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Diffraction, Fourier Optics

and Imaging

OKAN K. ERSOY

WILEY-INTERSCIENCE

A JOHN WILEY & SONS, INC., PUBLICATION

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any

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Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and speciﬁcally disclaim any implied warranties of merchantability or ﬁtness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of proﬁt or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

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Library of Congress Cataloging-in-Publication Data

Ersoy, Okan K.

Diffraction, fourier optics, and imaging / by Okan K. Ersoy. p. cm.

Includes bibliographical references and index. ISBN-13: 978-0-471-23816-4

ISBN-10: 0-471-23816-3

1. Diffraction. 2. Fourier transform optics. 3. Imaging systems. I. Title.

QC415.E77 2007
5350.42--dc22									2006048263
Printed in the United States of America
10	9	8	7	6	5	4	3	2	1

Contents

Preface	xiii
1. Diffraction, Fourier Optics and Imaging	1
1.1 Introduction	1

1.2Examples of Emerging Applications with Growing

	Signiﬁcance	2
	1.2.1 Dense Wavelength Division Multiplexing/Demultiplexing
	(DWDM)	3
	1.2.2 Optical and Microwave DWDM Systems	3
	1.2.3 Diffractive and Subwavelength Optical Elements	3
	1.2.4 Nanodiffractive Devices and Rigorous Diffraction Theory	4
	1.2.5 Modern Imaging Techniques	4
2. Linear Systems and Transforms		6
2.1	Introduction	6
2.2	Linear Systems and Shift Invariance	7
2.3	Continuous-Space Fourier Transform	10
2.4	Existence of Fourier Transform	11
2.5	Properties of the Fourier Transform	12
2.6	Real Fourier Transform	18
2.7	Amplitude and Phase Spectra	20
2.8	Hankel Transforms	21
3. Fundamentals of Wave Propagation		25
3.1	Introduction	25
3.2	Waves	26
3.3	Electromagnetic Waves	31
3.4	Phasor Representation	33
3.5	Wave Equations in a Charge-Free Medium	34

3.6Wave Equations in Phasor Representation

	in a Charge-Free Medium	36
3.7	Plane EM Waves	37
4. Scalar Diffraction Theory		41
4.1	Introduction	41
4.2	Helmholtz Equation	42

vi	CONTENTS
4.3 Angular Spectrum of Plane Waves	44

4.4Fast Fourier Transform (FFT) Implementation of the Angular

	Spectrum of Plane Waves		47
4.5	The Kirchoff Theory of Diffraction		53
	4.5.1	Kirchoff Theory of Diffraction	55
	4.5.2	Fresnel–Kirchoff Diffraction Formula	56
4.6	The Rayleigh–Sommerfeld Theory of Diffraction		57
	4.6.1	The Kirchhoff Approximation	59
	4.6.2 The Second Rayleigh–Sommerfeld Diffraction Formula		59

4.7Another Derivation of the First Rayleigh–Sommerfeld

Diffraction Integral

4.8The Rayleigh–Sommerfeld Diffraction Integral For

	Nonmonochromatic Waves	61
5. Fresnel and Fraunhofer Approximations		63
5.1	Introduction	63
5.2	Diffraction in the Fresnel Region	64
5.3	FFT Implementation of Fresnel Diffraction	72
5.4	Paraxial Wave Equation	73
5.5	Diffraction in the Fraunhofer Region	74
5.6	Diffraction Gratings	76
5.7	Fraunhofer Diffraction By a Sinusoidal Amplitude Grating	78
5.8	Fresnel Diffraction By a Sinusoidal Amplitude Grating	79
5.9	Fraunhofer Diffraction with a Sinusoidal Phase Grating	81
5.10	Diffraction Gratings Made of Slits	82
6. Inverse Diffraction		84
6.1	Introduction	84
6.2	Inversion of the Fresnel and Fraunhofer Representations	84
6.3	Inversion of the Angular Spectrum Representation	85
6.4	Analysis	86

7.Wide-Angle Near and Far Field Approximations

for Scalar Diffraction		90
7.1	Introduction	90
7.2	A Review of Fresnel and Fraunhofer Approximations	91
7.3	The Radial Set of Approximations	93
7.4	Higher Order Improvements and Analysis	95
7.5	Inverse Diffraction and Iterative Optimization	96
7.6	Numerical Examples	97
7.7	More Accurate Approximations	110
7.8	Conclusions	111

CONTENTS		vii
8. Geometrical Optics		112
8.1	Introduction	112
8.2	Propagation of Rays	112
8.3	The Ray Equations	117
8.4	The Eikonal Equation	118
8.5	Local Spatial Frequencies and Rays	120
8.6	Matrix Representation of Meridional Rays	123
8.7	Thick Lenses	130
8.8	Entrance and Exit Pupils of an Optical System	132

9.Fourier Transforms and Imaging with

Coherent Optical Systems		134
9.1	Introduction	134
9.2	Phase Transformation With a Thin Lens	134
9.3	Fourier Transforms With Lenses	136
	9.3.1 Wave Field Incident on the Lens	136
	9.3.2 Wave Field to the Left of the Lens	137
	9.3.3 Wave Field to the Right of the Lens	138
9.4	Image Formation As 2-D Linear Filtering	139
	9.4.1 The Effect of Finite Lens Aperture	141
9.5	Phase Contrast Microscopy	142
9.6	Scanning Confocal Microscopy	144
	9.6.1 Image Formation	144
9.7	Operator Algebra for Complex Optical Systems	147
10. Imaging with Quasi-Monochromatic Waves		153
10.1	Introduction	153
10.2	Hilbert Transform	154
10.3	Analytic Signal	157

10.4Analytic Signal Representation of a Nonmonochromatic

	Wave Field		161
10.5	Quasi-Monochromatic, Coherent, and Incoherent Waves		162
10.6	Diffraction Effects in a General Imaging System		162
10.7	Imaging With Quasi-Monochromatic Waves		164
	10.7.1	Coherent Imaging	165
	10.7.2	Incoherent Imaging	166

10.8Frequency Response of a Diffraction-Limited

Imaging System		166
10.8.1	Coherent Imaging System	166
10.8.2	Incoherent Imaging System	167
10.9 Computer Computation of the Optical Transfer Function		171
10.9.1	Practical Considerations	172
10.10 Aberrations		173
10.10.1	Zernike Polynomials	174

viii			CONTENTS
11. Optical Devices Based on Wave Modulation			177
11.1	Introduction		177
11.2	Photographic Films and Plates		177
11.3	Transmittance of Light by Film		179
11.4	Modulation Transfer Function		182
11.5	Bleaching		183
11.6	Diffractive Optics, Binary Optics, and Digital Optics		184
11.7	E-Beam Lithography		185
	11.7.1	DOE Implementation	187
12. Wave Propagation in Inhomogeneous Media			188
12.1	Introduction		188
12.2	Helmholtz Equation For Inhomogeneous Media		189
12.3	Paraxial Wave Equation For Inhomogeneous Media		189
12.4	Beam Propagation Method		190
	12.4.1 Wave Propagation in Homogeneous Medium with
		Index n	191
	12.4.2 The Virtual Lens Effect		192
12.5	Wave Propagation in a Directional Coupler		193
	12.5.1 A Summary of Coupled Mode Theory		193
	12.5.2 Comparison of Coupled Mode Theory and BPM
		Computations	194
13. Holography			198
13.1	Introduction		198
13.2	Coherent Wave Front Recording		199
	13.2.1	Leith–Upatnieks Hologram	201
13.3	Types of Holograms		202
	13.3.1 Fresnel and Fraunhofer Holograms		203
	13.3.2 Image and Fourier Holograms		203
	13.3.3	Volume Holograms	203
	13.3.4	Embossed Holograms	205
13.4	Computer Simulation of Holographic Reconstruction		205
13.5	Analysis of Holographic Imaging and Magniﬁcation		206
13.6	Aberrations		210

14.Apodization, Superresolution, and Recovery

of Missing Information			212
14.1	Introduction		212
14.2	Apodization		213
	14.2.1	Discrete-Time Windows	215
14.3	Two-Point Resolution and Recovery of Signals		217
14.4	Contractions		219
	14.4.1	Contraction Mapping Theorem	220

CONTENTS		ix
14.5	An Iterative Method of Contractions for Signal Recovery	221
14.6	Iterative Constrained Deconvolution	223
14.7	Method of Projections	225
14.8	Method of Projections onto Convex Sets	227
14.9	Gerchberg–Papoulis (GP) Algorithm	229
14.10	Other POCS Algorithms	229
14.11	Restoration From Phase	230
14.12	Reconstruction From a Discretized Phase Function
	by Using the DFT	232
14.13	Generalized Projections	234
14.14	Restoration From Magnitude	235
	14.14.1 Traps and Tunnels	237
14.15	Image Recovery By Least Squares and the
	Generalized Inverse	237
14.16	Computation of Hþ By Singular Value
	Decomposition (SVD)	238
14.17	The Steepest Descent Algorithm	240
14.18	The Conjugate Gradient Method	242
15. Diffractive Optics I		244
15.1	Introduction	244
15.2	Lohmann Method	246
15.3	Approximations in the Lohmann Method	247
15.4	Constant Amplitude Lohmann Method	248
15.5	Quantized Lohmann Method	249
15.6	Computer Simulations with the Lohmann Method	250
15.7	A Fourier Method Based on Hard-Clipping	254

15.8A Simple Algorithm for Construction of 3-D Point

	Images		257
	15.8.1	Experiments	259
15.9 The Fast Weighted Zero-Crossing Algorithm			261
	15.9.1	Off-Axis Plane Reference Wave	264
	15.9.2	Experiments	264
15.10 One-Image-Only Holography			265
	15.10.1	Analysis of Image Formation	268
	15.10.2	Experiments	270
15.11	Fresnel Zone Plates		272
16. Diffractive Optics II			275
16.1	Introduction		275
16.2	Virtual Holography		275
	16.2.1	Determination of Phase	276
	16.2.2	Aperture Effects	278
	16.2.3	Analysis of Image Formation	279

CONTENTS

16.2.4Information Capacity, Resolution, Bandwidth, and

Redundancy	282
16.2.5 Volume Effects	283

16.2.6Distortions Due to Change of Wavelength and/or Hologram Size Between Construction and

Reconstruction	284
16.2.7 Experiments	285

16.3The Method of POCS for the Design of Binary

DOE	287
16.4 Iterative Interlacing Technique (IIT)	289
16.4.1 Experiments with the IIT	291

16.5Optimal Decimation-in-Frequency Iterative Interlacing

Technique (ODIFIIT)	293
16.5.1 Experiments with ODIFIIT	297
16.6 Combined Lohmann-ODIFIIT Method	300

16.6.1Computer Experiments with the Lohmann-ODIFIIT

Method

301

17.Computerized Imaging Techniques I:

Synthetic Aperture Radar		306
17.1	Introduction	306
17.2	Synthetic Aperture Radar	306
17.3	Range Resolution	308
17.4	Choice of Pulse Waveform	309
17.5	The Matched Filter	311
17.6	Pulse Compression by Matched Filtering	313
17.7	Cross-Range Resolution	316
17.8	A Simpliﬁed Theory of SAR Imaging	317
17.9	Image Reconstruction with Fresnel Approximation	320
17.10	Algorithms for Digital Image Reconstruction	322
	17.10.1 Spatial Frequency Interpolation	322

18.Computerized Imaging II: Image

Reconstruction from Projections		326
18.1	Introduction	326
18.2	The Radon Transform	326
18.3	The Projection Slice Theorem	328
18.4	The Inverse Radon Transform	330
18.5	Properties of the Radon Transform	331

18.6Reconstruction of a Signal From its

	Projections	332
18.7	The Fourier Reconstruction Method	333
18.8	The Filtered-Backprojection Algorithm	335

CONTENTS			xi
19. Dense Wavelength Division Multiplexing			338
19.1	Introduction		338
19.2	Array Waveguide Grating		339
19.3	Method of Irregularly Sampled Zero-Crossings (MISZC)		341
	19.3.1 Computational Method for Calculating the
		Correction Terms	345
	19.3.2 Extension of MISZC to 3-D Geometry		346
19.4	Analysis of MISZC		347
	19.4.1	Dispersion Analysis	349
	19.4.2	Finite-Sized Apertures	350
19.5	Computer Experiments		351
	19.5.1	Point-Source Apertures	351
	19.5.2 Large Number of Channels		353
	19.5.3	Finite-Sized Apertures	355
	19.5.4 The Method of Creating the Negative Phase		355
	19.5.5	Error Tolerances	356
	19.5.6	3-D Simulations	356
	19.5.7	Phase Quantization	358
19.6	Implementational Issues		359

20.Numerical Methods for Rigorous

Diffraction Theory		361
20.1	Introduction	361
20.2	BPM Based on Finite Differences	362
20.3	Wide Angle BPM	364
20.4	Finite Differences	367
20.5	Finite Difference Time Domain Method	368
	20.5.1 Yee’s Algorithm	368
20.6	Computer Experiments	371
20.7	Fourier Modal Methods	374
Appendix A: The Impulse Function		377
Appendix B: Linear Vector Spaces		382
Appendix C: The Discrete-Time Fourier Transform,
	The Discrete Fourier Transform and
	The Fast Fourier Transform	391
References		397

Index

403

Preface

Diffraction and imaging are central topics in many modern and scientiﬁc ﬁelds. Fourier analysis and sythesis techniques are a unifying theme throughout this subject matter. For example, many modern imaging techniques have evolved through research and development with the Fourier methods.

This textbook has its origins in courses, research, and development projects spanning a period of more than 30 years. It was a pleasant experience to observe over the years how the topics relevant to this book evolved and became more signiﬁcant as the technology progressed. The topics involved are many and an highly multidisciplinary.

Even though Fourier theory is central to understanding, it needs to be supplemented with many other topics such as linear system theory, optimization, numerical methods, imaging theory, and signal and image processing. The implementation issues and materials of fabrication also need to be coupled with the theory. Consequently, it is difﬁcult to characterize this ﬁeld in simple terms. Increasingly, progress in technology makes it of central signiﬁcance, resulting in a need to introduce courses, which cover the major topics together of both science and technology. There is also a need to help students understand the signiﬁcance of such courses to prepare for modern technology.

This book can be used as a textbook in courses emphasizing a number of the topics involved at both senior and graduate levels. There is room for designing several one-quarter or one-semester courses based on the topics covered.

The book consists of 20 chapters and three appendices. The ﬁrst three chapters can be considered introductory discussions of the fundamentals. Chapter 1 gives a brief introduction to the topics of diffraction, Fourier optics and imaging, with examples on the emerging techniques in modern technology.

Chapter 2 is a summary of the theory of linear systems and transforms needed in the rest of the book. The continous-space Fourier transform, the real Fourier transform and their properties are described, including a number of examples. Other topics involved are covered in the appendices: the impulse function in Appendix A, linear vector spaces in Appendix B, the discrete-time Fourier transform, the discrete Fourier transform, and the fast Fourier transform (FFT) in Appendix C.

Chapter 3 is on fundamentals of wave propagation. Initially waves are described generally, covering all types of waves. Then, the chapter specializes into electromagnetic waves and their properties, with special emphasis on plane waves.

The next four chapters are fundamental to scalar diffraction theory. Chapter 4 introduces the Helmholtz equation, the angular spectrum of plane waves, the Fresnel-Kirchoff and Rayleigh-Sommerfeld theories of diffraction. They represent wave propagation as a linear integral transformation closely related to the Fourier transform.

xiii

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