introduction to fourier optics 2nd - j. goodman

Introduction to Fourier Optics McGraw-Hill Series in Electrical and Computer Engineering SENIOR CONSULTING EDITOR Stephen W. Director, Carnegie Mellon University Circuits and Systems Communications and Signal Processing Computer Engineering Control Theory Electromagnetics Electronics and VLSI Circuits Introductory Power and Energy Radar and Antennas PREVIOUS CONSULTING EDITORS Ronald N. Bracewell, Colin Cherry, James F. Gibbons, Willis W. Harman, Hubert Heffner, Edward W. Herold, John G. Linvill, Simon Ramo, Ronald A. Rohrer, Anthony E. Siegman, Charles Susskind, Frederick E. Terman, John G. Truxal, Ernst Weber, and John R. Whinnery Elec tromagnetics SENIOR CONSULTING EDITOR Stephen W. Director, Carnegie Mellon University Dearhold and McSpadden: Electromagnetic Wave Propagation Goodman: Introduction to Fourier Optics Harrington: Time - Harmonic Electromagnetic Fields Hayt: Engineering Electromagnetics Kraus: Electromagnetics Paul and Nasar: Introduction to Electromagnetic Fields Plonus: Applied Electromagnetics Introduction to Fourier Optics SECOND EDITION Joseph W. Goodman Stanford University THE McGRAW-HILL COMPANIES, INC. New York St. Louis San Francisco Auckland Bogot6 Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto [...]... angle 8 (with respect to the x axis) given by In addition, the spatial period (i.e the distance between zero-phase lines) is given by 8 Introduction to Fourier Optics \ \ FIGURE 2.1 - - - - - Lines of zero phase for the fur exp[j2.rr(fxn + fry)] - In conclusion, then, we may again regard the inverse Fourier transform as providing a means for decomposing mathematical functions The Fourier spectrum G of... of Photographic Emulsions 7.2 Spatial Light Modulators 7.2.1 Properties of Liquid Crystals / 7.2.2 Spatial Light Modulators Based on Liquid Crystals / 7.2.3 Magneto-Optic Spatial Light Modulators / 7.2.4 Deformable Mirror Spatial Light Modulators / 7.2.5 Multiple Quantum Well Spatial Light Modulators / 7.2.6 Acousto-Optic Spatial Light Modulators 7.3 Diffractive Optical Elements 7.3.1 Binary Optics. .. decomposition are, of course, just these 6 functions To find the response of the system to the input gl, substitute ( 2-4 3) in ( 2-4 1): Now, regarding the number gl(5, q ) as simply a weighting factor applied to the elementary function 6(x1 - 5, yl - q), the linearity property ( 2-4 2) is invoked to allow S { )to operate on the individual elementary functions; thus the operator S{} brought is within the integral, yielding... functions separable in rectangular coordinates Function e x - ( a +2 2 Transform ?- exp y 2 lab1 [ (2+ 2 )] -n - - I 6(ax,by) - exp[j.rr(ax+ by)] sgn(ax) sgn(by) ab 1 - 1 comb(ax) comb(by) -comb(fxla) comb(fylb) exp[jn(a 2 x 2+ b2y2)] - exp [-( alxl + blyl)l Circle function lab1 6(fx - al2, f~ - bl2) 1 lab1 1 circ( ) / , lab1 jnfx jnfy 1 lab1 2 / , = 2 + ( 2 nf ~ l a )1 ~ (27rfylb)2 + = 1 ( 0 otherwise... also like to thank the students in my 1995 Fourier Optics class, who competed fiercely to see who could find the most mistakes Undoubtedly there are others to whom I owe thanks, and I apologize for not mentioning them explicitly here Finally, I thank Hon Mai, without whose patience, encouragement and support this book would not have have been possible Joseph W Goodman Introduction to Fourier Optics CHAPTER... spite of what might appear to be a contrary implication of Eq ( 2-3 9) The dimensions of P are meters-2 18 Introduction to Fourier Optics This expression is separable in rectangular coordinates, so it suffices to find the onedimensional spectrum Lx Gx(fx)= ,jrPxZej2rfxx dX Completing the square in the exponent and making a change of variables of integration / from x to t = ,@ (x - $)yields This integral... and the inverse-transform operations Using the notation a { ) to represent the Fourier- Bessel transform operation, it follows directly from the Fourier integral theorem that at each value of r where gR(r) is continuous In addition, the similarity theorem can be straightforwardly applied (see Prob 2-6 c) to show that B{g~(ar))= a +o (s) When using the expression ( 2-3 1) for the Fourier- Bessel transform,... of Two-Dimensional Signals and Systems 13 FIGURE 2.2 Special functions Rectangle function 0 otherwise Sinc function sinc(x) = 1 sin(~x) TX x > o Signum function - 1 x < o Triangle function Comb function A(x) = cornb(x) = otherwise 6(x - n) 14 Introduction to Fourier Optics T A B L E 2.1 Transform pairs for some functions separable in rectangular coordinates Function e x - ( a +2 2 Transform ?- exp... correspond to a unique input, for as we shall see, a variety of input functions can produce no output Thus we restrict attention at the outset to systems characterized by many -to- one mappings A convenient representation of a system is a mathematical operator, S O , which we imagine to operate on input functions to produce output functions Thus if the function gl (xl, yl) represents the input to a system,... Bibliography 42 1 Index 433 PREFACE Fourier analysis is a ubiquitous tool that has found application to diverse areas of physics and engineering This book deals with its applications in optics, and in particular with applications to diffraction, imaging, optical data processing, and holography Since the subject covered is Fourier Optics, it is natural that the methods of Fourier analysis play a key role . elected to the National Academy of Engineering. In addition to Introduction to Fourier Optics, Dr. Goodman is the author of Statis - tical Optics (J. . would not have have been possible. Joseph W. Goodman Introduction to Fourier Optics CHAPTER 1 Introduction 1.1 OPTICS, INFORMATION, AND COMMUNICATION