Theory and simulations of singly resonant optical parametric oscillators

Optical parametric oscillators have been known and used for a long time as efficient sources of non-classical states of light both below threshold of oscillation, where they generate squeezed vacuum states and bi-partite entangled states, and above threshold of oscillation, where they generate intensity correlated twin beams. The singly-resonant cavity, where only one of the three field involved in the parametric amplification process is resonated (signal), is in principle a simpler configuration to realize experimentally but, to the best of our knowledge, theoretical investigations of non-classical features of the light from a singly-resonant OPO (SROPO) are missing. One of the reasons is that SROPOs operate with strongly non-degenerate frequencies while much of the literature on squeezing focuses on the degenerate or close to degeneracy cases. Recent interest in non-classical correlations of the strongly non-degenerate regime of parametric down-conversion makes the study of entanglement in SROPO important for the optimization of coherent sources with fluctuations below the shot-noise level. There are clear technical advantages for SROPO configurations: only resonance of the signal field has to be maintained, continuous temperature tuning and suppression of mode-hopping. As a matter of fact even if the doubly resonant configuration, where both the signal and the idler fields are resonated, has a much lower threshold pump power, the tuning behavior is complicated and is massively affected by changes of the crystal temperature or pump wavelength, causing the signal and idler wavelengths undergoing jumps, and the tuning is generally non-monotonous. This is because the operation wavelengths are determined primarily by the requirement for simultaneous resonance for signal and idler, and not only by a phase-matching condition as in the case of singly resonant configuration. It is in this spirit that in Chapter 4 we apply the input-output theory of optical cavities to formulate a quantum treatment of a continuous wave singly-resonant optical parametric oscillator. This case is mainly relevant to largely non-degenerate signal and idler modes. We show that both intensity and quadrature squeezing are present and that the maximum noise reduction below the standard quantum limit is the same at the signal and idler frequencies in a way similar to the doubly resonant case. As the threshold of oscillation is approached, however, the intensity-difference and quadrature spectra display a progressive line-narrowing which is absent in the balanced doubly-resonant case. By using the separability criterion for continuous variables, the signal-idler state is found to be entangled over wide ranges of the parameters. We show that attainable levels of squeezing and entanglement make singly-resonant configurations ideal candidates for two-colour quantum information processes because of their ease of tuning in experimental realizations. Another very interesting feature of SROPOs which, this time, has no counterpart in the doubly-resonant regime is described in Chapter 5 where model equations for the evolution of signal and idler pulses in a synchronously pumped optical parametric oscillator are derived and numerically integrated. A novel regime of giant sub-threshold pulses driven by quantum fluctuations is described through the analysis of stability eigenvalues, growth factors and pseudospectra. Subthreshold pulses driven by quantum fluctuations are found at various mirror reflectivities in the non degenerate regime where signal and idler have different group velocities. Giant sub-threshold pulses open the possibility of observing macroscopic continuous variable entanglement with nonclassical features. This important feature is peculiar to the singly-resonant configuration and has no counterpart in the doubly-resonant regime. Very interesting classical features of SROPOs light are investigated in Chapter 6 where we show that spatio-temporal dynamics of singly resonant optical parametric oscillators with external seeding displays hexagonal, roll and honeycomb patterns, optical turbulence, rogue waves and cavity solitons. We derive appropriate mean-field equations with a sinc² non-linearity and demonstrate that off-resonance seeding is necessary and responsible for the formation of complex spatial structures via self-organization. We compare this model with those derived close to the threshold of signal generation and find that back-conversion of signal and idler photons is responsible for multiple regions of spatio-temporal self-organization when increasing the power of the pump field.Optical parametric oscillators have been known and used for a long time as efficient sources of non-classical states of light both below threshold of oscillation, where they generate squeezed vacuum states and bi-partite entangled states, and above threshold of oscillation, where they generate intensity correlated twin beams. The singly-resonant cavity, where only one of the three field involved in the parametric amplification process is resonated (signal), is in principle a simpler configuration to realize experimentally but, to the best of our knowledge, theoretical investigations of non-classical features of the light from a singly-resonant OPO (SROPO) are missing. One of the reasons is that SROPOs operate with strongly non-degenerate frequencies while much of the literature on squeezing focuses on the degenerate or close to degeneracy cases. Recent interest in non-classical correlations of the strongly non-degenerate regime of parametric down-conversion makes the study of entanglement in SROPO important for the optimization of coherent sources with fluctuations below the shot-noise level. There are clear technical advantages for SROPO configurations: only resonance of the signal field has to be maintained, continuous temperature tuning and suppression of mode-hopping. As a matter of fact even if the doubly resonant configuration, where both the signal and the idler fields are resonated, has a much lower threshold pump power, the tuning behavior is complicated and is massively affected by changes of the crystal temperature or pump wavelength, causing the signal and idler wavelengths undergoing jumps, and the tuning is generally non-monotonous. This is because the operation wavelengths are determined primarily by the requirement for simultaneous resonance for signal and idler, and not only by a phase-matching condition as in the case of singly resonant configuration. It is in this spirit that in Chapter 4 we apply the input-output theory of optical cavities to formulate a quantum treatment of a continuous wave singly-resonant optical parametric oscillator. This case is mainly relevant to largely non-degenerate signal and idler modes. We show that both intensity and quadrature squeezing are present and that the maximum noise reduction below the standard quantum limit is the same at the signal and idler frequencies in a way similar to the doubly resonant case. As the threshold of oscillation is approached, however, the intensity-difference and quadrature spectra display a progressive line-narrowing which is absent in the balanced doubly-resonant case. By using the separability criterion for continuous variables, the signal-idler state is found to be entangled over wide ranges of the parameters. We show that attainable levels of squeezing and entanglement make singly-resonant configurations ideal candidates for two-colour quantum information processes because of their ease of tuning in experimental realizations. Another very interesting feature of SROPOs which, this time, has no counterpart in the doubly-resonant regime is described in Chapter 5 where model equations for the evolution of signal and idler pulses in a synchronously pumped optical parametric oscillator are derived and numerically integrated. A novel regime of giant sub-threshold pulses driven by quantum fluctuations is described through the analysis of stability eigenvalues, growth factors and pseudospectra. Subthreshold pulses driven by quantum fluctuations are found at various mirror reflectivities in the non degenerate regime where signal and idler have different group velocities. Giant sub-threshold pulses open the possibility of observing macroscopic continuous variable entanglement with nonclassical features. This important feature is peculiar to the singly-resonant configuration and has no counterpart in the doubly-resonant regime. Very interesting classical features of SROPOs light are investigated in Chapter 6 where we show that spatio-temporal dynamics of singly resonant optical parametric oscillators with external seeding displays hexagonal, roll and honeycomb patterns, optical turbulence, rogue waves and cavity solitons. We derive appropriate mean-field equations with a sinc² non-linearity and demonstrate that off-resonance seeding is necessary and responsible for the formation of complex spatial structures via self-organization. We compare this model with those derived close to the threshold of signal generation and find that back-conversion of signal and idler photons is responsible for multiple regions of spatio-temporal self-organization when increasing the power of the pump field