Diffraction, Fourier Optics and Imaging

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1 Diffraction, Fourier Optics and Imaging

3 Diffraction, Fourier Optics and Imaging OKAN K. ERSOY WILEY-INTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION

4 Copyright # 2007 by John Wiley & Sons, Inc. All rights reserved 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 form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) , fax (978) , or on the web at Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) , fax (201) , or online at go/permission. 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. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) , outside the United States at (317) or fax (317) Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at 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: ISBN-10: Diffraction. 2. Fourier transform optics. 3. Imaging systems. I. Title. QC415.E dc Printed in the United States of America

5 Contents Preface xiii 1. Diffraction, Fourier Optics and Imaging Introduction Examples of Emerging Applications with Growing Significance Dense Wavelength Division Multiplexing/Demultiplexing (DWDM) Optical and Microwave DWDM Systems Diffractive and Subwavelength Optical Elements Nanodiffractive Devices and Rigorous Diffraction Theory Modern Imaging Techniques 4 2. Linear Systems and Transforms Introduction Linear Systems and Shift Invariance Continuous-Space Fourier Transform Existence of Fourier Transform Properties of the Fourier Transform Real Fourier Transform Amplitude and Phase Spectra Hankel Transforms Fundamentals of Wave Propagation Introduction Waves Electromagnetic Waves Phasor Representation Wave Equations in a Charge-Free Medium Wave Equations in Phasor Representation in a Charge-Free Medium Plane EM Waves Scalar Diffraction Theory Introduction Helmholtz Equation 42 v

6 vi CONTENTS 4.3 Angular Spectrum of Plane Waves Fast Fourier Transform (FFT) Implementation of the Angular Spectrum of Plane Waves The Kirchoff Theory of Diffraction Kirchoff Theory of Diffraction Fresnel Kirchoff Diffraction Formula The Rayleigh Sommerfeld Theory of Diffraction The Kirchhoff Approximation The Second Rayleigh Sommerfeld Diffraction Formula Another Derivation of the First Rayleigh Sommerfeld Diffraction Integral The Rayleigh Sommerfeld Diffraction Integral For Nonmonochromatic Waves Fresnel and Fraunhofer Approximations Introduction Diffraction in the Fresnel Region FFT Implementation of Fresnel Diffraction Paraxial Wave Equation Diffraction in the Fraunhofer Region Diffraction Gratings Fraunhofer Diffraction By a Sinusoidal Amplitude Grating Fresnel Diffraction By a Sinusoidal Amplitude Grating Fraunhofer Diffraction with a Sinusoidal Phase Grating Diffraction Gratings Made of Slits Inverse Diffraction Introduction Inversion of the Fresnel and Fraunhofer Representations Inversion of the Angular Spectrum Representation Analysis Wide-Angle Near and Far Field Approximations for Scalar Diffraction Introduction A Review of Fresnel and Fraunhofer Approximations The Radial Set of Approximations Higher Order Improvements and Analysis Inverse Diffraction and Iterative Optimization Numerical Examples More Accurate Approximations Conclusions 111

7 CONTENTS vii 8. Geometrical Optics Introduction Propagation of Rays The Ray Equations The Eikonal Equation Local Spatial Frequencies and Rays Matrix Representation of Meridional Rays Thick Lenses Entrance and Exit Pupils of an Optical System Fourier Transforms and Imaging with Coherent Optical Systems Introduction Phase Transformation With a Thin Lens Fourier Transforms With Lenses Wave Field Incident on the Lens Wave Field to the Left of the Lens Wave Field to the Right of the Lens Image Formation As 2-D Linear Filtering The Effect of Finite Lens Aperture Phase Contrast Microscopy Scanning Confocal Microscopy Image Formation Operator Algebra for Complex Optical Systems Imaging with Quasi-Monochromatic Waves Introduction Hilbert Transform Analytic Signal Analytic Signal Representation of a Nonmonochromatic Wave Field Quasi-Monochromatic, Coherent, and Incoherent Waves Diffraction Effects in a General Imaging System Imaging With Quasi-Monochromatic Waves Coherent Imaging Incoherent Imaging Frequency Response of a Diffraction-Limited Imaging System Coherent Imaging System Incoherent Imaging System Computer Computation of the Optical Transfer Function Practical Considerations Aberrations Zernike Polynomials 174

8 viii CONTENTS 11. Optical Devices Based on Wave Modulation Introduction Photographic Films and Plates Transmittance of Light by Film Modulation Transfer Function Bleaching Diffractive Optics, Binary Optics, and Digital Optics E-Beam Lithography DOE Implementation Wave Propagation in Inhomogeneous Media Introduction Helmholtz Equation For Inhomogeneous Media Paraxial Wave Equation For Inhomogeneous Media Beam Propagation Method Wave Propagation in Homogeneous Medium with Index n The Virtual Lens Effect Wave Propagation in a Directional Coupler A Summary of Coupled Mode Theory Comparison of Coupled Mode Theory and BPM Computations Holography Introduction Coherent Wave Front Recording Leith Upatnieks Hologram Types of Holograms Fresnel and Fraunhofer Holograms Image and Fourier Holograms Volume Holograms Embossed Holograms Computer Simulation of Holographic Reconstruction Analysis of Holographic Imaging and Magnification Aberrations Apodization, Superresolution, and Recovery of Missing Information Introduction Apodization Discrete-Time Windows Two-Point Resolution and Recovery of Signals Contractions Contraction Mapping Theorem 220

9 CONTENTS ix 14.5 An Iterative Method of Contractions for Signal Recovery Iterative Constrained Deconvolution Method of Projections Method of Projections onto Convex Sets Gerchberg Papoulis (GP) Algorithm Other POCS Algorithms Restoration From Phase Reconstruction From a Discretized Phase Function by Using the DFT Generalized Projections Restoration From Magnitude Traps and Tunnels Image Recovery By Least Squares and the Generalized Inverse Computation of H þ By Singular Value Decomposition (SVD) The Steepest Descent Algorithm The Conjugate Gradient Method Diffractive Optics I Introduction Lohmann Method Approximations in the Lohmann Method Constant Amplitude Lohmann Method Quantized Lohmann Method Computer Simulations with the Lohmann Method A Fourier Method Based on Hard-Clipping A Simple Algorithm for Construction of 3-D Point Images Experiments The Fast Weighted Zero-Crossing Algorithm Off-Axis Plane Reference Wave Experiments One-Image-Only Holography Analysis of Image Formation Experiments Fresnel Zone Plates Diffractive Optics II Introduction Virtual Holography Determination of Phase Aperture Effects Analysis of Image Formation 279

10 x CONTENTS Information Capacity, Resolution, Bandwidth, and Redundancy Volume Effects Distortions Due to Change of Wavelength and/or Hologram Size Between Construction and Reconstruction Experiments The Method of POCS for the Design of Binary DOE Iterative Interlacing Technique (IIT) Experiments with the IIT Optimal Decimation-in-Frequency Iterative Interlacing Technique (ODIFIIT) Experiments with ODIFIIT Combined Lohmann-ODIFIIT Method Computer Experiments with the Lohmann-ODIFIIT Method Computerized Imaging Techniques I: Synthetic Aperture Radar Introduction Synthetic Aperture Radar Range Resolution Choice of Pulse Waveform The Matched Filter Pulse Compression by Matched Filtering Cross-Range Resolution A Simplified Theory of SAR Imaging Image Reconstruction with Fresnel Approximation Algorithms for Digital Image Reconstruction Spatial Frequency Interpolation Computerized Imaging II: Image Reconstruction from Projections Introduction The Radon Transform The Projection Slice Theorem The Inverse Radon Transform Properties of the Radon Transform Reconstruction of a Signal From its Projections The Fourier Reconstruction Method The Filtered-Backprojection Algorithm 335

11 CONTENTS xi 19. Dense Wavelength Division Multiplexing Introduction Array Waveguide Grating Method of Irregularly Sampled Zero-Crossings (MISZC) Computational Method for Calculating the Correction Terms Extension of MISZC to 3-D Geometry Analysis of MISZC Dispersion Analysis Finite-Sized Apertures Computer Experiments Point-Source Apertures Large Number of Channels Finite-Sized Apertures The Method of Creating the Negative Phase Error Tolerances D Simulations Phase Quantization Implementational Issues Numerical Methods for Rigorous Diffraction Theory Introduction BPM Based on Finite Differences Wide Angle BPM Finite Differences Finite Difference Time Domain Method Yee s Algorithm Computer Experiments 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

12 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

13 xiv PREFACE Chapter 5 discusses the Fresnel and Fraunhofer approximations that allow diffraction to be expressed in terms of the Fourier transform. As a special application area for these approximations, diffraction gratings with many uses are described. Diffraction is usually discussed in terms of forward wave propagation. Inverse diffraction covered in Chapter 6 is the opposite, involving inverse wave propagation. It is important in certain types of imaging as well as in iterative methods of optimization used in the design of optical elements. In this chapter, the emphasis is on the inversion of the Fresnel, Fraunhofer, and angular spectrum representations. The methods discussed so far are typically valid for wave propagation near the z- axis, the direction of propagation. In other words, they are accurate for wave propagation directions at small angles with the z-axis. The Fresnel and Fraunhofer approximations are also not valid at very close distances to the diffraction plane. These problems are reduced to a large extent with a new method discussed in Chapter 7. It is called the near and far field approximation (NFFA) method. It involves two major topics: the first one is the inclusion of terms higher than second order in the Taylor series expansion; the second one is the derivation of equations to determine the semi-irregular sampling point positions at the output plane so that the FFT can still be used for the computation of wave propagation. Thus, the NFFA method is fast and valid for wide-angle, near and far field wave propagation applications. When the diffracting apertures are much larger than the wavelength, geometrical optics discussed in Chapter 8 can be used. Lens design is often done by using geometric optics. In this chapter, the rays and how they propagate are described with equations for both thin and thick lenses. The relationship to waves is also addressed. Imaging with lenses is the most classical type of imaging. Chapters 9 and 10 are reserved to this topic in homogeneous media, characterizing such imaging as a linear system. Chapter 9 discusses imaging with coherent light in terms of the 2-D Fourier transform. Two important applications, phase contrast microscopy and scanning confocal microscopy, are described to illustrate how the theory is used in practice. Chapter 10 is the continuation of Chapter 9 to the case of quasimonochromatic waves. Coherent imaging and incoherent imaging are explained. The theoretical basis involving the Hilbert transform and the analytic signal is covered in detail. Optical aberrations and their evaluation with Zernike polynomials are also described. The emphasis to this point is on the theory. The implementation issues are introduced in Chapter 11. There are many methods of implementation. Two major ones are illustrated in this chapter, namely, photographic films and plates and electron-beam lithography for diffractive optics. In Chapters 9 and 10, the medium of propagation is assumed to be homogeneous (constant index of refraction). Chapter 12 discusses wave propagation in inhomogeneous media. Then, wave propagation becomes more difficult to compute numerically. The Helmholtz equation and the paraxial wave equation are generalized to inhomogeneous media. The beam propagation method (BPM) is introduced as a powerful numerical method for computing wave propagation in

14 PREFACE xv inhomogenous media. The theory is illustrated with the application of a directional coupler that allows light energy to be transferred from one waveguide to another. Holography as the most significant 3-D imaging technique is the topic of Chapter 13. The most basic types of holographic methods including analysis of holographic imaging, magnification, and aberrations are described in this chapter. In succeeding chapters, diffractive optical elements (DOEs), new modes of imaging, and diffraction in the subwavelength scale are considered, with extensive emphasis on numerical methods of computation. These topics are also related to signal/image processing and iterative optimization techniques discussed in Chapter 14. These techniques are also significant for the topics of previous chapters, especially when optical images are further processed digitally. The next two chapters are devoted to diffractive optics, which is creation of holograms, more commonly called DOEs, in a digital computer, followed by a recording system to create the DOE physically. Generation of a DOE under implementational constraints involves coding of amplitude and phase of an incoming wave, a topic borrowed from communication engineering. There are many such methods. Chapter 15 starts with Lohmann s method, which is the first such method historically. This is followed by two methods, which are useful in a variety of waves such as 3-D image generation, and a method called one-image-only holography, which is capable of generating only the desired image while suppressing the harmonic images due to sampling and nonlinear coding of amplitude and phase. The final section of the chapter is on the binary Fresnel zone plate, which is a DOE acting as a flat lens. Chapter 16 is a continuation of Chapter 15, and covers new methods of coding DOEs and their further refinements. The method of projections onto convex sets (POCS) discussed in Chapter 14 is used in several ways for this purpose. The methods discussed are virtual holography, which makes implementation easier, iterative interlacing technique (IIT), which makes use of POCS for optimizing a number of subholograms, the ODIFIIT, which is a further refinement of IIT by making use of the decimation-in-frequency property of the FFT, and the hybrid Lohmann ODIFIIT method, resulting in considerably higher accuracy. Chapters 17 and 18 are on computerized imaging techniques. The first such technique is synthetic aperture radar (SAR) covered in Chapter 17. In a number of ways, a raw SAR image is similar to the image of a DOE. Only further processing, perhaps more appropriately called decoding, results in a reconstructed image of a terrain of the earth. The images generated are very useful in remote sensing of the earth. The principles involved are optical and diffractive, such as the use of the Fresnel approximation. In the second part of computerized imaging, computed tomography (CT) is covered in Chapter 18. The theoretical basis for CT is the Radon transform, a cousin of the Fourier transform. The projection slice theorem shows how the 1-D Fourier transforms of projections are used to generate slices of the image spectrum in the 2- D Fourier transform plane. CT is highly numerical, as evidenced by a number of algorithms for image reconstruction in the rest of the chapter.

15 xvi PREFACE Optical Fourier techniques have become very important in optical communications and networking. One such area covered in Chapter 19 is arrayed waveguide gratings (AWGs) used in dense wavelength division multiplexing (DWDM). AWG is also called phased array (PHASAR). It is an imaging device in which an array of waveguides are used. The waveguides are different in length by an integer m times the central wavelength so that a large phase difference is achieved from one waveguide to the next. The integer m is quite large, such as 30, and is responsible for the large resolution capability of the phasar device, meaning that the small changes in wavelength can be resolved in the output plane. This is the reason why waveguides are used rather than free space. However, it is diffraction that is used past the waveguides to generate images of points at different wavelengths at the output plane. This is similar to a DOE, which is a sampled device. Hence, the images repeat at certain intervals. This limits the number of wavelengths that can be imaged without interference from other wavelengths. A method called irregularly sampled zero-crossings (MISZCs) is discussed to avoid this problem. The MISZC has its origin in one-image-only holography discussed in Chapter 15. Scalar diffraction theory becomes less accurate when the sizes of the diffracting apertures are smaller than the wavelength of the incident wave. Then, the Maxwell equations need to be solved by numerical methods. Some emerging approaches for this purpose are based on the method of finite differences, the Fourier modal analysis, and the method of finite elements. The first two approaches are discussed in Chapter 20. First, the paraxial BPM method discussed in Section 12.4 is reformulated in terms of finite differences using the Crank-Nicholson method. Next, the wide-angle BPM using the Pade approximation is discussed. The final sections highlight the finite difference time domain and the Fourier modal method. Many colleagues, secretaries, friends, and students across the globe have been helpful toward the preparation of this manuscript. I am especially grateful to them for keeping me motivated under all circumstances for a lifetime. I am also very fortunate to have worked with John Wiley & Sons, on this project. They have been amazingly patient with me. Without such patience, I would not have been able to finish the project. Special thanks to George Telecki, the editor, for his patience and support throughout the project.

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