OPTI 627

COMPUTER GENERATED HOLOGRAMS

TABLE OF CONTENTS ( Hyperlinked to Adobe Acrobat Format Chapters )

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1. The Fourier Transform

	Introduction
	1-D Fourier transform
	2-D Fourier transform
	Fourier series
	Discrete Fourier transform (DFT)
	Generalized harmonic analysis
	Appendix A: Three-dimensional Fourier transform
	Appendix B: Three-dimensional vector Fourier transform
	Appendix C: The n-dimensional vector Fourier transform
	Appendix D: The Fourier transform in polar coordinates
	Appendix E: The Fourier transform in spherical coordinates
	Appendix F: The Fourier transform in hyperspherical coordinates
	Appendix G: Relation of the FFT to the DFT
	Appendix H: The Fresnel transform
	Appendix I: The uncertainty principle
	Appendix J: The uncertainty principle and the Gaussian

2. Comb Math

Introduction

Relation of the Fourier series to the Fourier transform

Relation of the discrete Fourier transform (DFT) to the Fourier transform

Encoding phase in fringe shifts

Appendix A: An alternate derivation of the relation between the continuous Fourier transform and the discrete Fourier transformation

3. Sampling

Introduction

The Whittaker-Shannon sampling theorem

Sampling real functions: the Nyquist sampling rate

Over-sampling

Under-sampling and aliasing

The sampling theorem in two dimensions

Appendix A: Uniformly shifted sample points

Appendix B: Non-rectangular apertures

Appendix C: Sampling on non-rectangular lattices

Appendix D: A sampling theorem for a function and its derivative

Appendix E: A sampling theorem for the Fresnel transform

Appendix F: Cross artifacts and symmetry

Appendix G: The Lagrange sampling theorem

4. Carrier-Wave Modulation

Introduction

Cosine-carrier modulation

Squarewave-carrier modulation

The modulation image superposition

5. MATLAB

Introduction

Variables

Operators

Flow-control structures

Input/output commands

Disk commands

Other functions

Appendix A: Matrix filling examples

Appendix B: flow-control structure examples

Appendix C: Example M-file: DFT of a uniformly filled array

Appendix D: CGH program

Copyable M-file CGH script

6. Systems Theory

Systems diagrams

Operational calculus

Hilbert space

Power-series operators and Taylor series

First-order expansion and linear systems

Shift-invariance

LSI: Linear Shift-Invariant system

PART II: THE CGH

1. Holography in a Nutshell

What is a hologram?

Selected events in the history of holography

Waves

The parabolic approximation

Optical elements

The optical Fourier transform

Recording media characteristics

The hologram as a distorted diffraction grating

Fringe distortions in interferometric holograms

Binary holograms

Detour phase

Diffusers

Phase-only holograms

Depth effects

Appendix A: The parabolic approximation

2. Simulating the Optical Fourier Transform

Introduction

Application of comb math

Object space

Fourier space

Reduced coordinates

A remark on normalization

Diffraction efficiency

Bleached binary CGH's

3. Point-Oriented Versus Cell-Oriented CGH's

Introduction

Contrasting the two CGH families

Point-oriented example

Simulating a point-oriented CGH

Fabricating a point-oriented CGH

Cell-oriented example

Simulating a cell-oriented CGH

Fabricating a cell-oriented CGH

4. Point-Oriented CGH's

Introduction

The perfect CGH

The referenceless on-axis computer hologram (ROACH)

The graytone CGH

The squarewave CGH

The crossed-squarewave CGH

Appendix A: The sandwich hologram

Appendix B: The referenceless on-axis complex hologram (ROACH)

Appendix C: CGH's that reconstruct in higher diffraction orders

Appendix D: Multiplex CGH's

5. Cell-Oriented CGH's

Introduction

The Lohmann Type-III binary hologram

The gap and overlap problem: circular overflow solution

Appendix A: A useful identity

Appendix B: Fabricating a CGH

6. The Third Dimension

Introduction

Illumination

Transmission

Reflection

Propagation

Additional considerations for propagation

CGH's grouped by propagation region

CGH's with circular carriers

Appendix A: The wave equation

Appendix B: Characteristic bodies

Appendix C: Depth effects for Fresnel CGH's

Appendix D: Depth effects for near-Fourier CGH's

Appendix E: Depth effects for image-plane CGH's

Appendix F: Depth effects for near-field CGH'

Appendix G: Imaging in the parabolic approximation

Appendix H: Transforming the spherical wave

7. Computer Simulation of Non-Fourier Holograms

Introduction

The two-plane self-luminous object

Adjusting the viewing perspective

Depth effects in modulation images

PART III: ADVANCED TOPICS

1. Diffusers

Introduction

Desirable properties of the Fourier spectrum

Diffusers for interferometric holograms

Random vs. deterministic diffuser functions

General purpose (object independent) vs. specific (object dependent) diffuser functions

RADAR codes

Schroeder codes

Frank-Heimiller codes

Calabro-Wolf (hyperbolic) codes

Modern methods

Simulated annealing

Gerchberg-Saxton (phase diversity) procedure

Speckle suppression

2. Analysis of Imperfections

Complex amplitude distortions

Sampling and point-oriented CGH's

Fourier domain phase quantization

Shift effect of Fourier domain phase quantization

Depth effects of Fourier domain phase quantization

Effects on irradiance

3. Binarization of Holograms

Introduction

Thresholding

Wandering threshold

Halftoning

Dot density

Dithering or tesselating

Error diffusion

Dynamic programming (Bellman optimization)

PART IV: APPLICATIONS

1. Replication, Fanouts, and Interconnects

Introduction

Replication

Fanouts and interconnects

2. Image Display

Introduction

Information reduction

Shading models

Multiplexing and color display

Realtime display

3. Spatial Filtering

Introduction

Complex amplitude linear filters

Image enhancement

Image recognition

Encryption

Irradiance linear filters

4. Optical Testing

Introduction

Interferometry

Blanks ( rough surfaces)

Polarization

White Light

5. Diffractive Optical Elements

Introduction

Prisms

Lenses

Near-field optics

PART V: MISCELLANEOUS

References