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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

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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

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