.V. Liseykina1,2,* and D. Bauer11Institut für Physik, Universität Rostock, 18051 Rostock, Germany2Institute of Computational Technologies SD RAS, 630090 Novosibirsk, Russia(Received 26 September 2012; published 2 April 2013)We study the ionization dynamics in intense laser-droplet interaction using three-dimensional, relativ-istic particle-in-cell simulations. Of particular interest is the laser intensity and frequency regime forwhich initially transparent, wavelength-sized targets are not homogeneously ionized. Instead, the chargedistribution changes both in space and in time on a subcycle scale. One may call this the extreme nonlinearMie-optics regime. We find that—despite the fact that the plasma created at the droplet surface isoverdense—oscillating electric fields may penetrate into the droplet under a certain angle, ionize, andpropagate in the just generated plasma. This effect can be attributed to the local field enhancements at thedroplet surface predicted by standard Mie theory. The penetration of the fields into the droplet leads to theformation of a highly inhomogeneous charge density distribution in the droplet interior, concentratedmostly in the polarization plane. We present a self-similar, exponential fit of the fractional ionizationdegree which depends only on a dimensionless combination of electric field amplitude, droplet radius, andplasma frequency with only a weak dependence on the laser frequency in the overdense regime.DOI: 10.1103/PhysRevLett.110.145003 PACS numbers: 52.35.Mw, 32.80.Fb, 52.50.Jm, 52.65.RrIntroduction.—Spherical, wavelength-sized, homoge-neous dielectric or metal objects in plane-wave electro-magnetic radiation fall into the realm of Mie theory [1]and are of fundamental importance in optics. Standard Mietheory provides the electromagnetic field configurationinside and outside a homogeneous sphere of a given dielec-tric constant, assuming an incoming plane wave. However,nowadays available short and intense laser pulses interact-ing with matter create plasmas on a sublaser period timescale [2,3]. These plasmas, in turn, modify the furtherpropagation of the laser pulse. We call this the extremenonlinear optics regime, and in the case of (initially)spherical targets, nonlinear Mie optics.As the laser field propagation is determined by theelectron density distribution and the plasma is generatedby ionization, the charge state and density distributions areexpected to be sensitive to the ionization dynamics. In fact,even the strongest present-day lasers cannot directly fullyionize heavier elements so that the assumption of a pre-formed homogeneous plasma throughout the target with agiven dielectric constant, may be inadequate. Furthermore,the skin effect may prevent the laser from penetratinginto targets that turn overdense in the course of ionizationso that, in general, a richly structured space- and time-dependent charge distribution develops [4]. Such interac-tions of laser pulses with rapidly self-generated plasmashave already found applications, e.g., as ‘‘plasma mirrors,’’which are routinely used to increase the pulse contrast forintense laser-matter experiments [5,6].One expects that the part of the laser pulse that isscattered off an overdense target will be mainly determinedby the ratio of laser to plasma frequency at the surfacewhereas possible inhomogeneities inside the target do notplay a role. In fact, standard Mie scattering theory assum-ing a homogeneous, overdense plasma sphere was usedto characterize rare gas clusters in recent experiments ona shot-to-shot basis [7,8]. However, inside such a sphere,standard Mie theory would predict electric fields onlywithin a narrow skin layer while, in this Letter, we willshow that a highly inhomogeneous and temporally chang-ing charge density distribution may be created in thedroplet interior. In order to probe such inhomogeneousstructures inside the target, laser frequencies greaterthan the plasma frequency that corresponds to the maxi-mum plasma density in the target should be used. Indeed,Thomson scattering of present-day short-wavelength free-electron laser radiation (from, e.g., DESY in Hamburg,LCLS in Stanford, or SACLA in Japan) is employed toprobe overdense plasmas [9–11].Molecular dynamics is a powerful tool that is widelyused to describe the ionization dynamics in small laser-driven clusters [12–14]. However, for wavelength-sizedtargets such as droplets the influence of the target on thepropagation of the incident electromagnetic wave needsto be taken into account self-consistently. This requires thesolution of Maxwell’s equations together with the equa-tions of motion for the charged particles. In the case ofweakly coupled plasmas the problem can be reduced tothe solution of the Vlasov-Maxwell system of equations,which is efficiently achieved using particle-in-cell (PIC)codes [2].Numerically, we study the nonlinear Mie domain bymeans of a 3D relativistic PIC simulation with ionizationincluded. The code UMKA originated from the study inRef. [15]. We show that in a certain laser intensity regimethe droplet target is neither fully ionized nor are chargesPRL 110, 145003 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending5 APRIL 20130031-9007=13=110(14)=145003(5) 145003-1 2013 American Physical Society153 PLASMA-FORMATION DYNAMICS IN LASER-DROPLET INTERACTIONonly created at the droplet surface. Instead, fields penetrateunder a characteristic angle into the droplet, ionizingatoms in the polarization plane and triggering plasmawaves that collide in a focal spot. We present results forthe fractional ionization degree at various laser intensities,wavelengths, and densities, that turn out to follow a uni-versal scaling law.Simulations.—The ionization of an ion with charge stateZ 1 and ionization potential I due to the electric field E isimplementedusing the tunneling ionization rate formula [16]wðEÞ ¼2EchjEj2n k2@mjEjEchexp 2Ech3jEj; (1)with k¼ffiffiffiffiffiffi2mIp@, Ech¼@2k3me , n ¼ ZffiffiffiffiIHIq. Here,m is the electronmass and IH is the ionization potential of atomic hydrogen.When an ionization event takes place a free electron at restis created at the position of the ion. The energy needed forionization is taken out of the field via an ‘‘ionization current’’jion parallel to the electric field at the ion location. The valueof jion is such that jion E is the work spent on ionizationper time step [17,18]. Energy conservation is accountedfor during the whole process; if the remaining field energyin a cell is insufficient for further ionization, this cell is notconsidered anymore during the current time step [19].We start by presenting typical results from PIC simula-tions of the interaction of an intense, plane-wave laserpulse with an initially neutral He droplet [20]. A spatialresolution of x ¼ y ¼ z ¼ =100, 125 macroionsand 250 macroelectrons per cell were used. Absorbingboundary conditions for the fields and particles wereemployed in the propagation direction, periodic ones forthe other directions. The size of the simulation box wasalways chosen big enough to rule out any boundary effectson the observables of interest due to reflections or particlesleaving the box. A linearly (in the y direction) polarized10-cycle sin2-laser pulse of carrier frequency !0 enters thenumerical box through the boundary x ¼ 0 and propagatesinto the region x > 0. The dimensionless vector potentialamplitude a ¼ jeÂ=mcj ¼ jeÊ=m!0cj was 0.5, corre-sponding to a laser intensity I ’ 5:2 1017 W=cm2, thewavelength ¼ 2c=!0 was 800 nm (i.e., for a laserperiod TL ¼ 2:66 fs). The density of the 2R ¼ 4 ¼3:2 m diameter He-droplet was ¼ 0:14 g=cm3.If the droplet was completely preionized such a densitywould correspond to an electron density ne0 ¼ 24ncr,where ncr ¼ 1:8 1021 cm3 is the critical density for800-nm wavelength light. The droplet center was locatedat x ¼ 4, y ¼ z ¼ 10. In the simulations presented inthis Letter impact ionization was ‘‘switched-off.’’ Test runsshowed that for the intensities considered the effect ofcollisional ionization during the laser pulse is much smallerthan that of field ionization. Moreover, self-consistencyrequires that collisional absorption is taken into accountalong with the collisional ionization, as in the PIC codesdescribed in Refs. [21–23]. A recently introduced micro-scopic PIC code [24,25] bridges the gap between PIC andmolecular dynamics and is also capable of incorporatingcollisional ionization and collisional absorption, albeit sofar only for smaller targets.Results.—Figure 1 shows snapshots of the volume dis-tribution of electron and He2þ densities. In the beginningthe droplet is nonionized and thus transparent for the lead-ing part of the laser pulse. Later, as the field strength ofthe laser pulse increases in magnitude, ionization becomesmore efficient, and an overdense plasma is generated rap-idly on the droplet surface as the pulse propagates over it,leading finally to almost full ionization of a thin surfacelayer. Moreover, we observe that a highly inhomogeneousdensity distribution inside the droplet is formed, concen-trated mostly in the polarization plane. In particular, thereseems to be a focal spot [blue area in the polarization planein Figs. 1(b) and 1(c)]. The fractional ionization degreeIr ¼ 34R3ne0RneðrÞd3r of the droplet at the end of theinteraction is ’ 35%.He1þ, He2þ, and electron densities in the two perpen-dicular planes (k̂, Ê) and (k̂, B̂) at times t ¼ 7TL andFIG. 1 (color online). Electron density in the beginning (a) and at the end (b) of the interaction with the laser pulse. He2þ density (c)at the end of the interaction. For better visualization of the droplet interior a quarter of it was cut out. Laser and droplet parameters aregiven in the text. The laser propagation direction is indicated by an arrow in each panel.PRL 110, 145003 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending5 APRIL 2013145003-2154t ¼ 12TL are plotted in Fig. 2. Charge density builds upinside the droplet, starting from a certain region on thedroplet surface, most clearly seen in the He1þ-plot inFig. 2(a). At the later time in Fig. 2(b) the charge densityfronts merged already, creating the focal spot of He2þdensity. Comparing the charge densities in the two planesshows that the ionization dynamics mainly takes place inthe polarization plane (k̂, Ê). The corresponding distribu-tions of the electric field components (longitudinal Ex andin the polarization direction of the incident laser electricfield Ey) are presented in Fig. 3. It is seen that an oscillatingelectric field penetrates into the droplet, where in Fig. 2 thecharge density is created. This is another interesting ex-ample for an electric field propagating in a plasma that iscreated by it in the first place [26]. More precisely, it turnsout that the Mie-enhanced field at the surface (discussed inthe subsequent paragraph) first results in a deeper penetra-tion and thus more efficient ionization. In addition, theelectric field at the surface oscillates and thereby triggersplasma waves which propagate inwards up to the regionwhere plasma has not yet been created. The electric field ofthe plasma wave then ionizes further, which results in anionization front propagating inwards.Mie field enhancement.—We attribute the fact that thefield and ionization front dynamics originate from a sur-face region under a certain angle * =2 (with respectto k̂) to a local, time-dependent field enhancement on thedroplet surface. In order to corroborate this statement, weshow in Fig. 4 the radial electric field along the dropletsurface in the polarization plane vs time and as obtainedfrom the PIC simulation (a) and according to Mie theory[1] (b). Standard Mie theory is formulated for plane inci-dent waves. As Mie theory is linear we synthesized ourpulse via spectral decomposition and added the fieldscoherently. In the Mie simulation the droplet is assumedto be homogeneous and conducting, with a dielectric con-stant ¼ 1 ne0=ncr. Under such conditions Mie theorypredicts in the strongly overdense regime (where the skindepth is e ’ c=!p R with !p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2ne0=me0ptheelectron plasma frequency) that the electric field on thedroplet surface is perpendicular to it. In Fig. 4 the time axishas been shifted such that t ¼ 0 corresponds to the momentwhen the maximum of the incident laser pulse arrived atthe droplet center. Both the PIC and Mie result predictmaxima of the electric field on the droplet surface forangles = 2 ½0:4; 0:7. The slight disagreement in thefield distributions in the forward direction (small ) isdue to the fact that in the Mie calculation the droplet isassumed conducting (i.e., completely ionized) from thevery beginning, whereas in the PIC simulation there isnot yet plasma at the rear side of the droplet (see Fig. 2).The field enhancement predicted by Mie theory is inexcellent agreement with the PIC results (’1:9 times theincident field).Focused plasma waves.—The propagation direction ofthe field structures inside the droplet seen in Fig. 3(a) istilted with respect to k̂, leading to the observed focusingeffect. In order to interpret correctly these structures, weproject the field components inside the upper half of thedroplet onto , E ¼ Ex cos’þ Ey sin’, En¼Exsin’þEycos’, with ’ the angle between and k̂ [see Fig. 5(a)].FIG. 3 (color online). Electric fields (Ex top, Ey bottom) in twoperpendicular planes (k̂, Ê) (left) and (k̂, B̂) (right) at t ¼ 7TL(a) and t ¼ 12TL (b).FIG. 4 (color online). Radial electric field along the surface inthe polarization plane vs angle and time, obtained from the PICsimulation (a) and as predicted by Mie theory (b). The horizontalblack lines indicate the angle at which the electric field at thedroplet surface is highest.FIG. 2 (color online). Electron (bottom), He1þ (middle) andHe2þ (top) density in two perpendicular planes (k̂, Ê) (left) and(k̂, B̂) (right) at t ¼ 7TL (a) and t ¼ 12TL (b). Laser and dropletparameters are given in the text.PRL 110, 145003 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending5 APRIL 2013145003-3155 PLASMA-FORMATION DYNAMICS IN LASER-DROPLET INTERACTIONThe resulting field distributions for E and En are shownin Fig. 5(b). As the values of E are several times biggerthan the values of En, we identify the field structures as alongitudinal plasma wave. The necessary matching of theplasma wave to the electromagnetic field at the dropletsurface results in the tilt of with respect to k̂ because thephase velocity of the plasma wave is smaller than c. Thefrequency spectra of the electric field at two points x ¼ 3,y ¼ 11:5, and x ¼ 4:5, y ¼ 11 inside the dropletare shown in the Fig. 5(c). They peak at the frequencyapproximately equal to the local plasma frequency, whichmay be estimated from the plot of the local density vs timein Fig. 5(d).Fractional ionization degree.—Figure 6 collects allour simulation results for the final fractional ionizationdegree Ir. Introducing the dimensionless parameter ¼a=ðR=2eÞ, it turns out that for all the various clustersizes R> e, densities !2p !20, laser intensities andwavelengths simulated, Ir is well described by Ir ’ 1expð Þ. In our case of He we find ¼ 1560. Note thatthe species dependence only enters via the ionizationpotentials I in the tunneling ionization rate formula (1).Inserting the expression for the collisionless skin depthe¼c=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2p!20qwe obtain ¼eÊ=½2mRð!2p!20Þ’eÊ=ð2mR!2pÞ, showing that there is only a weakdependence on the laser frequency. Indeed, for tunnelingionization the electric field amplitude matters, not the laserfrequency. For small laser intensity and sufficiently bigdroplets, when only the thin skin layer on the dropletsurface gets ionized, one expects Ir¼4R2e4R3=3 R1. In theopposite limit of very high laser intensity or small dropletscomplete ionization Ir ¼ 1 is expected. Both limiting casesare contained in our formula. The chosen exponentialinterpolation between those two limiting cases matchesthe simulation results for the fractional ionization degreevery well.Summary.—A strong near-infrared or optical laser pulseinteracting with an initially neutral, wavelength-sized He-droplet may generate a charge density distribution thatneither is homogeneous throughout the droplet nor createdonly within a thin skin layer at the surface. Instead, electricfields may penetrate into the droplet interior for certainangles of incidence predicted by standard Mie theory.However, the time-dependent field and density distribu-tions inside the target are not accessible to standard Mietheory but fall into the realm of extreme nonlinear optics.The field penetration causes ionization inside the droplet,mainly confined to the polarization plane. The resultinginhomogeneous charge distribution may be probed viascattering of short-wavelength radiation and should betaken into account when studying typical laser-plasmainteraction applications such as ion acceleration or x-rayradiation from recombination in ionized droplets. A parti-cularly high abundance of He2þ is observed where theionization fronts and the trailing plasma waves collide.The fractional ionization degrees for various droplet andlaser parameters are found to be in good agreement with aself-similar exponential fit. At higher laser intensities aqualitatively similar ionization dynamics is expected forhigher-Z materials as well.FIG. 5 (color online). (a) The n ¼ 2:2ncr level of the electrondensity in the droplet central plane z ¼ 10 for successive times(t ¼ 5, 5.4, 6.1, 6.8, 7.4, 8.1, 8.8, 9.4, 10:1TL). The straight lineindicates the projection direction . (b) The distributions of theelectric field component E parallel and En perpendicular to att ¼ 8TL. (c) Frequency spectra of the electric field E at thepoints r1 ¼ ð3; 11:5; 10Þ and r2 ¼ ð4:5; 11; 10Þ inside the droplet.(d)ffiffiffiffiffiffiffiffiffiffiffiffiffiffine=ncrpvs time at the points r1 and r2.FIG. 6 (color online). Universal curve Ir ¼ 1 expð Þ(dotted) with ¼ a=ðR=2eÞ ¼ eÊ=½2mRð!2p !20Þ forthe final fractional ionization degree of the He droplet afterthe interaction with a plane-wave laser pulse. The total laserpulse duration in all cases was 26 fs. The symbols for different aare indicated in the plot. For some of them, runs with differentdensity or laser wavelength have been performed, as indicateddirectly by arrows. The numerical values for a, R=, ne0=ncr,and Ir are given in the Supplemental Material [20].PRL 110, 145003 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending5 APRIL 2013145003-4156This work was supported by the DFG within the SFB652. PIC simulations were performed using the computingresources granted by the John von Neumann-Institut fürComputing (Research Center Jülich) under the projectHRO01. We thank Professor Thomas Fennel for providingthe result of the Mie calculation.*tatyana.liseykina@uni-rostock.de[1] M. Born and E. Wolf, Principles of Optics (CambridgeUniversity Press, Cambridge, England, 2003).[2] P. Gibbon, Short Pulse Laser Interactions with Matter, anIntroduction (Imperial College Press, London 2005).[3] P. Mulser and D. Bauer, High-Power Laser-MatterInteraction (Springer, Berlin, Heidelberg, 2010).[4] E. C. Jarque, F. Cornolti, and A. Macchi, J. Phys. B 33, 1(2000).[5] H. C. Kapteyn, M.M. Murnane, A. Szoke, and R.W.Falcone, Opt. Lett. 16, 490 (1991).[6] C. Thaury, F. Quéré, J.-P. Geindre, A. Levy, T. Ceccotti,P. Monot, M. Bougeard, F. Réau, P. d’Oliveira, P. Audebert,R. Marjoribanks, and Ph. Martin, Nat. Phys. 3, 424 (2007).[7] C. Bostedt, E. Eremina, D. Rupp, M. Adolph, H. Thomas,M. Hoener, A. 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Mikaberidze, U. Saalmann, and J.M. Rost, Phys. Rev.Lett. 102, 128102 (2009).[15] V. A. Vshivkov, N.M. Naumova, F. Pegoraro, and S. V.Bulanov, Phys. Plasmas 5, 2727 (1998).[16] V. S. Popov, Phys. Usp. 47, 855 (2004).[17] S. C. Rae and K. Burnett, Phys. Rev. A 46, 2077(1992).[18] P. Mulser, F. Cornolti, and D. Bauer, Phys. Plasmas 5,4466 (1998).[19] A. J. Kemp, Y. Sentoku, T. Cowan, J. Fuchs, and H. Ruhl,Phys. Plasmas 11, L69 (2004).[20] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.110.145003 for discu-ssion of the case of focused pulses.[21] Y. Sentoku and A. J. Kemp, J. Comput. Phys. 227, 6846(2008).[22] R. A. Fonseca, L. O. Silva, F. S. Tsung, V. K. Decyk,W. Lu, C. Ren, W. B. Mori, S. Deng, S. Lee, T.Katsouleas, and J. C. Adam, Computational Science -ICCS 2002, Lecture Notes in Computer ScienceVol. 2331 (Springer, Berlin, 2002), p. 342.[23] E. Lefebvre, N. Cochet, S. Fritzler, V. Malka, M.-M.Aleonard, J.-F. Chemin, S. Darbon, L. Disdier, J. Faure,A. Fedotoff, O. Landoas, G. Malka, V. Meot, P. Morel,M. R. Le Gloahec, A. Rouyer, Ch. Rubbelynck, V.Tikhonchuk, R. Wrobel, P. Audebert, and C. Rousseaux,Nucl. Fusion 43, 629 (2003).[24] Ch. Varin, Ch. Peltz, Th. Brabec, and Th. Fennel, Phys.Rev. Lett. 108, 175007 (2012).[25] C. Peltz, C. Varin, T. Brabec, and T. Fennel, New J. Phys.14, 065011 (2012).[26] A. Debayle and V. T. Tikhonchuk, Phys. Plasmas 14,073104 (2007).PRL 110, 145003 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending5 APRIL 2013145003-5157 PLASMA-FORMATION DYNAMICS IN LASER-DROPLET INTERACTIONTime-resolved Thomson scattering on high-intensitylaser-produced hot dense helium plasmasP Sperling1, T Liseykina, D Bauer and R RedmerInstitut für Physik, Universität Rostock, D-18051 Rostock, GermanyE-mail: philipp.sperling@uni-rostock.deNew Journal of Physics 15 (2013) 025041 (13pp)Received 31 August 2012Published 27 February 2013Online at http://www.njp.org/doi:10.1088/1367-2630/15/2/025041Abstract. The introduction of brilliant free-electron lasers enables newpump–probe experiments to characterize warm and hot dense matter states, i.e.systems at solid-like densities and temperatures of one to several hundred eV.Such extreme conditions are relevant for high-energy density studies such as,e.g., in planetary physics and inertial confinement fusion. We consider herea liquid helium jet pumped with a high-intensity optical short-pulse laserthat is subsequently probed with brilliant soft x-ray radiation. The opticalshort-pulse laser generates a strongly inhomogeneous helium plasma whichis characterized with particle-in-cell simulations. We derive the respectiveThomson scattering spectrum based on the Born–Mermin approximation for thedynamic structure factor considering the full density and temperature-dependentThomson scattering cross section throughout the target. We observe plasmonmodes that are generated in the interior of the target and study their temporalevolution. Such pump–probe experiments are promising tools to measure theimportant plasma parameters density and temperature. The method describedhere can be applied to various pump–probe scenarios by combining opticallasers, soft x-rays and hard x-ray sources.1 Author to whom any correspondence should be addressed.Content from this work may be used under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the titleof the work, journal citation and DOI.New Journal of Physics 15 (2013) 0250411367-2630/13/025041+13$33.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft159 THOMSON SCATTERING ON LASER-PRODUCED HE-PLASMA2Contents1. Introduction 22. Theory for the dynamic structure factor 33. Plasma generation in helium droplets 53.1. A pump laser with an intensity of 1015 W cm−2 . . . . . . . . . . . . . . . . . 53.2. A pump laser with an intensity of 1018 W cm−2 . . . . . . . . . . . . . . . . . 63.3. A pump laser with an intensity of 1019 W cm−2 . . . . . . . . . . . . . . . . . 84. Thomson scattering 95. Conclusions 11Acknowledgments 12References 121. IntroductionMatter at high energy densities occurs in extreme conditions of pressure and/or temperature asrelevant for astrophysics, e.g. in the deep interior of planets, in brown dwarfs or in stars [1–3].Another important example is the concept of intertial confinement fusion, which relies on thefull understanding and control of cryogenic deuterium-tritium capsules driven to high energydensities by powerful lasers as used at the National Ignition Facility in Livermore [4] or, in thenear future, at the Laser Megajoule near Bordeaux; for a review, see [5].Such extreme states of matter can be generated using various techniques. For instance,high-energy particle beams (e.g. heavy-ion beams at GSI Darmstadt) or intense short-wavelength radiation from free electron lasers (FELs, e.g. FLASH Hamburg, LCLS Stanford,Elettra Trieste and SACLA Kobe), which are brilliant soft or hard x-ray radiation sources, canbe used to heat targets volumetrically. In contrast, optical lasers penetrate only into a thin surfacelayer (skin depth) of the target. The solid-density plasma that is generated due to rapid tunnelingionization acts as a mirror for optical wavelengths. The electrons can gain relativistic energiesin the laser pulse and induce important processes in the plasma such as impact ionization andbremsstrahlung [6–8]. With the planned installation of new high-energy and ultra-short-pulselaser facilities such as the Extreme Light Infrastructure, intensities up to 1024 W cm−2 will beaccessible in the near future. These intensities enable the production of hot dense matter (HDM)with densities close to solid density up to compressed matter well above solid density, but withelectron temperatures Te above 100 eV and hence above the warm dense matter regime.The investigation of such plasmas requires probe lasers that operate at frequencies higherthan the plasma frequency ω2pe = nee2/(ε0me) of the free-electron subsystem, with the free-electron density ne and the electron mass me. Therefore, efficient x-ray sources with highbrightness are inevitable for probing plasmas with densities at solid or even higher densities.Energetic optical lasers such as Omega (Rochester), Gekko (Osaka), Titan (Livermore), Vulcan(RAL) or Phelix (Darmstadt) can be used to generate intense but incoherent x-ray radiation.Alternatively, FELs provide brilliant and coherent radiation for such purposes. The FLASHin Hamburg [10, 11] operates in the soft x-ray region, while the LCLS in Stanford and thefuture European XFEL in Hamburg yield hard x-ray radiation. Simultaneously, x-ray Thomsonscattering is a promising tool for the diagnostics of dense strongly correlated plasmas [9] inthe HDM region. Therefore, this technique is already implemented at the Matter in ExtremeNew Journal of Physics 15 (2013) 025041 (http://www.njp.org/)1603Conditions (MEC) instrument at LCLS and will be installed at the high energy density (HED)matter experiments instrument at the future XFEL in Hamburg.Collective x-ray Thomson scattering experiments yield information on the density andtemperature of dense plasmas [12]. For a homogeneous density and temperature profilethroughout the target, these parameters can be determined directly from the plasmon dispersionand the ratio of the plasmon amplitudes via the detailed balance relation [13]. For laserfrequencies below the plasma frequency for solid targets, i.e. optical lasers and energiesof about 25–30 eV accessible with FLASH, the target is overdense and the absorption islimited to the skin depth, whereas the excitation by high-frequency FEL radiation generatessmoother gradients [14]. The scattering signal represents an average of the local density-and temperature-dependent scattering cross sections weighted with the respective density andtemperature profiles [15], which are generated not only by the pump pulse but also by theprobe pulse if intense x-ray radiation pulses are used as in the case of the FLASH and LCLSfacilities. Therefore, a consistent treatment of the light–matter interaction within pump–probeexperiments is crucial for the determination of the plasma density and temperature, their profilesthroughout the target and their temporal evolution. Furthermore, the electron–ion equilibrationrate can be extracted from the Thomson scattering spectrum by varying the time delay betweenthe pump and probe pulses.The derivation of an inhomogeneous Thomson scattering signal weighted with therespective density and temperature profiles was shown earlier by Thiele et al [16], whoinvestigated the averaged Thomson scattering signal of an inhomogeneous hydrogen plasmaproduced by an optical laser of wavelength λ = 800 nm and intensity I = 1015 W cm−2. Thetemporal evolution of a set of plasmon pairs characterizing different regions of the dropletcould be monitored. These plasmon pairs stem from the warm dense droplet front generatedby the optical laser, whereas a non-collective scattering signal representing the cold interior ofthe droplet was found to be caused by the FEL.The goal of the present paper is to demonstrate the capacity of x-ray Thomson scatteringexperiments for HED studies. Furthermore, the role of inhomogeneities that cannot be avoidedin laser–matter interaction on ultra-short time scales is investigated by using small heliumdroplets as the target material. For this purpose, we study here the interaction of helium dropletswith a high-intensity short-pulse laser at medium to high intensities of I = 1015, 1018 and1019 W cm−2, which generates HDM in different regimes. The x-ray Thomson scattering spectraare calculated in order to test whether the corresponding plasma parameters can be extractedfrom them.2. Theory for the dynamic structure factorWe start with the scattered power per solid angle d = sin θ dθ dϕ and per unit frequencyinterval dω which is experimentally accessible and given by the following expression [15]:d2 Pscddω=σTAradkfki∫∞−∞dω′2πG1ω(ω − ω′)∫dr l(r)See(k, ω′; T (r), n(r))ne(r). (1)Here, σT = 6.65 × 10−24 cm2 is the Thomson scattering cross-section, ki and kf are the initial andfinal photon wavenumbers, and the energy and momentum transfer are given by 1E = h̄ω =h̄ωf − h̄ωi and h̄k = h̄kf − h̄ki. The central quantity for the determination of the scattering signalNew Journal of Physics 15 (2013) 025041 (http://www.njp.org/)161 THOMSON SCATTERING ON LASER-PRODUCED HE-PLASMA4is the dynamic structure factor (DSF) See(k, ω), which can be calculated for given profiles ofelectron density ne(r), electron temperature Te(r), ion density ni(r) and ion temperature Ti(r),i.e. for the general case of an inhomogeneous target. The momentum transfer is related to thescattering angle θ in the limit h̄ω h̄ω0 according to k = 4πsin(θ/2)/λ0, with λ0 being theprobe wavelength. l(r) is the r-dependent power density of the probe beam taking into accountabsorption in the and target, Arad is the irradiated surface of the target. The DSF has to beconvoluted with the instrumental function G1ω(ω) that models the spectrometer’s finite spectralresolution as well as the probe’s spectral bandwidth. Usually, a normalized Gaussian distributionis employed with the full-width at half-maximum (FWHM) 1ω.The DSF can be written in terms of free–free, bound–free and bound–bound correlationsas proposed by Chihara [17, 18]:See(k, ω) = ZfreeS0ee(k, ω) + | fi(k) + q(k)|2Sii(k, ω) + Zc∫∞−∞dω′Sc(k, ω)Ss(k, ω − ω′). (2)Here, Zfree = ne/ni is the ionization degree of the plasma and Zc is the averaged number of coreelectrons. The three terms in equation (2) are discussed below.In this paper, we consider the contribution of free electrons S0ee(k, ω) (the first term inequation (2)), which is connected with the longitudinal dielectric function ε(k, ω) via thefluctuation–dissipation theoremS0ee(k, ω) = −ε0h̄k2πe2neIm ε−1(k, ω)1 − exp(−h̄ωkBTe) . (3)Considering free electrons without interactions the dielectric function is given by the randomphase approximation (RPA) for the one-component plasma. Including interactions between theparticles in the plasma via the dynamic electron–ion collision frequency ν(ω) [19], the moregeneral approach of Mermin [20] can be applied to the dielectric function, which then readsεM(k, ω)− 1 =(1 + i ν(ω)ω)[εRPA(k, ω + iν(ω)) − 1]1 + i ν(ω)ωεRPA(k,ω+iν(ω))−1εRPA(k,0)−1. (4)Calculating the electron–ion collision frequency in Born approximation defines theBorn–Mermin approximation (BMA) [13, 21–23].A further analysis of the DSF for free electrons and the Thomson scattering process on freeelectrons is possible via the scattering parameter α = κe/k, which relates the inverse screeninglength κe (given below) to the wavenumber k. For α < 1 the scattering is non-collective andwe can investigate short-range correlations, while long-range correlations are relevant forcollective scattering (α > 1). In the case of long-range correlations, the DSF S0ee(k, ω) showstwo particularly pronounced side maxima which are located symmetrically relative to the centralion feature described by the second term of equation (2). These peaks are directly related to thefree electron density via the plasmon frequency [12, 13, 21].For the calculation of the second term in the Chihara formula (2) which characterizesthe scattering on bound electrons, we use the atomic form factor fi(k) [24] and the simpleDebye–Hückel ion–ion structure factor for point charges [25]Sii(k) =k2 + κ2ek2 + κ2i + κ2e(5)New Journal of Physics 15 (2013) 025041 (http://www.njp.org/)1625Figure 1. Illustration of the three different spatial planes relative to the pumping,Gaussian-shaped laser pulse.with the inverse screening length κc =√e2cnc/(ε0kBTc) for species c = e (electrons) and c = i(ions). In the Debye–Hückel picture, the screening cloud can be given with the electron–ionstructure factor by q(k) =√ZfreeSei(k)/Sii(k) = Zfreeκ2e /(k2 + κ2e ).For the last term in equation (2), the contribution of bound–free transitions Sc, we usethe formalism of Schumacher et al [24, 26]. This part describes Raman-type transitions ofinner shell electrons to the continuum which are modulated by the ion motion contained inSs(k, ω) [27]. Here, this contribution is to be neglected for the relevant energy range of −10 to10 eV, where the Thomsons scattering signal of free electrons can be obtained.3. Plasma generation in helium dropletsAs a proposal for future experiments on inhomogeneous HDM states, we describe in this sectionthe interaction between cryogenic helium droplets and a high-intensity short-pulse optical pumplaser with different intensities. We consider for this purpose helium droplets of d = 6.4 µmdiameter at an initial temperature Tini = 20 K and density n = 2.2 × 1022 cm−3. The profiles ofelectron temperature and density generated in the strongly inhomogeneous helium plasma aresimulated with a particle-in-cell (PIC) code [28], where field ionization processes are includedvia a tunnel ionization rate [29]. The setup of the pump–probe experiment on the helium dropletis illustrated in figure 1.3.1. A pump laser with an intensity of 1015 W cm−2First, we consider a linearly polarized optical laser pulse of wavelength λ = 800 nm, pulseduration tFWHM = 30 fs, energy E = 3.2 µJ and focal spot diameter dFWHM = 3 µm whichirradiates a helium droplet. We apply the corresponding intensity of 1015 W cm−2 in order tocompare with the previous results for hydrogen given by Thiele et al [16]. Note that we use herea three orders of magnitude smaller laser energy than that in [16]. To investigate the interactionof the optical laser with the helium droplet, a PIC simulation in a cubic box of 8 × 8 × 8 µm3size was performed for a duration up to t = 400 fs after the maximum of the optical pumplaser pulse. Such long simulation times are necessary to obtain an equilibrated system for thetemperature determination. We used 500 × 500 × 500 grid cells with 64 heavy particles each.New Journal of Physics 15 (2013) 025041 (http://www.njp.org/)163 THOMSON SCATTERING ON LASER-PRODUCED HE-PLASMA6 x [µm] y [µm]024680 2 4 6 8Te [eV]050100150200Figure 2. Profiles of the ionization degree Zfree (left) and electron temperatureTe (right) at a time of t = 400 fs after the pulse maximum. The optical laserirradiates the droplet from the left along the x-axis. Due to the small laserenergy 3.2 µJ and intensity 1015 W cm−2, only a weakly ionized and thin front isobserved at the droplet surface.The resulting profiles for the electron temperature and the ionization degree are shown infigure 2 for the laser polarization plane at t = 400 fs after the maximum of the pump laser pulse.Here, a weakly ionized droplet front of several 10 nm thickness comparable to the skin depthis observed. Compared to the work of Thiele et al [16] for hydrogen a less ionized and lessextended plasma front is obtained for helium, which is due to the reduced laser energy and thehigher ionization potential of helium atoms. The temperature is determined on a reduced gridof 50 × 50 × 50 cells and adjusted to a Maxwell–Boltzmann statistics via the particle energies.3.2. A pump laser with an intensity of 1018 W cm−2To observe a higher ionized plasma we consider the laser–droplet interaction with a more intenselaser pulse of 3.2 mJ energy and 30 fs duration so that the intensity is 1018 W cm−2. These laserparameters are close to the relativistic regime where the ionized electrons reach relativisticvelocities. Such a laser is currently not available at FLASH, but for future experiments atthe XFEL in Hamburg or at LCLS in Stanford (where a corresponding short-pulse laser isalready installed in the MEC instrument) such setups would be possible. Again the simulationis performed in a cubic box of 8 × 8 × 8 µm3 size on 500 × 500 × 500 grid cells with 64 heavyparticles each for a duration up to a time of t = 440 fs after the maximum of the optical pumplaser pulse.The resulting profiles of the ionization degree are shown in figure 3 for three spatial planesat t = 80 fs after the maximum of the pump laser pulse: the laser polarization plane, the x–zplane perpendicular to the laser polarization plane and the y–z plane perpendicular to the laseraxis as illustrated in figure 1. The results for the different planes through the droplet centerindicate an inhomogeneous, asymmetrical ionization along the laser axis. In the x–z planeperpendicular to the laser polarization plane, we obtain an overcritical, highly ionized front ofseveral tens of nm thickness. In addition, in the polarization plane in the interior of the dropletan ionized plasma is observed due to the penetration of the laser into the droplet. This effect canNew Journal of Physics 15 (2013) 025041 (http://www.njp.org/)1647(a) (b)(c)Figure 3. Profile of ionization degree Zfree for different planes through thedroplet center at a time of t = 80 fs after the maximum of the optical pumplaser pulse with an intensity of 1018 W cm−2 and an energy of 3.2 µJ. Illustratedare (a) the laser polarization plane, (b) the x–z plane perpendicular to the laserpolarization plane and (c) the y–z plane perpendicular to the laser axis. Theoptical laser has irradiated the droplet from the left along the x-axis. We obtainan asymmetric ionization of the droplet which is caused by a focused penetrationof the laser into the droplet. The effect may be understood using Mie theory [30].be understood in terms of an enhanced electric field at the droplet surface (as predicted by Mietheory) and a plasma wave propagating from there into the target, ionizing also the interior ofthe droplet; this is investigated in more detail in [30].For the calculation of Thomson scattering spectra via the BMA temperature and densityprofiles throughout local thermal equilibrated droplets are necessary. Therefore, results for asubstantially longer time t = 440 fs after the maximum of the pump laser pulse are illustratedin figure 4. Here, the ionization degree is given for the laser polarization plane and theplane perpendicular to the laser axis on the left side of figure 4. The corresponding electrontemperature profiles were derived from a reduced grid of 50 × 50 × 50 boxes, see the right sideof figure 4. For such longer times we observe a highly ionized droplet front and, in contrastNew Journal of Physics 15 (2013) 025041 (http://www.njp.org/)165 THOMSON SCATTERING ON LASER-PRODUCED HE-PLASMA8 x[µm] y[µm]024680 2 4 6 8Te [eV]050010001500 x[µm] z[µm]024680 2 4 6 8Te [eV]050010001500(a)(b)Figure 4. The ionization degree Zfree (left) and electron temperature Te (right)for different droplet planes at a time of t = 440 fs after the pulse maximum.Illustrated are (a) the laser polarization plane and (b) the x–z plane perpendicularto the polarization plane. The optical laser of intensity 1018 W cm−2 irradiatesthe droplet from the left along the x-axis. For this time, we observe an almosthomogeneously ionized interior of the droplet with a thin highly ionized front atthe surface. Here, we assume an equilibrated system, where a slight decrease ofthe electron temperature from the front surface to the back surface of the dropletis observed.to shorter times of t = 80 fs, a more homogeneous interior of the helium droplet. The focusingeffect obtained for shorter times is still visible in the droplet front. The temperatures show adecrease of several hundreds of eV from the droplet front to the back of the droplet.3.3. A pump laser with an intensity of 1019 W cm−2As a third case, we study a pump laser with an even higher intensity of 1019 W cm−2 andan energy of 32 mJ. Due to this high intensity an emission of ions and electrons from thedroplet is expected. Therefore, we used a bigger simulation box of 8 × 18 × 18 µm−3 size onNew Journal of Physics 15 (2013) 025041 (http://www.njp.org/)1669 x [µm] y [µm]024680 2 4 6 8Te [eV]02000400060008000(a)(b)Figure 5. Profiles of (a) ionization degree Zfree for two different droplet planesat a time of t = 50 fs after the pulse maximum and (b) ionization degree Zfreeand electron temperature Te at a time of t = 90 fs after pulse maximum. Theoptical laser irradiates the droplet from the left along the x-axis. In this case of alaser energy 32 mJ and intensity 1019 W cm−2 a high ionization of the droplet isobserved.500 × 1125 × 1125 grid cells with 64 heavy particles per cell. In this way boundary effects areavoided but the simulation time is substantially increased. Hence, the laser–matter interactionand thereby the ionization degree profile could so far be simulated only up to 90 fs afterthe laser pulse maximum; the results are shown in figure 5. In contrast to the simulationof the laser–matter interaction with intensities of 1018 W cm−2, an accelerated focused laserpenetration into the droplet is observed. This can be explained by the increased laser intensity,which generates faster electrons penetrating the droplet. Moreover, a higher ionization degreeat the front as well as the interior of the droplet is obtained due to the higher laser energy.4. Thomson scatteringWe have calculated the overall Thomson scattering spectrum of the helium droplet via theBMA according to section 2 for a probe laser wavelength of λ0 = 13.5 nm. We use the plasmaparameter profiles from the PIC simulations with a laser intensity of 1018 and 1019 W cm−2,which are illustrated in figures 4 and 5. For the lower intensity of 1015 W cm−2 the plasmonsignal is very weak because of the low degree of ionization is and completely concealed by theNew Journal of Physics 15 (2013) 025041 (http://www.njp.org/)167 THOMSON SCATTERING ON LASER-PRODUCED HE-PLASMA10-15 -10 -5 0 5 10 15ΔE [eV]00.20.40.60.81scattering signal [arbit. units] t = 440 fs, I = 1018 W/cm2t = 190 fs, I = 1018 W/cm2t = 90 fs, I = 1019 W/cm2θ = 20oθ = 90o-10 -5 0 5 10ΔE [eV]00.050.10.15scattering signal [arbit. units] θ = 20o, t = 440 fs, I = 1018 W/cm2θ = 200, t = 190 fs, I = 1018 W/cm2θ = 20o, t = 90 fs, I = 1019 W/cm2overall spectrum forthe simulated dropleteffective spectrum forthe homogeneous targetFigure 6. Thomson scattering spectrum as a function of the probe laser energyshift for a scattering angle of θ = 20◦ and 90◦ and a laser intensity of 1018 and1019 W cm−2 at a time of t = 90, 190 and 440 fs after the maximum of the pumplaser pulse. On the left side the normalized Thomson scattering spectrum isshown and on the right side the Thomson scattering of free electrons is displayed.The solid lines illustrate the overall Thomson scattering spectrum. The brokenlines represent the effective Thomson scattering spectrum of a homogeneousplasma with plasma parameters as derived from the plasmon signal of the overallThomson scattering spectrum for the inhomogeneous droplet at θ = 20◦. Thebroken lines are normalized to the maximal value of the plasmon pair of thecorresponding overall Thomson scattering spectrum.ion feature. In the following, the ion density is fixed at the initial density, the ion temperature isassumed to be equal to the electron temperature and a normalized Gaussian distribution functionwith the FWHM 1ω = 1 eV as the instrumental function is employed. For the BMA calculation,we have used the reduced equidistant grid of 50 × 50 × 50 boxes as for the determination of theelectron temperature profile. The overall Thomson scattering spectrum of the helium droplet isderived as the equally weighted sum of the Thomson scattering spectra for each box. Each box isassumed to contain a homogeneous plasma, because FEL absorption effects are neglected due tothe high penetration depth of the FEL and the high thermal energy in the system in comparisonto the FEL energy.The overall Thomson scattering spectrum is shown in figure 6 for a laser intensity of1018 W cm−2 for two different scattering angles and times t = 190 and 440 fs after the maximumof the pump laser pulse and for a laser intensity of 1019 W cm−2 for one scattering angle andtime t = 90 fs after the maximum of the pump laser pulse. On the left side of figure 6 theoverall Thomson scattering spectrum is covered, whereas on the right side the plasmon peaksrepresenting collective Thomson scattering of free electrons are shown enlarged. Here, weinvestigate for a scattering angle of θ = 90◦ the non-collective and for a scattering angle ofθ = 20◦ the collective scattering regime. In addition, we observe for each time and intensityat the scattering angle θ = 20◦, one plasmon pair which determines the dominating plasmaparameters of the droplet. For a laser intensity of 1018 W cm−2, the plasmon pair at time t =190 fs at an energy shift |1E | ≈ 4.25 eV represents a plasma with an effective electron densityne ≈ 1.3 × 1022 cm−3 and temperature Te ≈ 75 eV, whereas at time t = 440 fs the plasmonNew Journal of Physics 15 (2013) 025041 (http://www.njp.org/)16811pair at an energy shift |1E | ≈ 4.6 eV specifies a plasma with an effective electron densityne ≈ 1.4 × 1022 cm−3 and temperature Te ≈ 260 eV. These plasma parameters stem from theinterior of the droplet and indicate an evolution of the plasma in which the HDM expandsinto the droplet. This behavior can also be seen in figures 3 and 4. For a laser intensity of1019 W cm−2, the plasmon pair at an energy shift |1E | ≈ 6.46 eV identifies a plasma with aneffective electron density ne ≈ 2.15 × 1022 cm−3 and temperature Te ≈ 2030 eV. At a time ofabout 90 fs after the pump pulse maximum a similar homogeneous interior as in the case of apump laser with an intensity of 1018 W cm−2 is observed. However, the ionization degree is inthis case about unity and, therefore, two pronounced plasmon peaks are obtained at an energyof |1E | ≈ 6.46 eV. For comparison, the Thomson scattering spectrum for a homogeneousplasma with these effective plasma parameters is illustrated for θ = 20◦ in figure 6. Here,the plasmon peak is normalized to the maximal plasmon peak of the corresponding overallThomson scattering spectrum. Compared to the effective Thomson scattering spectrum of thehomogeneous target, we observe a broadened plasmon peak and higher values for smallerenergy shifts 1E in the overall Thomson scattering spectrum. This indicates that the plasmonpair signal consists of plasma regions with different plasma parameters, which all contribute tothe overall Thomson scattering spectrum.5. ConclusionsIn this paper, we have calculated the overall Thomson scattering signal of inhomogeneoushelium droplets which are pumped by a short-pulse optical laser of different intensities andprobed with brilliant FEL radiation. We have obtained the density and temperature profiles forthe pump phase with a 1015 W cm−2 laser at t = 400 fs, a 1018 W cm−2 laser at t = 80, 190 and440 fs and a 1019 W cm−2 laser at t = 50 and 90 fs after the maximum of the pump laser pulsewith a PIC code. The Thomson scattering signal was derived from these density and temperatureprofiles only for 1018 and 1019 W cm−2 via the BMA, because for 1015 W cm−2 the plasmon pairsare concealed by the ion feature.We have shown that the inhomogeneous ionization process in the helium droplets irradiatedwith a high-intensity short-pulse laser predicted for short time scales in [30] evolves in theinterior of the droplet for larger time scales of several hundreds of fs to a plasma witha homogeneous ionized interior and a highly ionized front surface. The evolution of theionized droplet interior is reflected in the resulting overall Thomson scattering spectrum ofthis system, where only one plasmon pair dominates. The temporal resolution of this plasmonpair characterizes the evolution of the plasma parameters throughout the helium droplet. Theoverall Thomson scattering spectrum shows that in contrast to hydrogen droplets irradiated withsmaller intensities of I ≈ 1015 W cm−2 as studied by Thiele et al [16], the Thomson scatteringspectrum is not dominated by the surface of the droplet. In the present work, a plasmon pair canbe resolved which stems from the interior of the droplet. The plasma is produced by an intenselaser of I ≈ 1018 W cm−2 which can be provided at the MEC station at the LCLS at SLAC or inthe future at the HED instrument at the XFEL in Hamburg.Our calculations show that Thomson scattering on inhomogeneous helium dropletsirradiated with high-intensity short-pulse lasers can spatially and temporally resolve the interiorheating process in the HDM region. This technique can be extended by applying the calculationof the non-equilibrium DSF to derive the Thomson scattering spectrum in the short-scale heatingprocess of the helium droplet where a strongly inhomogeneous target is observed. At these timesNew Journal of Physics 15 (2013) 025041 (http://www.njp.org/)169 THOMSON SCATTERING ON LASER-PRODUCED HE-PLASMA12separated plasmon pairs are predicted. In addition, the long-time behavior of the droplet coolingcan be observed if a more-dimensional radiation-hydrodynamics code is applied, with the PICdensity and temperature profiles as the input.AcknowledgmentsThis work was supported by the DFG within the SFB 652 ‘Strong correlations and collectiveeffects in radiation fields: Coulomb systems, clusters and particles’ and the Federal Ministry ofEducation and Science (BMBF) under grant no. FSP 301-FLASH, project no. 05KS7HRA.PIC simulations were performed with the computing resources granted by the John vonNeumann-Institut für Computing under project numbers HRO01 and HRO02.References[1] Remington B A, Cavallo R M, Edwards M J, Ho D D-M, Lasinski B F, Lorenz K T, Lorenzana H E, McnaneyJ M, Pollaine S M and Smith R F 2005 Accessing high pressure states relevant to core conditions in thegiant planets Astrophys. Space Sci. 298 235[2] Fortney J J, Glenzer S H, Koenig M, Militzer B, Saumon D and Valencia D 2009 Frontiers of the physics ofdense plasmas and planetary interiors: experiments, theory, and applications Phys. Plasmas 16 041003[3] Drake R P 2009 Perspectives on high-energy-density physics Phys. Plasmas 16 055501[4] Moses E I, Boyd R N, Remington B A, Keane C J and Al-Ayat R 2009 The National Ignition Facility: usheringin a new age for high energy density science Phys. 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Phys. 8 1428[25] Sperling P, Thiele R, Holst B, Fortmann C, Glenzer S H, Toleikis S, Tschentscher Th and Redmer R 2011Two-color Thomson scattering at FLASH High Energy Density Phys. 7 145–9[26] Gregori G, Glenzer S H, Rogers F J, Pollaine S M, Landen O L, Blancard C, Faussurier G, Renaudin P,Kuhlbrodt S and Redmer R 2004 Electronic structure measurements of dense plasmas Phys. Plasmas11 2754[27] Sahoo S, Gribakin G F, Shabbir Naz G, Kohanoff J and Riley D 2008 Compton scatter profiles for warmdense matter Phys. Rev. E 77 046402[28] Liseykina T V, Pirner S and Bauer D 2010 Relativistic attosecond electron bunches from laser-illuminateddroplets Phys. Rev. Lett. 104 095002[29] Popov V S 2004 Tunnel and multiphoton ionization of atoms and ions in a strong laser field (Keldysh theory)Phys. Usp. 47 855[30] Liseykina T V and Bauer D 2012 Plasma formation in intense laser–droplet interaction arXiv:1209.5948New Journal of Physics 15 (2013) 025041 (http://www.njp.org/)171 THOMSON SCATTERING ON LASER-PRODUCED HE-PLASMA173 NUMERICAL SUPPORT TO THE EXPERIMENTAL STUDIES6. Numerical support to the experimentalstudiesThe list of publications:S. Kar, M. Borghesi, C. A. Cecchetti, L. Romagnani, F. Ceccherini, T. V. Liseykina, A. Macchi, R.Jung, J. Osterholz, O. Willi, L. A. Gizzi, A. Schiavi, M. Galimberti and R. Heathcote, Dynamics ofcharge-displacement channeling in intense laser-plasma interactions, New J. Phys. 9, 402 (2007)K. Quinn, P. A. Wilson, C. A. Cecchetti, B. Ramakrishna, L. Romagnani, G. Sarri, L. Lancia, J.Fuchs, A. Pipahl, T. Toncian, O. Willi, R. J. Clarke, D. Neely, M. Notley, P. Gallegos, D. C. Carroll,M. N. Quinn, X. H. Yuan, P. McKenna, T. V. Liseykina, A. Macchi, and M. Borghesi, Laser-drivenultrafast field propagation on solid surfaces, Phys. Rev. Lett. 102 194801 (2009)L. Romagnani, A. Bigongiari, S. Kar, S. V. Bulanov, C. A. Cecchetti, T. Zh. Esirkepov, M. Gal-imberti, R. Jung, T. V. Liseykina, A. Macchi, J. Osterholz, F. Pegoraro, O. Willi, and M. Borghesi,Observation of magnetized soliton remnants in the wake of intense laser pulse propagation throughplasmas, Phys. Rev. Lett. 105, 175002 (2010)G. Sarri, A. Macchi, C. A. Cecchetti, S. Kar, T. V. Liseykina, X. H. Yang, M. E. Dieckmann, J.Fuchs, M. Galimberti, L. A. Gizzi, R. Jung, I. Kourakis, J. Osterholz, F. Pegoraro, A. P. L. Robinson,L. Romagnani, O. Willi, and M. Borghesi, Dynamics of self-generated, large amplitude magneticfields following high-intensity laser matter interaction, Phys. Rev. Lett. 109, 205002 (2012)T h e o p e n – a c c e s s j o u r n a l f o r p h y s i c sNew Journal of PhysicsDynamics of charge-displacement channeling inintense laser–plasma interactionsS Kar1,9, M Borghesi1, C A Cecchetti1, L Romagnani1,F Ceccherini2, T V Liseykina2,7, A Macchi2,8, R Jung3,J Osterholz3, O Willi3, L A Gizzi4, A Schiavi5, M Galimberti6and R Heathcote61 School of Mathematics and Physics, the Queen’s University of Belfast,Belfast BT7 1NN, UK2Dipartimento di Fisica ‘E. Fermi’, Università di Pisa, Pisa, Italy3 Institut für Laser-und Plasmaphysik, Heinrich-Heine-Universität,Düsseldorf, Germany4 Intense Laser Irradiation Laboratory, IPCF-CNR, Pisa, Italy5Dipartimento di Energetica, Università di Roma 1 ‘La Sapienza’, Roma, Italy6 Rutherford Appleton Laboratory, Central Laser Facility, Chilton,OX11 0QX, UK7 Institute for Computational Technologies, SD-RAS, Novosibirsk, Russia8 PolyLAB, CNR-INFM, Pisa, ItalyE-mail: s.kar@qub.ac.ukNew Journal of Physics 9 (2007) 402Received 31 July 2007Published 7 November 2007Online at http://www.njp.org/doi:10.1088/1367-2630/9/11/402Abstract. The dynamics of transient electric fields generated by the interactionof high intensity laser pulses with underdense plasmas has been studiedexperimentally with the proton projection imaging technique. The formation ofa charged channel, the propagation of its front edge and the late electric fieldevolution have been characterized with high temporal and spatial resolution.Particle-in-cell simulations and an electrostatic, ponderomotive model reproducethe experimental features and trace them back to the ponderomotive expulsionof electrons and the subsequent ion acceleration.9 Author to whom any correspondence should be addressed.New Journal of Physics 9 (2007) 402 PII: S1367-2630(07)56421-91367-2630/07/010402+10$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft175 DYNAMICS OF CHARGE-DISPLACEMENT CHANNELING2Contents1. Introduction 22. Experimental set-up 33. Experimental results 44. Data analysis by numerical simulations 65. Conclusion 9Acknowledgments 9References 91. IntroductionThe study of the propagation of intense laser pulses in underdense plasmas is relevant toseveral highly advanced applications, including electron ([1] and references therein) and ionacceleration [2, 3], development of x- and γ -ray sources [4], and fusion neutron production [5].It is also of fundamental interest, due to the variety of relativistic and nonlinear phenomenawhich arise in the laser–plasma interaction [6]. Among these, self-focusing and self-channelingof the laser pulse arise in this regime from the intensity dependence of the relativistic index ofrefraction [7, 8].Strong space charge electric fields are generated during the early stage of the propagationof a superintense laser pulse through an underdense plasma as the ponderomotive force (PF)acts on electrons, pushing them away from the axis. Thus, for a transient stage the pulse maypropagate self-guided in a charged channel [9], while the space-charge field in turn drags andaccelerates the ions to MeV energies [3]. So far, experiments have provided evidence of channelformation and explosion using optical diagnostics [9]–[14], or by detecting radially acceleratedions [2, 3, 5, 14], while a direct detection of the space-charge fields has not been obtainedyet. The development of the proton projection imaging (PPI) technique [15] has provided avery powerful tool to explore the fast dynamics of plasma phenomena via the detection ofthe associated transient electric field structures. The technique is based on the use of laser-accelerated multi-MeV protons ([16, 17] and references therein) as a charged probe beam oftransient electromagnetic (EM) fields in plasmas, a possibility allowed by the low emittanceand high laminarity of the proton source [18, 19], as well as by its ultra-short duration andthe straightforward synchronization with an interaction laser pulse. The experimental PPIimplementation takes advantage from the broad energy spectrum of protons, since in a time-of-flight arrangement protons of different energy will probe the plasma at different times, andthus an energy-resolved monitoring of the proton probe profile allows single-shot, multi-frametemporal scans of the interaction to be obtained [15]. PPI and the related ‘proton deflectometry’technique permit spatial and temporal maps of the electric fields in the plasma to be gathered,and therefore have proven to be an unique tool to explore the picosecond dynamics of laser–plasma phenomena [20]–[22] via the associated space-charge fields.In this paper, we report on an experiment using the PPI technique to study the formationand subsequent evolution of a charge-displacement channel in an underdense plasma. Theseinvestigations have led to the first direct experimental detection of the transient electric fieldsin the channel, providing an insight of the fundamental physical processes involved. TheNew Journal of Physics 9 (2007) 402 (http://www.njp.org/)17633 mmOptica lpr obeCPA2CPA1Gas Jet3 cm(a)MultilayerRCF stack(d)Pr otonpr obe(b)n e(1021/cc)(b)Figure 1. (a) Schematic diagram of the experimental set-up. (b) Top:interferogram of the plasma at 25 ps before the arrival of CPA1 at its focalplane x = 0. The CPA1 peak intensity was 1.5× 1019Wcm−2. Bottom: thecorresponding electron density profile along the y = 0 axis.comparison of the experimental data with two-dimensional (2D) EM particle-in-cell (PIC)simulations and a simple 1D electrostatic (ES) PIC model allows to characterize in detail theelectric field dynamics at different stages of its evolution.2. Experimental set-upThe experiment was carried out at the Rutherford Appleton Laboratory, employing theVULCANNd-glass laser system [23], providing two chirped pulse amplified (CPA) pulses, with1.054µm wavelength, synchronized with picosecond precision. Each of the beams deliveredapproximately 30 J on target in 1.3 ps (full width at half maximum, FWHM) duration. Byusing f/6 off-axis parabolas, the beams were focused to spots of ∼8µm (FWHM) achievingpeak intensities up to 1.5× 1019Wcm−2. The short pulses were preceded by an amplifiedspontaneous emission (ASE) pedestal of 300 ps duration and a contrast ratio of ∼106. One ofthe beams (CPA1) was directed to propagate through He gas from a supersonic nozzle, havinga 2mm aperture, driven at 50 bar pressure. The interaction was transversely probed by theproton beam produced from the interaction of the second CPA beam (CPA2) with a flat foil(a 10µm thick Au foil was typically used), under the point projection imaging scheme [15].The schematic of the experimental set-up is shown in figure 1(a). Due to the Bragg peak energydeposition properties of the protons, the use of multilayered stacks of radiochromic film (RCF)detector permits energy-resolved monitoring of the proton probe profile, as each layer willprimarily detect protons within a small energy range. This allows to obtain single-shot, multi-frame temporal scans of the interaction in a time-of-flight arrangement [15]. The spatial andtemporal resolution of each frame were of the order of a few picoseconds, and of a few microns,respectively, while the magnification was 11.The interaction region was also diagnosed by Nomarsky interferometry, employing afrequency doubled CPA pulse of low energy. The reconstructed electron density profile alongNew Journal of Physics 9 (2007) 402 (http://www.njp.org/)177 DYNAMICS OF CHARGE-DISPLACEMENT CHANNELING4Figure 2. PPIs of the interaction region at different times obtained in twodifferent laser shots. The x and y coordinates refer to the object (interaction)plane, which intersects the probe axis at (x, y) = (0, 0). The images (a) and(b) have been obtained from the same shot at an intensity I = 4.0× 1018Wcm−2,while (c) and (d) correspond to a shot with I = 1.5× 1019Wcm−2. The signal inthe frames is mainly due to protons (of energies E = 13, 12.5, 13 and 12MeVin (a)–(d), respectively) reaching the Bragg peak within their active layers. Thetime labels give the probing time of the protons propagating along the probe axis[t0(E)], relative to the time of arrival of the peak of interaction pulse at the planex = 0µm. White (dark) regions correspond to lower (higher) proton flux thanthe background. The labels indicate the most prominent features: (I) the bullet-shaped leading edge and (II) the central region of the ‘white’, positively chargedchannel; (III) the ‘black’ line along the axis, indicating a region of field inversioninside the channel.the propagation axis before the high-intensity interaction (see figure 1(b)), broadly consistentwith the neutral density profile of the gas jet [24] suggests complete ionization of the gas by theASE prepulse.3. Experimental resultsFigures 2(a)–(d) show four sequential PPIs of the interaction region. The laser pulse propagatesfrom left to right. A ‘white’ channel with ‘dark’ boundaries is visible at early times (figures 2(a)and (b)). The leading, ‘bullet’ shaped edge of the channel, indicated by the label I in figures 2(a)and (b), is seen moving along the laser axis. In the trail of the channel, the proton fluxdistribution changes qualitatively (figures 2(c) and (d)), showing a dark line along the axisNew Journal of Physics 9 (2007) 402 (http://www.njp.org/)1785ObjectPlaneSource0τL(x,E)x0 L(x)RCFFigure 3. Schematic diagram showing the evaluation of the channel front speed.Protons of energy E from the point-like source cross the channel axis at adistance x from the centre of the object plane at the time τ(x, E) = t0(E)+[L(x) − L0]/vp, where L0 is the distance between the source and the objectplane, vp =√2E/mp is the proton velocity, L(x) =√x2 + L20, and t0(E) is thetime at which the protons of energy E directed perpendicularly to the objectplane cross the latter, relative to the instant at which the laser pulse peak reachesthe focal plane (x = 0).(indicated by the label III), which is observed up to tens of picosecond after the transit of thepeak of the pulse. The ‘white’ channel reveals the presence of a positively charged region aroundthe laser axis, where the electric field points outwards. This can be interpreted as the result ofthe expulsion of electrons from the central region. The central dark line observed at later timesin the channel suggests that at this stage the radial electric field must change its sign, i.e. pointinwards, at some radial position. As discussed below, this field inversion is related to the effectsof ion motion.Due to multi-frame capability of the PPI technique with picosecond temporal resolution, ithas been possible to estimate the propagation velocity v of the channel front. Critically, one hasto take into account that, due to the divergence of the probe beam, the probing time varies alongthe pulse propagation axis (see figure 3) as τ(x, E) ' t0(E)+ τ0(E)(√1 + x2/L20− 1), whereL0 ' 0.3 cm is the distance between the plane and the proton source and τ0 = L0/√2E/mp '220 ps/√E/MeV is the proton time of flight from the source to the center of the object plane(see figure 3). The reference time t0(E) is relative to the instant at which the laser pulse peakcrosses the focal plane x = 0. We divide the displacement of the tip of the channel leadingNew Journal of Physics 9 (2007) 402 (http://www.njp.org/)179 DYNAMICS OF CHARGE-DISPLACEMENT CHANNELING6+500 +1000−1000 −500 0+100−1000x (µm)y(µ)ma)b)c)d)Figure 4. (a) Simulated proton image, obtained from particle tracing simulationsof 12MeV protons (as shown in the data figure 2(d)) in the electric field patternE(r, z, t) given by the 1D particle simulations based on the ponderomotive, ESmodel. (b) Profiles of electric field Er and ion density ni (c) and (d): phase spacedistributions of ions fi(r, pr) and electrons fe(r, pr), respectively, from the 1Dsimulation, at various times (t = 0 refers to the laser pulse peak).front from frame (a) to (b), 1X = Xb − Xa ' (−300 + 500) µm= 200µm by the difference1τ in the corresponding probing times 1τ = τ(Xb, Eb) − τ(Xa, Ea) ' 0.66 ps to obtainv = 1X/1τ ' 3× 108ms−1. Based on the nominal reference time, we estimate the peak ofthe pulse to be approximately 0.6 ps behind the tip of the channel front.4. Data analysis by numerical simulationsIn order to unfold the physical mechanisms associated with the dynamics of the charged channel,a 1D ES PIC model in cylindrical geometry was employed, in which the laser action is modeledsolely via the PF of a non-evolving laser pulse. A similar approach has been previously used byother authors [12, 14]. The code solves the equation of motion for plasma particles along theradial direction, taking into account the ES field obtained from Poisson’s equation and the PFacting on the electrons: Fr = −mec2∂r[1 + a2(r, t)/2]1/2 [25]. Here, a(r, t) = a0e−r2/2r02f (t) andf (t) define the temporal envelope of the laser pulse. For the latter, a ‘sin2’ profile was used (theuse of a Gaussian profile did not yield significant differences).Figure 4(b) shows the electric field Er(r, t) and the ion density ni(r, t) obtained from asimulation with r0 = 7.5λ, a0 = 2.7, τ0 = 300λc−1 and an initial density 0.01nc (where nc isthe critical density and nc = 1021 cm−3 for λ = 1µm). The initial depletion of electrons and thelater formation of an ambipolar electric field front are clearly evident. In order to achieve adirect comparison with the experimental data, a 3D particle tracing simulation, employing theNew Journal of Physics 9 (2007) 402 (http://www.njp.org/)1807PTRACE code [22], was carried out to obtain the proton images for an electric field given byE(x, r, t) = r̂Er(r, t − x/c). The experimental proton source characteristics (e.g. spectrum anddivergence), the detector configuration and dose response were taken into account. A simulatedproton image, reproducing well the main features observed in the experiment, is shown infigure 4(a). The tip of the channel front is located 0.75 ps ahead of the pulse peak, in fairagreement with the previous estimate based on the data.An essential theoretical description of the dynamics observed in the 1D simulations canbe given as follows (a detailed description is reported [26]). In the first stage, Fr pushes partof the electrons outwards, quickly creating a positively charged channel along the axis and aradial ES field which holds the electrons back, balancing almost exactly the PF, i.e. eEr ' Fr;thus, the electrons are in a quasi–equilibrium state, and no significant electron heating occurs.Meanwhile, the force ZeEr ' ZF r accelerates the ions producing a depression in ni aroundthe axis (see figure 4(b)); at the end of the pulse (t ' 2 ps), we find Er ' 0 (see figure 4(b)),indicating that the ions and electrons have moved in order to restore the local charge neutrality.However, the ions retain the velocity acquired during the acceleration stage. For r > rmax, wherermax is the position of the PF maximum, the force on the ions, and thus the ion final velocity,decrease with r ; as a consequence, the ions starting at a position ri(0) > rmax are ballisticallyfocused towards a narrow region at the edge of the intensity profile, and reach approximatelythe same position (r ' 15µm) at the same time (t ' 3 ps, i.e. 2 ps after the peak of the laserpulse); here they pile up producing a very sharp peak of ni. Correspondingly, the ion phasespace shows that the fastest ions (with energies ' 400 keV, of the order of the time-averagedponderomotive potential) overturn the slowest ones and hydrodynamical breaking of the ionfluid occurs. Using a simple model [26], the ‘breaking’ time and position can be estimatedto be τb ' tp + (π/2√2)e3/4√(A/Z)(mp/me)(r0/a0c) (where tp = 1 ps is the time at which thepulse has maximum amplitude) and rb ' (3/2)3/2r0, yielding τb− tp ' 1.3 ps and rb ' 14µmfor the simulation in figure 4, in good agreement with the numerical results. As inferred fromthe ion phase space plot at t = 4.7 ps, a few ions acquire negative velocity after ‘breaking’;they return toward the axis and lead to the formation of a local density maximum at r = 0 aftert ' 15 ps.The electron phase space shows that at breaking the electrons are strongly heatedaround the ion density peak, generating a ‘hot’ electron population with a ‘temperature’Th ' 13 keV and a density nh ' 4× 1019 cm−3, corresponding to a local Debye length isλD = (Th/4πnhe2)1/2 ' 0.13µm. A modeling of the sheath field thus generated around thedensity spike [26] (whose thickness d ' 0.1µm is less than both λD and the sheath width L)yields a peak field Es ' 2πenhd ' 6× 1010Vm−1 and a sheath width L ' 4λ2D/d ' 0.7µm,consistently with the simulation results. The ambipolar field at the ‘breaking’ location can thusbe interpreted as the sheath field resulting from the local electron heating.The good agreement of the simulated image with the experimental ones indicates thatthe 1D ponderomotive, ES model contains the essential physics of self-channeling and relatedelectric field dynamics, despite the exclusion of nonlinear pulse evolution due, e.g. to self-focusing, or of the plasma inhomogeneity. To address these issues, we performed EM PICsimulations of the laser–plasma interaction. The simulation were 2D in planar geometry—a fully 3D simulation with spatial and temporal scales close to the experimental ones andadequate numerical resolutions is a way beyond present days computational power. The2D results can be considered to be qualitative since, for instance, the diffraction length ofthe laser beam or the scaling with distance of the ES field are different in 3D. DespiteNew Journal of Physics 9 (2007) 402 (http://www.njp.org/)181 DYNAMICS OF CHARGE-DISPLACEMENT CHANNELING8Figure 5. Top frame: ion density (ni) and electric field components (Ez andEy) at t = 2.0 ps. Bottom frame: lineout of Ey (blue) and ni (red) along they-axis at two different x-positions, showing the transition in the radial fieldpattern. The density is normalized to nc = 1021cm−3 and the fields to meωc/e =3.2× 1011Vm−1. The initial density profile reproduces the experimental one infigure 1 (b). The laser pulse propagates from left to right along the x-axis and has1ps duration. The peak amplitude is a0 = 2 in dimensionless units. The laserpulse is s-polarized (i.e. the polarization is perpendicular to the simulationplane). In this configuration Ez is representative of the amplitude of thepropagating EM pulse and Ey is generated by the space-charge displacement.The observed increase in Ez by a factor of ∼ 1.2 with respect to the peak valuein vacuum is due to self-focusing.of these limitations, the main features of the channel observed in the experimental dataare qualitatively reproduced in 2D simulations for a range of parameters close to theexperiment. In the simulation of figure 5, the laser pulse has a Gaussian intensity profileboth in space and time, with peak dimensionless amplitude a0 = 2, radius r0 = 4 λ andduration τ0 = 300 λ c−1, where λ is the laser wavelength. For λ = 1µm the pulse durationand intensity correspond to 1 ps and 5.5× 1018Wcm−2, respectively. The charge-to-massratio of ions is Z/A = 1/2. The electron density grows linearly along the x-axis from zeroto the peak value n0 = 0.1 nc over a length of 400 λ, and then remains uniform for 200 λ.A 6500× 1200 numerical grid, a spatial resolution 1x = 1y = λ/10 and 16 particles per cellfor both electrons and ions were used.Figure 5 shows the ion density (ni) and the components Ey and Ez of the electric fieldat the time t = 600 λ c−1 ' 2.0 ps. In this simulation, the laser pulse is s-polarized, i.e. thepolarization is along the z-axis, perpendicular to the simulation plane. Thus, in figure 5 Ezis a representative of the amplitude of the propagating EM pulse, while Ey is generated by thespace-charge displacement. Simulations performed for the case of p-polarization showed nosubstantial differences in the ES field pattern. The simulation clearly shows the formation of anelectron-depleted channel, resulting in an outwardly directed radial space-charge electric fieldNew Journal of Physics 9 (2007) 402 (http://www.njp.org/)1829whose peak value is 6.7× 1010Vm−1 (see the lineout at x = 410 λ in figure 5). In the regionbehind the peak of the pulse, two narrow ambipolar fronts (one on either side of the propagationaxis) are observed. The ambipolar fields have peak values of '6× 1010Vm−1 (see the lineoutat x = 325 λ in figure 5). As shown above, such a radial electric field profile produces a patternin the proton images similar to that observed in the region (III) of figure 2.5. ConclusionWe have reported the first direct experimental study of the electric field dynamics in a charge-displacement channel produced by the interaction of a high intensity laser pulse with anunderdense plasma. The field profiles observed clearly identify different stages of the channelevolution: the electron depletion near the axis due to the PF, and the following ion accelerationcausing a field inversion along the radius. The features observed are reproduced and interpretedby means of 1D ES and 2D EM PIC simulations, followed by a reconstruction of the protonimages employing a 3D particle tracing code.AcknowledgmentsThis work has been supported by an EPSRC grant, Royal Society Joint Project and Short VisitGrants, British-Council-MURST-CRUI, TR18 and GRK1203 networks, and MIUR (Italy) viaa PRIN project. Part of the simulations were performed at CINECA (Bologna, Italy) sponsoredby the INFM super-computing initiative. We acknowledge useful discussions with F Cornoltiand F Pegoraro and the support of J Fuchs and the staff at the Central Laser Facility, RAL (UK).References[1] Malka V et al 2005 Plasma Phys. Control. Fusion 47 B481[2] Krushelnick K et al 1999 Phys. Rev. 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Lett. 92 055003New Journal of Physics 9 (2007) 402 (http://www.njp.org/)183 DYNAMICS OF CHARGE-DISPLACEMENT CHANNELING10[19] Cowan T et al 2004 Phys. Rev. Lett. 92 204801[20] Borghesi M et al 2002 Phys. Rev. Lett. 88 135002[21] Borghesi M et al 2005 Phys. Rev. Lett. 94 195003[22] Romagnani L et al 2005 Phys. Rev. Lett. 95 195001[23] Danson C et al 1998 J. Mod. Opt. 45 1653[24] Jung R et al Central Laser Facility Annual Report 2004/2005, RAL Report No. RAL-TR-2005-025, p 23available from http://www.clf.rl.ac.uk/reports/2004-2005/pdf/16.pdf[25] Bauer D, Mulser P and Steeb W H 1995 Phys. Rev. Lett. 75 4622[26] Macchi A et al 2007 Preprint physics/0701139New Journal of Physics 9 (2007) 402 (http://www.njp.org/)184Laser-Driven Ultrafast Field Propagation on Solid SurfacesK. Quinn,1,* P. A. Wilson,1 C.A. Cecchetti,1,† B. Ramakrishna,1 L. Romagnani,1 G. Sarri,1 L. Lancia,2 J. Fuchs,2A. Pipahl,3 T. Toncian,3 O. Willi,3 R. J. Clarke,4 D. Neely,4 M. Notley,4 P. Gallegos,4,5 D. C. Carroll,5 M.N. Quinn,5X.H. Yuan,5 P. McKenna,5 T.V. Liseykina,6,‡ A. Macchi,7 and M. Borghesi11Department of Physics and Astronomy, Queen’s University Belfast, Belfast BT7 1NN, United Kingdom2Laboratoire pour l’Utilisation des Lasers Intenses, École Polytechnique, 91128 Palaiseau, France3Institut für Laser–und Plasmaphysik, Heinrich-Heine-Universität, D-40225 Düsseldorf, Germany4Central Laser Facility, Rutherford Appleton Laboratory, Chilton, Oxfordshire OX11 0QX, United Kingdom5SUPA, Department of Physics, University of Strathclyde, Glasgow G4 0NG, United Kingdom6Max Planck Institute for Nuclear Physics, Heidelberg, Germany7CNR/INFM/polyLAB, Dipartimento di Fisica ‘‘E. Fermi,’’ Pisa, Italy(Received 28 January 2009; published 14 May 2009)The interaction of a 3 1019 W=cm2 laser pulse with a metallic wire has been investigated usingproton radiography. The pulse is observed to drive the propagation of a highly transient field along thewire at the speed of light. Within a temporal window of 20 ps, the current driven by this field rises to itspeak magnitude 104 A before decaying to below measurable levels. Supported by particle-in-cellsimulation results and simple theoretical reasoning, the transient field measured is interpreted as a charge-neutralizing disturbance propagated away from the interaction region as a result of the permanent loss of asmall fraction of the laser-accelerated hot electron population to vacuum.DOI: 10.1103/PhysRevLett.102.194801 PACS numbers: 41.75.Jv, 52.38.Kd, 52.57.KkThe processes by which the relativistic electrons gener-ated in intense (I > 1018 W=cm2) laser-solid interactionstransfer their energy from the focal spot to the bulk targetremain elusive, yet further investigation into such transportmechanisms is vital, not least of all, to the realization offast ignition [1]. The incidence of a high-intensity pulseonto a cone-wire target has been used to study the mecha-nisms by which ignition energy might be delivered to thecore [2]. The production of hot electron currents in solidtargets is also the basis for the growing field of laser-drivenion acceleration [3]. It has been shown that the adaptationof the divergent, broadband proton beams generated viatarget normal sheath acceleration (TNSA) [3] to a range ofapplications [3,4] might be facilitated by the developmentof one of two schemes for the optimization of a laser-driven proton beam to its minimal bandwidth and di-vergence angle [5,6], both of which are related to thelaser-induced ultrafast (picosecond-time scale) chargingof a solid. The development of fast rise-time laser-drivenZ pinches, meanwhile, also demands a thorough knowl-edge of the transient currents induced in laser-wire inter-actions [7].While the global charging of a solid irradiated at highintensity has been reported previously [8,9], the propaga-tion mechanism of the related charging fields remainsunclear, and local current flows have not yet been directlyobserved.In this Letter, we present the experimental observationof the highly transient laser-driven current induced in theinteraction of an intense (3 1019 W=cm2) pulse with ametallic wire target. The velocity at which the charge pulsemoves along the wire is confidently measured by a novelexperimental arrangement to be c. The measurementsare consistent with a current which rises to a peak magni-tude of 8 kA before decaying to below measurable levelsover 20 ps. All measurements were performed by employ-ing a TNSA proton beam as a charged particle probe of theelectromagnetic fields set up by the interaction pulse in theregion of the target. Some 3 1011 electrons are calcu-lated to have been drawn toward the interaction region pastthe cross section of measurement, a figure consistent with asimple model of the target charging.The proton radiography or imaging technique [10] oper-ates on the principle that the Lorentz deflections impingedupon the constituent protons of the probe beam act as ameasure of the E and B fields set up in the interactionregion. Dosimetric radiochromic film (RCF) [11] is com-monly employed transverse to the incident beam as aproton detector, providing spatially and energetically re-solved detection of the incident proton beam when used ina layered configuration. Each RCF layer may then berelated to a proton energy Ep by calculating the distanceinto the film pack at which the Bragg peak occurs as afunction of Ep. The probing time assigned to a given layeris then given by tprobe ¼ dðmp=2EpÞ1=2, where d is thedistance from the proton source to the point of interest atthe target and mp is the proton mass. The evolution of theelectric and magnetic fields set up as a result of the inter-action pulse may hence be inferred with mps spatiotem-poral resolution by comparing the experimental protondensity behavior at the detector with the results of three-dimensional (3D) particle-tracing simulations [10].The experiment was conducted on the VULCANPetawatt laser system [12]. One-tenth of the cross sectionPRL 102, 194801 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending15 MAY 20090031-9007=09=102(19)=194801(4) 194801-1 2009 The American Physical Society185 ULTRAFAST FIELD PROPAGATION ON SOLID SURFACESof the main 260 J, picosecond-duration, 60 cm beam enter-ing the target chamber was diverted in situ via a 235 mm-diameter pickup mirror to provide the interaction pulseCPA2 [13], which was focused down to an 8 4 m2spot to give an on-target intensity of 3 1019 W=cm2.The remainder of the main beam (CPA1) was focuseddown to an area of 5 4 m2 by a 1 m off-axis parabolaonto a 20 m Au foil to accelerate the proton beam forprobing of the interaction. A 50 ps optical delay wasapplied to CPA2 so that the target was completely envel-oped in the probe beam at the time of the interaction. Thecutoff energy of the proton beam accelerated by the 41020 W=cm2 CPA1 pulse was roughly 25 MeV so that,when combined with the RCF stack, observation of thewire was provided over 40 ps.We first studied the interaction of CPA2 with a125 m-diameter gold wire in a vertical orientation [shotA; see Fig. 1(a)]. Measurements were performed at the zerolevel of the probe beam (the level at which protons emittedfrom the source have no vertical component of velocity).As shown in Fig. 2, over a temporal window of some 20 ps,the extent of the deflections imposed upon the probe pro-tons as they pass the wire increases to its maximum beforedecaying. This feature is interpreted as being caused by thetransient charging and subsequent discharging of the wireas a result of the CPA2 interaction. That this electricalcharging is positive is revealed by the fact that protondensity ‘‘pileups’’ are visible on either side of the wireimage while the central region is depleted (protons havebeen deflected away from the wire).The fields set up as a result of the laser-wire interactionwere modeled in the particle tracer. If Es is the strength ofthe outwards-pointing radial electric field at the wire sur-face and rw the wire radius, then by assuming that(i) positive charge is contained exclusively within thewire and (ii) the negative charge density outside the wireis low compared to the positive charge density inside, theelectric field strength at a given radial distance r from theaxis may be approximated by Esðrw=rÞ for r rw. Theelectric field is largely absent for r < rw, meanwhile,falling to 1=e of its maximum strength within a skin depthof the wire surface. The azimuthal magnetic field associ-ated with current propagation along the wire is in turncomputed via application of Ampère’s law, althoughparticle-tracing simulations demonstrated that deflectionsto the probe protons could be attributed exclusively to theE field as long as the maximum B-field strength did notexceed 100 T.For each RCF layer, the value of Es was repeatedlyvaried in the particle tracer until a match was obtainedbetween the simulated proton density behavior at the de-tector and that recorded experimentally. The maximumvalue of Es was hence determined to be 8 109 V=m.Because of the fact that the wire is oriented vertically, ddoes not vary significantly within the field of view of theprobe beam, and each RCF layer in the data set for shot Acorresponds effectively to a discrete probing time. Hence,although there is some indication in Fig. 2 that a chargingfront might be propagating along the wire, such a front isnot clearly resolved.Relativity dictates that, for the whole target to havebecome positively charged, such a front must have trav-ersed the wire at some finite velocity vf c. On shot B, asmall but crucial change was made to the experimentalsetup to enable the resolution of such a charging front [seeFig. 1(b)]. The experiment was repeated under identicalFIG. 1 (color online). (a) Schematic describing the experimen-tal setup used on shot A. (b) The corresponding setup used onshot B. In both cases, CPA2 strikes into the page.FIG. 2 (color online). Samples of RCF data from shot A de-scribing the CPA2 interaction on a vertical 125 mAu wire. Thedashed red line marks the zero level of the probe proton beam.Corresponding proton density lineouts taken across the zerolevel are shown below each RCF image. Relative to the timeof arrival of the CPA2 interaction pulse, the probing timesdepicted by each layer are (i) 0, (ii) 10, and (iii) 25 ps.PRL 102, 194801 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending15 MAY 2009194801-2186conditions in terms of CPA2 intensity and pulse durationand wire material and diameter. The wire itself, however,was modified so as to lie an angle ¼ 30 0:5 to thevertical. As a result, the protons forming the bulk of thesignal on a given RCF layer, all traveling at the samevelocity, will have probed different points along the wireat different times based on their initial elevation anglerelative to the zero level . In this way, continuous obser-vation of the target is provided.As shown in Fig. 3, the motion of such an electric fieldfront up the wire is easily resolved on shot B. It movesaway from the CPA2 interaction point at vf ¼ð0:95 0:05Þc. The data presented here, hence, representthe first experimental measurement of the velocity at whichfield spreads over a target in an intense laser-solid interac-tion. With its single-shot, multiframe capabilities, protonradiography is arguably the only diagnostic currently avail-able with which this measurement could have been made.The observation of the propagation of this field frontaway from the CPA2 interaction point at c enables theresult of shot A, in describing the evolution of the magni-tude of the radial electric field at the wire surface EsðtÞ, tobe interpreted as being caused by the flow of a transientcurrent past the zero level, the magnitude of which may beshown by application of Gauss’s law and the continuityequation to be given by IðtÞ ¼ 20rwvfEsðtÞ. This cur-rent, then, is calculated to rise to its peak value of 8.1 kAbefore falling to below measurable levels (1 kA) over atemporal window of some 20 ps. At later times, IðtÞ willrelax further as laser-driven hot electrons recombine withthe wire plasma; ultimately, though, global neutralizationof the target is facilitated by the flow of negative chargefrom the effectively infinite electron reservoir of the targetmount [6].Integration of IðtÞ over time reveals the net total numberof electrons moving downwards past the zero level to beN0 ’ 3 1011. This contrasts starkly with the total numberof hot electrons predicted to have been accelerated byCPA2 at the beam focus. The total number of hot electronsaccelerated can be estimated by the energy balance relationNtotal ¼ fEðkBThÞ1 [14], where f is the fraction of laserenergy absorbed by hot electrons, E the pulse energy, andkBTh the hot electron temperature which is predicted byI2 scaling to be 1.4 MeV [15]. By assuming 20% ab-sorption of laser energy into hot electrons, then [16], the30 J CPA2 pulse is estimated to accelerate some 3 1013hot electrons at its focus. A large positive electrostaticpotential, however, will develop in the region of the inter-action as the laser-accelerated hot electron populationstreams to vacuum. Only the most energetic electronswill escape the developing potential well of the target,with the remainder returning to the wire under space-charge separation.An estimation of the number of electrons escaping tovacuum may be made by developing a simple model of theelectrostatic forces at work in this system. If a Maxwellianelectron population is released from an initially neutralsphere of radius r0, the sphere will develop a progressivelylarger electrostatic potential as more electrons stream tovacuum. By equating the minimum electron energy re-quired for permanent escape to the depth of the electro-static potential energy well at the sphere surface, thefraction 1 of the Ntotal laser-accelerated electrons lost tovacuum may be estimated byln11¼ rcr0mec2kBThNtotal;where rc ¼ 2:82 1015 m is the classical electron radius[17,18]. The main source of error in the calculation of 1comes from the choice for r0. With this in mind, by settingthe value of r0 to the average of its maximum and mini-mum possible values (the 62:5 m wire radius and 5 mCPA2 spot radius, respectively), the total number of elec-trons escaping to vacuum is estimated to be 1011. Inagreement with measurements performed under similarexperimental conditions [19], the model predicts that1% of the hot electrons accelerated at the CPA2 focusare subsequently lost to vacuum. At 3 1011, then, thetotal number of electrons measured to have flowed past thezero level toward the interaction point is consistent withFIG. 3 (color online). (a) Selection of experimental (top) andsimulated (bottom) RCF data detailing the propagation of thecharging front away from interaction point on shot B. The frontposition is indicated by the red arrows. The CPA2 interactionplasma is visible to the bottom right of each experimental imageas a circular area of proton depletion. (b) Measurement ofdistance from interaction point (d2) as a function of absoluteprobing time (relative to the acceleration of the proton beam atthe source). The velocity at which the charging front movesalong the wire is measured to be ð0:95 0:05Þc.PRL 102, 194801 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending15 MAY 2009194801-3187 ULTRAFAST FIELD PROPAGATION ON SOLID SURFACESthe degree of global charge neutralization required follow-ing the permanent escape of a small fraction of the laser-accelerated hot electron population to vacuum.The simulation of a scaled-down model experimentcan prove fruitful in investigating the ultrafast electromag-netic dynamics of such a system. A 2D-Cartesian particle-in-cell (PIC) code [20] was hence used to simulate theinteraction of a laser pulse of ¼ 1 m wavelength,1:2 1019 W=cm2 peak intensity, 100 fs duration, and4 m spot diameter normally incident with a plane plasmaslab of electron density 1022 cm3. The target plasmaextends over 0< x< 5 m, while the pulse strikes fromthe left-hand side at x ¼ 0, y ¼ 0.Figure 4 shows the contour levels of Ex and Jy (thecomponents of the electric field and current density per-pendicular and parallel to the target surface, respectively)at the rear side of the target. Only the y > 0 region is shownsince Ex (Jy) is symmetrical (antisymmetrical) about y ¼0. A propagation of both Ex and Jy along the target surface,i.e., along y, is observed. Two fronts are visible: an innerfront propagating at 0:4c over the time interval investigatedand an outer one propagating at ’c. The inner front en-compasses a region of strong charge separation and currentrecirculation, features interpreted as being caused by therefluxing of hot electrons in and around the target. In theouter region, however, Jy is positive and is localized to athin target surface layer. A similar feature is observed onthe front side, i.e., near x ¼ 0 (not shown). In our inter-pretation, this current, corresponding to the flow of nega-tive charge toward the interaction region, is driven by theantennalike electromagnetic disturbance generated bytransient charge separation in the region of the interaction.The disturbance propagates freely in vacuum but pene-trates only the skin layer of the target. Its observed propa-gation velocity (’c) and the related vacuum features of Eyand Bz (omitted for brevity) are wholly consistent with thisdescription.In summary, our measurements quantitatively supportpreviously made postulations that the positive charging ofsolid targets irradiated at relativistic laser intensities mightbe attributed to hot electron escape [6–8]. Furthermore, thespread of charge in intense laser-solid interactions, a sig-nificant phenomenon for many applications in this inten-sity regime [1,5–7], has now been resolved directly.Importantly, supported by the results of PIC simulations,a mechanism by which such ultrafast target charging mightoccur has been identified.This research was supported by: EPSRC Grants No. EP/E048668/1, No. EP/E035728/1, and No. EP/C003586/1;DFG No. TR18 and No. GK1203; the British CouncilAlliance Program; QUB Internationalization Funds; aDEL/AWE plc CAST grant; the STFC Direct AccessScheme; and finally, by an EPSRC/Andor TechnologyDorothy Hodgkin Postgraduate grant. The support of thestaff of the Central Laser Facility at RAL is gratefullyacknowledged. The PIC simulations were performed atCINECA (Italy) and were sponsored by the CNR/INFMsupercomputing initiative.*kquinn09@qub.ac.uk†Present address: Intense Laser Irradiation Laboratory,IPCF, CNR, Pisa, Italy.‡On leave from the Institute of ComputationalTechnologies, SD-RAS, Novosibirsk, Russia.[1] M. Tabak et al., Phys. Plasmas 1, 1626 (1994).[2] J. S. Green et al., Nature Phys. 3, 853 (2007).[3] M. Borghesi et al., Fusion Sci. Technol. 49, 412 (2006),and references therein.[4] V. Malka et al., Nature Phys. 4, 447 (2008).[5] T. Toncian et al., Science 312, 410 (2006).[6] S. Kar et al., Phys. Rev. Lett. 100, 105004 (2008).[7] F. N. Beg et al., Phys. Rev. Lett. 92, 095001 (2004).[8] M. Borghesi et al., Appl. Phys. Lett. 82, 1529 (2003).[9] P. McKenna et al., Phys. Rev. Lett. 98, 145001 (2007).[10] L. Romagnani et al., Laser Part. Beams 26, 241 (2008).[11] http://www.gafchromic.com.[12] C. N. Danson et al., Nucl. Fusion 44, S239 (2004).[13] D. Strickland and G. Mourou, Opt. Commun. 56, 219(1985).[14] A. Hauer and R. J. Mason, Phys. Rev. Lett. 51, 459 (1983).[15] S. Wilks and W. Kruer, IEEE J. Quantum Electron. 33,1954 (1997).[16] K. B. Wharton et al., Phys. Rev. Lett. 81, 822 (1998).[17] E. E. Fill, Phys. Plasmas 12, 052704 (2005).[18] V. T. Tikhonchuk, Phys. Plasmas 9, 1416 (2002).[19] G. Malka and J. L. Miquel, Phys. Rev. Lett. 77, 75 (1996).[20] G. I. Dudnikova et al., Comp. Technol. 10, 37 (2005).FIG. 4 (color online). PIC simulation results detailing Ex andJy contours near the target rear surface for two different times (infemtoseconds) relative to the arrival of the pulse peak at thetarget front surface. The rear surface is located at x ¼ 5. Ex is inunits of 3:05 108 V=cm and Jy in units of 4:8 1013 A=cm2.PRL 102, 194801 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending15 MAY 2009194801-4188Dynamics of Self-Generated, Large Amplitude Magnetic Fields FollowingHigh-Intensity Laser Matter InteractionG. Sarri,1 A. Macchi,2,3 C. A. Cecchetti,4 S. Kar,1 T.V. Liseykina,5 X.H. Yang,1 M. E. Dieckmann,1J. Fuchs,6 M. Galimberti,7 L. A. Gizzi,2 R. Jung,8 I. Kourakis,1 J. Osterholz,8 F. Pegoraro,2,3A. P. L. Robinson,7 L. Romagnani,6 O. Willi,8 and M. Borghesi11School of Mathematics and Physics, The Queen’s University of Belfast, Belfast, BT7 1NN, United Kingdom2Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche, research unit ‘‘Adriano Gozzini’’, 56124 Pisa, Italy3Dipartimento di Fisica ‘‘E. Fermi’’, Largo B. Pontecorvo 3, I-56127 Pisa, Italy4Light4Tech s.r.l, 50018 Scandicci, Italy5Institut für Physik, Universität Rostock, D-18051 Rostock, Germany6LULI, Ecole Polytechnique, CNRS, CEA, UPMC, 91128 Palaiseau, France7Rutherford Appleton Laboratory, Central Laser Facility, Chilton, OX11 0QX, United Kingdom8Institute for Laser and Plasma Physics, Heinrich Heine University, Dusseldorf, Germany(Received 9 February 2012; revised manuscript received 30 May 2012; published 13 November 2012)The dynamics of magnetic fields with an amplitude of several tens of megagauss, generated at bothsides of a solid target irradiated with a high-intensity (1019 W=cm2) picosecond laser pulse, has beenspatially and temporally resolved using a proton imaging technique. The amplitude of the magnetic fieldsis sufficiently large to have a constraining effect on the radial expansion of the plasma sheath at the targetsurfaces. These results, supported by numerical simulations and simple analytical modeling, may haveimplications for ion acceleration driven by the plasma sheath at the rear side of the target as well as for thelaboratory study of self-collimated high-energy plasma jets.DOI: 10.1103/PhysRevLett.109.205002 PACS numbers: 52.25.Xz, 52.38.Fz, 52.70.NcThe generation of magnetic fields in plasmas is a phe-nomenon of great relevance for a wide range of physicalscenarios and it has been studied in laser-produced plasmassince the introduction of high-power lasers, with particularemphasis on their role in inertial confinement fusion [1]. Inthis scenario, involving nanosecond laser pulses of inten-sity IL 1014–1016 W=cm2, magnetic field generation isgenerally described by hydrodynamic modeling [2] and ithas been recently characterized by temporally and spatiallyresolved measurements [3,4]. In the case of shorter( ps) and more intense (IL 1018 W=cm2) laser pulses,the production of large currents of high energy ‘‘hot’’electrons requires a more complex modeling [5] and avery intense magnetic field may be generated by, e.g.,Weibel-like instabilities [6] or electron recirculation atthe plasma boundary (fountain effect) [7]. In this lattercase, magnetic fields may influence the emission ofmulti-MeV proton [8,9] and positron [10] beams.Magnetic fields generated in relativistic laser-producedplasmas are also of central importance in reproducingconditions resembling large-scale astrophysical processes[11] on a laboratory scale such as the self-collimation ofrelativistic leptonic jets [12] or the upstream-downstreammixing in supernova remnant shocks [13].Previous experimental work has detected effects in-duced by such magnetic fields on external optical beams[14] or on the polarization of self-generated harmonics[15]. However, these measurements suffered from limita-tions, in terms either of the range of plasma densityaccessible [14] or of spatial and temporal resolution [15].Furthermore, only the fields generated at the front (laser-irradiated) side of the target have been investigated; themagnetic field generation at the rear side, where protonacceleration in the expanding fast electron sheath takesplace [16], is thus yet to be experimentally characterized.In this Letter, we report on simultaneous measurementsof the magnetic fields generated at the front and rear side ofa solid target irradiated by a short and intense laser pulse,using a spatially and temporally resolved proton imagingtechnique [17]. Clear evidence is given of the generationof toroidal magnetic fields [maximum amplitude of 50megagauss (MG)] that decay in time on a picosecond timescale. Their spatial distribution and amplitude is consistentwith the recirculation of the laser-accelerated electronsaround the target and they are sufficiently intense to con-fine the radial plasma expansion.The experiment was carried out at the Vulcan lasersystem in the Rutherford Appleton Laboratory [18] usingtwo laser beams both with a central wavelength L ¼1:05 m, energy EL ¼ 50 J, and duration L ¼ 1 ps.Both beams were preceded by a lower intensity plateau(IP 1012 W=cm2, duration of 300 ps), due to ampli-fied spontaneous emission. The first laser beam (CPA1)was focussed, down to a focal spot with radius rL 10 m, to a peak intensity of IL 1019 W=cm2 (dimen-sionless intensity a0 2:7) onto a d ¼ 10 m thick alu-minum foil, with a 45 angle of incidence. Hydrodynamicsimulations [19] indicate that the laser prepulse ablates aPRL 109, 205002 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending16 NOVEMBER 20120031-9007=12=109(20)=205002(5) 205002-1 2012 American Physical Society189 SELF-GENERATED LARGE AMPLITUDE MAGNETIC FIELDSsubmicron layer of the aluminum foil that expands, in aplasma state, with an exponentially decreasing densityprofile (scale-length of 2.5 microns) and an electron tem-perature of 100 eV. The second laser beam (CPA2) wasfocussed onto a 20 m thick gold foil to generate, viatarget normal sheath acceleration [16], a proton beamwith a Boltzmann-like spectrum [temperature of TP ¼3:0 0:2 MeV, and cutoff energy of Ep 20 MeV, seeFigs. 1(g) and 1(o) for its spatial distribution]. This beamwas used as a charged-particle probe [17] and was recordedonto a stack of calibrated radiochromic films (RCFs)[20] giving a pointlike projection of the interaction witha geometrical magnification M ðlþ LÞ=l 11 withl 3 mm and L 3 cm the distances between the Auand Al foils and between the Al foil and the RCF respec-tively, as sketched in Figs. 1(h) and 1(p).Previous theoretical modeling of high-intensity irradia-tion of thin solid targets [9] indicates the generation ofmagnetic fields having a toroidal structure with azimuthalsymmetry and field lines parallel to the target surface. Theintense electric fields generated in the expanding plasmasheath are almost normal to the target surface and havebeen directly detected using a probe proton beam parallelto the surface [21]. In the present experiment, in order tomaximize the probe proton deflections due to magneticfields over those due to electric fields, the propagation axisof the probe beamwas normal to the target surface. In orderto ascertain the magnetic nature of the deflecting fields,two configurations were adopted: the proton beam firstencountered either the rear, unirradiated side of the target[direct configuration, Fig. 1(h)] or the front, irradiated side[reverse configuration, Fig. 1(p)]. For a given polarity ofthe magnetic field distribution, the two configurationsshould induce opposite deflections.A typical set of RCF images, obtained in the directconfiguration, are displayed in Figs. 1 [series (a)–(f) and(i)–(k)]. All images depict the presence of two main fea-tures: an outer ring (radius of the order of 100–200 m) andan inner dot (radius of the order of 30–40 m) of protonaccumulation. The outer ring is seen to slowly expand intime while roughly preserving its amplitude, whereas theinner dot becomesweaker and eventually disappears as timeprogresses. The reverse configuration [Figs. 1(l)–1(n)]induces an inverse deflection pattern, the central dot beingmuch darker than the outer ring. In the direct configuration,the rear magnetic field focusses the protons propagatingnear the axis, whereas the front field enhances thedivergence of the protons propagating at a wider angle[see Fig. 2(a)]. In the radiographs, this leads to the forma-tion of an accumulation dot produced by the rear field and ofan outer ring produced by the front field. In the reverseconfiguration, the same structures should be expected, yetwith a different relative amplitude [Fig. 2(b)], in qualitativeagreement with Fig. 1.For a quantitative analysis of the magnetic field, particletracing (PT) [22] simulations have been performed. Thecode follows the propagation of a proton beam, asemployed in the experiment, through a prescribed mag-netic field distribution and it includes the response of anRCF detector providing synthetic RCF images. The mag-netic field is assumed, at both the front and rear sides of thetarget, as having a cylindrically symmetric toroidal distri-bution localized at the target surfaces, analogous to themodel reported in Ref. [3]. In this model, the magnetic fieldat each of the two surfaces is described via the followingparameters: B (field amplitude), R (radius of the fieldFIG. 1 (color online). (a)–(f) Proton imaging snapshots in the direct configuration [sketched in frame (h)] and typical image of theunperturbed proton beam (g). (i)–(k) Additional set of images, still obtained in the direct configuration, from a different laser shot.(l)–(n) Proton imaging snapshots in the reverse configuration [sketched in the frame (p)] and typical image of the unperturbed protonbeam (o). In all images, the spatial scale refers to the interaction plane and time is relative to the arrival of the peak of the CPA1 pulseon target.FIG. 2 (color online). Sketch of the proton deflections inducedby the magnetic fields in the direct (a) and reverse(b) configuration.PRL 109, 205002 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending16 NOVEMBER 2012205002-2190distribution), and Lr and Lz (field scale length in directionsparallel and perpendicular to the target surface, respec-tively). These parameters were varied independently untila satisfactory match was reached between experimentaland synthetic data, in terms of position and amplitude ofthe optical density peaks observed on RCF in correspon-dence to the ring and dot proton accumulation regions (seeFig. 3). PT simulations indicate that variations in B and Rat both surfaces affect strongly peak amplitude and radius,respectively, while the field scale lengths have only asecondary effect on the synthetic RCF profile. By keepingLz ¼ Lr ¼ 10 m [consistent with particle-in-cell (PIC)simulations, see later] and varying independently B and Rat front and rear, profile matching as shown in Fig. 3 couldbe obtained, in both the direct and reverse configuration.The matching was considered satisfactory whenever theposition and peak value of the simulated optical densitymaxima reproduced those of the experiment within5 m and 5%, respectively. These values are of the orderof the intrinsic spatial resolution of the proton backlighting[17] and of a typical small-scale nonhomogeneity of theproton density on a beam cross section.By iteratively applying this method to each RCF layer, ithas been possible to simultaneously extract the temporalevolution of the amplitude and radius of the magnetic fieldat each side of the target (see Fig. 4). In correspondence tothe falling edge of the laser pulse, both fields are seen toincrease their radius and decrease their amplitude in time.After the laser irradiation (t 1 ps in Fig. 4) the front fieldis seen to rapidly drop down to an almost constant ampli-tude of BF 10–12 MGwhile its radius increases up to anapproximately constant value of 90 m. On the otherhand, the rear field exponentially decreases in amplitudewith a typical time scale of the order of 7 ps. Meanwhile,its radius increases in time with decreasing radial velocity.The longer persistence of magnetic fields at the frontsurface might be related to the presence of the underdensepreplasmawhich is able to better support the magnetic fieldlines [23].The presence of megagauss magnetic fields of oppositepolarity at both sides of the target, is also supported bytwo-dimensional (2D) PIC simulations. A density profilecomposed of an exponential ramp reproducing the abovementioned preplasma, followed by a plasma bulk withelectron density ne ¼ 40nc and charge-to-mass ratioZ=A ¼ 9=26 is assumed. The laser pulse has a Gaussiantransverse profile (FWHM ¼ 5 m), a duration of 250 T(with the laser period T ¼ 3:3 fs for 1 m wavelength),dimensionless peak amplitude a0 ¼ 2:7 on axis and it isincident at 20 with respect to the normal of the targetsurface. The simulation ran up to 600 T ’ 2 ps, in order tooverlap with the proton radiographs at earlier times.Figure 5 shows the generation of magnetic fields at thefront and rear surfaces of the target, both having an ap-proximately antisymmetrical distribution with respect tothe axis, and polarity opposite to each other. Near the peakof the laser pulse, the fields reach a maximum amplitude ofthe order of 50 MG [Fig. 2(a)] and, during the rise of thelaser pulse, they propagate in the transverse (y) directionwith a constant velocity of ’ 2:7 108 m=s (consistentwith the scenario experimentally investigated in Ref. [24])while, as the laser intensity falls down, the field distribu-tion drifts at a much lower velocity of2 107 m=s untilFIG. 3 (color online). Comparison between the experimentalproton dose deposition as recorded by the RCF and that resultingfrom PT simulations assuming magnetic field distributions as theones depicted in Fig. 5. Spatial scales refer to the interactionplane. Frames (a) and (b) refer to the direct configurationwhereas frames (c) and (d) to the reverse one.FIG. 4 (color online). Temporal evolution of the magnetic fieldamplitude and width at the rear (a) and front (b) side of the targetas extracted from matching PT simulations, compared with thePIC results. Time refers to the arrival of the peak of CPA1 ontarget.PRL 109, 205002 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending16 NOVEMBER 2012205002-3191 SELF-GENERATED LARGE AMPLITUDE MAGNETIC FIELDSstagnation is reached (t 10 ps). Within the intrinsicapproximations that a PIC model unavoidably introduces(such as two-dimensional geometry and noncollisionality)a fair agreement is found between the experimental andnumerical results (see Fig. 4). The PIC simulations alsoindicate that the electrostatic fields at the target’s surfaces(not shown for brevity) are almost normal to the originalsurface (i.e., parallel to the main axis of propagation ofthe probing proton beam), in agreement with reportedexperimental observations [21], and allow us to estimatetheir intensity. The inclusion of such electrostatic fieldsdoes not affect significantly the PT images and can thus beneglected.In principle, strong magnetic fields may also be gener-ated inside the target due to resistive return currents whichmust balance the fast electron flow. In order to evaluatethese fields, 3D simulations of the propagation of an elec-tron beam through aluminium at an initial temperatureof 1 eV were performed, using the code ZEPHYROS [25].The simulation assumed suitable parameters for the elec-tron beam (electron energy of 0.6 MeV, beam density of4 1020 cm3, initial radius rs ¼ 8:5 m, divergenced ’ 25 [26]) and a background-temperature dependenttarget resistivity [27]. Simulation results show the growthof small-scale filaments [28] with magnetic fields ofamplitude up to ’ 40 MG and a characteristic spatial scaleof 4 m [Fig. 5(e)] which is below the proton imagingresolution [17]. Indeed, including magnetic fields in thetarget bulk with these simulated amplitude and spatialdistribution induces in PT images the superposition ofrandom fluctuations with amplitude below 5%. In thesespecific experimental conditions, the proton deflections arethus predominantly induced by the magnetic fields gener-ated at the surfaces of the target.It is of particular interest to analyze in more detail thefield dynamics at the rear side of the target, which are muchless explored, and are of direct relevance to the sheath ionacceleration. Here, the intense magnetic fields are gener-ated by hot electron currents which, when crossing the rearsurface, can not be balanced anymore by a counterpropa-gating return current. The expected temperature of hotelectrons produced in the interaction is Th ’ 0:6 MeV,corresponding to a velocity vh ’ 0:9c and to a relativisticfactor ’ 2; the hot electron density may be thus roughlyestimated by a balance of energy fluxes, fIL ¼ nhvhThyielding, for an absorption fraction f 0:1 [26], nh ’4 1020 cm3. The total current due to hot electronsflowing through the target may be estimated as Ih ¼enhvhS ’ 5 106 A, where S ¼ r2L ¼ 3 106 cm2is the area of the laser focal spot. The generation of amagnetic field requires a diverging electron flow since, ifit were collimated, the large back-holding electric field Eresulting from charge displacement would cause an equaland antiparallel displacement current JE ¼ "0@tE ofrefluxing electrons. In this case, the source term for themagnetic field would exactly vanish. A divergent flowallows part of the current to flow in the radial directionforming loops which fall back to the target where a surfacereturn current may close the circuit. We developed [29] asimple geometrical model of such a ‘‘fountain effect’’ toestimate the peak magnetic field as Bmax ’ dB0 whereB0 ¼ 0Ih=ð2r0Þ, d ’ 25 is the divergence of the flow,r0 ’ 15 m the radius of the electron emitting area, and ’ 8Th=ðeEr0Þ with E the typical value of the electricfield. By estimating E ’ 1012 Vm1, as suggested both bythe PIC simulations and by proton emission data in similarconditions [21], we obtain a peak value of Bmax ’ 70 MG,in fair agreement with the experimental results. The cor-responding value of the Larmor radius mevh=eB ’3B110 m (where B10 is the field in units of 10 MG) issmall enough to indicate that electrons would be stronglymagnetized in the regions of peak field. In these conditions,the magnetic field lines are frozen in the electron fluid,and the magnetic forces tend to confine the plasma. Thistendency may account for the ‘‘negative acceleration’’ inthe torus radius versus time d2rF;R=dt2 < 0 observedin Fig. 4. The magnetic energy density (um ¼ B2=20 ¼4 105B210 J cm3) becomes in fact comparable to themaximum plasma thermal energy density expected at thepeak of the laser pulse (ut ¼ nhTh ’ 4 107 J cm3).FIG. 5 (color online). (a)–(c): PIC simulation results. Theframes show the transverse magnetic field (Bx) at t ¼ 442 Tin units of B0 ¼ me!c=e ¼ 107 MG (a), and a zoom of thefront (b) and rear (c) regions corresponding to the dashedrectangles in (a) at t ¼ 442, 492, and 592 T. The black arrowsindicate the position of the field maxima. (d)–(e): transportsimulation results. The frames show the hot electron densityinside the target in logarithmic scale and units of cm3 (d) andthe related magnetic field distribution in units of MG (e), both att ¼ 1 ps after the peak of the pulse.PRL 109, 205002 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending16 NOVEMBER 2012205002-4192Since the typical times for collisional dissipation and mag-netic diffusion (of the order of tens of ns) are much longerthan the time scales of the observation, the magnetic fieldvalue should mainly decay in time because of the sheathexpansion and thus be roughly inversely proportional to thesquare of the observed radius, in agreement with theobservations.In conclusion, temporally and spatially resolved protonimaging indicates the generation of toroidal magnetic fieldshaving tens of megagauss strength on both sides of a foilirradiated by an intense laser pulse. The magnetic fields arestrong enough to effectively confine the radial expansion ofthe plasma region where they are generated, thus possiblyaffecting the ion acceleration in the expanding sheath.Moreover, the self-confining effect qualitatively resemblesthe collimation of leptonic astrophysical jets, suggestingthat the present framework is suitable to investigate similarmechanisms in down-scaled laboratory experiments.We acknowledge the support of the RAL-CLF staff. Thiswork has been supported by the Engineering and PhysicalSciences Research Council [Grants No. EP/E035728/1(LIBRA consortium), No. EP/D06337X/1, and No. EP/J002550/1], by British Council-MURST-CRUI, by theItalian Ministry for University and Research (FIRBProject ‘‘SULDIS’’) and by the Leverhulme TrustFellowship No. ECF-2011-383. The PIC simulationswere performed using the computing resources grantedby the VSR of the Research Center Jülich under ProjectNo. HRO01. L. R. acknowledges support from theULIMAC grant from the Triangle de la Physique RTRAnetwork.[1] J. A. Stamper, E. A. McLean and B.H. Ripin, Phys. Rev.Lett. 40, 1177 (1978); A. Raven, O. Willi, and P. Rumsby,ibid.41, 554 (1978).[2] M.G. Haines, Can. J. Phys. 64, 912 (1986).[3] C. A. Cecchetti et al., Phys. Plasmas 16, 043102 (2009).[4] R. D. Petrasso et al., Phys. Rev. Lett. 103, 085001 (2009);J. R. Rygg et al., Science 319, 1223 (2008); L. Willingaleet al., Phys. Rev. Lett. 105, 095001 (2010).[5] R. J. Mason and M. Tabak, Phys. Rev. Lett. 80, 524(1998).[6] U. Wagner et al., Phys. Rev. E 70, 026401 (2004).[7] Y. Sakagami, H. Kawakami, S. Nagao, and C. Yamanaka,Phys. Rev. Lett. 42, 839 (1979).[8] S. Miyazaki, R. Sonobe, S. Kawata, and T. Kikuchi, Phys.Rev. E 71, 056403 (2005).[9] A. Pukhov, Phys. Rev. Lett. 86, 3562 (2001).[10] H. Chen et al., Phys. Rev. Lett. 105, 015003 (2010).[11] B. A. Remington, D. Arnett, R. P. Drake, and H. Takabe,Science 284, 1488 (1999).[12] M. Honda and Y. S. Honda, Astrophys. J. 569, L39(2002).[13] S. F. Martins, R. A. Fonseca, L. O. Silva, and W.B. Mori,Astrophys. J. 695, L189 (2009).[14] M. Borghesi, A. J. MacKinnon, A. R. Bell, R. Gaillard,and O. Willi, Phys. Rev. Lett. 81, 112 (1998).[15] M. Tatarakis et al., Nature (London) 415, 280 (2002).[16] R. A. Snavely et al., Phys. Rev. Lett. 85, 2945 (2000).[17] G. Sarri et al., New J. Phys. 12, 045006 (2010).[18] C. N. Danson et al., J. Mod. Opt. 45, 1653 (1998).[19] J. T. Larsen and S.M. Lane, J. Quant. Spectrosc. Radiat.Transfer 51, 179 (1994).[20] J. F. Dempsey, D. A. Low, S. Mutic, J. Markman, A. S.Kirov, G. H. Nussbaum, and J. F. Williamson, Med. Phys.27, 2462 (2000).[21] L. Romagnani et al., Phys. Rev. Lett. 95, 195001(2005).[22] L. Romagnani, Ph.D. thesis, The Queen’s University ofBelfast, 2005.[23] A. Frutchman, Phys. Fluids B 3, 1908 (1991).[24] K. Quinn et al., Phys. Rev. Lett. 102, 194801 (2009).[25] S. Kar, A. Robinson, D. Carroll, O. Lundh, K. Markey, P.McKenna, P. Norreys, and M. Zepf, Phys. Rev. Lett. 102,055001 (2009).[26] J. Fuchs et al., Nat. Phys. 2, 48 (2005).[27] J. R. Davies, Phys. Rev. E 65, 026407 (2002).[28] X. H. Yuan et al., New J. Phys. 12, 063018 (2010).[29] A. Macchi, arXiv:1202.0389v1.PRL 109, 205002 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending16 NOVEMBER 2012205002-5193 SELF-GENERATED LARGE AMPLITUDE MAGNETIC FIELDSObservation of Magnetized Soliton Remnants in the Wake of IntenseLaser Pulse Propagation through PlasmasL. Romagnani,1,2 A. Bigongiari,1,* S. Kar,1 S. V. Bulanov,3 C. A. Cecchetti,1,7 T. Zh. Esirkepov,3 M. Galimberti,4R. Jung,5 T.V. Liseykina,6,† A. Macchi,7,8 J. Osterholz,5 F. Pegoraro,8 O. Willi,5 and M. Borghesi11Centre for Plasma Physics, School of Mathematics and Physics, The Queen’s University of Belfast, Belfast BT7 1NN, United Kingdom2LULI, École Polytechnique, CNRS, CEA, UPMC, route de Saclay, 91128 Palaiseau, France3APRC, JAEA, Kizugawa, Kyoto 619-0215, Japan4Central Laser Facility, Rutherford Appleton Laboratory, Chilton, United Kingdom5Institut für Laser-und Plasmaphysik, Heinrich-Heine-Universität, Düsseldorf, Germany6Max-Planck-Institut für Kernphysik, Heidelberg, Germany7Istituto Nazionale di Ottica, CNR, Pisa, Italy8Dipartimento di Fisica ‘‘E. Fermi,’’ Università di Pisa, Pisa, Italy(Received 15 January 2010; published 19 October 2010)Slowly evolving, regularly spaced patterns have been observed in proton projection images of plasmachannels drilled by intense (* 1019 Wcm2) short ( 1 ps) laser pulses propagating in an ionized gas jet.The nature and geometry of the electromagnetic fields generating such patterns have been inferred bysimulating the laser-plasma interaction and the following plasma evolution with a two-dimensionalparticle-in-cell code and the probe proton deflections by particle tracing. The analysis suggests theformation of rows of magnetized soliton remnants, with a quasistatic magnetic field associated withvortexlike electron currents resembling those of magnetic vortices.DOI: 10.1103/PhysRevLett.105.175002 PACS numbers: 52.35.Sb, 52.38.Fz, 52.65.Rr, 52.70.NcThe generation of coherent and ordered structures is oneof the most prominent features in the dynamics of non-linear many-body systems [1]. Theoretical and experimen-tal studies have shown that plasmas interacting with laserpulses at relativistic intensities provide uniquely favorableconditions to investigate a broad class of nonlinear phe-nomena, the most known examples being arguably stimu-lated Raman and Brillouin scattering, laser filamentationand self-focusing, or the excitation of large amplitudewake plasma waves (see [2] and references therein). Adifferent but not unrelated class of phenomena which hasmore recently attracted a great deal of attention includesthe generation of organized nonlinear entities such as theso-called electromagnetic (EM) solitons [3–5] and electronvortices [6,7].EM solitons have been extensively investigated bothnumerically [3] and in the frame of analytical models [4],and their macroscopic remnants [postsolitons (PSs)]have been experimentally observed to develop followingthe interaction of an intense laser pulse with a rarefiedplasma [5]. Rows of electron vortices [6] and solitary mag-netic dipole vortices (MDVs) [7] have been predicted toform in the trail of an intense laser pulse propagating in anunderdense plasma. Besides being per se relevant as abenchmark for nonlinear plasma theoretical models, theexperimental investigation of these phenomena might alsohave practical implications, as such nonlinear entities maycontain a sizable fraction of the initial laser pulse energy[3–5] or be the signature of the development of plasmainstabilities [6].In this Letter we present the experimental observation,employing proton projection imaging (PPI) [8], of slowlyevolving, localized EM structures, generated following theinteraction at relativistic intensities of a picosecond laserpulse with an underdense plasma. These structures initiallyappear as a quasiperiodical pattern aligned along the low-density channel drilled by the laser pulse, and evolve ona time scale much longer than the pulse duration, re-maining visible for more than 100 ps after the interaction.Simulations of the laser-driven plasma evolution carriedout with a two-dimensional (2D) particle-in-cell (PIC)code show the development of EM solitons and theirevolution into PSs inside laser-generated plasma channels[9]. Most noticeably the simulations indicate that, besidesexhibiting properties typical of solitons (e.g., trapping ofEM radiation in their inside and spatial localizationconsistent with the laser-pulse-depletion generationmechanism), these structures are also accompanied byvortexlike electron currents and quasistatic magnetic fieldpatterns similar to those of MDVs. Extrapolating the re-sults from 2D PIC simulations to the three-dimensional(3D) space, we can infer that in 3D the quasistatic magneticfield would assume a toroidal configuration (see also [3,7]).Particle tracing (PT) simulations [8] of the probe protondeflections show that such a 3D field distribution producessynthetic proton images consistent with the experimentalobservation.The experiment was carried out at the RutherfordAppleton Laboratory (RAL), employing the VULCANNd-glass laser system operating in the chirped pulsePRL 105, 175002 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending22 OCTOBER 20100031-9007=10=105(17)=175002(4) 175002-1 2010 The American Physical Society195 MAGNETIZED SOLITON REMNANTS IN THE WAKE OF INTENSE LASER PULSEamplification (CPA) mode. A first laser pulse (1:054 mwavelength, 1.2 ps duration, and delivering 30 J energyon target in linear polarization, hereafter named CPA1)was focused at intensities * 1019 Wcm2 onto a super-sonic helium (He) jet (2 mm aperture diameter nozzledriven at 50 bar pressure). The main pulse was alwayspreceded by a pedestal ( 300 ps duration and 106 con-trast ratio) capable of preionizing the interaction region.The electron density profile (linearly ramping along thelaser axis up to its peak value 1:5 1020 cm3 over adistance of 400 m and remaining constant after that,see also [10]) was diagnosed by in situ frequency-doubledoptical interferometry with picosecond time resolution.Comparison of the electron density profile with a separatecharacterization of the neutral gas density in the gas jet[10] indicates full preionization of the gas.The EM fields generated in the He plasma in the trail ofthe CPA1 laser pulse were diagnosed employing a laser-driven transverse proton probe, arranged in a point pro-jection geometry with single-shot temporal multiframecapabilities [8]. The proton beamwas accelerated by focus-ing a second laser pulse (CPA2) (with similar parametersto CPA1) onto a thin metal foil (typically a 10 m thick Aufoil, proton target) and it was detected employing a multi-layer stack of radiochromic films (RCFs). The distanceof the proton target from the CPA1 propagation axis wasl 3 mm, and from the RCF pack it was L 3 cm, givinga projection magnificationM ’ L=l 10. Spatial and tem-poral resolutions of a fewm and of a few ps, respectively,were achieved.Typical PPI experimental data are shown in Fig. 1. In theimages, regions of a darker (lighter) color compared to thebackground proton signal correspond to regions of accu-mulation (depletion) of the probe protons, and for ourexperimental conditions the proton density variations re-flect the EM field gradients in the probed plasma. The laserpulse, linearly polarized in the z direction (i.e., along thenormal to the page, z being the symmetry axis of the protonbeam), is impinging from the left and by the earlier probingtime t 3:5 ps [1(a)] it has already exited the field of viewto the right-hand side. In the proton images the channeldrilled by the CPA1 laser pulse into the ionized He jet isdisplayed as a lighter color region along x delimited by twodark lines [1(a)]. An additional dark line is visible near thechannel axis on the left-hand side of the images.Longitudinal (i.e., reflecting a dependence of the deflectingfields on x) modulations first appear in the channel atprobing times t * 5 ps [1(b)], and later they evolve intoa row of localized bubblelike structures aligned along theplasma channel and particularly evident on the right-handside of the image 1(c). The localized structures slowlyexpand, remaining visible until the latest observationtimes (t * 140 ps) [Fig. 2(a)], when they have evolvedinto a cloud of irregularly distributed bubbles. Wheneverthe plasma channel has split into secondary filaments,bubble structures are also observed inside some of thefilaments [2(b)].In order to infer on the nature of the observed patterns,the interaction of the laser pulse with the He plasma andthe following plasma evolution were modeled with 2D PICsimulations [9]. The simulations were performed in a rangeof plasma densities and laser pulse parameters close to theexperimental ones. In the following, lengths are in units ofthe laser wavelength , times in units of the laser periodTL ¼ =c ¼ 2=!, densities in units of the critical den-sity nc ¼ me!2=4e2, and field amplitudes are expressedin terms of the dimensionless parameter a ¼ eE=me!c(see [9] for conversions to standard units). The largestgrid employed in the simulations was a 7750 2400mesh with a spatial resolution of =10, and the simulationswere running up to a time t ¼ 1500TL. The plasma densitywas linearly ramping from zero at x ¼ 25 to its peakvalue ne ¼ 0:1nc at x ¼ 425, remaining constant afterthat. Ions with Z=A ¼ 1=2 were assumed. The laser pulse(propagating along x) had a duration of 330TL, it was Spolarized (electric field in the z direction, normal to the(x; y) simulation plane), and its field peak amplitude wasa ¼ 2:7.In the simulation a low-density channel is ponderomo-tively bored into the plasma by the laser pulse, with thechannel breaking into a number of secondary narrowerchannels in the higher background density region [9].Following the initial evolution [9,10] and after the laserpulse has exited the simulation box, both the ion andFIG. 1 (color online). Typical PPI data (acquired in a singlelaser shot). The probing times, relative to the arrival of the peakof the interaction pulse at x ¼ y ¼ 0, are (a) t ¼ 3:5 ps,(b) t ¼ 7 ps, and (c) 17 ps [probe proton energies12 MeV (a), 11 MeV (b) and 8 MeV (c)].FIG. 2 (color online). (a) Detail of PPI data showing bubblestructures at late times (t 140 ps, probe proton energy8 MeV). (b) Detail of a secondary filament exhibiting a mul-tiple bubble pattern (t 37 ps, probe proton energy 11 MeV).PRL 105, 175002 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending22 OCTOBER 2010175002-2196electron densities are left with a depression on the channelaxis while peaking at the channel edges, giving rise to aspace-charge separation electric field, mainly in the ydirection, Ey. At this stage (t > 650TL) Ey has evolvedinto two ambipolar fronts on each side of the channel; i.e.,it points outwards outside the channel and inwards insideit. An electron current is generated along the channel axisin the wake of the laser pulse (electrons flowing in the pulsepropagation direction) and it is compensated by two cur-rent sheaths in the opposite direction along the channeledges. Such currents persist until the latest simulationtimes and produce a quasistatic magnetic field Bz, withBz < 0 (i.e., entering the simulation plane) above thechannel axis and Bz > 0 below it.At even later times (t > 750TL) the simulations show theonset of localized modulated patterns in the particle, cur-rent, and fields’ distributions inside the main and secondarychannels. A detail of the EM field distribution in a singlelocalized structure is shown in Fig. 3. The frequency-resolved analysis of the fields reveals an oscillating EMfield component (Bx, By, Ez) which has a frequency justbelow the plasma frequency of the surrounding plasma andis therefore trapped inside the structure [3(b) and 3(c)]. Thecurrent density forms an antisymmetric double loop, givingrise to a double lobe of quasistatic magnetic field Bz, withBz < 0 above the channel axis and Bz > 0 below it [3(d)].We stress here that this is a constant field, associated with astationary electron current, differently from the oscillatingfield observed in other works [3] and also here. The com-bined effect of the magnetic pressure associated with thestatic Bz component and of the radiation pressure associ-ated with the oscillating fields drills a density depression incorrespondence with each localized structure. The result-ing space-charge separation gives rise to a quasistaticelectric field ðEx; EyÞ in the radial direction from eachstructure center [3(a)]. Such field is less pronounced nearthe channel axis (due to the fact that the plasma density islower here), and at the latest times (t > 875TL) it tends tooverlap with the ambipolar field at the channel edges.It should be noted here that the presence of a trappedoscillating field is typical of EM solitons and PSs, while thecurrent and quasistatic magnetic field distributions re-semble those of MDVs. As these features coexist and thestatic and oscillating fields are of similar strength, it isdifficult to identify the observed structures with eithersolitons or vortices. However, their position in the plasmachannels provides a hint of the mechanism leading to theirformation and hence of their primary nature. The localiza-tion of the structures towards the end of the channels (bothin the experiment and PIC simulations) is qualitativelyconsistent with estimations of the laser-pulse-depletionlength (ldepl cTLðnc=neÞ mm range) [3], indicatingthat frequency downshift of the laser pulse and the conse-quent trapping of EM radiation is likely to be the relevantgeneration mechanism [3,4]. Hence the observed structuresshould be regarded as PSs. A likely scenario is that theelectron currents readjust according to the density distribu-tion associated with the preformed soliton structure, there-fore giving rise to the vortexlike pattern. As these currentswill always be present in thewake of a laser pulse propagat-ing in an underdense plasma, a staticmagnetic field could bean unavoidable feature of laser-excited solitons.To verify that the field distributions observed in the 2DPIC simulation, extrapolated to a 3D geometry, may pro-duce the observed proton images, PT simulations [8] havebeen carried out. In PTs the deflections of test protonscrossing a given 3D EM field distribution are numericallycomputed and the particle density in the proton detector iscalculated. Only static fields have been considered, as forour experimental conditions the contribution of the oscil-lating components to the proton deflection is canceled outby integration along the particle trajectories. The input 3Dfield distribution is extrapolated from the 2D PIC outputsby assuming an azimuthal symmetry around the channelaxis, and a reference frame with cylindrical coordinates (,, x) (x being the channel axis) is considered. The electricfield is chosen to be oriented along the radial ̂ directionand the magnetic field to form closed loops in the azimu-thal ̂ direction.We first simulated the deflections given by the fieldsassociated with the plasma channel before the longitudinalmodulations appear. In the simulation the electric fieldis taken of the form E ¼ fðÞ þ fþðÞ, withfðÞ ¼ 2:33E0½ð Þ= expf½ð Þ=2g, andthe magnetic field of the form B ¼ 2:33B0ð=Þexp½ð=Þ2, where the numerical factor 2.33 is chosenin order for E0 and B0 to represent the peak fields’ ampli-tudes. The parameters , , and can be related to theFIG. 3 (color online). Details of the fields’ distribution in asingle PS from PIC simulations. (a) Quasistatic electric field (Ex,Ey) at t ¼ 875TL, (b) oscillating electric field Ez, (c) oscillatingmagnetic field (Bx, By), and (d) quasistatic magnetic field Bz att ¼ 800TL.PRL 105, 175002 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending22 OCTOBER 2010175002-3197 MAGNETIZED SOLITON REMNANTS IN THE WAKE OF INTENSE LASER PULSEspatial characteristics of the PPI image of the channel,yielding ’ ’ ð15–20Þ m and ’ ð5–10Þ m. ThePPI data were best reproduced for E0 ’ ð1–2Þ 109 Vm1 and B0 ’ ð150–300Þ T [Fig. 4(a)], in fair agree-ment with the values obtained from the PIC simulations.Inspection of the PT results clarifies how the experimentalproton images form. The ambipolar Ey field tends to pileup the protons on the channel axis and at the channel edges[see deflection map 4(b)]. The vpxBz component of thevp B (vp being the proton velocity, with vpx mainlyarising from the proton beam divergence) force tends tofocus the protons on the axis for x < 0 (where vpx < 0,while Bz < 0 for y > 0 and Bz > 0 for y < 0), and at thechannel edges for x > 0 (where vpx > 0) [4(c)]. Hence Eygives rise to the dark lines delimiting the channel andcontributes to the central dark line visible for x < 0. Bzcontributes to the central dark line for x < 0, while forx > 0 it cancels the piling up on the channel axis given byEy and contributes to the external dark lines.We next introduce a modulation [described by a sin2ð2x=lÞ weight function, where l ’ 60 m] of B alongthe x direction. Such a field distribution describes a row oftori and represents the simplest possible extrapolation to a3D geometry of the magnetic field associated with thevortexlike patterns observed in the 2D PIC simulations.The resulting simulated proton image [Fig. 4(d)] is in quali-tative agreement with the experimental ones [Figs. 1(c) and2]. The PT also indicates that the proton deflection in they direction arises from the same effects described above,with additional longitudinal modulations in the proton den-sity introduced by the B dependence on x. We stress herethat the presence of a modulated magnetic field, introducedfor consistency with the 2D PIC simulations, is critical inorder to reproduce the experimental data and that the protondeflection induced by the electric field alone cannot explainthe experimental results.In conclusion, we have shown that stable modulatedpatterns observed via PPI inside laser-plasma channelscan be identified with the late time remnants of EM solitonsobserved in PIC simulations. PIC simulations also revealthe simultaneous presence of a quasistatic magnetic fieldassociated with a vortexlike current distribution, and PTsimulations evidence that such a magnetic field is essentialfor their experimental detection. The peculiar features ofthese structures, such as their magnetic nature, their orga-nization into periodical patterns, and their detailed 3Dtopology, should be stimulating for further theoretical andnumerical investigations.We acknowledge the support of the RAL/CLF staff. Thiswork has been supported by EPSRC Grants No. EP/E035728/1 (LIBRA consortium) and No. EP/C003586/1,by British-Council-MURST-CRUI, ESF-COST, TR18, andGRK1203 networks. Part of the simulations were per-formed at CINECA (Bologna, Italy) sponsored by CNR/INFM.*Present address: CEA/DSM/IRAMIS/LSI, ÉcolePolytechnique, 91128 Palaiseau, France, and LULI,Université Paris VI, 3 rue Galilée, 94200 Ivry-sur-Seine,France.†Present address: Institut für Physik, Universität Rostock,Rostock, Germany.[1] T. Dauxois and M. Peyrard, Physics of Solitons(Cambridge University Press, Cambridge, England, 2006).[2] G. A. Morou et al., Rev. Mod. Phys. 78, 309 (2006).[3] S. V. Bulanov et al., Phys. Fluids B 4, 1935 (1992); S. V.Bulanov et al., Phys. Rev. Lett. 82, 3440 (1999); N.M.Naumova et al., Phys. Rev. Lett. 87, 185004 (2001); T.Esirkepov et al., Phys. Rev. Lett. 89, 275002 (2002); T.Esirkepov et al., Phys. Rev. Lett. 92, 255001 (2004).[4] V. A. Kozlov et al., Sov. Phys. JETP 49, 75 (1979); P. 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Relativistic laser-plasma interaction

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Relativistic laser-plasma interaction

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