Dispersion CompensationforReflection HolographybyArno KleinB.S., Perception and Cognition StudiesUniversity of Michigan in Ann Arbor, 1993Submitted to the Program in Media Arts and Sciences,School of Architecture and Planningin partial fulfillment of the requirements for the degree ofMaster of Science in Media Arts and Sciencesat theMassachusetts Institute of TechnologySeptember 1996@1996 Massachusetts Institute of TechnologyAll rights reservedSignature of AuthorProgram in kedia Arts and SciencesJune 15, 1996Certified byStephen A. BentonAllen Professor of Media Arts and SciencesProgram in Media Arts and SciencesThesis SupervisorAccepted byF 7!vStephen A. BentonChairDepartmental Committee on Graduate StudentsProgram in Media Arts and Sciences2Dispersion Compensationfor Reflection HolographybyArno KleinSubmitted to the Program in Media Arts and Sciences,School of Architecture and Planningon June 15, 1996 in partial fulfillment of the requirements for theDegree of Master of Science in Media Arts and SciencesABSTRACTThe work outlined in this thesis is the exploration, application, and evaluation ofthe technique of dispersion compensation to reduce color blurring of reflectionholograms. In particular, the technique of illuminating a hologram with a predispersinggrating is applied to the development of full-color, full-parallax, reflection holographicstereograms in various display formats, from a large, open-air viewstation to a compact,light guide-mounted hologram. The effects of wave-front curvature are assessed forreducing the grating size for more compact displays.A blur equation is derived and experimentally contrasted with Benton's blurequations with the use of spectrophotometer measurements of an experimental grating,and raytracing computer programs included in the appendices.Finally, two approaches were developed that provide a consistent means ofdesigning dispersion-compensation gratings tailored to realizable, desired geometries.Both approaches are effectively reverse-raytracing methods, beginning from the pupil andending at the extended light source. Compensation is optimized for image points along aprescribed viewing axis, and depending on the geometry, along all lines of sight parallelto this axis. The first approach solves for a grating design where the grating isreconstructed with the same angles to which it was exposed. The second, more exactapproach, is an optimization of a matrix expression containing the relevant angular and/orvertical focus equations, using the Moore-Penrose pseudoinverse function. This lattermethod will enable the reader to design a grating that will play out at the desired anglesfor component wavelengths at as close to the desired distances as possible.Thesis Supervisor: Stephen A. BentonTitle: Allen Professor of Media Arts and Sciences4Dispersion Compensation for Reflection HolographyThe following people served as readers for this thesis:Certified byEdward AdelsonAssociate Professor of Vision ScienceCertified byStephen D. FantoneSenior Lecturer, Mechanical EngineeringPresident, Optikos Corp., Cambridge, MACertified by _Emmett N. LeithProfessor of Electrical Engineeringand Computer ScienceUniversity of Michigan, Ann Arbor6%a ouy othe and kathe48AcknowledgmentsThis section is inevitably going to be woefully incomplete, so I will mention onlythe names of those who have had a direct impact on the production of this thesis.First, I would like to thank Steve Benton, for the opportunity to work in theMedia Laboratory, and for his extremely helpful e-mail correspondence.Sincere thanks also goes to my other thesis readers for reviewing this document:Professor Ted Adelson, Professor Stephen Fantone, and Professor Emmett Leith.The Spatial Imaging Group, as of spring, 1996:Mike Klug, the other primary individual constituting the "we" that is usedthroughout the thesis. Mike shot the holograms illuminated by the gratings,and pulled together all elements to complete the viewstations for our sponsors.My reluctant lab mentor: "Don't you have a thesis to work on?"Ravi Pappu, my colleague and cohort, for fielding my math questions, howeverfar afield: "On what planet does sine of zero equal one?!" Thank you forbearing the innumerable intrusions!Paul Christie, my 6th and longest-lasting officemate, for sharing my abode:"It would probably be safer if you'd sleep farther away from the door."Have a wonderful wedding!Melissa Yoon, for countless administrative feats: "The package should have arrived inEngland this morning - Anything more you'd like to Fedex today?"Take care and good luck with your (near) future endeavors...Wendy Plesniak, for inspirational work that adds touch to an otherwise ghostly medium.Carlton Sparrell: "You didn't know? I'm leaving in two weeks."Michael HalleOur undergraduate researchers; in particular, Adam Kropp and Benjie Chen, for theircomputer graphics work instrumental to providing the subjects of our holograms.John Sutter, our Japanese affiliateBarrett Comiskey, director of darkroom #2, my friend, roommate, and fellowholographer: "Your thesis is my thesis."Mitch Henrion, for edgelit enthusiasm.Everyone in the Physics and Media Group, for providing me with a friendly place towork on my thesis round the clock.Steve Mackara, Kirk Steijn, and Bill Gambogi of DuPont.Steve Mackara was kind enough to shoot the master grating and some contact copies forthe final viewstations of Chapter 5.Kirk Steijn was very helpful in providing TK Solver information.Our sponsors: Honda R&D Co., IBM, and NEC, for their gratious support.Sincere gratitude to those who have supported my work in the past:My familyJason Smith, my close friend and holography partner since high school,for his shared vim, vigor, and vitality.Kashiko Kodate and Katsuma-sensei, who made it possible for me to do independentholography work in Japan.Professor Emmett Leith once again, for his generous offer to use his facilities during mystay at the University of Michigan: "Here comes the night shift."Helpful discussions, distractions:Professor Seth Teller, for discussions, not distractions.Ranjana Mitra, my mutual distraction: "I know you've got a thesis, but do you want toscale some buildings tonight?"David James, my personal D.J., for providing musical interludes from afar,such as The Blur! They helped me to get focused...Kory and Ian - Congratulations, and I wish you all the best for your future together!Text in this thesis was written on Microsoft Word for Windows 95, Version 7.0.Mathematics segments were written with Microsoft Equation 2.0,and for Chapter 2, Mathcad 5.0+ (MathSoft Inc.).Figures were drawn with Microsoft PowerPoint for Windows 95, Version 7.0.Graphs in Chapter 1 were made using Microsoft Excel for Windows 95, Version 7.0.Computer programs were written in:TK Solver+, Version 1.2 (Universal Technical Systems),for the first program in Appendix 4.Matlab (The Math Works, Inc.), for the programs of Appendices 5 and 6,and the second program in Appendix 4.Dictation was performed for parts of the thesis, using Dragon Dictate (Classic Edition)for Windows, Version 2.01 (Dragon Systems, Inc.).Table of ContentsPage: 13 Extended Table of ContentsPage: 17 List of Figures and TablesPage: 21 IntroductionPage: 23 Dispersion compensation flowchartPage: 25 1. Blur in holographic imagesPage: 49 2. Trigonometric derivation of a single-plane blur equationPage: 65 3. Introduction to dispersion compensationand a brief review of its application to holographic displaysPage: 77 4. Full-parallax reflection holographic stereogramsPage: 85 5. Example design of a dispersion-compensated,full-parallax holographic viewstationPage: 97 6. Wave-front shapes, compact displays, and designing gratingsPage: 125 Conclusions and future workPage: 129 Al. Single-plane trigonometric raytracing formulasPage: 133 A2. Derivation of a vector blur equationand its reduction to the blur equation of Chapter 2Page: 141 A3. TK Solver+ and Matlab programsapplying the derived equationsPage: 155 A4. A Matlab program for the design ofdispersion-compensation gratings (Chapter 6)Page: 159 A5. A Matlab program attempting to solve for a pre- and post-dispersing grating set flush against a hologram (Chapter 6)Page: 161 References12Extended Table of ContentsPage: 17 List of Figures and TablesPage: 21 IntroductionPage: 23 Dispersion compensation flowchartPage: 25 1. Blur in holographic imagesPage: 25 1.1 Blur in holographic imagesPage: 26 1.2 Basic principles of holographyPage: 29 1.3 The Gabor zone platePage: 32 1.4 Longitudinal and lateral dispersionPage: 35 1.5 Overlapping zone platesPage: 35 1.6 Perception of color blur: acuity and stereoacuityPage: 38 1.7 Resolution vs. detection / perceived vs. absolute color blurPage: 40 1.8 Spectral intensity filter #1: The retinaPage: 41 1.9 Spectral intensity filter #2: Diffraction efficiencyof the hologram(s)Page: 42 1.10 Spectral intensity filter #3: The illuminantPage: 44 1.11 Resolution of color-blurred reflection holographic imagesPage: 47 1.12 Factor for perceived blur sizePage: 49 2. Trigonometric derivation of a single-plane blur equationPage: 49 2.1 Derivation of a color blur equationPage: 53 2.2 Inclusion of source-size blur into a final blur equationPage: 55 2.3 Approximation to the final blur equationPage: 55 2.4 Benton's color blur equation derivationPage: 57 2.5 Benton's achromatic angle derivationPage: 59 2.6 Benton's source-size blur equation derivationPage: 60 2.7 Benton's final blur equationPage: 61 2.8 Summary and comparison of the blur equationsPage: 62 2.9 Blur measurements using a spectrophotometerPage: 65 3. Introduction to dispersion compensationand a brief review of its application to holographic displaysPage: 65 3.1 The first dispersion-compensation displays:single view angle transmission hologram / gratingPage: 70 3.2 Single viewpoint dispersion compensation: BurckhardtPage: 71 3.3 Diffractive and refractive mediaPage: 71 3.4 Compact single view angle displays: BurckhardtPage: 72 3.5 Other transmission displays: Boj, et al., and BentonPage: 73 3.6 Reflection displays: Bazargan, KubotaPage: 77 4. Full-parallax reflection holographic stereogramsPage: 77 4.1 Making horizontal-parallax-only holographic stereogramsPage: 79 4.2 The transfer hologramPage: 81 4.3 Full-parallax holographic stereogramsPage: 85 5. Example design of a dispersion-compensated,full-parallax holographic viewstationPage: 86 5.1 The exposure wavelengthPage: 86 5.2 The holographic recording materialPage: 87 5.3 The exposure and reconstruction geometriesPage: 93 5.4 Selecting an illumination sourcePage: 95 5.5 The resulting displayPage: 97 6. Wave-front shapes, compact displays, and designing gratingsPage: 97 6.1 Sources of blur in edgelit hologramsPage: 99 6.2 Pulling the grating and hologram closer togetherby using a light guide: the steep angle formatPage: 101 6.3 Compensation gratings with diverging wave frontsPage: 102 6.4 Grating playout distancesPage: 104 6.5 Dispersion compensationand grating output wave-front shapesPage: 105 6.6 The planar wave front casePage: 107 6.7 The diverging wave front case (phase-conjugate)Page: 108 6.8 The diverging wave front case (non-phase-conjugate)Page: 110 6.9 Consistent method for designing gratingsPage: 110 6.10 Solving for optimized anglesPage: 112 6.11 Solving for optimized distancesPage: 112 6.12 Designing a diverging wave-front gratingfor a compact displayPage: 114 6.13 Designing a perfectly reconstructing diverging gratingPage: 116 6.14 Designing a diverging grating that plays out ideal anglesand optimized distancesPage: 121 6.15 An attempt to design a pre- and post-dispersingcompensation gratingPage: 125 Conclusions and future workPage: 129 Al. Single-plane trigonometric raytracing formulasAl. 1 The X- and Z- raytracing equationsA1.2 The Welford and X-equationsA1.3 The Z-equationPage: 130 A 1.4 Spectrophotometer dataPage: 132 A1.5 The horizontal and vertical focus equationsPage: 145 A2. Derivation of a vector blur equationand its reduction to the blur equation of Chapter 2Page: 133 A2.1 Color blur equation derivationPage: 135 A2.2 Approximation proofPage: 138 A2.3 Source-size blur derivationPage: 141 A3. TK Solver+ and Matlab programsapplying the derived equationsPage: 151 Alternative Matlab programPage: 155 A4. A Matlab program for the design ofdispersion-compensation gratings (Chapter 6)Page: 159 A5. A Matlab program attempting to solve for a pre- and post-dispersing grating set flush against a hologram (Chapter 6)Page: 161 References16List of Figures and TablesPlease note that in the figures, holographic images that seem to float outside of thesolid angle subtended by the observer's eyes and the hologram are merely for illustrativepurposes. Visible image points must lie along a line of sight from the eye to thehologram without the aid of smoke, mirrors, or re-emitting gases.Figure 1.1: Successive wave fronts emanating from a point source of lightFigure 1.2: The Moire fringe patternFigure 1.3: The formation of a reflection hologram (after Jeong 1980)Figure 1.4: The reconstruction of a reflection holographic image (Jeong 1980)Figure 1.5: Interference pattern forming a Gabor zone plateFigure 1.6: Spatial frequenciesFigure 1.7: Formation of a Gabor zone plate, with more closely spaced fringes toward its periphery(after Greivenkamp 1995)Figure 1.8: Illumination of a Gabor zone plate with a single wavelength, with resultingvirtual and real image pointsFigure 1.9: Phase-conjugate illumination of a Gabor zone plate with a single wavelengthFigure 1.10: Illumination of a Gabor zone plate with two wavelengths, and the resultinglongitudinal dispersionFigure 1.11: The formation of an off-axis reflection hologram (after Jeong 1980)Figure 1.12: Reconstruction of an off-axis reflection hologramFigure 1.13: Dispersion in an off-axis reflection hologram: illumination by two wavelengthsFigure 1.14: Examples of different fringe patterns (Geisler and Banks, fig. 7.25.36-19)Figure 1.15: Different acuity tests (Geisler and Banks, fig. 7.25.36-19)Figure 1.16: StereoacuityFigure 1.17: Airy disk diffraction pattern (Riggs 1965, fig. 11.10)Figure 1.18: Rayleigh's criterion for just resolvable points (Riggs 1965, fig. 11.11)Figure 1.19: Two blurred points that are just resolvableFigure 1.20: Photopic curve data from Wyszecki and Stiles (1967, table 4.2)Figure 1.21: Illuminant A curve (Wyszecki and Stiles 1967; MIT 1936)Table 1.1: Data from the spectral luminous efficiency (Wyszecki and Stiles 1936)and Illuminant A relative intensity (MIT 1936) curvesFigure 1.22: Intensity filters: the retina and illuminantFigure 1.23: Just resolvable points defined by the diffraction efficiency curveFigure 1.24: Perceived blur size factorFigure 2.1: Spectral components from different points on the hologram contributing to blurFigure 2.2: Dispersed focus at the pupil, perceived as color blur, from Appendix 2Figure 2.3: Perceived color blur and source-size blur heightFigure 2.4: Dispersion for the cases of m = +1 or m = -1 order image pointscombined in one diagramFigure 2.5: The achromatic angle for the cases of m = +1 or m = -1 order image pointscombined in one diagramFigure 2.6: Wavelength-dependent focusing power of a hologram,and the achromatic angle (Saxby 1992)Figure 2.7: The perceived blur angle according to Benton (1994)Figure 2.8: (from Appendix 1)Figure 2.9: Measurement 3 setupTable 2.1: Blur length data for the vertical focus of 543.1 and 632.8 nm wavelengthsFigure 3.1: DeBitetto's original diagram discriminating the effectsof diffraction orders on relative dispersionFigure 3.2: Forward and reverse raytracing the spectral components through a transmission hologramFigure 3.3: Raytracing the dispersed rays from the hologram through a transmission gratingFigure 3.4: A schematic representing the original geometry of the three researchers in reverse(Latta 1972)Figure 3.5: Single view angle dispersion compensationThis schematic represents the blurring of image points off of the local normalfor a collimated reference /collimated illumination geometry.Figure 3.6: Burckhardt's (1966) use of a louver film to block the zero order of the hologram in asingle view angle dispersion-compensation displayFigure 3.7: Burckhardt's single viewpoint dispersion compensation technique(Collier, Burckhardt, and Lin 1971)Figure 3.8: Example of the complexity of modern head-up displays incorporatingdispersion-compensation gratings (Roberts, et al. 1995)Figure 3.9: Bazargan's display, manufactured by Holoplex Systems, Ltd. (Syms 1990, fig. 10.5-2)Figure 3.10: Benton's patent drawing of the white-light transmission hologram transfer step (1972)Figure 3.11: Benton's patent drawing of the special achromatizing gratingused in an exposure step (1985)Figure 3.12: (a) Distortions of a square image across the visible spectrum (400-700 nm)(b) Single view angle dispersion compensation for reducing lateral chromatic aberration(Bazargan 1986, fig. 7.20)Figure 3.13: Bazargan's suggested approach for making a full-color, dispersion-compensated displayFigure 3.14: Kubota's reflection /reflection dispersion-compensated displayFigure 3.15: Intensity profile of the reconstructed image pointsin Kubota's single view angle dispersion-compensation displayFigure 4.1: Sequential images are taken, and later projected to exposea hologram with an appropriately scaled geometry (Redman 1968).Figure 4.2: DeBitetto's original drawing of the spatial multiplexing techniquemost commonly used in display holographic stereograms today (DeBitetto 1968)Figure 4.3: A horizontal-parallax-only holographic stereogram (HPO HS, or Hi)Figure 4.4: The first (a) and last (b) vertical strip exposures of a holographic stereogram (top view)Figure 4.5: Transfer setupFigure 4.6: Phase-conjugate illumination of the HPO HS H2,and reconstruction of an orthoscopic, real imageFigure 4.7: Benton's pseudocolor technique for producing achromatic or full-color images (1983, 1988)Figure 4.8: An example of different perspective views of a full-parallax hologram(image care of Michael Klug)Figure 5.1: Uncompensated and compensated dispersion of a holographic imageFigure 5.2: Schematic of the final viewstation with tipped gratingFigure 5.3: Schematic of the exposure (2250) and illumination (450) geometriesof the dispersion-compensated H2 (from Chapter 4)Figure 5.4: The desired grating playout (450)Figure 5.5: The required master grating exposure geometry (56.32' degrees),taking into account shrinkage of the photopolymerFigure 5.6: The master grating exposure holographic table setupFigure 5.7: The transfer grating exposure holographic table setupFigure 5.8: filament orientationFigure 5.9: Bulb with an ellipsoidal, dichroic reflectorFigure 5.10: Plane-wave grating dispersion-compensation display (photos care of Michael Klug)Figure 6.1: Non-chromatic blurring of the edgelit hologramFigure 6.2: Exposure of an edgelit hologram illuminated with the same source distanceand folded beam path (Upatnieks 1992)Figure 6.3: Reduction of chromatic and non-chromatic blurring of the edgelit hologramusing a wave-front-shaping steep angle HOEFigure 6.4: Upatnieks' achromatic edgelit hologram (1992)Figure 6.5: A comparison of Birner's reflection HOE/transmission steep angle H2with the author's reflection/reflection displaysFigure 6.6: Maximum acceptable perceived longitudinal blur lengthFigure 6.7: Collimated, phase-conjugate, single view angle dispersion-compensated hologramwith maximum depths (3 arc minutes of resolution):Figure 6.8: Uncollimated, phase-conjugate, single view angle dispersion-compensated hologramwith maximum forward compensated depths (1 arc minute of resolution)Figure 6.9: Uncollimated, non-phase-conjugate, single view angle dispersion-compensatedhologram with extreme field curvature (diverging/diverging geometry)Figure 6.10: Ideal predispersed hologram illumination angles and distancesshould be equivalent to the ideal predispersed grating output angles.Figure 6.11: Exposure of the transfer (H2) hologram with a diverging reference sourceFigure 6.12: Reconstruction of the H2 with a diverging illumination sourceFigure 6.13: The final table setup for the 750/750 diverging/diverging geometryFigure 6.14: A graph of the exposure and illumination angles for a range of central output angles.This graph is the output from the Matlab 'plot' command in the program of Appendix 4.Figure A1.1: Grating vector cloud (Goodman, 1996)Figure A1.2: Dispersion measurements with an Oriel spectrophotometerTable A1.1: Measurements of dispersion with an Oriel spectrophotometerFigure A1.3: Diagram accompanying Benton's vertical focus equation derivation (Benton, 1994Figure A2.1: Magnitudes and directions for the vector derivation of a blur equationFigure A2.2: Trigonometric diagramFigure A2.3: A dispersed focus at the pupil creating the perception of blur height20Introduction"True observers of nature, however they may differ in opinion in other respects, will agree that allwhich presents itself as appearance, all that we meet with as phenomenon, must either indicate an originaldivision which is capable of union, or an original unity which admits of division, and that the phenomenonwill present itself accordingly. To divide the united, to unite the divided, is the life of nature; this is theeternal systole and diastole, the eternal collapsion and expansion, the inspiration and expiration of the worldin which we live and move."-- Goethe, Theory of Colours (1810, 293-4)It is the sincere hope of the author that the following body of work, the result of ayear of study, may serve as a helpful starting point for future researchers interested in theproblems of image blur in holographic displays, and grating design. In order to expeditethe reader's inquiry, this Introduction will serve as a brief outline of the contents of eachchapter. The "Conclusions and future work" section will parallel this summary.Chapter 1 consists of two parts: The first is an introduction to holography anddispersion, or color blur. In the second part, we will estimate the degree to which ourperception of blur affects the resolution of a holographic image.In Chapter 2, we derive and experimentally test a trigonometric blur equation forthe absolute raytraced size of a blurred image. We then apply the perceived blur sizefactor from Chapter 1 to find the perceived size of this image. The author's blur equationis compared with Benton's blur equation.In Chapter 3, we introduce dispersion compensation, a method to correct forcolor blur, and we give a brief history of the application of dispersion compensation todisplay holography.Chapter 4 gives an introduction to full-parallax holographic stereograms, theparticular type of hologram we will be illuminating in Chapter 5's display.Chapter 5 is the culmination of the previous chapters' findings, applied to aplane-wave viewstation display. Laboratory techniques for forming a dispersioncompensating grating are outlined here.Chapter 6 compares the effectiveness of compact dispersion-compensationdisplays that do and do not use a plane-wave grating. Then, a consistent means fordesigning dispersion-compensation gratings is put forward in the chapter and in theaccompanying computer programs of Appendices 3 and 4. It is shown that the methodpresented here (the use of the Moore-Penrose pseudoinverse) compares favorably withthe perfect reconstruction geometries implemented in the viewstations of Chapters 5 and6 and is well-suited to finding optimized angles, and also approximate distances, with lessprecision. It therefore is tailored to compensate for lateral and longitudinal dispersion asa function of eye position and orientation.22Dispersion compensation flowchartHologram view angle withoptimized compensationHologram illumination anglesfor a given bandwidthIdeal grating outputangles, distancesExposure geometry forperfect reconstruction,ideal and with grating tilt3.1, 6.9, 6.13, A36.9, 6.13, A3Optimizing gratingdesigns for ideal angularand distance playout(imperfect reconstruction)5.3, 6.13, A36.9-6.11, 6.14, A45.1-5.2Ch. 5, 65.4 6.1-6.26.3, 6.5-6.8, 6.12, A4The numbers next to the boxes represent relevant sections.5.3, 6.12-6.1324Chapter 1Blur in holographic images1.1 Blur in holographic imagesThe primary sources of blur in holographic images are the spectral width andphysical size of the illumination source, as well as non-ideal properties of the exposuremedium and substrate. For full-parallax1 holographic displays with controlled lightingconditions, the spectral width usually constitutes the greatest source of image blur, and assuch will comprise the bulk of our discussions. As most optical aberrations have achromatic component, we will refer to the blur due to polychromatic illumination simplyas "color blur." In this first chapter, we will introduce blur as we cover basic principlesof holography. Then we will determine how our perception of blur affects the resolutionof a holographic image.Because holograms rely on diffraction, and diffraction is wavelength dependent(that is, red wavelengths will deflect at greater angles through diffractive media than willblue wavelengths), a single holographically recorded object replays in a different locationfor each component wavelength of the illumination. This separation of componentwavelengths, or dispersion, is analogous to the phenomenon of dispersion in refractivemedia, such as a prism. The extent of the spectral width of the source determines theextent to which light will disperse through a given medium, and therefore a broadbandwhite light source can render an image an indistinguishable rainbow blur.One trivial solution to the problem of color smearing of images is to illuminatethe hologram instead with a very narrow spectral band source, such as a laser or an arclamp with an interference filter. These are impractical for most viewing situations and,unless one sets up three such lamps, offer only monochromatic playback. Anothersolution is to produce better recording materials or to find better processing methods forproducing more narrow band playout. In either case, whether a hologram is illuminatedby a narrow spectral band source or a hologram with a narrow band playout is illuminatedby an intense broad band source, much of the illumination light is wasted. To producebrighter, full-color, full-parallax holograms that are sharp and deep, correcting for colorblur in white light illumination displays is vital.1 "Full-parallax" refers to the change in perspective vertically as well as horizontally. Synthetic full-parallax holograms are the subject of Chapter 5.Chapter 1: Blur in holographic images1.2 Basic principles of holographyTo better understand color blur, we will take a brief look at some basicinterference patterns, and then invoke the limiting case of the first type of hologram thatwas invented by Dennis Gabor in 1947 (Gabor 1948). If we have a source point acting asa harmonic oscillator, radiating waves in three dimensions, we can represent theinstantaneous wave fronts emanating from the center as concentric spheres, whose radiiare incremented by one-half of a wavelength (/2). The direction of travel for any pointon the wave front is simply the direction of expansion, the local normal to the sphere(Figure 1.1). We will call the path the light's direction takes a ray, and tracing its path iscalled raytracing./2Figure 1.1: Successive wave fronts emanating from a point source of lightChapter 1: Blur in holographic imagesThe superposition of two such wave fronts results in an interference pattern ofhyperboloidal surfaces:Figure 1.2: The Moire fringe pattern is a helpful visual aid to understanding interference effects 2.Here we have the oscillating source above in isolation, and interfering with another source.(a) The spherical wave fronts from a single light source(b) The superposition of wave fronts from two light sources(c) The interference pattern created by the two sources (Jeong 1980)If the two sources are mutually coherent then this interferencepattern will remain constant over time, a "standing wave." We may place aphotosensitive recording material anywhere in the volume of space around or betweenthese two sources to record the interference pattern. If placed between the two sources,the recording material will subsequently form partially reflective, hyperboloidal surfacesspaced through its volume:2 For and in-depth look into the analogy between moire fringe patterns and interference fringes,Abramson's book (1981) is an invaluable guide. A summary table may be found on page 9 of the book.Chapter 1: Blur in holographic imagesFigure 1.3: The formation of a reflection hologram (after Jeong 1980)If the fringe pattern in this recording material retains its structural integrity and isthen re-illuminated by a wave front from one of these two sources, this wave front will bepartially reflected off each hyperboloidal surface and will interfere with itself. Theinterference determines the intensity of the light reflected off these surfaces in differentdirections. Light will be reinforced more in some directions than in other directions.Light traveling in directions corresponding to the directions light traveled from the othersource during exposure are constructively interfered, and a reflection holographic imageof the other source position will be formed (Figure 1.4).Figure 1.4: The reconstruction of a reflection holographic image (Jeong 1980)The illumination source point is to the right, the reconstructed image to the left.Chapter 1: Blur in holographic images1.3 The Gabor zone plateThe particular interference pattern pictured in Figure 1.3, formed by source pointson a line (the optical axis) intersecting the recording material, is called a "Gabor zoneplate," or "in-line" hologram. If we had set the recording material along the optical axis,but not in between the two source points, a different form of Gabor zone plate would beformed, an in-line transmission hologram:Figure 1.5: Interference pattern forming a Gabor zone plateThe interference pattern of the in-line transmission hologram is composed ofconcentric circles centered around the axis. Because the angle between the linecontaining the source points and the axis is zero, so too is the corresponding fringedensity, or spatial frequency 3 of the interference pattern. The angle between these twopoints with respect to a point on the hologram off-axis increases toward the edges, andthe corresponding spatial frequency increases (See Figures 1.6 and 1.7).3 Spatial frequency refers to the reciprocal of the distance between elements of a repeated pattern, such asholographic fringes: f = sin(Ore)- , in cycles/mm.A = wavelengthChapter 1: Blur in holographic images(a) (b)(b)-. zFigure 1.6: As the apparent distance between source points increases, so toodoes the spatial frequency of the fringes. In this figure, we are looking at the twosources from two different points on the recording material, one point close to theoptical (z-) axis (a), another farther off-axis (b).Chapter 1: Blur in holographic imageszFigure 1.7: Formation of a Gabor zone plate, with more closelyspaced fringes toward its periphery (after Greivenkamp 1995)Because the center of the zone plate has zero spatial frequency 4, light illuminatingthe hologram on axis will pass through undiffracted. Diffractive, or focusing powerincreases with increasing spatial frequency, so the zone plate's edges deflect incidentlight the most, in the direction of the original wave front from the other source point, aswell as toward a focused, real image point in front of the observer. This second image inline with the first, is sometimes referred to as a "twin image." In Figure 1.8, the twinimage is a real, focused image.4 There may or may not be a fringe in the center of a transmission in-line hologram, depending on therelative phase of the two exposure beams. The reflection in-line hologram has fringes parallel with thesurface. In either case, the center of the zone plate has no diffractive power for on-axis rays.Chapter 1: Blur in holographic imageswavefrontnormalsGabor zone platevirtual image real image(order m = +1) (order m = -1)Figure 1.8: Illumination of a Gabor zone plate with a single wavelength, with resultingvirtual and real image pointsPhase-conjugate illumination is essentially a reversal of the directions of the raysthat exposed the hologram. This "time-reversed" illumination results in a real imagepoint projected where the original object point was located. Here, the twin image is avirtual image behind the hologram plane from the observer (Figure 1.9).real image virtual image(order m = -1) (order m = +1)Figure 1.9: Phase-conjugate illumination of a Gabor zone plate with a single wavelength1.4 Longitudinal and lateral dispersionAs the spacing of between the fringes determines what wavelengths willconstructively interfere at different angles, it is possible to produce another image point ata different location by illumination with a different wavelength. The focusing power istherefore not only a function of spatial frequency, but also of wavelength, and thereforethe wavelengths illuminating a Gabor zone plate focus to different points along the axis.The color blur of the longitudinally displaced images of the source points is termedlongitudinal color blur and is not noticeable by an observer situated along this axis. If,however, the observer moves off the axis, the color blur becomes quite apparent.Chapter 1: Blur in holographic imagesBR Rdispersedvirtual imagepointsdispersedreal imagepointsFigure 1.10: Illumination of a Gabor zone plate with two wavelengths, red (R) and blue (B),and the resulting longitudinal dispersionLikewise, if we take a source point off-axis during recording of the hologram,white-light illumination of the hologram will result in lateral color blur. This off-axishologram, invented by Leith and Upatnieks (1962), can eliminate the twin image so thatthe observer sees only one image.Figure 1.11: The formation of an off-axis reflection hologram(after Jeong 1980)Chapter 1: Blur in holographic imagesR B- .11.Figure 1.12: Reconstruction of an off-axis reflection hologram,where the twin image is eliminated. The illumination source pointis to the right, and the virtual image is to the left.Figure 1.13: Dispersion in an off-axis reflection hologram:The broad band illumination source point to the right results inRed and Blue image points to the left.Chapter 1: Blur in holographic imagesThe lateral color blur may be explained by the fact that the source points arelaterally displaced, creating a different interference pattern, and therefore the off-axiswhite light illumination of the hologram will focus points of different colors so that theyare laterally displaced, as well as longitudinally displaced (as in the in-line case). Theresulting image not only suffers from spherical aberration, but also from coma, and mostsignificantly from astigmatism (Latta December 1971).1.5 Overlapping zone platesAn object illuminated by one of these sources may itself be considered acollection of a vast number of radiating sources, with spherical wave fronts emanatingfrom each point location on its surface. We may therefore consider a hologram formedby an illuminated object to be a composite of a vast number of overlapping zone plates, orgratings. We will assume that diffraction by each of these gratings is independent ofdiffraction by the other gratings. As the points on the object's surface may be in disparatelocations, the resulting fringes formed have different orientations and spacings. The moreextreme the distances between these object points become, the closer their fringe patternswill resemble those of different hologram geometries (Figure 1.14).(a) (b)(c) (d)Figure 1.14: Examples of different fringe patterns (Goodman 1996):(a) plane-wave transmission grating(b) general transmission grating(c) plane-wave reflection grating(d) general reflection grating1.6 Perception of color blur: acuity and stereoacuityWe will now attempt to quantify the amount of color blur we perceive when welook at a holographic image. George and McCrickerd (1969) derived the ultimate angularresolution of a hologram and of a holographic stereogram (to be described in Chapter 4)Chapter 1: Blur in holographic imagesaccording to depth-of-focus and diffraction blurring considerations, and found the formerto be 1 (L = length of the hologram) and the latter to be (1 = length of the exposureLaperture5). As we are concerned with blur of a considerably more insidious nature, due toa non-ideal illumination source, we will neglect George and McCrickerd's Gaussianbeam analysis, and instead derive a filter factor that will determine the perceived imagepoint size from an absolute raytraced size. The result of this analysis could have asignificant effect on the raytraced blur equation derived in the following chapter.For this analysis, it would be helpful to introduce some basic concepts and datafrom the psychophysics literature. In particular, data6 on measurements of resolution,visual acuity and stereoacuity, will provide us with ideal quantities with which we maycompare the resolution of dispersed holographic images. We will refer to visual acuity asa measurement of one eye's ability to just resolve two laterally spaced points, whereasstereoacuity will be a measurement of both eyes' ability to just resolve two longitudinallyspaced points. Both measurements will be in terms of the reciprocal of the minimumangle of resolution, so a high visual acuity means that the minimum angle of resolution issmall. The measurements will rely to some degree on parameters such as stimuliluminance and choice of stimuli (Figure 1.15).In the case of acuity under ideal conditions, the average threshold for resolution inthe grating task (resolving closely spaced lines) is considered to be about one arc minute7(1 arc minute = 1*), and under optimal conditions, the Landolt ring and letter acuity60tasks result in an average threshold of about 30 arc seconds 8 (1 arc sec = - arc minute).60For high intensities of light (above 4000 meter-candles), acuity scores as low as 24 arcseconds have been measured. For vernier acuity, a hyperacuity task (where resolution ofthe image projected on the retina exceeds the intercone spacing), resolution can be as lowas 2 arc seconds.5 The exposure aperture is an aperture in a mask through which a small area of a hologram may beexposed. The significance of the exposing small adjacent areas of a holographic recording material willbecome apparent in Chapter 4.6 Unless footnoted, the psychophysics data presented here is repeated from Riggs (1965)7 Grating task data is taken from Riggs (1965, 326)64 arc sec (Lister)50 arc sec (Hirschmann)52 arc sec (Bergmann)64 arc sec (Helmholtz)56 arc sec (Uhtoff)64 arc sec (Kobb)35 to 40 arc sec (Shlaer (1937), Keesey (1960))8 Handbook of Optics, vol. I, pp. 25.36-7 (1995)Chapter 1: Blur in holographic imagesrating vernierLandolt letterFigure 1.15: Different acuity tests(Geisler and Banks, fig. 7.25.36-19)Stereoacuity can vary depending on interpupillary and fixation distances. For anaverage interpupillary distance of 65 mm, the mean retinal disparity threshold has beenfound to be approximately 20 to 40 arc seconds for shorter distances (40 cm)9, and 4 to 10arc seconds for longer distances (65 to 130 cm). 0 "1mean retinaldisparity thresholdFigure 1.16: stereoacuityAs we will be calculating resolution of image points subjected to color blurprimarily in the vertical direction due to a vertically offset illumination source, the valueof one arc minute of lateral resolution will be sufficient for much of our work. Alsoimportant to note is that acuity depends on wavelength for low to moderate levels ofillumination. Acuity is best for the middle of the visual spectrum (yellow-green) anddegrades toward the longer (red), and is worst for the shorter (blue) wavelengths. Ourprototype viewstation at the time of writing displays holograms exposed andreconstructed with a central wavelength of 514.5 nm. This wavelength is roughly in thecentral region of the visible spectrum, and therefore our ability to resolve image points forsuch an image should be better than for a full-color image in such a display.1.7 Resolution vs. detection / perceived vs. absolute color blur9 G. Heron, et.al (1985), repeated in Charman (1995)10 McFadden paper: 2-12 arc sec range amongst his six subjects11 Repeated from Graham (Chapter 18, 1966) in Charman (1995)Chapter 1: Blur in holographic imagesTo simply assess the perceived size of a single blurred image point by calculatingthe directions of the light rays that we can detect could prove inadequate. The reason forthis is that our ability to detect a stimulus is dependent on many factors, including notonly properties of the stimulus itself such as contrast and luminance, but also the state ofthe observer. The threshold for perceiving light ranges across a logarithmic scaledepending on our level of light adaptation, and under optimal conditions, we can detect alight source when only a few photons impinge on our photoreceptor cells. Also, becausethe diffracted image of a point source is a set of concentric rings like those of the Gaborzone plate about an "Airy disk," very small stimuli (critical widths less than ten secondsof arc 12) appear to be the same size.©Figure 1.17: Airy disk diffraction pattern. The lines merely ascribe areas of the diffracted patternto points on the corresponding intensity curve (Riggs 1965, fig. 11.10).We will therefore rely not on absolute detectability or brightness discrimination ofa single blurred image point to calculate its apparent size. Instead, we will calculate theresolvability of two closely situated points by defining a maximum overlap. One modelto help us better understand resolvability is Rayleigh's criterion. Rayleigh's criterionstates that the images of two points are considered just resolvable if the distance betweenthem is at least the distance from the center of one Airy disk to its first minimum.Figure 1.18: Rayleigh's criterion for just resolvable points. Here are two overlapping Airy disk patterns,the images of two closely spaced points. As in Figure 1.17, the lines merely ascribe areas of the diffractedpattern to points on corresponding intensity curves (Riggs 1965, fig. 11.11).Equation (1.1): Rayleigh's criterion is given by the relation (Riggs 1965): O = 1.22AdChapter 1: Blur in holographic images12 (Riggs 1965, 322)where o is equal to the minimum resolvable angular separation between the two centers,X is the wavelength, and d is the diameter of the lens, in this case the pupil diameter.Although we could predict from this expression that an increase in pupil diameter shouldresult in a corresponding linear decrease in the minimum angle of resolution, when theaperture size gets larger than 2 mm or so, resultant optical aberrations compromise thelinear relation. Experimentally, the Dawe's Limit, w = -, seems to be corroborated fordpupil diameters of less than 1 mm. A maximum acuity is attained for pupil sizes of 2.5to 4 mm, although the value does not increase dramatically from 2 to 5 mm. As we areinterested in an average pupil size of 3 mm or so, it would be most appropriate to useRayleigh's Limit with d = 2.5 mm.We will apply the concept underlying Rayleigh's Limit equation not to Airy disksof concentric rings of diffracted maxima and minima, but instead to blurred points ofdecreasing intensity toward their peripheries. The minimum resolvable distance betweentwo points will then be from the point's center of maximum intensity to its periphery ofsubmaximal intensity, to be defined by a contrast specification below.The primary confounding factor in defining resolution is, of course, noise in theimage. We will assume a background noise that negligibly affects the relative intensityacross blurred images, and is sufficient to preclude the possibility of "super resolution,"where the resolving power exceeds Rayleigh's, Sparrow's (Smith 1990, 152), and Dawe'scriteria. The holographic image contrast suffers from ambient illumination, scatteringand dispersion of light in the recording material and substrate, intermodulation noise("cross-talk" between component gratings), and other artifacts that compromise theobserver's ability to discriminate intensity differences in an image.If were interested in radiometric image brightness (Caulfield 1979, 235-6), thenwe would need to include factors such as the relative size of the image to the object, or inthe case of the stereogram (Chapter 4), the magnification of the viewzone, as this willaffect how spread out the light from the hologram becomes 13. However, as we have ruledout the consideration of absolute intensity thresholds of the eye for blur sizedetermination, we will instead make a photometric estimation of relative intensities basedon the three wavelength-dependent intensity filtersThe perceived intensity dropoff determining the boundaries of the blurred imageswill be calculated using three filters: (1) the spectral luminous efficiency curve of the eye,13 In actuality, the image size is dependent on wavelength, and the viewzone will be magnified for longerwavelengths. We will assume that this has a negligible effect on the relative intensities for differentwavelengths of the blurred image. The topic of a hologram's viewzone will be taken up in Chapter 3 ("Full-parallax reflection holographic stereograms") and in Chapter 6 ("Wavefront shapes and compact displays").It will prematurely be stated that this magnification is equal to the ratio of the area of the viewzone to thearea of the master hologram. For the collimated grating in the viewstation of Chapter 5 ("Design of a full-parallax holographic viewstation"), this magnification is unity. For a diverging illumination geometry, thisvalue drops below one as the image of the master hologram, and therefore the viewzone, becomesmagnified.Chapter 1: Blur in holographic images(2) the diffraction efficiency curve of the hologram, and (3) the relative intensity curve forthe illumination source.I minimum resolvabledistanceFigure 1.19: Two blurred points that are just resolvable1.8 Spectral intensity filter #1: The retinaThe ultimate filter for these color-blurred images will be the eye, so it would beappropriate to first outline the sensitivity of the eye to different wavelengths. The CIEgraph usually used to demonstrate the relative response of the eye is termed the spectralluminous efficiency curve. There are actually two curves, one for the light-adapted eye,the "photopic curve," and one for the dark-adapted eye, the "scotopic curve." Thephotopic curve peaks at about 550 nm, attributable to the yellow-green sensitivity of thecone photoreceptors, whereas the scotopic curve peaks at 507 nm, the "Purkinje shift"toward the blue, attributable to the blue-sensitive rods. We will use the data from thephotopic curve, as it is our aim to produce displays that are viewable in even brightly litsurrounds.Spectral Luminnus Efciency of the Retina (Photopic Curve)0.9S0.8 -| 0.70.61 0.50.4-0.3 -0.20"X0.1 .008 0 00 000 00 0 00000 (M I c w CO 04 CD CO CIJ e(J 00 N c, WWWavelength (nm, Visible Region of the SpectnmsFigure 1.20: Photopic curve data from Wyszecki and Stiles (1967, table 4.2)Chapter 1: Blur in holographic images1.9 Spectral intensity filter #2: Diffraction efficiency of the hologram(s)The eye will only receive the wavelengths at the intensities the hologram candeliver to it, so the next filter we will apply will be that of the hologram as aninterference filter. The filter limits the diffraction efficiency of different outputwavelengths, and may be calculated using Kogelnik's coupled wave equations (Kogelnik1967, 1969).For our analysis, we will assume that the diffraction efficiency distribution of thespectral components, and therefore the intensity distribution of a blurred image pointcomposed of these spectral components, is gaussian and centered at the playoutwavelength of maximum diffraction efficiency. The intensity distribution of a gaussianbeam is described by-2r 22Equation (1.2) (Smith 1990, 155-6): I(r) = 10 e "where Io is the intensity on axis (the playout angle at the central wavelength),e is 2.17..., r is the radial distance (distance from the center of the blurred image),and w is the beam width. Beam width is defined as the radial distance at which theintensity is O (13.5% of the central value). If the intensity distribution is thenenormalized, with the peak intensity equaling one, then fractional values would beascribed to the relative visibility along r in Equation (1.2). This fractional value wouldneed to be raised to the power of the number of holograms in the system, assuming thereis no wasted light and each hologram reconstructs with the same bandwidth.Chapter 1: Blur in holographic images1.10 Spectral intensity filter #3: The illuminantThere is one final factor affecting the relative intensity of different wavelengths inour consideration. Just as we needed to know what light intensities the hologramimparted to the retina, we need to specify the relative intensities of the differentwavelengths illuminating the hologram. For an example of such data, we will use theInternational Commission on Illumination's choice source for a colorimetry standard:"Illuminant A," a tungsten lamp at a temperature of 2848K (MIT 1936). The choice of atungsten filament is appropriate, as one of the viewstations of Chapter 5 is illuminated byone. However, the modern standard color temperature for quartz-halogen lamps,including those of Chapter 5, is either 3200 or 3400K. If we were to choose a highercolor temperature, we will find that this choice further substantiates the final filter factorwe are deriving.Relative Spectral Intensities for theCIE Standard Illuminant A (Tungsten)0.8-0.7*0.6-0.5N0.4Z 000000000Wavelength (nm, Visible Region of the Spectrum)Figure 1.21 (Wyszecki and Stiles 1967; MIT 1936)The relative intensities of the wavelengths have been normalized in Table 1.1below. Each Illuminant A normalized value may be multiplied by the values for thecorresponding wavelengths from the spectral luminous efficiency and diffractionefficiency curves. The resulting product of the corresponding Illuminant A and retinalefficiency data are listed in the rightmost column and represent what fraction of the lightof that wavelength is delivered from the tungsten source to the hologram, and from thehologram to the eye, neglecting diffraction efficiencies.42 Chapter 1: Blur in holographic imagesRetinal efficiency, Illuminant A Product of thenormalized (CIE tungsten, retinal efficiency &normalized) Illuminant A380 0 0.04051 0390 0.0001 0.050027 5E-06400 0.0004 0.060868 2.43E-05410 0.0012 0.073158 8.78E-05420 0.004 0.086895 0.000348430 0.0116 0.102081 0.001184440 0.023 0.118757 0.002731450 0.038 0.136922 0.005203460 0.06 0.156494 0.00939470 0.091 0.177391 0.016143480 0.139 0.199652 0.027752490 0.208 0.223073 0.046399500 0.323 0.247693 0.080005510 0.503 0.273348 0.137494520 0.71 0.299996 0.212997530 0.862 0.32743 0.282245540 0.954 0.35565 0.33929550 0.995 0.38445 0.382528560 0.995 0.413787 0.411718570 0.952 0.443497 0.422209580 0.87 0.473538 0.411978590 0.757 0.503703 0.381303600 0.631 0.533951 0.336923610 0.503 0.564158 0.283771620 0.381 0.594281 0.226421630 0.265 0.624116 0.165391640 0.175 0.653701 0.114398650 0.107 0.682873 0.073067660 0.061 0.711549 0.043404670 0.032 0.739728 0.023671680 0.017 0.767286 0.013044690 0.0082 0.794182 0.006512700 0.0041 0.820375 0.003364710 0.0021 0.845823 0.001776720 0.001 0.870443 0.00087730 0.0005 0.894277 0.000447740 0.0003 0.917201 0.000275750 0.0001 0.939297 9.39E-05760 0.0001 0.960442 9.6E-05770 0 0.980718 0780 0 1 0Table 1.1: Data from the spectral luminous efficiency (Wyszecki and Stiles 1936) and Illuminant A relativeintensity (MIT 1936) curves, and their corresponding productsChapter 1: Blur in holographic imagesSuperposition of the two spectral component intensity filters: theretina and the illumination source1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41Wavelength (380-780nm)Figure 1.22: Intensity filters: the retina and illuminant1.11 Resolution of color-blurred reflection holographic imagesThe data used to make Figure 1.22 could have been multiplied by theircorresponding diffraction efficiency values for a complete determination of the relativeintensities of the spectral components. However, the above analysis is only reasonablefor application to transmission holograms with a wide spectral bandwidth (on the order of200 to 300 nm), and is more extensive than it needs to be for thicker, reflectionholograms having a narrow spectral band (of about 20 to 30 nm).For a narrow segment of the visual spectrum, the spectral luminous efficiency andrelative illumination intensity curves are approximately linear and of small slope for theillumination source that was chosen. The diffraction efficiency curve, on the other hand,is very narrow about the central wavelength, and therefore of very steep slope. We willtherefore consider the retinal and illuminant distributions to have a negligible effect onthe perceived size of blurred image points. Instead, we will consider the perceived size tobe wholly dependent on the diffraction efficiency distribution. First, we will define theresolvability of two blurred images, and then we will define the perceived size of a singleblurred image point.To define the resolvability of two blurred images, we will overlap the intensitydistributions of two very closely situated images (as in section 1.7). A correspondenceexists between distance from the center of a blurred image and playout wavelength, so theChapter 1: Blur in holographic imagesspectral intensity distribution will enable us to determine the size of the image above aspecified cutoff intensity. The distance from the center to the "boundary" of a blurredimage is simply the length along the blur from the focused wavelength of highestdiffraction efficiency to the focused wavelengths of some cutoff diffraction efficiency.We will define the minimum resolvable distance between two images as half of thedistribution width at 50 percent intensity, the point on the Gaussian profile withoutcurvature (Figure 1.23). We will further define the absolute bandwidth, AX, as thosewavelengths contributing 95 percent of the intensity, from Equation (1.2):Equation (1.3): I(r =- ± 095, with ra equal to half the absolute image sizeIo e3For a 50 percent spectral intensity cutoff, the minimum resolvable distance maybe calculated from the absolute size of the blurred image, using the total calculatedbandwidth.minimum resolvable distance50% intensity- I(r) = 0.5mrinFigure 1.23: Just resolvable points defined by the diffraction efficiency curve. Here are twooverlapping blurred images. As in Figure 1.18, the lines merely ascribe areas of the blurred imageto points on corresponding intensity curves.First, the bandwidth of the hologram may be defined using Equation (1.4)(Syms1990, 7)14:AX = cot(Oill), where A is the fringe spacing, A is the illumination wavelength,t2t2 is the thickness of the grating, and Oill is the illumination angle.14 Leith (1992) uses a completely different method for deriving the bandwidth, and defines the bandwidthto equal the illumination wavelength divided by the number of fringes, N:A A_-AAA ~ -, Ths is equivalent to Sym's for the conformal (untilted) fringe case with ill= 45 degrees.N t2Chapter 1: Blur in holographic imagesFor most of the thesis, we will use a geometry with Oill = 450, and X = 514.5 nm.For t2 = lOm:From Benton (1994, 1996), we have Equation (1.5): A 2 out- o ill2. i -smil2Therefore, AX = 34.59 nm.The extreme wavelengths of this bandwidth may then be used in the calculations ofChapter 2 as the extreme raytracing wavelengths. For consistency, however, we willassume from this point forward a bandwidth of 20 nm for the different recordingmediums we will be using.Now we will calculate a "resolution factor," R, to determine the minimumresolvable distance between two points from the absolute blurred image height, hblur(Chapter 2):Mimimum resolvable distance = absolute image size x resolution factor (R),rso%Equation (1.6): R =rAA- w21n(0.5)From Equation (1.2): r5o% = 2 = 0.5887w,and beam width w = 2r, 2, 0.8165rAInL1e 3Therefore, R = 0.4807An ideal quantity to which resulting minimum resolvable distance calculationsmay be compared may be obtained from Equation (1.2), Rayleigh's criterion:1.22Ao = d . With a pupil diameter d=2.5mm and wavelength X=514.5 nm, the idealdtheoretical resolution is then 2.51 X 10-4 radians, about 5.2 arc seconds.hblurmin x 0.4807With a mimimum resolvable angle of 2.51 x 10- = ,Deyethe minimum absolute blur height, hblurmin, is equal to 0.261 mm at a viewing distanceof 500 mm, or about 1.8 arc minutes.For the rest of the thesis, we will be concerned primarily with determining theextent of the blur in holographic images, not with their resolution. Therefore, we need toperform a similar calculation as Equation (1.6) to find the perceived blur size.Chapter 1: Blur in holographic images1.12 Factor for perceived blur sizeThe perceived size of a blurred image is the absolute blurred image size, hblur(Chapter 2), multiplied by some perceived blur size factor, B. We will estimate thisperceived blur size factor to be the ratio of the perceived height, hperceivedblur to the totalheight, hblur. Just as the total width of the gaussian distribution was marked by a cutoffintensity that also determined the boundary of the blurred image point, the distancefrom the center to the perceived boundary of a blurred image is the radial distance fromthe center to a perceived cutoff intensity. As the beam width, defined above in section1.9, contains 86.5 percent of the beam power, we will define our perceived bandwidth asthe wavelengths playing out within the beam width. Now we are prepared to define a theperceived blur factor, B:Perceived image size = absolute image size x perceived blur factor (B),wEquation (1.7): R = -From Equation (1.2), beam width w = -2, =0.8165rAr1Ine 3Therefore, B = 0.81651/e2 = 86.5%1/e3= 95%Figure 1.24: Perceived blur factor, B=0.8165Chapter 1: Blur in holographic imagesThis perceived blur factor should be a reasonable approximation, considering themultiple sources of contrast degradation and noise mentioned above in section 1.7. In thenext chapter, we will derive the absolute blur extent in terms of height, as well as anglesubtended from the eye. The perceived blur factor may be applied to this absolute blurextent as an estimate of the perceived extent of a blurred image.Chapter 1: Blur in holographic imagesChapter 2Trigonometric derivation of a single-plane blur equationIn the previous chapter, we first looked at color blur from a qualitative standpoint,and applied successive intensity filters to determine the apparent size of a blurred imagepoint as perceived by an observer. We found that we could define the extent of theperceived blur as well as resolution of blurred image points by simply approximating thediffraction efficiency of the recording material as a normal distribution about a meanwavelength, and defining the wavelengths at two standard deviations as the cutoffperceived wavelengths. In this chapter, we will take a more mathematical approach todefine the absolute extent of this blur. We will then experimentally test the results withspectrophotometer measurements. Finally, we will apply last chapter's perceived blurfactor to the forthcoming derived blur equation.The trigonometric derivation below gives us a formula for determininghologram color and source-size blur. We will compare the resulting formula withBenton's blur equations. A vector blur equation is derived in Appendix 2, and reduces tothe trigonometric form developed here.2.1 Derivation of a color blur equationFigure 2.1 will provide us with the physical model we will be using in thederivation and future discussions. Spectral components of the illuminant reconstructingdisplaced images produce the blur. These components diffract at the hologram and createa "dispersed focus" at the pupil of the eye:white light illuminationorder m = +1 (virtual image)R BB Rorder m = -1 (real image)Figure 2.1: Spectral components from different points on the hologram contributing to blurChapter 2: Trigonometric derivation of a single-plane blur equationWe will start our derivation with the "X-equation" of Appendix 1. TheX-equation is a raytracing equation that describes the relationship between the exposureand reconstruction geometries at some position (1) on the hologram.The method we will use for the derivation of a blur equation is to substituteunknown angles at some position (2) with known position (1) angles. Position (1) is thepoint on the hologram in direct line of sight of the viewer and reconstructs at the central("green") wavelength. Position (2) is another point on the hologram that will reconstructat another wavelength. In the following figures, this is a longer ("red") wavelength. Thekey to this technique is that position (1) variables are in terms of the central wavelength,denoted by " 2 ," whereas position (2) variables are in terms of an extreme wavelength fora given spectral bandwidth illuminating the hologram. Although these extremewavelengths will be denoted by "12red" and "k2biue," the color names merely describe theirwavelengths relative to the central wavelength. For reflection holograms with verynarrow spectral bandwidths, these wavelengths may all appear, for example, to be red orblue.Rill2hRout2-0tperceived.. blur angle .illi.- -. Rref2Deye2 Robj2 -Robjl ',IDeyel Rref 1Figure 2.2: Dispersed focus at the pupil, perceived as color blur, from Appendix 2The angles in the text are denoted with the same subscripts as the vector magnitudes. For example, Robj2corresponds to an object distance of Robj2 exposing the holographic recording material at the angle 9obj2.For clarity, only Gout 2 of the longer playout wavelength is included in the right diagram.Equation (2.1): sin (Oout 2) :=m (sin (Oobj 2) sin (eref 2)) + sin (Oill 2 )For small h, R, is approximately R 2: Equation (2.2): sin (9 2)R sin (0 1) + hFor a proof of the validity of the above approximations, please refer to Appendix 2.The resulting sine values to be substituted into the above equation follow.Chapter 2: Trigonometric derivation of a single-plane blur equationEquation (2.3):Equation (2.4):Equation (2.5):(O Robj -sin(Oobj 1) + hsin (Oobj 2)RbI Robj 1Rref 1-sin(eref 1) +hsin (Oref 2)Z_ Re1r f 1Rill isin (eill i) + hsin(ill2RillRegardless of the reconstruction geometry (reflection or transmission, orderm = ± 1), the difference between the sines of the output angles from the viewer's eye topositions (1) and (2) on the hologram (sin(Ooutj)-sin(Oout 2)) is opposite that of thedifference between the sines of the other angles from their respective source positions tothe same two positions on the hologram. In other words, notice that h is not added withthe other term in the numerator, as is the case in the other sine variables listed above.Equation (2.6): sin( It D e2e) -sin (Oout 1) hD eyeiWhen we substitute the four sine equations into the modified X-equation, weobtain a value for h, the length of the hologram contributing to color blur for a particulareye position and spectral bandwidth:Equation (2.7):A_ 2 sin(Oobj I) sin(Oref 1)1 2 1 1 1 1-m -- + - -- +k1 Robj Rref 1 Rilli D eyeiThis may be simplified to the following expression, with either a long (red) orshort (blue) peripheral X2 resulting in either a red or blue horizontal focus (denoted by thesubscript "HORIZ") and associated height h, or h blu along the hologram in question:Equation (2.8): h r 2redhred 2green)sin (obj I) sin (Orefi)R 1 1 DRout IlredHORIZ D eyelThe final color blur height, as seen by a viewer at a given distance Deye along theOO9t angle, is then scaled by the distance of the blurred point from the hologram(Deye+RoutgreenERT, where Deye is taken to be positive and the latter term is the verticalfocus at the central wavelength) and by the output angle from position (1):Chapter 2: Trigonometric derivation of a single-plane blur equationDeyel +- Rout lgreenVERT (ot~( leEquation (2.9): h colorblur : D eeCos (out 1)- (h red + h blue)DeyelWhen expanded, the final expression for color blur is the sum of the followingwith its blue counterpart (or simply twice the following as an approximation):Equation (2.10):D eyel + Rout lgreenVERTD eyei + Rout 1redHORZRout lredHORZcos (Oout i)-(2red - 2green -(sin (Oobj ) sin (Oref i))From this form, it is more apparent that for this perceived blur function there is anonlinear effect for small viewer distances. There is also a discontinuity when thechromatically dispersed focus Deye coincides with either RoutredHORIZ or RoutblueHORIZ, thatis, when the pupil is situated at one of the horizontal foci. This discontinuity makesphysical sense, as a pupil situated at these foci will have light filling the retina; the imageshould appear to be very large.Chapter 2: Trigonometric derivation of a single-plane blur equation2.2 Inclusion of source-size blur into a final blur equationTo include the effect of source size on the vertical blur of an image point, a newhred and hblee are found from the effective translation of the illumination source by half ofthe height of the source, CDil, in opposite directions for the two extreme wavelengths. Thelonger, "red" wavelength determines the maximum hred, and the shorter, "blue"wavelength determines the maximum hblue* The final blur equation follows the inclusionof the change in illumination angle from the extreme ends of the source into the originalequation. The central illumination angle, 0ill, is that from the center of the source toposition (1) on the hologram. 2greenSin(Ooutgreen) x sin(aef)>t2blue ,Z2bluesin(Ooutblue) - 1 sin(Oii)Al AlHowever, because the first and second columns of the coefficient (middle) matrixare multiples of one another, a simple matrix equation with an inverse function will notwork to solve for the unknown (object, reference, and illumination) angles. Instead, tosimultaneously solve for an optimized solution for the three unknowns, we will perform aMoore-Penrose pseudoinverse 48 operation, Equation (6.6):47 The angle and distance equations are presented in Appendix 1.48 X = pinv(A) gives the Moore-Penrose pseudoinverse, a matrix of the same dimensions as A' satisfyingthe following four conditions: A*X*A = AX*A*X = XA*X and X*A are HermitianChapter 6: Wave-front shapes, compact displays, and designing gratings2 2red ~ Z2redsin(Oobj) A4 A 1 sin(Ooutred)212green 012greenSin(aef) A A - 1 x sin(outgreen)pinv AIAsin(Oii) _ 1 2blue - 1 2biue 1 sin(Ooutblue)If a range of central wavelength output angles are substituted into the aboveexpression, a set of different optimized solutions for the unknowns may be found. Thesesolutions correspond to different potential exposure and illumination geometries thatprovide the desired (ideal) blur from the predispersing grating.Unfortunately, we can not obtain an exact solution for our matrix expressionAx=b representing [coefficients] [sin(exposure angles)]= [sin(output angles)]. This maybe proven by taking the original expression Ax = b and performing Gaussian eliminationto form a zero submatrix in the modified A*, and determining that A*x can not equal b*.However, the pseudoinverse technique does provide an optimal solution x- to theleast squares problem of Ax=b. This is verified by x~ meeting the two conditions:(1) x~ is in the row space of A, and (2) Ax~ must be a projection of b onto the columnspace of A. We will use this pseudoinverse technique to find optimized grating outputdistances and will work through several geometries to assess the degree of error in theoptimization of both output angles and distances.6.11 Solving for optimized distancesSolving for optimized distances requires the inclusion of a second set of threevertical focusing equations:(cos(Ooutred)) 2 A 2 red f(cos(oj))2 (cos(6rer))2= m - +Routred 2 i Robj Rref Ri(cos(Ooutgreen)) 2 12green (cos(Oobj)) 2 (cos(Oref)) 2 (cos(Oii)) 2= m -re _____ +Routgreen Al Robj Rref Riin(cOS(outblue)) 2 = 2 hue (cOS(Oobj)) 2 (cOS(Oref)) 2 (cos(Oii))2Routblue i , Robj Rref RiAfter we have substituted the most practical angles from the set of angle vectorsolutions into the distance matrix expression above, the remaining three unknowns are theobject, reference, and illumination distances.Chapter 6: Wave-front shapes, compact displays, and designing gratings1 12Equation (6.7) is a pseudoinverse expression solvinggiven ideal output distances:(coS(9obi)) 2Robs(coS(Oref)) 2Rref(coS(Ojin))2Rin=pinv2red2greenA I/2blue2 2red2 2greenA I12bluefor optimal exposure distances,(Co S(6outred)) 2Routred(cos(outgreen)) 2Routgreen(co S(9outblue)) 2Routblue6.12 Designing a diverging wave-front grating for a compact displayFigure 6.9 is the model of the prototype diverging/diverging viewstation we madeto evaluate the ease and effectiveness of non-phase-conjugate geometries for futurereduced-size gratings. The standard procedure for the making of an HPO HS with a widefield of view was used for the first full-parallax trials. This procedure involves a 300 mmHl/H2 transfer separation and a 500 mm HI image playout distance. The onlycompromise that the H2 underwent was to have a matching short reference andillumination distance of 750 mm, to limit warping of the HI aperture image as well assatisfy the vertical focus equation. The vertical focus formula is much more sensitive tochanges in distance than is the more familiar horizontal focus equation: a slight changein one distance could significantly alter the other distances used to expose and reconstructthe image.referencebeam: 225 degrees;Rref = 750mmHI H2Figure 6.11: Exposure of the transfer (H2) hologram with a divergingreference sourceChapter 6: Wave-front shapes, compact displays, and designing gratingsHI image(distorted illuminationviewzone) beam: 45 degrees;Rill = 750mmFigure 6.12: Reconstruction of the H2 with a diverging illumination sourceThe TK Solver+ program of Appendix 3 calculates with basic trigonometry thatfor a grating arbitrarily set 450 mm from the given 750 mm/750 mm 300 mm-high H2, itssize may be reduced from the 300 mm required for a plane-wave grating to 130 mm.This reduced size is sufficient to illuminate all parts of the hologram with at least thecentral wavelength playing out of the grating. The size of the grating may be furtherreduced if its playout distance is smaller.6.13 Designing a perfectly reconstructing diverging gratingThe TK program also solves for an equivalent exposure and reconstructiongeometry that will compensate for a given hologram's dispersion, based on a reverseraytrace of the blur extent required by the hologram, to the blur extent provided by thegrating. This program has the constraint that the exposure and reconstruction angles areidentical for the purpose of minimizing distortion. The grating and H2 were made withoutregard for the astigmatism that one obtains in such a diverging/diverging procedure,however. The resulting H2 in the present display has a vertical focus at the intended 500mm, whereas its horizontal focus is thrown out to 1500 mm.In the first step, the hologram's predispersed illumination angles had to bedetermined. These are the angles for their respective wavelengths (in nanometers) that allplay out on the normal for single view angle dispersion compensation:6llred = 43.874AOillHredgreen = 1.1260Ollgreen = 45AOillHbluegreen = 1.10400llblue = 46.104Chapter 6: Wave-front shapes, compact displays, and designing gratings114The second step is to determine the playout angles of the grating. As mentionedearlier, the ideal angles for individual wavelengths are not always attainable across thebandwidth, and a compromise must be made between the ideal angular differencesbetween the central wavelength and longer wavelength playout angles and between thecentral wavelength and shorter wavelength playout angles. The A~outIDEAL's are foundthrough basic trigonometric calculations. In order to illuminate the grating close to thenormal (an arbitrary choice), a tip angle is necessary during exposure if the samegeometry is to be used during reconstruction. Following are three possible geometries,none of which are ideal:A0outGredgreenIDEAL = 3.5 0AOoutGbluegreenIDEAL = 2.40Ooutgreen = Oobj = 650 Oobj = 660 Oobj = 670AOoutGredgreen = 2.510 AOoutGredgreen = 2.64' AOoutGredgreen = 2.79'AOoutGbluegreen = 2.29' AOoutGbluegreen = 2.390 AOoutGbluegreen = 2.50The exposure and central reconstruction wavelength is 514.5, a blue-green, so thecenter geometry was chosen as the closest approximation.Please refer to Appendix 3; the TK Solver+ Variables menu contains the dataderived here, as it was computed. With a grating/hologram separation of 450 mm, thegrating perfectly reconstructs an image point 300 mm from its center, at an angle of 660to its normal. The grating is therefore tipped 240 toward the H2 to illuminate the H2 at its450 illumination angle. The tilt angle is simply to increase the blur of the grating requiredby the hologram at the close distance chosen. If the grating's playout distance were to beincreased, then the grating would not have to be tipped as much and horizontal dispersionwould be negligible. This would also, however, create a severe mismatch between the H2reference and illumination distances and require a very short H2 reference beam, 375 mmin the limiting case of collimated illumination with the above procedure.Of course, the tilt angle of the grating may be reduced, or sometimes eliminated, ifit is illuminated for imperfect reconstruction (exposure angles different from thereconstruction angles), although this will lead to some distortion of the input blur to thehologram and will usually require an off-axis illumination source.We made an exposure with the above geometry, and the final table setup isdiagrammed below in Figure 6.13. The primary difficulty in the shooting of the gratingwas obtaining uniform intensities across its area. For the scale of the display, theproximity of the exposure sources were very small, and widely diverging lenses wererequired. A 10OX objective (focal length of about 5 mm) and a ball lens of roughly thesame focal length were used in the setup. To further increase the uniformity of the spreadof the beams, meniscus lenses were placed in the object and reference beam paths toshorten the diverging optics' apparent focal lengths.Chapter 6: Wave-front shapes, compact displays, and designing gratings 115180 degrees(f ~=grey glass600mm. silver-halide plate(emulsion toward ball lens)reference beam* meniscus lens (f = 500mm)object beamM sl d 50/50 beamsplitterMsilveredrattenuator Argon-ion laser (514.5nm)Figure 6.13: The final table setup for the 750/750 diverging/diverging geometry6.14 Designing a diverging grating that plays out ideal anglesand optimized distancesWe used the resulting 24' tip angle in a prototype display to good effect.However, we were intent upon obtaining grating angles as close to the ideal angles aspossible. If we follow the above pseudoinverse procedure, and input the desired gratingoutput angles and distances into the Matlab program of Appendix 4, then we receive a listof potential geometries.Chapter 6: Wave-front shapes, compact displays, and designing gratings116First, here is the list of input variables (Actually, the Matlab program of Appendix 4calculates the grating's output angles and distances from hologram input variables.):%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% INPUT VARIABLES: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%deltaGR = 62.909265 - 60;deltaGB = -57.326903 + 60;lambdal = 514.5lambdaR = 524.5;lambdaG 514.5;lambdaB = 504.5;outGmin = 0;outGmax = 70;RoutRideal = 218.7;RoutGideal = 300;RoutBideal = 395;Rillmax = 1000;% angular spread of output wavelengths (green-red)% angular spread of output wavelengths (green-blue)% exposure wavelength% output long wavelength -- "Red"% output central wavelength -- "Green"% output short wavelength -- "Blue"% output angle range minimum (Green)% output angle range minimum (Green)% ideal grating output distance -- Red% ideal grating output distance -- Green% ideal grating output distance -- Blue% maximum grating illumination distance%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%The output file consists of a long list of potential geometries. The following threewere selected each for a different reason. The first grating's playout angle is almost 45*,so the tip angle for the grating is almost 0*. The middle geometry (2) was chosenbecause the illumination and central output angles match; this means that the grating canbe tipped with respect to the light source the same degree as to the hologram (as in Figure5.2). Also, the illumination distance is not too large. The third geometry was alsoselected as an example because its playout distances are closest to the ideal distances.outR = 49.909,obj = 58.377,(1) Robj = 2094.202,RoutRideal = 218.700,RoutR = 257.400,outR = 64.909,obj = 35.830,(2) Robj = -1160.218,RoutRideal = 218.700,RoutR = 217.400,outR = 72.909,obj = 25.204,(3) Robj = -1303.140,RoutRideal =218.700,RoutR = 237.400,outG = 47.000,ref = 238.377,Rref= -2094.202,RoutGideal = 300.000,RoutG = 300.000,outG = 62.000,ref = 215.830,Rref = 1160.218,RoutGideal = 300.000,RoutG = 300.000,outG = 70.000,ref = 205.204,Rref= 1303.140,RoutGideal = 300.000,RoutG = 300.000,outB = 44.327ill = 255.981Rill = 44.304RoutBideal = 395.000RoutB = 320.000outB = 59.327ill = 196.685Rill = 472.451RoutBideal = 395.000RoutB = 300.000outB = 67.327ill = 174.935Rill = 603.367RoutBideal = 395.000RoutB = 360.000Chapter 6: Wave-front shapes, compact displays, and designing gratingsThese geometries were then recalculated (by hand and by the program ofAppendix 3, in boldface type) to test the pseudoinverse function's validity as anoptimization procedure. It was found that the output angles are within a tenth of a degreeof those calculated.514.5 11514.5 12514.6 1 m35.83 0obj215.83 OrefOout196.68 OillRoutRRoutGRoutBdegreesdegrees62.10 degreesdegrees217.400 mm300.000 mm300.000 mmexposure wavelength (vacuum)reconstruction wavelength (vacuum)diffraction order: m=+1 for virtualbetween 0&90' object beam anglebetween 900 and 270' reference anglebetween ±90' (for m =+ 1) output anglebetween 90* and 270' (for m =+1) illumoutput distance -- Redoutput distance -- Greenoutput distance -- BlueThe distances, however, are less precise and vary in precision in the Matlab program'soutput. Recalculated quantities are again set in boldface:outR = 49.909,49.841(1) RoutRideal = 218.700,RoutR = 257.400,261.256outR = 64.909,(2) 64.860RoutRideal = 218.700,RoutR = 217.400,229.006outR = 72.909,(3) 72.881RoutRideal = 218.700,RoutR = 237.400,235.177outG = 47.000,47.123RoutGideal = 300.000,RoutG = 300.000,291.717outG = 62.000,62.086RoutGideal = 300.000,RoutG = 300.000,270.884outG = 70.000,70.0466RoutGideal = 300.000,RoutG = 300.000,305.546outB = 44.32744.271RoutBideal = 395.000RoutB = 320.000324.146outB = 59.32759.290RoutBideal = 395.000RoutB = 300.000313.414outB = 67.32767.3076RoutBideal = 395.000RoutB = 360.000376.597Chapter 6: Wave-front shapes, compact displays, and designing gratings118Exposure and illumination angles for the 750/750 grating300 1250- X reference200- X X )Killumination K)beam150-100-object50-0o beam£ L _ I .I45 50 55 60 65 70Central wavelength output angleFigure 6.14: A graph of the exposure and illumination angles for a rangeof central output angles. This graph is the output from the Matlab plotcommand in the program of Appendix 4.These reconstruction geometries aie far closer to the ideal than is the case with theperfectly reconstructing angles in section 6.13. This improvement would suggest thatperhaps the pseudoinverse function could improve dispersion compensation in theviewstations of Chapter 5 as well. At first this seems unlikely, because it is usuallyassumed that a plane-wave grating provides predispersed light from wavelengthcomponent points infinitely far away. However, this is not the case, as we shall see. Wewill follow the same procedure again, first computing the appropriate blur angle toilluminate the hologram with the TK Solver+ program variables of section 6.13. Thegrating/hologram separation will again be 450 mm:Chapter 6: Wave-front shapes, compact displays, and designing gratings 119OillROillGOillBdObluri450 sepdeblur_46.12474543.89692.22785.5658degreesdegreesdegreesdegreesmmdegreesH ill. angle for long (Red) 12H ill. angle for middle (Green) 12H ill. angle for short (Blue) 12blur angle illuminating HdcGrating/H separationnecessary dcG blur angleThe above hologram illumination angles are used to find the necessary grating outputangles. Below, the exposure and illumination distances are used to find the ideal gratingoutput distances:RobjBen300 RrefBen300 Rill dcRoutRBeRoutGBeRoutBBeRoutRveRoutGveRoutBve9E9-9354.5559E98532.3305-4372.7119E94560.6652mmmmmmmmmmmmmmmmmmdcG DISTANCES(ideal: = RoutGvert dcG)obj distance(ideal: = RilldcG) ref. distancedcGrating illumination distancedcG playout distance: ReddcG playout distance: GreendcG playout distance: BluedcG vertical playout distance: Red(=Rill-sep; ideal output =RobjdcG)dcG vertical playout distance: BlueThe ideal grating playout distances (Rout's) are all large compared with the dimensionsof a normal hologram, but are not infinity. Now we will use the Matlab program ofAppendix 4:%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% INPUT VARIABLES: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%deltaGR = 46.125 - 45;deltaGB = 45 - 43.897;lambdal = 514.5lambdaR = 524.5;lambdaG = 514.5;lambdaB = 504.5;outGmin =0;outGmax = 70;RoutRideal = 7414.56 - 450;RoutGideal = -9e9;RoutBideal = -8014.58 - 450;Rillmax = 1000;% angular spread of output wavelengths (green-red)% angular spread of output wavelengths (green-blue)% exposure wavelength% output long wavelength -- "Red"% output central wavelength -- "Green"% output short wavelength -- "Blue"% output angle range minimum (Green)% output angle range minimum (Green)% ideal grating output distance -- Red% ideal grating output distance -- Green% ideal grating output distance -- Blue% maximum grating illumination distanceChapter 6: Wave-front shapes, compact displays, and designing gratings120ideal distances perfectly. The same imprecision for distances as above applies, so the therecalculated values (boldface quantities) are included:outR = 49.909,49.746obj = 58.479,(1) Robj = 178.118,RoutRideal = 6964.560,RoutR = 6964.560,7381.409outR = 49.909,49.746obj = 58.479,(2) Robj = 177.070,RoutRideal = 6964.560,RoutR = 6964.560,7375.484outR = 49.909,49.746obj = 58.479,(3) Robj = 176.010,RoutRideal = 6964.560,RoutR = 6964.560,7342.012outG = 47.000,47.029ref = 238.479,Rref = -178.118,RoutGideal = -9e9,RoutG = -9e9,-4.6e6outG = 47.000,47.029ref = 238.479,Rref = -177.070,RoutGideal = -9e9,RoutG = -9e9,-1.2e6outG = 47.000,47.029ref = 238.479,Rref = -176.010,RoutGideal = -9e9,RoutG = -9e9,-9.7e5outB = 44.32744.179ill = 256.704Rill = -17.233RoutBideal = -8464.580RoutB = -8564.580-8198.115outB = 44.32744.179ill = 256.704Rill = -17.130RoutBideal = -8464.580RoutB = -8464.580-8112.68outB = 44.32744.179ill = 256.704Rill = -17.027RoutBideal = -8464.580RoutB = -8364.580-8053.654We find that only in the case of a highly converging illumination beam can weattain the proper distances. However, the angles themselves are quite different from thesimple perfect reconstruction geometry that is universally applied. This discrepancysuggests that in order to attain the proper distribution of wavelengths within thepredispersed blur illuminating the hologram, a more careful approach such as thepseudoinverse function might prove advantageous, particularly for very deep images.6.15 An attempt to design a pre- and post-dispersing compensation gratingThe ultimate compact reflection dispersion-compensation display would consistnot even of a small, unobtrusive grating at some distance from the hologram, but wouldideally consist of a single material, or a double-layered film without an intervening waveguide or louver film of questionable optical quality. We will attempt here (and in thecomputer program of Appendix 5) to assess the viability of a pre&post-dispersing,compensation grating. We will again work backwards from the desired, dispersion-compensated condition where output angles = 0 through the grating (m = - 1 in the 2ndpass), to the hologram, and back through the grating (m = +1 in the 1 't pass). With theangle matrix above, we have:Chapter 6: Wave-front shapes, compact displays, and designing gratingsEquation (6.8): 2 "D pass through the grating (de): sin(.,,) = 0; m = T 1-dcILL2 ] = [ -dcCOEFFA2red A2red--sin(O2 ildcred) Aldc Aldc_ Z2green 02green- sin( 2 idcgreen) _ + +)ldc /01dc/12biue A 2bluesin(O2 idcblue) t blu+ ZlueEquation (6.9): hologram:_sin(6uh) = sin( 2illdc)[ dcILL2 ] =sin(O2 ildcred)sin(O2 ildcgreen) =sin(O2 ildcblue)] x [ dcUK ](Sin abjdc)(sin aefdc)hCOEFF + hILL ] x [ hUK ]iZ2redm'%hm 12greenA lhm 2blueA lh,%2red-m21hA2greenM-mA IhA2blueS-hAlhsin(ilredh)sin(illgreenh)sin(Qllblueh)Equation (6.10): 1t pass through the grating (de): sin(lutdc)= sin(Ollh); M = ± 1hILL ]=[F ,2redSin(Ollredh)Sin(Ollgreenh)Sin(Ollblueh)+dcCOEFFZ2redAldcA 2greenA l dcA2bluetldc+ 1] x [A 2greenAldcZ2bluei ldcdcUK ]sin(objdc)sin(aefdc)sin(11dc)After substitutions, we have Equation (6.11):-[-dcCOEFF x dcUK] = [± dcCOEFF+([dcCOEFF+1] x dcUK)] x [hUK]Chapter 6: Wave-front shapes, compact displays, and designing gratingssin(Oobjh)sin(aefh)1122The values for the sines of the object and reference angles were incrementedthrough a range of values in the example program of Appendix 5. However, it wasconsistently found to be the case that a single or double pre&post-dispersing transmissiongrating layer set in front of a reflection hologram cannot by itself compensate for its owndispersion as well as dispersion from the hologram. It has not yet been determinedwhether the introduction of very thin refractive optics between the layers could make thisdisplay viable.In this final chapter, we have explored increasingly compact dispersion-compensation formats while evaluating their effectiveness. Two methods for HOEdesign have been developed here and in the accompanying appendices: (1) calculation ofexposure and reconstruction angles based on a reverse-engineered analysis of perceivedblur used for the grating /hologram dispersed focus, instead of the hologram/pupildispersed focus, and (2) the Moore-Penrose pseudoinverse function for solving for a setof optimized geometries for ideal angles, with the further calculation of optimizeddistances as an option.Chapter 6: Wave-front shapes, compact displays, and designing gratings 123124Conclusions and future work"If the reader, in this spirit, recalls what has been stated by us throughout, generally and in detail,with regard to colour, he will himself pursue and unfold what has been here only lightly hinted at. He willaugur well for science, technical processes, and art, if it should prove possible to rescue the attractivesubject of the doctrine of colours from the atomic restriction and isolation in which it has been banished, inorder to restore it to the general dynamic flow of life and action which the present age loves to recognise innature. These considerations will press upon us more strongly when, in the historical portion, we shall haveto speak of many an enterprising and intelligent man who failed to possess his contemporaries with hisconvictions."-- Goethe, Theory of Colours (1810, 297-8)This final section of the main body of the text will be kept very succinct tofacilitate the reader's interpretation of the contents of the thesis. A brief mention will bemade of the highlights and shortcomings of each chapter.In Chapter 1, we found that for most illumination sources and reflectionholograms with reasonably narrow bandwidth, the diffraction efficiency curve should besufficient to determine the extent of blur an observer will perceive. Applying a modifiedform of Rayleigh's criterion, we defined the resolution of a holographic image, assuminguniform background noise. Both of these determinations rested on an approximation ofthe diffraction efficiency curve as a normalized distribution, with the absolute bandwidthconstituting three standard deviations on either side of the curve, and the perceivedbandwidth as two standard deviations on either side. The resulting perceived blur factorof 0.8165, to be multiplied by an absolute blur extent, needs to be verified withpsychophysics experiments on human subjects.In Chapter 2, our spectrophotometer data suggest that the absolute blur equationderived here measures favorably compared with Benton's blur equation for viewingdistances in front of the image beyond the nearpoint of the eye. The difference betweenthe two lies in the less mathematically intuitive, but more physically realistic,interpretation of the perception of blur in the author's equation. Although for on-axisobject/image geometries the two equations render very close solutions, for less commongeometries, their solutions deviate considerably. This equation gives absolute, raytracedblur extent, but until the perceived blur factor of Chapter 1 is verified or revised,perceived blur extent will not be fully characterized.Chapter 3 is predominantly a brief history of the application of dispersioncompensation to display holography, so some trends may be apparent to the reader.Certainly, the use of a predispersing, precompensating grating is much preferred over theoriginal use of a post-compensating grating. An exception to this preference may befound in the compact geometry and narrow bandwidth conferred by reflection gratingsrelaying dispersion-compensated light in head-up displays. The incidence of theapplication of the technique to reflection holography has been extremely sparse, butrecent, so its future is uncertain. However, a review of the literature gives one theConclusions and future workimpression that there is a general desire to obtain full-color dispersion-compensatedimages. Reflection holography is well-suited to this task, as it provides sufficientlynarrow bandwidth for deeper sharp images of higher hue saturation.Chapter 4 provides the rationale for designing dispersion-compensation gratingsfor full-parallax, reflection holographic stereograms: The degree of improvement inimage sharpness conferred by dispersion compensation relative to an uncompensateddisplay is greater for full-parallax than for horizontal-parallax-only holographicstereograms because it can reduce blurring of vertical perspective views in the former.It is recommended that care be taken when sampling perspective views in the making of afull-parallax holographic stereogram. Although coarse sampling in the vertical directioncan reduce residual color blurring for a display with a vertically offset illuminationsource, the effect may sometimes be likened to a sporadic animation when the observermoves up and down.As the device displaying the successive synthetic views is most likely going tohave a resolution much coarser than the spacing of the just-resolvable blurred imagepoints of Chapter 1, an expression accounting for object resolution should beincorporated into the perceived blur equation.Chapter 5 presents an example of a viewstation with a plane-wave grating. Someof the advantages of a planar wave front for dispersion compensation will be mentionedin the Chapter 6 summary. The practical advantages in the display of a hologramilluminated by a plane-wave grating include the fact that the grating and hologram can beset at any unobtrusive distance from one another, and the resulting image can be made toexhibit no warping or astigmatism.The single-most important factor when selecting a recording material, anddeciding on exposure and reconstruction wavelengths and geometries, is how bright thefinal image will be. DuPont photopolymers were chosen because they reconstruct withvery high diffraction efficiency. A high-wattage, bare tungsten-halogen lamp with itssmall filament oriented to present a solid, radiating area to the grating, can provide justenough source-size blur to a holographic stereogram so that its image may appear brightand sharp while the image of its master hologram is rendered an unnoticeable blur.The grating edge farthest from the light source may be rotated toward the hologram forincreased predispersing capability to compensate for shrinkage of the hologram. Thegrating may also be rotated if it itself has shrunk and is playing out at the wrong angles.The next step in plane-wave dispersion compensation is certainly to produce a bright,full-color viewstation.Chapter 6 compares the effectiveness of compact dispersion-compensationdisplays that do and do not use a plane-wave grating. Phase-conjugate-illuminated plane-wave gratings provide the broadest range of viewer positions with perfect dispersioncompensation for a single view angle. Phase-conjugate-illuminated diverging wave-frontgratings present a wider field of view with deeper peripheral points for an observerConclusions and future work126situated at the ideal viewer location and orientation. However, at non-ideal viewerpositions, dispersion compensation degrades and the corresponding maximum imagepoint distances reduce considerably. Non-phase-conjugate-illuminated diverging wave-front gratings present a wider field of view of even deeper off-axis image points in frontof and behind the hologram, again when viewed from the ideal location. Maximumimage point distances are more asymmetric about the ideal view axis than in the phase-conjugate cases above.Although perfectly reconstructing gratings do not suffer from distortion, they donot necessarily approximate ideal predispersed angles or distances very closely.Imperfectly reconstructing gratings can be designed with optimized exposure andreconstruction angles and distances that are much better approximations of the idealquantities.The procedure the author has adopted to optimize grating playout angles anddistances is a reverse-raytracing method using the Moore-Penrose pseudoinverse functionof a matrix expression containing the grating and vertical focus equations. Appendix 4'sMatlab program is an implementation of the pseudoinverse method. Although the authorhas proven that there is no exact solution to the system of linear angle and distanceequations, the pseudoinverse does provide an optimal solution. It is shown that theparticular viewstation geometries of Chapters 5 and 6 are better approximated using thismethod. However, further experimental testing is required to validate the procedure'seffectiveness.A pre-and post-dispersing grating, set flush against the hologram, does not seemcapable of compensating for the hologram's, in addition to its own, dispersion. (Appendix5 is a trial Matlab program). It remains to be seen whether or not thin refractive opticsmight make such a display possible. The diverging wave-front grating has yet to beincorporated in the steep angle format; this could produce the next best approach towardvery compact dispersion-compensated displays.The main body of the thesis is a treatment of vertical dispersion and does notaccount for horizontal dispersion, even in the analysis of compact displays. A morerigorous vector analysis would provide a more complete picture of blur extent in threedimensions. This would also enable one to model a maximum image envelope,composed of image points at maximum distances in front of the hologram within theacuity of the eye, as the eye position and orientation changes.I extend my sincere encouragement to my fellow researchers who further the workoutlined in this thesis.Conclusions and future work128Appendix 1Single-plane trigonometric raytracing formulasA1.1 The X- and Z- raytracing equationsTo begin to understand the significance of the complementary roles of the X- andZ-equations, imagine an infinitely thin recording material set between two sources ofinterfering light. The resulting grating would form an equivalent transmission hologram,and as such, would exhibit the same diffraction characteristics as its transmissioncounterpart.If the recording material is thick, the surface grating structure alone still definesthe diffraction pattern of light leaving the material, in terms of angular playout (notintensity) of any given wavelength. However, the fringe structure (the tip angle, spacing,and number of Bragg planes) will effectively act as a tipped interference filter that filtersthe diffracted light from the surface grating. The surface grating is governed by the X-equation, and the interference filter by the Z-equation. A simultaneous solution of thetwo equations means that one wavelength will constructively interfere at its primaryoutput angle (as determined by the surface grating, or X-equation), and will therefore playout with the highest diffraction efficiency.A1.2 The Welford and X-equationsA2n'1Welford's Equation (A1.1) (1975) ^)=fA)n'2reduces to the trigonometricX-Equation (A1.2) (Benton 1994) sindut - sindn =m12 ni (sinabj - sin6ef)A i n2A1.3 The Z-equationt 1A2n'1Z-Equation (A1.3) (Benton 1994) cos6'out - cos'M = m t (cos6'obj- cos6'ref)This filtering equation will have some degree of uncertainty in its filteringcapability. A final term reflecting this uncertainty in off-Bragg reconstruction may beadded. The following term, Equation (A1.4), is a simplified form of Goodman's blurredK- (grating) vector analysis (Goodman 1996) and defines the spread of the angular andbandwidth playout:Appendix 1 129Grating-vector cloud 0KgEquation (A1.4)A 2t2 n'2HologramFigure A1.1: Grating vector cloud (Goodman, 1996)For our analyses, however, we will assume, calculate, or measure a spectral bandwidthand perform calculations for primary (on-Bragg) playout angles.A1.4 Spectrophotometer dataThe author evaluated a grating exposed to two source points with a Krypton laserat 647 nm on Agfa-Gevaert's 8E75 silver halide emulsion. The grating was illuminatedwith white light focused 600 mm from the film, at 450, so that it would play out achromatically blurred real image of the object point on axis. The exposed fiber end of anOriel spectrophotometer was carefully translated along this spectral line, and recordingswere made of the playout wavelengths and corresponding angles.white lightsourceillumination = 600mm, 225 degrees(reference source = 1200mm, 45 degrees)Oriel spectrophotometer,with its exposed fiber translatedalong the blurred image pointsgratingoutput focus(m = -1) = -159mm, 0 degrees(object source distance = 125mm, 180 degrees)Figure A1.2: Dispersion measurements with an Oriel spectrophotometer translatedalong the blur (achromatic) angle while oriented parallel to the hologram normalAppendix 1130In Table Al.1, the first column lists the recorded wavelengths, and in the adjacentcolumn, measured angles. From the wavelength measured along the normal, a shrinkageof 8.66% was calculated (assuming negligible change in index of refraction).Wavelength Measured Z: X-equation(nm) (degrees)680.2 186.1 182.1670 181.4660 185.1 180.8650 184.1 180.2647 183.8 180640 183.2 179.6632.8 182.6 179.1630 182.4 178.9619.9 181.6 178.3610 181.1 177.7599.9 180.2 177.1590.1 180 176.4580.1 179.9 175.8570.1 178.8 175.2560 178.5 174.6550 177.5 173.9543.1 176.3 173.5Table A1.1: Measurements of dispersion with an Oriel spectrophotometerAppendix 1A1.5 The horizontal and vertical focus equationsTwo additional equations that we will be using throughout the text are distance, orfocusing equations, the "1/R," or horizontal focus equation and the "cos 2O," or verticalfocus equation.1 1 A 2 1 1The horizontal focus Equation (A1.5) is: R 111 obji RRui Rini A i(Roj RreflWe will conclude this appendix with Benton's trigonometric derivation for the verticalfocus equation. This equation is essentially a derivation for the horizontal focus equationas well, but takes into account the effect of wave-front curvature on relative phase.Figure A1.3: (Benton, 1994)N N' ' .4%'.4 \Routi rout2 +rout2 =(hcos(61))2_( o(i)) rout2 = Routiiouti + hcos(Oi)fouti + 900Routi(Routi) routi + Routihcos(6i)fiouti + 900(Rout1) 2 + (hcos(1)) 2(Rout1) 2 sin(Oi)(Routi)2 + (hcos(O1)) 2h(cos(O1)) 2= sin(O1) + RoutiEquation (A1.6)Rout1(h)2cos(01)(Routi) 2 + (hcos(Oi))2(cos(Outi))2 (cos(61n))2 A2 (cos(Obj 1))2 (cos(aef))2Routi Riin A1 Robji RrefnAppendix 1132Appendix 2Derivation of a vector blur equationand its reduction to the single-plane,trigonometric blur equation of Chapter 2A2.1 Color blur equation derivationIn this appendix, we will raytrace multiple wavelengths about a hologram so as toderive a vector blur equation. We will compare the result with the trigonometric result ofChapter 2. The key to the following derivation is the same as that of the trigonometricanalysis: Position (1) values are in terms of the central wavelength, whereas position (2)values are in terms of an extreme wavelength for a given spectral bandwidth.Rill2rill2Rill1rill1Rout2rout2. *Deye2 .*Deyel Robjirobji,Robj2robj2Rref2rref2Rreflrref1Figure A2.1: Magnitudes and directions for the vector derivation of a blur equationAppendix 2IWe will substitute the following variables into a raytracing equation just as we didin the trigonometric version in Chapter 2. As we hope to compare the result with theresult from the trigonometric analysis of the x,z-plane geometry, we will begin withWelford's Equation (Al. 1):2n X (fout - in) = m n X (fobj - iref)Equations (A2.1):A Deyeifout1 + hSny x Deye2134Deye2 rfout2= Deyei fouti + hhRin12 in12= Rini fian + hhRobj2 robj2= Robj irobji +hhRrea ifref2 = Rref 1refn + hhDeyeifouti + hhrfout2 Deye2Riii fini + hh-> i112 = R12Robji .obji + hhRobj2Rref1 rrefI +hh= r~ref2 = Rref2hhIR2r2RIr02Figure A2.2The result of the substitutions of Equations (A2. 1) into (Al.1) is Equation (A2.2):Riini ii1 + hhRi12A 2 ,m-n' 1IRobji robji + hhRobj2Rref1 rref 1 + hhRref2Appendix 2A2.2 Approximation proofWe will now prove that the denominators with the subscript 2 may beapproximated as their counterparts of subscript 1. Then we will proceed with ourderivation.R .sin (0 1) + h Xredsin(9 2) R 2R 1-sin (9 1) + h XredFor small h, R1 is approximately equal to R 2sin(e 2approx) .=- ) IIn order to determine how reasonable an approximation the above equation is,one needs to know the maximum discrepancies between the two angles and the twohypotenuses.The difference between the angle and its approximation must lie withinthe acuity of the eye:AO 2max' 0 2approx - 0 2 degree60This difference would then give us a maximum difference betweenthe two hypotenuses: 8 max'= 2The determination is made using trigonometric identities:sin 2approx- sinG 2 = 2.sin 0 2approx- 0 2 -Cos / 2approx+ 9 22apro 22 222 1= 2-sinI -cos 2 60 = 2.91-10 -(cos (G 2) - 1.45104-sin( 2))2)DI-COS (01)Cos (20 2 /4 D c(cos (0()sinG 2approx sine 2 = 2.9110 - D 2Appendix 2Equating this quantity with the difference between the two sines results in:sinO 2approx - sino 2 = (D -sin ( 1) + h). I- )2.91-10~4-D 1-cos (0 1) (DD I+ max2.91.10~4-D 1-cos ( 0)3max h + sin ( 1)D I(D I.sin(O 1) + h) 1)1 D max 1lsin(91) +h) .I-DI DI+8maD I-Cos (01)3436sin (0 1)An extreme viewing condition testing the approximation is a viewing angleof 45 degrees and viewing distance at the near point (200 mm):-m:= 200 cos (45) = 0.0582 mmma 3436sin(45)cos (0 2) . D 1-cos (0 1)D I + 8 max200cos (45)200 + 0.0582sn( 2 D 1-sin (9 1) + h m axsin(' 2) D -si0 D -isin (01) +h maxS (l~ 2approx) I DI= 0.7069 0 2:= 45.017 degreesh max:=0.085 mm0 2approx := 45.034 degreesThe difference is then seen to be within the acuity of the eye for this extremeviewing condition, so the approximation is a reasonable one to make for thederivation of the blur equation:9 2approx- 0 2 degree60Appendix 2136To resume the derivation, if we ascribe a central wavelength k 2green to position (1)different from that of position (2), then we may reduce the vector expression to thesingle-plane condition, Equation (A2.3):hred hred A2red hred hredsin(at1(2,=)+- -sin(eiii)- -11 sin(abji)+ -sin(eri)- IIEbye1 Riini 21 ldjo1 1 A2rd( 1 1 22redhr& - -n =-si(6t1(X,-)+sin(ain)+nr (sin(aji )- sin(aen))Ily)ei Rii i Rren )1II(A2red - X2grem)ni 1 (s(iQj)-sin(Oen)).hrd -,Ilye1 lat1HORujreThe above Equation (A2.3) exactly matches the trigonometric color blur Equation(2.8) of Chapter 2, with a negative value for Rout lHORIZred. Our final derivation willadd the source-size component to image blur.Appendix 2 137A2.3 Source-size blur derivation+in--. Rill2ril12RillrillRout2rout24 4hblDeye1Figure A2.3: A dispersed focus at the pupil creating the perception of blur heightAppendix 2CD/2 CDhh138Rilfill2 =- D + hh + Rillifilli(e x iilli) x h = (filli x f) x E( - )iiili - (filli -i)O = (h -filli)S - (h -h)fill1(E - Tilli + (h - h)fill1 + ( -)fill1(illi -E)(h - plays out w/acceptable blur -- front and backmaximum output distance withspecified acceptable blur-- front and backmax. output (front) -- Bentonmax. output (back) -- Benton771.631.63514.5micronsmicronstl dcGt2_dcGshrink_n1_dcGn2 dcG11_dcG0illR0illGOillBdObluri450 sepdOblur_0objBenOrefBenOillBenOoutRBe45 0outGBeOoutBBeOtiltBe 046.8682624543.0192993.84896323.848963445179.96792179.9679246.86675743.020841degreesdegreesdegreesdegreesdegreesmmdegreesdegreesdegreesdegreesdegreesdegreesDISPERSION COMPENSATIONGRATING (dcG) INPUT(necessary input from above:everything up through"bandwidth,"&"Rill"&"Deye")H'\\ll//'Il|@=011> <--------- IlIl[I < . Il IIDeye .//I///1/<------*/ RilldcGdcG* B* RoutGvert-dcG (m=+1)* RH ill. angle for long (Red) 12H ill. angle for middle (Green) 12H ill. angle for short (Blue) 12blur angle illuminating HdcGrating/H separationnecessary dcG blur angledcG ANGLESPerfect reconstructionInput OoutG-dcG, Guess Oill-dcGdcG object angle (between ±900)dcG reference angle (900 to 2700)dcG illumination angledcG output angledcG output angle (=0i11 if perfect=0)dcG output angledcG tilt angleAppendix 3142RobjBen9E9 mm300 RrefBen300 Rill dcRoutRBe -9354.555RoutGBe 9E9RoutBBe 8532.3305RoutRve-4372.711 mmRoutGve 9E9RoutBve4560.6652 mm300 LholoLdcGBen 299.99999dcG DISTANCES(ideal: = RoutGvertdcG)obj distancemm (ideal: = RilldcG) ref. distancemm dcGrating illumination distancemm dcG playout distance: Redmm dcG playout distance: Greenmm dcG playout distance: BluedcG vertical playout distance: Redmm (=Rill-sep; ideal output =Robj-dcG)dcG vertical playout distance: BluemmmmMINIMUM dcG SIZEvertical Length of the hologrammin. recom. vertical Length -- BentonAppendix 3 143144 Appendix 3"dispersion compensation © Arno Ktlfu nl I ng iifln n o ln ln n a te f lffflinginiitiff i llann iffillif till f l t ? " README" This TK Solver+ program is extracted from MATLAB programs designed to" measure color blur with and without a dispersion-compensated system." First, the amount of uncompensated dispersion from a hologram will be" calculated. Then, a dispersion compensation system will be included." The technique that will be employed by this program is to solve for the" amount of blur created by a dispersion compensation grating (dcG) that is" required to introduce the extreme wavelengths of the bandwidth to the" hologram (H) at angles that will play out along the normal of H toward the" on-axis viewer. From this calculated blur, the exposure and ideal" and non-ideal reconstruction geometries of the dcG will be determined."" Please take NOTE of the bug report below if you intend to use this" TK Solver program." SOME ABBREVIATIONS" Unless appended with '_dcG,' all measurements are w/ resp. to H.'Lblur' refers to the length of the blurred image point, and" 'Lholo' and 'LdcG' refer to the length along the hologram/dcGrating"f contributing to the perceived blur." R,G, and B refer to long (Red), central (Green), & short (Blue)"f wavelengths. The color names are merely labels." All R 's are distances from the hologram; Deye = distance of the observer." UNCOMPENSATED DISPERSION" The playout wavelengths will be determined by the smaller spectral" bandwidth of H and dcG, assuming equal central playout wavelengths, 12G."12G = 1212R - 12B = bandwidthOoutG = Oout" For constant OillG (no dcG), we have the dispersed HOLOGRAM output angles."sind(90-m*90+m*0outR) = m*12R/l1*(sind(Oobj) - sind(180-Oref))+sind(90+m*90-m*0ill)sind(90-m*90+m*0outG) = m*12G/l I *(sind(Oobj) - sind(1 80-Oref))+sind(90+m*90-m*0ill)sind(90-m*90+m*0outB) = m* 12B/l1 *(sind(Oobj) - sind(1 80-Oref))+sind(90+m*90-m*0ill)dOout = abs(OoutR - OoutB)" Vertical focus playout distances -- Routvert calculations"(cosd(OoutR))^2/RoutRvert = m*12R/l1 *((cosd(Oobj))^2/Robj - (cosd(Oref))^2/Rref+ (cosd(Oill))^2/Rill(cosd(OoutG))A2/RoutGvert = m* 12G/l I *((cosd(Oobj))A2/Robj-(cosd(Oref))A2/Rref)+(cosd(Oill))^2/Rill(cosd(OoutB))A2/RoutBvert = m* 12B/11*((cosd(Oobj))A2/Robj -(cosd(Oref))A2/Rref)+ (cosd(Oi11))A2/RillAppendix3 145" This block calculates blur according to A.A.Ward's diagram.""Horizontal focus playout distances -- Rout_ calculations"l/RoutR = m*l2R/l1*(1/Robj - 1/Rref) + 1/Rill1/RoutG = m*12G/l1*(1/Robj - 1/Rref) + 1/Rill1/RoutB = m*12B/l1*(1/Robj - l/Rref) + 1/RillLholoBcolor = abs((m*12G/1-m*12B/l1)*(sind(Oobj)-sind(Oref))*Deye*RoutB/(Deye+RoutB))LholoRcolor = abs((m* 12R/l 1 -m* 12G/1 )*(sind(Oobj)-sind(Oref))*Deye*Ro.utR/(Deye+RoutR))Lholo Bssize = abs(ssize*cosd(Oill)*Deye*RoutB/(2*Rill*(Deye+RoutB)))LholoRssize = abs(ssize*cosd(Oill)*Deye*RoutR/(2*Rill*(Deye+RoutR)))Lholoblur = LholoBcolor+LholoBssize+LholoRcolor+LholoRssizeLblur = Lholoblur*abs(cosd(OoutG))*(Deye+RoutGvert)/Deyed0blur*piO/180 = Lblur/(Deye+RoutGvert)" Maximum depth (front and back): Oblur = Oacuity"" USAGE: Type 'G' for 'Guess' in the 'St' field on the left."t Then type a small value in the Input field and Solve."d0acuity*pio/180 = Lholo blurmaxfront*abs(cosd(OoutG))/DeyedOacuity*pio/180 = Lholoblurmaxback*abs(cosd(OoutG))/DeyeLholo blurmaxfront =Lholo Bcolormaxfront+LholoBssizemaxfront+LholoRcolormaxfront+LholoRssizemaxfrontLholo blurmaxback =LholoBcolormaxback+LholoBssizemaxback+LholoRcolormaxback+LholoRssizemaxbackLholoBcolormaxfront = abs((12G/l 1 -12B/11)*(sind(0obj)-sind(Oref))*Deye*RoutBmaxfront/(Deye+RoutBmaxfront))LholoRcolormaxfront = abs((12R/l 1-12G/l 1)*(sind(Oobj)-sind(Oref))*Deye* RoutRmaxfront/(Deye+RoutRmaxfront))LholoBcolormaxback = abs((l2G/ll-12B/ll)*(sind(Oobj)-sind(Oref))*Deye*RoutBmaxback/(Deye+RoutBmaxback))LholoRcolormaxback = abs((12R/l 1-12G/I )*(sind(Oobj)-sind(Oref))*Deye*RoutRmaxback/(Deye+RoutRmaxback))Lholo Bssizemaxfront = abs(ssize*cosd(Oill)*Deye*RoutBmaxfront/(2* Rill*(Deye+RoutBmaxfront)))Lholo Rssizemaxfront = abs(ssize*cosd(Oill)*Deye* RoutRmaxfront/(2*Rill*(Deye+RoutRmaxfront)))Lholo Bssizemaxback = abs(ssize*cosd(Oill)*Deye* RoutBmaxback/(2*Rill*(Deye+RoutBmaxback)))Lholo Rssizemaxback = abs(ssize*cosd(Oil)* Deye*RoutRmaxback/(2* Rill*(Deye+RoutRmaxback)))1/RoutRmaxfront = -12R/l1*(1/Robjmaxfront - 1/Rref) + 1/Rill1/RoutBmaxfront = -12B/l1*(1/Robjmaxfront - 1/Rref) + 1/Rill1/RoutRmaxback = 12R/l1*(1/Robjmaxback - 1/Rref) + 1/Rill1/RoutBmaxback = 12B/l1*(1/Robjmaxback - 1/Rref) + 1/Rill-abs(cosd(OoutG))^2/Routfront =abs(-12G/lI *((cosd(0obj))A2/Robjmaxfront - (cosd(Oref))^2/Rref) + abs(cosd(Oill))^2/Rill)abs(cosd(OoutG))^2/Routback =abs(12G/l 1 *((cosd(Oobj))^2/Robjmaxback - (cosd(Oref))^2/Rref) + abs(cosd(Oill))^2/Rill)This block calculates blur according to S.A.Benton's diagram."dOBentoncolor = 180/pio*(bandwidth/12G)*abs((sind(OoutG)-sind(Oill))/cosd(OoutG))dOBentonssize = 180/pio*ssize*abs(cosd(Oill)/(Rill*cosd(OoutG)))d0BentonA2 = d0Benton colorA2 + dOBenton ssize^2LblurBenton = abs(RoutGvert*d0Benton*piO/180)d0blurBenton*piO/1 80 = LblurBenton/(RoutGvert+Deye)Appendix3146" Maximum depth (front and back) -- Benton:"RoutBentonfront = -Deye*tand(dOacuity/2)/(tand(d0Benton/2)+tand(d0acuity/2))RoutBentonback = Deye*tand(d0acuity/2)/(tand(d0Benton/2)-tand(d0acuity/2))" DISPERSION COMPENSATIONRratio = 12R/11 dcGGratio = 12G/l1 dcGBratio = 12B/11_dcG" For sin0out = zero, we first calculate the HOLOGRAM illumination angles."-sind(90+m*90-m*0illR) = m*12R/l 1 *(sind(0obj) - sind(I 80-Oref))-sind(90+m*90-m*0illG) = m* 12G/l I *(sind(Oobj) - sind(I 80-Oref))-sind(90+m*90-m*0ilIB) = m* 12B/l1 *(sind(Oobj) - sind(1 80-Oref))" The difference between these angles defines" the blur illuminating the hologram."dOblurill = abs(OillR - OillB)" We can now determine the necessary blur created by the dcG (dOblur-dcG)."" With the specified grating/hologram separation (sep) and" the HOLOGRAM illumination distance (Rill) at the central wavelength,RoutGvertdcG = Rill - sepabs(RoutGvertdcG)*tand(dOblur-dcG/2) = Rill*tand(d0blurill/2)" We will work backwards from the desired dcGrating blur to determine" the output and illumination (as well as tilt) angles of the dcGrating.""f NOTE" The Ward calculations in this section are presently getting stuck insome kind of loop in the iterative solver. Although this section is stillable to provide correct angles and distances for the geometries the authorhas tried, these values disappear from the screen when solving.The usual TK solution is to include redundant equations to help theiterative solver.The accompanying Variables section in this appendix, therefore, liststhe blur calculations of Benton's equations. Both Ward's and Benton'ssolutions are in very close accordance for this geometry, in any case.This block calculates angles from blur according to A.A.Ward's diagram."d0blur dcG*pio/180 = Lblur dcG/(sep+RoutGvertdcG)LblurdcG = LdcGblur*cosd(0outGdcG)*(sep+RoutGvertdcG)/sepLdcG blur= LdcGBcolor+LdcGBssize+LdcGRcolor+LdcGRssizeLdcGBcolor =abs((Gratio-Bratio)*(sind(OobjdcG)-sind(Oref dcG))*sep*RoutB-dcG/(sep+RoutB_dcG))LdcGRcolor =abs((Rratio-Gratio)*(sind(0obj dcG)-sind(Oref dcG))*sep*RoutR-dcG/(sep+RoutR-dcG))LdcGBssize = abs(ssize*cosd(0ill dcG)*sep*RoutB dcG/(2*Rill_dcG*(sep+RoutB_dcG)))LdcGRssize = abs(ssize*cosd(OilldcG)*sep*RoutR-dcG/(2*RilldcG*(sep+RoutRdcG)))"Horizontal focus playout distances -- Rout_ calculations"1/RoutRdcG = 12R/IlldcG*(/Robj-dcG - 1/RrefdcG) + 1/RilldcG1/RoutGdcG = 12G/ll dcG*(1/Robj-dcG - 1/RrefdcG) + 1/Rill-dcG1/RoutB-dcG = 12B/lI dcG*(1/Robj-dcG - 1/RrefdcG) + 1/Rill-dcGAppendix3 147(cosd(OoutR-dcG))A2/RoutRvertdcG = 12R/lldcG*((cosd(0obj-dcG))A2/Robj-dcG -(cosd(Oref-dcG))A2/Rref-dcG) + (cosd(Oill dcG))2/RilldcG(cosd(0outG-dcG))A2/RoutGvertdcG = 12G/l1_dcG*((cosd(0objjdcG))A2/Robj-dcG -(cosd(OreftdcG))A2/Rref dcG) + (cosd(Oill_dcG))2/RilldcG(cosd(0outB-dcG))A2/RoutBvertdcG = 12B/lldcG*((cosd(0obj_dcG))A2/Robj_dcG -(cosd(Oref-dcG))A2/Rref-dcG) + (cosd(Oill-dcG))A2/RilldcG" The ANGLES for the different wavelengths playing out of the" dcG are all the result of a constant illumination angle (Oill-dcG).sind(OoutR dcG )-sind(1 80-Oill dcG ) = Rratio*(sind(OobjdcG ) - sind(1 80-Oref-dcG))sind(OoutG dcG )-sind(180-Oill dcG ) = Gratio*(sind(Oobj-dcG ) - sind(1 80-Oref dcG))sind(OoutB-dcG )-sind(1 80-0ill-dcG ) = Bratio*(sind(Oobj-dcG ) - sind(1 80-Oref dcG))" This block calculates angles from blur according to S.A.Benton's diagram."d0blurdcGA2 = d0blurBentoncolorA2 + d0blurBentonssizeA2dOblurBentoncolor = 180/pio*(bandwidth/12G)*abs((sind(OoutGBenton dcG)-sind(OillBentondcG))/cosd(OoutGBentondcG))d0blurBentonssize = 180/pio*ssize*abs(cosd(OillBenton-dcG)/(Rill dcG*cosd(OoutGBenton-dcG)))RillBentondcG = RilldcG1/RoutRBentondcG = 12R/lldcG*(1/RobjBenton-dcG - 1/RrefBenton-dcG) + 1/RilldcG1/RoutGBentondcG = 12G/lldcG*(1/RobjBenton-dcG - 1/RrefBenton-dcG) + 1/RilldcG1/RoutBBentondcG = 12B/ll1dcG*(1I/RobjBenton-dcG - 1/RrefBenton-dcG) + 1/RilldcG(cosd(OoutRBenton-dcG))2/RoutRvertBentondcG =12R/lldcG*((cosd(0objBenton-dcG))A2/RobjBentondcG -(cosd(OrefBenton dcG))A2/RrefBenton-dcG) + (cosd(OillBentondcG))A2/Rill-dcGRoutGvertBenton-dcG = RoutGvertdcG(cosd(OoutGBenton-dcG))2/RoutGvertBentondcG =12G/l 1_dcG*((cosd(0objBentonjdcG))A2/RobjBentonJdcG -(cosd(OrefBentondcG))A2/RrefBenton-dcG) + (cosd(OillBentondcG))A2/Rill-dcG(cosd(0outBBenton_dcG))A2/RoutBvertBenton_dcG =12B/l1_dcG*((cosd(0objBentondcG))A2/RobjBenton-dcG -(cosd(OrefBentondcG))A2/RrefBenton-dcG) + (cosd(OillBentondcG))A2/Rill-dcGsind(OoutRBenton-dcG) =Rratio*(sind(0objBenton-dcG) - sind(1 80-OrefBentondcG)) + sind(1 80-0illBenton-dcG)sind(OoutGBentonjdcG) =Gratio*(sind(OobjBenton-dcG) - sind(OrefBenton-dcG)) + sind(OillBentondcG)sind(OoutBBenton-dcG) =Bratio*(sind(OobjBenton-dcG) - sind(180-OrefBentondcG)) + sind(180-OillBenton-dcG)" For the case of perfect reconstruction of the dcG (reconstruction angles =" exposure angles), the dcG will be tilted with respect to Hso that the dcG creates sufficient blur.This tilt angle will be the illumination angle of the dcG, so that the 12Goutput angle of the dcG will equal the central illumination angle of H."Input Oout-dcG or Oill-dcG, Guess the other.Otilt-dcG = Oill - (m*90+90) - 0outG-dcGOref-dcG = Oill-dcGOobj-dcG = OoutGdcGOtiltBentondcG = Oill - (m*90+90) - OoutGBentondcGOrefBentondcG = OillBentondcGOobjBentondcG = OoutGBentondcGAppendix 3148" MINIMUM dcG SIZE" Because the dcG acts as a window through which H is illuminated," we will determine a reasonable minimum dcG size based on previouscalculations. The size could be smaller, depending on the blur length,but this calculation assumes that we want the whole H area to beilluminated by at least the central wavelength.Lholo = length of the hologram; LdcG = length of the dcGratingtand(Olow)= (Rill*sind(OillG)-Lholo/2)/(Rill*cosd(OillG))tand(Ohigh)= (Rill*sind(OillG)+Lholo/2)/(Rill*cosd(OillG))dOspread = Ohigh-OlowLdcG = abs(d0spread*pio/180 * RoutGvert dcG/(cosd(OillG)*cosd(Otilt-dcG)))LdcGBenton = abs(d0spread*pio/180 * RoutGvertBenton-dcG/(cosd(OillG)*cosd(OtiltBenton-dcG)))Appendix 3 149150 Appendix 3Alternative Matlab program% WARDvsSABblur.m color blur% MATLAB program to compare color blur values for Benton's and Ward's diagramsclgclearfid = fopen('WARD.dat', 'w');HBLURWARD =HBLURSAB = [ ;DVIEWER = [ ;HTOTALWARD =BLURANGLEWARD = [ ;BLURANGLESAB = [ ;ROBJG=[];ROUTGVERT =[];objG = 68.179171;refG = 180.02732;illG = 180.02732;RobjG = 9E9;RrefG = 9E9;RillG = 9E9;ssize = 0;tlt2 =7/7;lambdal = 647;lambdaR = 657;lambdaG = 647;lambdaB = 637;m= 1;ROBJG = [ROBJG RobjG];lambdaRratio = m*lambdaR/lambdal;lambdaGratio = m*lambdaG/lambdal;lambdaBratio = m*lambdaB/lambdal;sinobjG = sin(objG*pi/180);cosobjG = cos(objG*pi/180);sinrefG = sin(refG*pi/180);cosrefG = cos(refG*pi/180);sinillG = sin(illG*pi/180);cosillG = cos(illG*pi/180);sinoutG = lambdaGratio*(sinobjG - sinrefG)+ sinillG;outG = asin(sinoutG)* 180/pi;if abs(objG) > 90 & m >= 1outG = 180 - outG;endAppendix 3cosoutG = cos(outG*pi/180);RoutRhorz = 1/(lambdaRratio*(1/RobjG - 1/RrefG) + 1/RilIG);RoutGhorz = 1/(lambdaGratio*(1/RobjG - 1/RrefG) + 1/RilIG);RoutBhorz = 1/(lambdaBratio*(1/RobjG - 1/RrefG) + 1/RilIG);RoutGvert = cosoutG*cosoutG/(IambdaGratio*(cosobjG*cosobjG/RobjG - cosrefG*cosrefG/RrefG) +cosillG*cosillG/RillG);ROUTGVERT = [ROUTGVERT RoutGvert];fprintf(fid, 'objG = %4.3f, refG = %4.3f, ilG = %4.3f\noutG = %4.3f\nRrefG = %4.3f, RillG =%4.3f\nRoutRhorz = %4.3f, RoutBhorz = %4.3f\nlambdaGratio = %4.3f/%4.3f\n',objG,refG,illG,outG,RrefG,RillG,RoutRhorz,RoutBhorz,lambdaG,lambdal);DMIN = 200;if m == -1DMIN = -RoutBhorz + 200;endfor D = DMIN: 10: 1000;hRcolorWARD = (lambdaRratio-lambdaGratio)*(sinobjG-sinrefG)*D*RoutRhorz/(D + RoutRhorz);hBcolorWARD = (lambdaBratio-lambdaGratio)*(sinobjG-sinrefG)*D*RoutBhorz/(D + RoutBhorz);hRssizeWARD = ssize*cosillG*D*RoutRhorz/(2*RillG*(D + RoutRhorz));hBssizeWARD = ssize*cosillG*D*RoutBhorz/(2*RillG*(D + RoutBhorz));htotalWARD = abs(hRcolorWARD+hRssizeWARD) + abs(hBcolorWARD+hBssizeWARD);hblurWARD = htotalWARD*abs(D+RoutGvert)*abs(cosoutG)/D;blurangleWARD = hblurWARD/(D+RoutGvert) * 180/pi;DVIEWER = [DVIEWER D];HTOTALWARD = [HTOTALWARD htotalWARD];HBLURWARD = [HBLURWARD hblurWARD];BLURANGLEWARD = [BLURANGLEWARD blurangleWARD];hcolorSAB = ((lambdaR-lambdaB)/lambdaG)*(sinoutG-sinilG)/cosoutG;hssizeSAB= ssize*cosillG/(RillG*cosoutG);hblurSAB = abs(RoutGvert*sqrt(hcolorSAB*hcolorSAB + hssizeSAB*hssizeSAB));blurangleSAB2 = sqrt(hcolorSAB*hcolorSAB + hssizeSAB*hssizeSAB) * 180/pi;blurangleSAB = hblurSAB/(D+RoutGvert) * 180/pi;HBLURSAB = [HBLURSAB hblurSAB];BLURANGLESAB = [BLURANGLESAB blurangleSAB];fprintf(fid, 'RobjG = %4.3f, RoutGvert = %4.3f, D = %4.3f\nhblurWARD = %4.3f, hblurSAB =%4.3f\nblurangleWARD = %4.4f, blurangleSAB = %4.4f\n\n',RobjG,RoutGvert,D,hblurWARD,hblurSAB,blurangleWARD,blurangleSAB);endendendfclose(fid)subplot (3,1,1), plot (ROUTGVERT, HBLURWARD, 'go');Appendix 3152title ('perceived blur height with image output distance');hold onplot (ROUTGVERT, HBLURSAB, 'y*');xlabel('Image output distance (mm)');ylabel('perceived blur height (mm)');subplot (3,1,2), plot (ROUTGVERT, BLURANGLEWARD, 'go');title ('perceived blur angle with image output distance');hold onplot (ROUTGVERT,BLURANGLESAB, 'y*');xlabel('image output distance (mm)');ylabel('perceived blur angle (degrees)');subplot (3,1,3), plot (ROUTGVERT, HTOTALWARD, 'g-');title ('Height of hologram region contributing to blur');xlabel('image output distance (mm)');ylabel('hologram height -- blurWARD');hold offAppendix 3 153154Appendix 4A Matlab program for the design ofdispersion-compensation gratings (Chapter 6)% gratingdesigner.m% This program first calculates the angles and distances required for three different wavelengths to% reconstruct a holographic image (H2) along a viewer axis. Illumination of the hologram from these% calculated positions compensates for chromatic dispersion that would have occurred had a single,% broadband point source of light illuminated the hologram.% Then the program takes the specified separation between this hologram and a dispersion% compensation grating (HI), and calculates what the playout angles and distances for the three different% wavelengths would have to be from HI to compensate for dispersion of H2.% Finally, the program outputs a list of potential exposure and reconstruction geometries into the% file: "gratingdesigner.dat".clearfid = fopen('gratingdesigner.dat', 'w');OBJ =REF =ILL =OUTG=[];%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%//%%%%%%%%%%%%% INPUT VARIABLES: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%lambdal 514.5; % exposure wavelength (nm)lambdaR 524.5; % output long wavelength -- "Red"lambdaG -514.5; % output central wavelength -- "Green"lambdaB = 504.5; % output short wavelength -- "Blue"H2out = 180; % degrees % dispersion-compensated H2 output angleH2obj = 0; % degrees % H2 object (signal) beam angleH2ref= 225; % degrees % H2 reference beam anglemH2 =-; % H2 diffractive orderH2Rout = -500; % H2 output distanceH2Robj = 300; % H2 object distanceH2Rref = 750; % H2 reference point distancemHl = 1; % H1 diffractive orderHIRillmax = inf; % maximum grating illumination distanceH I H2separation = 450; % grating/hologram separationH IoutGmin = 30; % output angle range minimum (Green)HIoutGmax = 70; % output angle range minimum (Green)deltaHIRout 50; % maximum discrepancy of HI output distances%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% H2 ANGLES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%H2out = H2out*pi/l 80;H2obj = H2obj*pi/180;H2ref = H2ref*pi/180;HloutGmin = HloutGmin*pi/180;Appendix4 155H IoutGmax = H IoutGmax*pi/180;lambdaRratio = lambdaR/lambdal;lambdaGratio = lambdaG/lambdal;lambdaBratio = lambdaB/lambdal;H2iIIR = asin(sin(H2out) - lambdaRratio*mH2*(sin(H2obj)-sin(H2ref)));H2illG = asin(sin(H2out) - lambdaGratio*mH2*(sin(H2obj)-sin(H2ref)));H2ilIB = asin(sin(H2out) - lambdaBratio*mH2*(sin(H2obj)-sin(H2ref)));RH2G = abs(H2iIIR - H2iIIG); % angular H2 illumination (green-red)BH2G = abs(H2iIlG - H2iIIB); % angular H2 illumination (green-blue)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% H2 DISTANCES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ideal H2 illumination distances -- Red, Green, and BlueRH2 = cos(H2iIIR)^2/(cos(H2out)^2/H2Rout - lambdaRratio*mH2*(cos(H2obj)^2/H2Robj -cos(H2ref)^2/H2Rref));GH2 = cos(H2illG)^2/(cos(H2out)^2/H2Rout - lambdaGratio*mH2*(cos(H2obj)^2/H2Robj -cos(H2ref)^2/H2Rref));BH2 = cos(H2ilIB)^2/(cos(H2out)^2/H2Rout - lambdaBratio*mH2*(cos(H2obj)^2/H2Robj -cos(H2ref)^2/H2Rref));%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% HI DISTANCES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% H I ideal playout (Green)H IG = GH2 - H 1H2separation;% HI ideal playout (Blue, Red) (cosine law)H I Bideal = sqrt(BH2^2 + H I H2separation^2 - 2*BH2*H I H2separation*cos(BH2G));H I Rideal = sqrt(RH2^2 + H I H2separation^2 - 2*RH2*H I H2separation*cos(RH2G));%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% HI EXPOSURE ANGLES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%coeffl = [lambdaRratio*mH1 -lambdaRratio*mH IIlambdaGratio*mH1 -lambdaGratio*mH1 1lambdaBratio*mH 1 -lambdaBratio*mHl 1];% HI playout angles: sin(H2HIBideal)/BH2 = sin(BH2G)/HIBideal (sine law)H2H I B = pi - asin(BH2*sin(BH2G)/HIBideal);BHIG =pi-H2HIB;H2H1R = pi - asin(RH2*sin(RH2G)/H IRideal);RHIG =pi-H2HR;outG = H loutGmin;for outG = HIoutGmin : 1*pi/180 : HIoutGmaxoutR = outG + RH IG;outB = outG - BHIG;sinoutR = sin(outR);sinoutG = sin(outG);sinoutB = sin(outB);sinout = [sinoutR; sinoutG; sinoutB];Appendix4156sinobjrefill = pinv(coeffl)*sinout;sinobj = sinobjrefill(1);sinref= sinobjrefill(2);sinill = sinobjrefill(3);if sinobj-1 & sinref>-1 & sinill>-lobj = asin(sinobj);ref pi-asin(sinref);ill = pi-asin(sinill);OBJ = [OBJ obj*180/pi];REF = [REF ref*180/pi];ILL = [ILL ill*180/pi];OUTG = [OUTG outG*180/pi];%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% HI EXPOSURE DISTANCES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%coeff2=[lambdaRratio*mHI*cos(obj*pi/180)^2 -lambdaRratio*mHI*cos(ref*pi/180)^2 cos(ill)^2lambdaGratio*mHI*cos(obj*pi/180)^2 -lambdaGratio*mHI*cos(ref*pi/180)^2 cos(ill)^2lambdaBratio*mH1*cos(obj*pi/180)^2 -lambdaBratio*mHI*cos(ref*pi/180)A2 cos(ill)A2];% Non-ideal HI output distances may be required to satisfy the vertical focus equations.% HlR= HlRideal;% HIB = HIBideal;for HIR = HIRideal - deltaH IRout/2 :1: HIRideal + deltaH l Rout/2for H I B = H I Bideal - deltaH l Rout/2 :1: H I Bideal + deltaH I Rout/2cossqoverH 1 R = cos(outR)A2/H I R;cossqoverH IG = cos(outG)^2/H IG;cossqoverH I B = cos(outB)^2/H 1 B;cossqoverH lout = [cossqoverH 1 R; cossqoverH 1 G; cossqoverH I B];oneoverRobjrefill = pinv(coeff2)*cossqoverH lout;Robj = 1/oneoverRobjrefill(1);Rref = 1/oneoverRobjrefill(2);Rill = 1/oneoverRobjrefill(3);% if (Robj<-1000 I Robj>0) & (Rref<-1000 I Rref>O) & Rill>O & Rill <= HiRillmaxfprintf(fid, 'outR = %4.3f, outG = %4.3f, outB = %4.3f\nobj = %4.3f, ref = %4.3f,ill = %4.3f\nRobj = %4.3f, Rref =%4.3f, Rill = %4.3f\nRH2 = %4.3f, GH2 =%4.3f,BH2=%4.3f\nHlRideal=%4.3f, HIGideal=%4.3f, HIBideal=%4.3f\nHIR=%4.3f, HIG=%4.3f, HIB = %4.3f\n\n', outR*180/pi, outG*180/pi, outB*180/pi, obj*180/pi, ref*180/pi, ill*180/pi,Robj, Rref, Rill, RH2, GH2, BH2, HIRideal, HIG, HIBideal, HIR, HIG, HIB);% endendendendendplot (OUTG, OBJ, 'go');hold on;title ('Exposure and Illumination Angles for the Diverging Grating: 750/750');plot (OUTG, REF, 'g-');plot (OUTG, ILL, 'y*');xlabel('central lambda output angle, outG (degrees)');ylabel('obj(o), ref(-), and ill(*) angles (degrees)');Appendix4158Appendix 5A Matlab program attempting to solve for a pre- and post-dispersing grating set flush against a hologram (Chapter 6)% Hologram/grating sandwich% -----------------------------------------------------------% fdcg = abs(sinobjdcg - sinrefdeg)/lambdal;% fH = abs(sinobjH - sinrefH)/lambdal;%-------------------------------------------------------------------------------% 1st pass through the dcg%-------------------------------------------------------------------------% sinoutldcg = sinillldeg + mldcg*lambda2*fdcg;% ------------------------------------------------------------------------% Reflection hologram% -------------------------------------------------------------------------% sinoutH = sinoutldcg + mH*lambda2*fH;% sinoutH = sinillldcg + mldcg*lambda2*fdcg + mH*lambda2*fH;%-------------------------------------------------------------------------% 2nd pass through the dcg%-------------------------------------------------------------------------% sinout2dcg = 0 = sinoutH + m2dcg*lambda2*fdcg;% 0 = sinillldcg + mldcg*lambda2*fdcg + mH*lambda2*fH + m2dcg*lambda2*fdcg;% --------------------------------------------------------------------------% Substitution:% ------------------------------------------------------------------------% sinillldcg = -m1dcg*lambda2*fdcg - mH*1ambda2*fH - m2dcg*lambda2*fdcg;% fdcg = (sinobjdcg - sinrefdcg)/lambdal;% refdcg = illdcg% sinillldcg = (-(mldcg*lambda2 + m2dcg*lambda2)*objdcg-% mH*lambdal *lambda2*fH)% /(lambdal - mldcg*lambda2 - m2dcg*lambda2)% ---------------------------------------------fid = fopen('bumsteadDcgH.dat','w');fid2 = fopen('bumsteadDcgH.dat','a');fprintf(fid,'\n');lambda1G = .000000514;lambda2R = .000000524;lambda2G = .000000514;lambda2B = .000000504;for mldcg = -1 2: 1for m2dcg = -1 :2: 1formH =-1:2:1for objdcg = 110 : 10 : 180for objH = -70: 10: 250for refH = -70: 10 : 250sinobjdcg = sin(pi/180*objdcg);fH = abs(sin(objH*pi/180) - sin(refH*pi/180))/lambda1G;Appendix 5numG = (-(m1dcg*1ambda2G + m2dcg*lambda2G)*sinobjdcg - mH*lambdalG*lambda2G*fH);numR = (-(mldcg*lambda2R + m2dcg*lambda2R)*sinobjdcg - mH*lambdalG*1ambda2R*fH);numB = (-(mldcg*1ambda2B + m2dcg*lambda2B)*sinobjdcg - mH*lambdalG*lambda2B*fH);denomG = (lambdalG - mldcg*lambda2G - m2dcg*lambda2G);denomR = (lambdalG - mldcg*lambda2R - m2dcg*lambda2R);denomB = (lambdalG - mldcg*lambda2B - m2dcg*lambda2B);sinillldcgG = numG/denomG;sinillldcgR = numR/denomR;sinillldcgB = numB/denomB;illdcg = 180 - 180/pi*asin(sinillldcgG);% just to make sure that the zero order does not strike the viewer: outH>30 or outH<30degfdcg = abs(sinobjdcg - sinillldcgG)/lambda1G;sinoutldcg = sinillldcgG + mldcg*lambda2G*fdcg;sinoutH = sinoutldcg + mH*lambda2G*fH;outH = 180/pi*asin(sinoutH);if outH>30 I outH<-30if sinillldcgG >-1 & sinillldcgG <1if illdcg>100 & illdcg<260if sinillldcgR == sinillldcgGif sinillldcgB == sinillldcgGif (objH>-70 & objH<70 & refH>1 10 & refH<250) I (objH>1 10 & objH<250 & refH>-70 & refH<70)fprintf(fid2,'mldcg = %4.5f, m2dcg = %4.5f, mH = %4.5f\n', mldcg,m2dcg,mH);fprintf(fid2,'refdcg = %4.5f, or %4.5f\n', illdcg, -180 - illdcg);fprintf(fid2,'objdcg = %4.5f, refdcg = %4.5f, or %4.5f\n', objdcg,illdcg,-180-illdcg);fprintf(fid2,'objH = %4.5f, refH = %4.5f\n\n', objH, refH);endendendendendendendendendendendendfclose(fid)Appendix 5160ReferencesAbramson, Nils. 1981. 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In situ color control for reflection holography

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