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Kyoji Matsushima

Introduction to ComputerHolographyCreating Computer-Generated Hologramsas the Ultimate 3D Image

123

Kyoji MatsushimaFaculty of System EngineeringKansai UniversityOsaka, Japan

ISSN 2509-5900 ISSN 2509-5919 (electronic)Series in Display Science and TechnologyISBN 978-3-030-38434-0 ISBN 978-3-030-38435-7 (eBook)https://doi.org/10.1007/978-3-030-38435-7

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Preface

It is an undoubted fact that the evolution of digital computers impacts our lifestylesas well as science and technology. It is also reasonable to assume that people will besurprised to see three-dimensional (3D) images produced by a well-made opticalhologram, and that they believe that the 3D image is perfect and accurate in itsrepresentation. Unfortunately, hologram galleries that exhibit holograms for art anddecoration have been disappearing recently, and holograms we see in our daily livesare limited to those on bills and credit cards. Such holograms are interestingbecause the appearance of the image changes when the observer changes his or herviewing angle, but they do not look like 3D images. This decline of holography for3D imaging is attributed to the fact that holography has not yet evolved into adigital technology. It is not possible to store the digital data of an optical hologramin a digital medium and it is not possible to transmit it through digital networks.

The idea of creating and handling 3D holographic images using computers has along history. In fact, the origin of the idea goes back to the days right after theactualization of 3D imaging by optical holography. However, computer-generatedholograms (CGH) comparable to optical holograms were not developed untilrecently. This is entirely due to the tremendous data sizes and computational effortsrequired to produce CGHs capable of reconstructing perfect 3D images. The cre-ation of outstanding large-scale CGHs is, in a sense, a fight against the availabilityof computer resources at a time. The required computational capabilities oftenexceed those of state-of-the-art computers. Algorithms and techniques available inthe literature are often ineffective because they are too time and/or resourceconsuming.

This book does not intend to provide all the various techniques proposed incomputer holography extensively. The techniques discussed herein are instead onesthat have been confirmed and proven to be useful and practical for producing actuallarge-scale CGHs whose 3D images are comparable to those of optical holography.Some of these techniques are of an atypical nature as well. It is both unfortunate,and a little delightful for researchers like me that such techniques are still far fromcompletion. Thus, I intended this book to provide just a snapshot of these devel-oping technologies.

v

When we produced the first large-scale CGH, “The Venus” in 2009, approxi-mately 48 h of computation time was needed. An expensive computer mounted on aserver rack was used. The computer was too noisy to be kept in a room where onestudies, owing to the cooling mechanism. Now, the same CGH can be calculated inapproximately 20 min using a quiet desktop computer. This is mainly due to thecontinuous development of computer hardware and partially due to the develop-ment of our technique with time.

This book is for any researcher, graduate or undergraduate student, who wants tocreate holographic 3D images using computers. Some of the chapters (e.g., Chaps. 3and 4) deal with basic topics and researchers already familiar with them can skipthose. Some techniques described in this book (e.g., Chaps. 6, 9, and 13) are usefulnot only for computer holography but also for all fields of research that requiretechniques for handling wavefields of light. In addition, because of the shortness ofwavelength, spatial data of light tends to be of gigantic sizes, exceeding memorysizes of computers. Some techniques introduced in this book (e.g., Chap. 12) givehints on how to handle such large-scale wavefields on computers.

I am sincerely grateful to Prof. Nakahara, who was a colleague at KansaiUniversity, co-author of many papers, and someone with the same enthusiasm forcreating CGHs as me. Many CGHs introduced in this book were fabricated by him.We visited many places together to present our work on CGHs. I would also like tothank Dr. Claus E. Ascheron for his strong recommendation to write a book oncomputer holography. He was an executive editor of Springer-Verlag when I methim for the first time and is now retired. This book would never have been written ifhe had not patiently persuaded me for over 3 years. I regret that I could notcomplete this book before his retirement. I am also grateful to Dr. Petr Lobaz whoreviewed my manuscript carefully and gave me many suggestions based on hisprofound knowledge and enthusiasm for holography. I am also thankful to myformer and current students: Dr. Nishi, Mr. Nakamoto and Mr. Tsuji, who didreviews of the text and formulae. I am grateful to them, and any mistakes that mayhave crept in are entirely mine.

Finally, I would like to thank my colleagues at the Department of Electrical andElectronic Engineering at Kansai University. I am particularly thankful to all myPh.D., masters, and undergraduate students, an assistant, and everyone whobelonged to my laboratory over the past 20 years at Kansai University. Their ideas,efforts, and passion has led to our success in computer holography.

Osaka, JapanOctober 2019

Kyoji Matsushima

vi Preface

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Computer Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Difficulty in Creating Holographic Display . . . . . . . . . . . . . . . . 31.3 Full-Parallax High-Definition CGH . . . . . . . . . . . . . . . . . . . . . 5

2 Overview of Computer Holography . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Optical Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Computer Holography and Computer-Generated Hologram . . . . 152.3 Steps for Producing CGHs and 3D Images . . . . . . . . . . . . . . . . 162.4 Numerical Synthesis of Object Fields . . . . . . . . . . . . . . . . . . . . 17

2.4.1 Object Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.2 Field Rendering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4.3 Brief Overview of Rendering Techniques . . . . . . . . . . 19

2.5 Coding and Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Introduction to Wave Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1 Light as Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1 Wave Form and Wave Equation . . . . . . . . . . . . . . . . 253.1.2 Electromagnetic Wave . . . . . . . . . . . . . . . . . . . . . . . 273.1.3 Complex Representation of Monochromatic

Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.4 Wavefield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Plane Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.1 One-Dimensional Monochromatic Wave . . . . . . . . . . 313.2.2 Sampling Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.3 Plane Wave in Three Dimensional Space . . . . . . . . . . 333.2.4 Sampled Plane Wave . . . . . . . . . . . . . . . . . . . . . . . . 363.2.5 Maximum Diffraction Angle . . . . . . . . . . . . . . . . . . . 373.2.6 More Rigorous Discussion on Maximum

Diffraction Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

vii

3.3 Spherical Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3.1 Wave Equation and Solution . . . . . . . . . . . . . . . . . . . 403.3.2 Spherical Wavefield and Approximation . . . . . . . . . . 413.3.3 Sampled Spherical Wavefield and Sampling

Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4 Optical Intensity of Electromagnetic Wave . . . . . . . . . . . . . . . . 46

4 The Fourier Transform and Mathematical Preliminaries . . . . . . . . 494.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 The Fourier Transform of Continuous Function . . . . . . . . . . . . 49

4.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2.2 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.2.3 Several Useful Functions and Their Fourier

Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3 Symmetry Relation of Function . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3.1 Even Function and Odd Function . . . . . . . . . . . . . . . 544.3.2 Symmetry Relations in the Fourier Transform . . . . . . 55

4.4 Convolution and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 564.5 Spectrum of Sampled Function and Sampling Theorem . . . . . . . 584.6 Discrete Fourier Transform (DFT) . . . . . . . . . . . . . . . . . . . . . . 614.7 Fast Fourier Transform (FFT) . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.7.1 Actual FFT with Positive Indexes . . . . . . . . . . . . . . . 654.7.2 Use of Raw FFT with Symmetrical Sampling . . . . . . 664.7.3 Discrete Convolution Using FFT . . . . . . . . . . . . . . . . 70

5 Diffraction and Field Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 755.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.1.1 Field Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.1.2 Classification of Field Propagation . . . . . . . . . . . . . . . 77

5.2 Scalar Diffraction Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.2.1 Angular Spectrum Method . . . . . . . . . . . . . . . . . . . . 785.2.2 Fresnel Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . 835.2.3 Fraunhofer Diffraction . . . . . . . . . . . . . . . . . . . . . . . . 86

5.3 Optical Fourier Transform by Thin Lens . . . . . . . . . . . . . . . . . 885.3.1 Wave-Optical Property of Thin Lens . . . . . . . . . . . . . 885.3.2 Wavefield Refracted by Thin Lens . . . . . . . . . . . . . . 90

5.4 Propagation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.4.1 Propagation Operator as System . . . . . . . . . . . . . . . . 925.4.2 Backward Propagation . . . . . . . . . . . . . . . . . . . . . . . 93

6 Numerical Field Propagation Between Parallel Planes . . . . . . . . . . 956.1 Far-Field Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.1.1 Discrete Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.1.2 Destination Sampling Window . . . . . . . . . . . . . . . . . 97

viii Contents

6.1.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . 986.1.4 Sampling Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.2 The Fourier Transform by Lens . . . . . . . . . . . . . . . . . . . . . . . . 1016.3 Single-Step Fresnel Propagation . . . . . . . . . . . . . . . . . . . . . . . . 102

6.3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.3.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . 1036.3.3 Sampling Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.4 Convolution-Based Technique: Band-Limited AngularSpectrum Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.4.1 Discrete Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.4.2 Sampling Problem of Transfer Function . . . . . . . . . . . 1076.4.3 Problem of Field Invasion . . . . . . . . . . . . . . . . . . . . . 1116.4.4 Discussion on Band Limiting . . . . . . . . . . . . . . . . . . 1146.4.5 More Accurate Technique . . . . . . . . . . . . . . . . . . . . . 116

7 Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177.1 Optical Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177.2 Thin Hologram and Volume Hologram . . . . . . . . . . . . . . . . . . 1187.3 Types of Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207.4 Mathematical Explanation of Principle . . . . . . . . . . . . . . . . . . . 1217.5 Spatial Spectrum of Amplitude Hologram . . . . . . . . . . . . . . . . 1237.6 Conjugate Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267.7 Theory and Examples of Thin Hologram . . . . . . . . . . . . . . . . . 127

7.7.1 Hologram with Plane Wave . . . . . . . . . . . . . . . . . . . 1287.7.2 Hologram with Spherical Wave . . . . . . . . . . . . . . . . . 1327.7.3 Fourier Transform Hologram . . . . . . . . . . . . . . . . . . . 147

8 Computer Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1538.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1538.2 Viewing Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1538.3 Space-Bandwidth Product Problem . . . . . . . . . . . . . . . . . . . . . 1558.4 Full-Parallax and Horizontal-Parallax-Only CGH . . . . . . . . . . . 1578.5 Coding and Optimization of Fringe Pattern . . . . . . . . . . . . . . . 1588.6 Amplitude CGH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

8.6.1 Amplitude Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 1598.6.2 Brightness and Noise . . . . . . . . . . . . . . . . . . . . . . . . 1618.6.3 Binary-Amplitude CGH . . . . . . . . . . . . . . . . . . . . . . 164

8.7 Phase CGH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1658.7.1 Phase Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1668.7.2 Example of Phase CGH . . . . . . . . . . . . . . . . . . . . . . 1678.7.3 Binary-Phase CGH . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Contents ix

8.8 Spatial Frequency of Fringe Pattern . . . . . . . . . . . . . . . . . . . . . 1708.8.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1708.8.2 Example of Fringe Frequency . . . . . . . . . . . . . . . . . . 1738.8.3 Fringe Oversampling . . . . . . . . . . . . . . . . . . . . . . . . 176

8.9 Fourier-Transform CGH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1788.9.1 Higher-Order Diffraction Images . . . . . . . . . . . . . . . . 1788.9.2 Generation of Fringe Pattern . . . . . . . . . . . . . . . . . . . 1808.9.3 Amplitude Fringe Pattern Based on Hermitian

Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1818.10 Single-Sideband Method in Amplitude CGH . . . . . . . . . . . . . . 182

8.10.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1838.10.2 Generation of Fringe Pattern . . . . . . . . . . . . . . . . . . . 184

9 The Rotational Transform of Wavefield . . . . . . . . . . . . . . . . . . . . . 1879.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1879.2 Coordinate Systems and Rotation Matrices . . . . . . . . . . . . . . . . 1879.3 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1899.4 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

9.4.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 1929.4.2 Paraxial Approximation . . . . . . . . . . . . . . . . . . . . . . 194

9.5 Numerical Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1959.5.1 Sampling Distortion . . . . . . . . . . . . . . . . . . . . . . . . . 1959.5.2 Shifted Fourier Coordinates . . . . . . . . . . . . . . . . . . . . 1979.5.3 Actual Procedure to Perform the Rotational

Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1989.5.4 Resample of Uniformly Sampled Spectrum . . . . . . . . 202

9.6 Numerical Examples and Errors . . . . . . . . . . . . . . . . . . . . . . . . 2039.6.1 Edge Effect and Sampling Overlap . . . . . . . . . . . . . . 2039.6.2 The Rotational Transform with Carrier Offset . . . . . . . 2069.6.3 Examples of the Rotational Transform

in Practical Wavefield . . . . . . . . . . . . . . . . . . . . . . . . 209

10 The Polygon-Based Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21110.1 Surface Source of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

10.1.1 Generation of Scattered Light . . . . . . . . . . . . . . . . . . 21110.1.2 Theoretical Model of Polygonal Surface Source

of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21310.2 Basic Theory for Rendering Diffused Surface . . . . . . . . . . . . . . 214

10.2.1 Surface Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 21510.2.2 Spectrum Remapping by Incident Plane Wave . . . . . . 21610.2.3 Rotation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21810.2.4 Rotational Transform of Remapped Spectrum . . . . . . 22010.2.5 Short Propagation to Object Plane . . . . . . . . . . . . . . . 22110.2.6 Superposition of Polygon Fields and Propagation

to Hologram Plane . . . . . . . . . . . . . . . . . . . . . . . . . . 222

x Contents

10.3 Practical Algorithm for Rendering Diffused Surface . . . . . . . . . 22310.3.1 Input Data and Controllable Parameters . . . . . . . . . . . 22310.3.2 Tilted and Parallel Frame Buffers . . . . . . . . . . . . . . . 22410.3.3 The Fourier Transform of Surface Function . . . . . . . . 22410.3.4 Basic Procedure for the Rotational Transform

and Short Propagation . . . . . . . . . . . . . . . . . . . . . . . . 22510.3.5 Maximum Diffraction Area of Polygon . . . . . . . . . . . 22710.3.6 Determination of Sampling Interval of Surface

Function by Probing Sample Points . . . . . . . . . . . . . . 22910.3.7 How to Determine Sizes of PFB and TFB . . . . . . . . . 23110.3.8 Back-Face Culling . . . . . . . . . . . . . . . . . . . . . . . . . . 23510.3.9 Overall Algorithm for Rendering Diffused

Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24010.3.10 Variation of Probing Sample Points . . . . . . . . . . . . . . 242

10.4 Band Limiting of Polygon Field . . . . . . . . . . . . . . . . . . . . . . . 24310.4.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24310.4.2 Limit of Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . 24410.4.3 Modification of Algorithm . . . . . . . . . . . . . . . . . . . . 246

10.5 Computation Time of Object Field . . . . . . . . . . . . . . . . . . . . . . 24710.6 Shading and Texture-Mapping of Diffused Surface . . . . . . . . . . 249

10.6.1 Brightness of Reconstructed Surface . . . . . . . . . . . . . 24910.6.2 Amplitude of Surface Function . . . . . . . . . . . . . . . . . 25210.6.3 Shading of Diffused Surfaces . . . . . . . . . . . . . . . . . . 25310.6.4 Texture-Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

10.7 Rendering Specular Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 25810.7.1 Spectrum of Diffuse and Specular Reflection . . . . . . . 25910.7.2 Phong Reflection Model . . . . . . . . . . . . . . . . . . . . . . 26010.7.3 Spectral Envelope of Specular Component . . . . . . . . . 26110.7.4 Generation of Specular Diffuser for Surface

Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26210.7.5 Fast Generation of Specular Diffuser by Shifting

Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26510.7.6 Flat Specular Shading . . . . . . . . . . . . . . . . . . . . . . . . 26810.7.7 Smooth Specular Shading . . . . . . . . . . . . . . . . . . . . . 27210.7.8 Examples of High-Definition CGHs with Specular

Shading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

11 The Silhouette Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28111.1 Occlusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28111.2 Processing of Mutual Occlusion . . . . . . . . . . . . . . . . . . . . . . . . 283

11.2.1 Silhouette Method . . . . . . . . . . . . . . . . . . . . . . . . . . 28311.2.2 Formulation of Object-by-Object Light-Shielding

for Multiple Objects . . . . . . . . . . . . . . . . . . . . . . . . . 284

Contents xi

11.2.3 Actual Example of Object-by-ObjectLight-Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

11.2.4 Translucent Object . . . . . . . . . . . . . . . . . . . . . . . . . . 28711.3 Switch-Back Technique for Processing Self-Occlusion

by the Silhouette Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28911.3.1 Principle of Polygon-by-Polygon Light-Shielding

and Associated Problem . . . . . . . . . . . . . . . . . . . . . . 28911.3.2 The Babinet’s Principle . . . . . . . . . . . . . . . . . . . . . . . 29011.3.3 Light-Shielding by Use of Aperture Instead

of Mask . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29411.3.4 Formulation for Multiple Polygons . . . . . . . . . . . . . . 29611.3.5 Practical Procedure for Computation of Object

Field with P-P Shielding . . . . . . . . . . . . . . . . . . . . . . 29811.3.6 Inductive Explanation of the Switch-Back

Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29911.3.7 Numerical Technique and Sampling Window

for Switch-Back Propagation . . . . . . . . . . . . . . . . . . . 30011.3.8 Emulation of Alpha Blend of CG . . . . . . . . . . . . . . . 30111.3.9 Acceleration by Dividing Object . . . . . . . . . . . . . . . . 30211.3.10 Integration with the Polygon-Based Method . . . . . . . . 30311.3.11 Actual Examples of P-P Light-Shielding

and Computation Time . . . . . . . . . . . . . . . . . . . . . . . 30411.4 Limitation of the Silhouette Method . . . . . . . . . . . . . . . . . . . . . 306

12 Shifted Field Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30912.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

12.1.1 What is Shifted Field Propagation . . . . . . . . . . . . . . . 30912.1.2 Rectangular Tiling . . . . . . . . . . . . . . . . . . . . . . . . . . 310

12.2 Mathematical Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31112.2.1 Fractional DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31112.2.2 Scaled FFT for Symmetric Sampling . . . . . . . . . . . . . 313

12.3 Shifted Far-Field Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 31612.3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31612.3.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . 31912.3.3 Sampling Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 319

12.4 Shifted Fresnel Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . 32012.4.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32012.4.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . 32112.4.3 Sampling Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 321

12.5 Shifted Angular Spectrum Method . . . . . . . . . . . . . . . . . . . . . . 32612.5.1 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . 32612.5.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32712.5.3 Band Limiting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

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12.5.4 Actual Procedure for Numerical Calculation . . . . . . . . 33312.5.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . 33512.5.6 Discussion on the Limit Frequency . . . . . . . . . . . . . . 335

13 Simulated Reconstruction Based on Virtual Imaging . . . . . . . . . . . 33713.1 Need for Simulated Reconstruction . . . . . . . . . . . . . . . . . . . . . 33713.2 Simulated Reconstruction by Back Propagation . . . . . . . . . . . . 338

13.2.1 Examples of Reconstruction by Back-Propagation . . . 33913.2.2 Control of DOF Using Aperture . . . . . . . . . . . . . . . . 34213.2.3 Control of View-Direction Using Aperture . . . . . . . . . 343

13.3 Image Formation by Virtual Lens . . . . . . . . . . . . . . . . . . . . . . 34413.3.1 Sampling Problem of Virtual Lens . . . . . . . . . . . . . . . 34413.3.2 Equal Magnification Imaging by Virtual Lens . . . . . . 34613.3.3 Reduced Imaging by Virtual Lens . . . . . . . . . . . . . . . 34713.3.4 Change of Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . 350

13.4 Simulated Reconstruction from Fringe Pattern . . . . . . . . . . . . . 35313.4.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35413.4.2 Comparison Between Simulated and Optical

Reconstructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35413.5 Simulated Reconstruction in Color . . . . . . . . . . . . . . . . . . . . . . 356

13.5.1 Production of Full-Color Reconstructed Image . . . . . . 35613.5.2 Examples of Simulated Reconstruction in Color . . . . . 358

14 Digitized Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36314.1 Concept of Digitized Holography . . . . . . . . . . . . . . . . . . . . . . . 36314.2 Digital Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

14.2.1 Phase-Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36514.2.2 Lensless-Fourier Digital Holography

for Converting Sampling Interval . . . . . . . . . . . . . . . 36714.2.3 Synthetic Aperture Digital Holography

for Capturing Large-Scale Wavefield . . . . . . . . . . . . . 37214.3 Capture of Object-Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

14.3.1 Monochromatic Object Field . . . . . . . . . . . . . . . . . . . 37614.3.2 Object Fields in Full-Color . . . . . . . . . . . . . . . . . . . . 378

14.4 Occlusion Processing Using the Silhouette Method . . . . . . . . . . 38114.4.1 The Silhouette Method Including Captured

Object Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38114.4.2 Making Silhouette Masks . . . . . . . . . . . . . . . . . . . . . 382

14.5 Examples of Optical Reconstruction . . . . . . . . . . . . . . . . . . . . . 38414.5.1 Monochrome CGH . . . . . . . . . . . . . . . . . . . . . . . . . . 38414.5.2 Full-Color CGH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

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14.6 Resizing Object Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38914.6.1 Resizing by Change of Sampling Intervals . . . . . . . . . 38914.6.2 Resizing by Virtual Imaging . . . . . . . . . . . . . . . . . . . 38914.6.3 Resizing by Shifted Fresnel Propagation . . . . . . . . . . 391

15 Fabrication of High-Definition CGH . . . . . . . . . . . . . . . . . . . . . . . . 39515.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39515.2 Fringe Printers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

15.2.1 Spot-Scanning Fringe Printer . . . . . . . . . . . . . . . . . . . 39715.2.2 Image-Tilling Fringe Printer . . . . . . . . . . . . . . . . . . . 400

15.3 Laser Lithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40115.3.1 Photomasks as a Binary-Amplitude CGH . . . . . . . . . . 40215.3.2 Structure of Photomasks . . . . . . . . . . . . . . . . . . . . . . 40315.3.3 Process to Fabricate Photomasks . . . . . . . . . . . . . . . . 40415.3.4 Pattern Drawing by Laser Writer . . . . . . . . . . . . . . . . 40615.3.5 Actual Processes of Development and Etching . . . . . . 40715.3.6 Creation of Phase CGHs . . . . . . . . . . . . . . . . . . . . . . 409

15.4 Wavefront Printer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41215.4.1 Principle and Difference from Holographic

Printer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41215.4.2 Optical Systems for Generating Object Fields . . . . . . 41415.4.3 Calculation of Object Fields and Encoding

of Fringes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41815.4.4 Denysyuk-Type Wavefront Printer . . . . . . . . . . . . . . . 419

15.5 Full-Color Reconstruction of HD-CGHs Using OpticalCombiner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

15.6 Full-Color CGH Using RGB Color Filters . . . . . . . . . . . . . . . . 42315.6.1 Principle and Structure . . . . . . . . . . . . . . . . . . . . . . . 42415.6.2 Fringe Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42715.6.3 Design Parameters of RGB Color Filters . . . . . . . . . . 42815.6.4 Examples of Optical Reconstruction . . . . . . . . . . . . . 429

15.7 Full-Color Stacked CGVH. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43015.7.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43215.7.2 Compensation for Thickness and Refractive

Index of Substrates . . . . . . . . . . . . . . . . . . . . . . . . . . 43515.7.3 Fabrication of Stacked CGVH . . . . . . . . . . . . . . . . . . 43815.7.4 Optical Reconstruction of Stacked CGVH . . . . . . . . . 440

Appendix: Data of Major HD-CGHs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

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