Contemporary Optical Image Processing with MATLAB

Contemporary Optical Image Processing with MATLAB |

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Contemporary Optical Image Processing with MATLAB | Ting-Chung Poon Bradley Department of Electrical and Computer Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA

Partha P. Banerjee Department of Electrical and Computer Engineering, University of Dayton, Dayton, OH, USA

2001

ELSEVIER A M S T E R D A M . L O N D O N . NEW Y O R K - O X F O R D . PARIS. SHANNON- TOKYO

ELSEVIER SCIENCE Ltd The Boulevard, Langford Lane Kidlington, Oxford OX5 1GB, UK 9 2001 Elsevier Science Ltd. All rights reserved. This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier Science Global Rights Department, PO Box 800, Oxford OX5 I DX, UK; phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: [email protected]. You may also contact Global Rights directly through Elsevier's home page (http://www.elsevier.nl), by selecting 'Obtaining Permissions'. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (978) 7508400, fax: (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London WIP 0LP, UK; phone: (+44) 207 631 5555; fax: (+44) 207 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier Science is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier Science Global Rights Department, at the mail, fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. MATLAB is a registered trademark of The MathWorks, Inc. For MATLAB product information, please contact: The MathWorks Inc., 3 Apple Hill Drive, Natick, MA 01760-2098, USA. Tel. +1 508 647 700; Fax: +1 508 647 7101; E-mail: [email protected] Web: www.mathworks.com First edition 2001 British Library Cataloguing in Publication Data Poon, Ting-Chung Contemporary optical image processing with MATLAB 1. M A T L A B ( C o m p u t e r file) 2. I m a g e p r o c e s s i n g - Computer programs I.Title II. B a n e r j ee, P a r t h a P. 6 2 1. 3 ' 67 '0 2 8 5 5 3 6 9 ISBN

0080437885

Library of Congress Cataloging in Publication Data Poort, Ting-Chung. Contemporary optical image processing with MATLAB / Ting-Chung Poort, Partha P. Banerjee.- 1st ed. p. cm. ISBN 0-08-043788-5 Cnardeover)

1. Image proe~sing. 2. MATLAB. I. Banerjee, Partha P. II. Title. TA1637 .P65 2001 621.36'7--de21

2001023232

ISBN: 0-08-043788-5 Q The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands.

PREFACE This book serves two purposes: first to introduce the readers to the concepts of geometrical optics, physical optics and techniques of optical imaging and image processing, and secondly, to provide them with experience in modeling the theory and applications using a commonly used software tool MATLAB. It is a comprehensively revised and updated version of the authors' previous book Principles of Applied Optics. A sizeable portion of this book is based on the authors' own in-class presentations, as well as research in the area. Emphasis is placed on physical principles, on innovative ways of analyzing ray and wave propagation through optical systems using matrix and FFT methods that can be easily implemented using MATLAB. The reason MATLAB is emphasized is because of the fact that it is now a widely accepted software tool, which is very routinely used in signal processing. Furthermore, MATLAB is now commonly available in PC or workstation clusters in most universities, and student versions of it (Version 5) are available at the price of an average hardback textbook. Although student versions of MATLAB do have limitations compared with the professional version, they are nevertheless very powerful for array and matrix manipulation, for performing FFTs and for easy graphing. MATLAB code is very concise and complex mathematical operations can be performed using only a few lines of code. In our book we provide several examples of analysis of optical systems using MATLAB and list MATLAB programs for the benefit of readers. Since optical processing concepts are based on two dimensional linear system theories for the most part, we feel that this approach provides a natural bridge between traditional optics specialists and the signal and digital image processing community. We stress however that we have chosen to use MATLAB as a supplement rather than a replacement of traditional analysis techniques. Along with traditional problems, we have included a set of computer exercises at the end of each chapter. Taking this approach enables instructors to teach the concepts without committing to the use of MATLAB alone. The book is primarily geared towards a senior/graduate level audience. Since the purpose of the book is to bring out the systems aspect of optics, some of the traditional theories in physical optics such as the classical derivation of the Fresnel diffraction formula have been omitted. Instead we emphasize the transfer function approach to optical propagation wherever possible, discuss the coherent and optical transfer function of an image processing system etc. In geometrical optics, we take the readers quickly to the matrix formalism, which can be easily used to simulate ray propagation in the absence of diffraction. Emphasis is also placed on Gaussian beam optics, and the qformulation is derived in a straightforward and simple way through the transfer function concept. Holography and complex spatial filtering is introduced simultaneously since they are essentially similar. Also novel in the book is the ray theory of hologram construction and reconstruction, which is elegant and simple to use in order to determine quickly the location and characteristics of the reconstructed image. Of course the ray theory of holograms has its roots in the rigorous wave theory, this is pointed out clearly in the text.

vi

Preface

Another novel feature in the book is the discussion of optical propagation through guided media like optical fibers and self-induced guiding using optical nonlinearities. In each case, there are ample MATLAB simulations to show beam propagation in such media. The reason for introducing nonlinearities in this book is because an increasingly large number of applications of optical nonlinearities exist in image processing, e.g., edge enhancemem and image correlation through phase conjugation. Contemporary topics such as this, as well as scanning holography, bipolar incoherent image processing, image processing using acousto-optics, and dynamic holographic techniques for phase distortion correction of images are discussed in the book. A comment concerning units and notation: we mainly use the MKS system of units and the engineering convention for wave propagation, to be made more precise in the text. Instructive problems and MATLAB assignments are included at the end of each Chapter. Note that some of the examples given in the text may not work with the student version because of the size of the matrix. We hope that the book will adequately prepare the interested readers to modem research in the area of image processing. T.-C. Poon would like to thank his wife Eliza Lau and his children Christina and Justine for their encouragement, patience and love. P.P. Banerjee would like to thank his wife Noriko and his children Hans and Neil for their encouragement and support. The authors would like to thank Taegeun Kim and Christina Poon for their help in writing some of the MATLAB codes, Christina Poon and Justine Poon for help in typing parts of the manuscript and drawing some of the figures, and Bill Davis for his assistance on the use of the word processing software. We would like to acknowledge all our students who have contributed substantially to some of the work reported in this book, especially in Chapters 4 and 7. We would also like to express our gratitude to Professor Adrian Korpel of the University of Iowa for instilling in us the spirit of optics. Last, but not least, we would like to thank our parems for their moral encouragement and sacrifice that made this effort possible.

vii

CONTENTS

Chapter 1: Introduction to Linear Systems 1.1 One and Two-dimensional Fourier Transforms 1.2 The Discrete Fourier Transform 1.3 Linear Systems, Convolution and Correlation

Chapter 2: Geometrical Optics 2.1 2.2 2.3 2.4

Fermat's Principle Reflection and Refraction Refraction in an Inhomogeneous Medium Matrix Methods in Paraxial Optics 2.4.1 The ray transfer matrix 2.4.2 Illustrative examples 2.5 Ray Optics using MATLAB

9 10 11 15 18 19 25 32

Chapter 3: Propagation and Diffraction of Optical Waves

39

3.1 Maxwell's Equations: A Review 3.2 Linear Wave Propagation 3.2.1 Traveling-wave solutions 3.2.2 Intrinsic impedance, the Poynting vector, and polarization 3.3 Spatial Frequency Transfer Function for Propagation 3.3.1 Examples of Fresnel diffraction 3.3.2 MATLAB example: the Cornu Spiral 3.3.3 MATLAB example: Fresnel diffraction of a square aperture 3.3.4 Fraunhofer diffraction and examples 3.3.5 MATLAB example: Fraunhofer diffraction of a square aperture 3.4 Fourier Transforming Property of Ideal Lenses 3.5 Gaussian Beam Optics and MATLAB Example 3.5.1 q-transformation of Gaussian beams 3.5.2 Focusing of a Gaussian beam 3.5.3 MATLAB example: propagation of a Gaussian beam

40 43 43

Chapter 4 : Optical Propagation in Inhomogeneous Media 4.1 Introduction: The Paraxial Wave Equation 4.2 The Split-step Beam Propagation Method 4.3 Wave Propagation in a Linear Inhomogeneous Medium 4.3.1 Optical propagation through graded index fiber 4.3.2 Optical propagation through step index fiber 4.3.3 Acousto-optic diffraction 4.4 Wave Propagation in a Nonlinear Inhomogeneous Medium 4.4.1 Kerr Media

48 53 57 61 63 67 76 80 84 85 87 88 97 97 99 101 102 108 111 117 118

viii

Contents

4.4.2 Photorefractive Media

Chapter 5 Single and Double Lens Image Processing Systems 5.1 5.2 5.3 5.4 5.5

Impulse Response and Single Lens Imaging System Two-Lens Image Processing System Examples of Coherent Image Processing Incoherent Image Processing and Optical Transfer Function MATLAB Examples of Optical Image Processing 5.5.1 Coherent lowpass filtering 5.5.2 Coherent bandpass filtering 5.5.3 Incoherent spatial filtering

124 133 133 138 140 146 150 150 155 160

Chapter 6: Holography and Complex Spatial Filtering

169

6.1 6.2 6.3 6.4 6.5 6.6

170 173 186 189 193 199

Characteristics of Recording Devices The Principle of Holography Construction of Practical Holograms Reconstruction of Practical Holograms and Complex Filtering Holographic Magnification Ray Theory of Holograms: Construction and Reconstruction

Chapter 7: Contemporary Topics in Optical Image Processing

207

7.1 Theory of Optical Heterodyne Scanning 7.1.1 Bipolar incoherent image processing 7.1.2 Optical scanning holography 7.2 Acousto-Optic Image Processing 7.2.1 Experimental and numerical simulations of 1-D image processing using one acousto-optic cell 7.2.2 Improvement with two cascaded acousto-optic cells 7.2.3 Two-dimensional processing and four-corner edge enhancement 7.3 Photorefractive Image Processing 7.3.1 Edge enhancement 7.3.2 Image broadcasting 7.3.3 All-optical joint transform edge-enhanced correlation 7.4 Dynamic Holography for Phase Distortion Correction of Images

208 214 219 226

245 248

Subject Index

257

227 231 235 239 241 243

Chapter 1 Introduction to Linear Systems

1.1 One and Two-dimensional Fourier Transforms 1.2 The Discrete Fourier Transform 1.3 Linear Systems, Convolution and Correlation In this Chapter, we introduce readers to mathematical basics that are often used throughout the rest of the book. First, we review some of the properties of the Fourier transform and provide examples of twodimensional Fourier transform pairs. Next we introduce readers to discrete Fourier transforms since this serves as the basis for Fast Fourier transform algorithms that will be used for simulations using MATLAB. Finally, we discuss properties of linear systems and the concept of convolution and correlation.

1.1 One and Two-dimensional Fourier Transforms The one-dimensional (I-D) spatial Fourier transform of a squareintegrable function f (z) is given as [Banerjee and Poon (1991)]

F(k~) - f _ ~ f ( z ) e x p ( j k ~ z ) d z -

f~{f(z)}.

(1.1-1)

The inverse Fourier transform is

f ( x ) - ~ f ~ o F(k~)exp( - jk~x) dx - ~ ~ {F(/c~)}.

(1.1-2)

2

1 Introduction to Linear Systems

The definitions for the forward and backward transforms are consistent with the engineering convention for a traveling wave, as explained in Banerjee and Poon (1991). If f ( x ) denotes a phasor electromagnetic quantity, multiplication by exp(jwt) gives a collection or spectrum of forward traveling plane waves. The two-dimensional (2-D) extensions of Eqs. (1.1-1), (1.1-2) are

F(kx,kv) - f _ ~ f ~ o ~ f (x, y ) e x p ( j k x X + jkyy) d x d y (1.1-3)

= ,Txy{f(x, Y)}, 1

O<3

f (x,y) - ~ f_o~F(k~, kv) exp( - jk~x - jkyy) d x d y (1.1-4)

=

In many optics applications, the function f ( x , y ) represents the transverse profile of an electromagnetic or optical field at a plane z. Hence in Eqs. (1.3-3) and (1.3-4), f ( x , y ) and F(kx,kv) have z as a parameter. For instance, Eq. (1.1.4) becomes 1

OO

f (x,y;z) - -g~ f_o F(kx, ky, z) exp( - jk~x - jkyy) dxdy.

(1.1-5)

The usefulness of this transform lies in the fact that when substituted into the wave equation, one can reduce a three-dimensional (3-D) partial differential equation (PDE) to a one-dimensional ordinary differential equation (ODE) for the spectral amplitude F(k~, kv; z). Typical properties and examples of two-dimensional Fourier transform appear in the Table below.

Function in (x, y)

Fourier transform in (kx, ky)

1. f(x,y)

F(~x,~)

2. f (x-xo, y -

Yo) 3. f (ax, by); a, b complex constants 4. f*(x,y)

F(kx,ky)exp(jkxXO + jkyyo) 1 F(kz "

s.r(~,v)

47r2f(- kx,- ky)

6. Of(x, y)/Ox 7. deltafunction

r * ( - kx,- k~) - jk~F(~z,k~)

1.2 The Discrete Fourier Transform

1

3

oo

(5(:c, Y)= ~--~2f-oo e+Jk~x+jk~Ydzdy

1

4 7c26 (kx ,ky ) sinc function

8.1

9. rectangle function rect(x, y) - rect(x)rect(y), where rect(x)= ( 0,l'lx <1/2 otherwise \

sinc( kx , ~k~ ) - sinc( ~kx )sinc( ~),k~ where sinc(x) - sin(Trx)

/

7rz

10. Gaussian

Gaussian

e x p [ - c~(x 9 + ye)]

~-exp[~ - - 7 S ]

2

11. comb function comb ~x, y

2

kx + k v

(x) ~ comb

-comb

comb function N ,

where comb ~X =~-2~~(x - nxo)

comb ~0' ~

'

where kxo=27r/xo, kyo=27r/yo

t~=-oo

Table 1.1 Properties and examples of some two-dimensional Fourier Transforms.

1.2 The Discrete Fourier Transform Given a discrete function f(nA), n - 0,...N-I, where A is the sampling interval in x, a corresponding periodic function fp(nA) with period NA can be formed as [Antoniou (1979)]" OO

fp(nA) - ~ f ( n A + r N A )

(1.2-1)

r=-oo

The discrete function f(nA) may be formed by the discrete values of a continuous function f ( x ) evaluated at the points x - nL. The discrete Fourier transform (DFT) of fp(nA) is defined as N-1

Fp(mK)-

E fp(nA)exp(jmnKA)

,

K- -

2~

NA"

(1.2-2)

n=0

The inverse DFT is defined as N-1

fp(nA) - ~ Fp(mK)exp( - j m n K A ) .

(1.2-3)

rn=0

For properties of the DFT, e.g., linearity, symmetry, periodicity etc., as well as relationship to the z-transform, the Fourier transform and the Fourier series, the readers are referred to any standard book on digital signal processing [Antoniou (1979)].

4

1 Introduction to Linear Systems

For the purposes of this book, the DFT is a way of numerically approximating the continuous Fourier transform of a function. The DFT is of interest because it can be efficiently and rapidly evaluated by using standard fast Fourier transform (FFT) packages. Note that the direct evaluation of the DFT requires N complex multiplications and N - 1 complex additions for each value of Fp(mK), and since there are N values to determine, N 2 multiplications and N ( N - 1) additions are necessary. However, by using FFT algorithms, such as decimation in time or decimation in frequency, the number of multiplications can be reduced to (N/2)log2N. For example, if N > 512, the number of multiplications is reduced to less than 1% of that required by direct evaluation. Details of FFT algorithms can be found in any standard digital signal processing text, see for instance, Antoniou (1979). We will use FFT concepts in beam propagation problems in Chapter 4 and in image processing, in Chapters 5-7. The direct connection between the continuous Fourier transform and the DFT is given below. For a function f(x) and its continuous Fourier transform F (kx ) ,

I KI <

(1.2-4)

In Eq. (1.2-4), Fp(mK) is defined, as in Eq. (1.2-2), to be the DFT of fp(nA). The equality holds for the fictitious case when the function is both approximately space and spatial frequency limited.

1.3 Linear Systems, Convolution and Correlation A system is the mapping of an input or set of inputs into an output or set of outputs. A convenient representation of a system is a mathematical operator [Poularikas and Seely (1991)]. For instance for a single-input single-output system,

fo(x, v) -

v)},

(1.3-1)

where fi and fo are the input and output, respectively, and where P~y is the operator. A system is linear if for all complex constants a and b,

1.3 Linear Systems, Convolution and Correlation

5

Pxv{afil (x, y) + bfi2(x, y)} = a P x y { f i l (x, y ) ) +

bP~y{f~2(x, y)},

(1.3-2)

that is, the overall output is the weighted sum of the outputs due to inputs fil and fi2. This feature is particularly useful in constructing the output for a given input, knowing the output for an elementary input like the delta function. For a delta function input of the form ~5(x- x ' , y - y~), the output Pzy {6(x - x', y - y')} - h(x, y, x', y') is called the impulse response of the linear system. Using the sifting property of delta functions, we know that an arbitrary function fi(x, y) can be represented A

as

(1.3-3) i.e., fi(x, y) can be regarded as a linear combination of weighted and shifted delta functions. We can then write the output fo(X, y) of the linear system as

- L ~ L ~ f ~ ( ~ ' , y')h(~, y, ~', y') d~'ay'.

(1.3-4)

Now, a linear system is called space-invariant if the impulse response h(x, y, x', y') depends only on x - x', y - y', that is, A

(~.3-5)

h ( ~ , u, ~', y') - h ( ~ - ~', y - y').

Thus for linear space-invariant systems, the output (1.3-4) can be rewritten as fo(~, y) - L ~ L ~ f ~ ( ~ ' ,

fo(X, y)

y ' ) h ( ~ - ~', y - y') a ~ ' d y ' .

from Eq.

(1.3-6)

6

1 Introduction to Linear Systems

Now the convolution defined as

9(x, y) of two functions 91 (x, y) and g2(x, y) is

9(x, y) -- f_~oof_~o~gl(x', y')g2(x - x', y - y') dx'dy' = g~ (x, v),9~(x, y)

(1.3-7)

Using this definition, Eq. (1.3-6) can be reexpressed as (1.3-8)

fo(~, v) - f~(x, v),h(~, v) - h(~, v), f~(~, v)

It can be readily shown that the Fourier transform G(kx, ky)of 9(x, y)is related to the Fourier transforms G1,2(kz, ky) of gl,2(x, y) as (1.3-9) Hence, using this property, it follows that

Fo(kx, k~) =f /(kx, k~)H(kx, k~),

(1.3-10)

Fo(k~, kv), F~(k~, ky) and H(kx, lcy) are the Fourier transforms of fo(X, y), fi(x, y), and h(x, y), respectively. H(k~, ky) is called the transfer function of the system. The correlation s(x, y)of two functions sl(x, y) and s2(x, y) is

where

defined as

= ~ (x, v) |

(1.3-11)

~ (~, v).

Note upon comparing with Eq. (1.3-7) that this can be rewritten as 8(X, y) -- 81(X, y) @ 82(X, y) = ~ (x, v ) , ~

( - x, - v) - ~ (

- x, - v ) , ~

(~, v),

(1.3-12)

where the last step in the above relation can be verified by writing down the entire integral and making a simple substitution. Now, using the properties of Fourier transforms it follows that

1.3 L i n e a r S y s t e m s , C o n v o l u t i o n a n d C o r r e l a t i o n

S(kx,

7

=s; (kx, k )S:(kx,

(1.3-13)

S(kx, kv), Sl(]cx, ~y) and S2(k~, ky) s(x, y), s~ (x, y) and s 2(x, y), respectively.

are the Fourier transforms of

where

We will use properties of convolution in our discussion on transfer functions for propagation in Chapter 3, and properties of correlation in connection with matched filtering and image processing in Chapter 6.

Problems 1.1

Verify the Fourier transform pairs 9, 10 and 11 in Table 1.1.

1.2

From first principles find the Fourier transforms of: (a) the signumfunction sgn(x, y) - sgn(x)sgn(y), where sgn(x) - 1, x > 0, = 0, x - 0, = - 1, x < 0.; (b) the triangle function a ( x , y) - A(x)A(y), where A ( x ) - 1 - I~1, < 1, ---- 0, otherwise; (c) sech(x/Xo)Sech(y/yo ).

1.3

Verify that

1.4

Find the Fourier transforms of (i) [91(x, y)*92(x, y)]g3(x, y), (ii) [91(x, y)g2(x, g)]*g3(x, y). Express your results in terms of the Fourier transforms of 91,2,3.

1.5

Prove ee

fi(x, y).h(x, y) - h(x, y). fi(x, y).

Parseval's theorem: oe

1

oe

oe

f_~f_~lf(x, y)[2dxdy - ~--~f_~f_~[F(kx, kyl2dkxdk,.

8

1.6

1 Introduction to Linear Systems

Find the DFTs of the following periodic functions, defined over a period as" (a) fv(nA) = 1 for n - 0,2,4,6 = 0 for n - 1,3,7; (b) fv (nA) = 1 for n - 0,1,2,3 = 0 for n - 4,5,6,7. Write a M A T L A B program to find the FFTs of the above functions, and compare with your analytical results.

References 1.1 1.2 1.3

Banerjee, P.P., and T.-C. Poon (1991). Principles of Applied Optics. Irwin, Boston. Antoniou, A. (1979). Digital Filters. Analysis and Design. Mc. Graw Hill, N.Y. Poularikas A.D., and S. Seely (1991). Signals and Systems. PWSKent, Boston.

Chapter 2 Geometrical Optics

Fermat's Principle Reflection and Refraction Refraction in an lnhomogeneous Medium Matrix Methods in Paraxial Optics 2.4.1 The ray transfer matrix 2.4.2 Illustrative examples 2.5 Ray Optics using MATLAB 2.1 2.2 2.3 2.4

In geometrical optics, we view light as particles of energy traveling through space. The trajectory of these particles follows along paths that we call rays. We can describe an optical system comprising elements such as mirrors and lenses by tracing the rays through the system. In vacuum or free space, the speed of light particles is a constant approximately given by c - 3 x 10 8 m/s. The speed of light in a transparent homogeneous material, which we term v, is again a constant but less than c. This constant is a physical characteristic or signature of the material. The ratio c/v is called the refractive index n of the material. We can derive the laws of geometrical optics, namely reflection and refraction, using a simple axiom known as Fermat's principle. This is an extremum principle from which we can trace the rays in a general optical medium. Based on the laws of reflection and refraction, we will introduce a matrix approach to analyze ray propagation through an optical system. Geometrical optics is a special case of wave or physical optics, which will be mainly our focus through the rest of the Chapters in the

10

2 Geometrical Optics

book. Specifically, it can be shown that we can recover geometrical optics by taking the limit in which the wavelength of light approaches zero. In this limit, diffraction and the wave nature of light is absent.

2.1 Fermat's Principle In classical mechanics, Hamilton's principle of least action provides a recipe to find the optimum displacement of a conservative system from one coordinate to another [Goldstein (1950)]. Similarly, in optics, we have Fermat's principle which states that the path a ray of light follows is an extremum in comparison with the nearby paths. In Section 2.2, we will use Fermat's principle to derive the laws of geometrical optics. We now give a mathematical enunciation of Fermat's principle. Let n(x, y, z) represent a position-dependent refractive index. Then ds C /1"~

nds ~

C

represents the time taken to traverse the geometric path ds in a medium of refractive index n as c is the speed of light in free space. Thus, the time taken by the ray to traverse a path C between points A and B (see Figure 2.1) is ! fc n(x, y z) ds C

~

"

B

J

C

A Figure 2.1 A ray of light traversing a path C between points A and B.

2.2 Reflection and Refraction

11

The integral above is called the optical path length (OPL). According to Fermat's principle, the ray follows the path for which the OPL is an extremum" 6 ( O P L ) - 6f c

n(x, y, z) ds -

O.

(2.1-1a)

The variation of the integration means that we find the partial differentials of the integral with respect to the free parameters in the integral. This will become clear in the next Section where we derive the laws of reflection and refraction when we have a common boundary between two media of different refractive indices. In a homogeneous medium, i.e., in a medium with a constant refractive index, the rays are straight lines. We can also restate Fermat's principle as a principle o f least time. To see this, we divide Eq. (2.1-1 a) by c to get c~ (Sf c

n(x, Y , z ) ds - O.

(21" - lb)

We remark that Eq. (2.1-1b) is incorrectly called the least time principle. To quote Feynmann [Feynmann (1963)], Eq. (2.1-1 b) really means that "if we make a small change ... in the ray in any manner whatever, say in the location at which it comes to the mirror, or the shape of the curve, or anything, there will be no first order change in the time; there will be only a second order change in the time."

2.2 Reflection and Refraction When a ray of light is incident on the boundary MM' separating two different media, as in Figure 2.2, observation shows that part of the light is reflected back into the first medium, while the rest of the light is refracted as it enters the second medium. The directions taken by these rays are described by the laws of reflection and refraction. We will now use Fermat's principle to derive the two laws.

12

2 Geometrical Optics

Incident r a y ~ N ~

M

.,q o;/

Medium 1 Medium 2

~ R e f l e c t e d ray

,~

M' o,

efracted ray

Figure 2.2 Reflected and refracted rays for light incident at the interface of two media. Consider a reflecting surface as shown in Figure 2.3. Light from point A is reflected from the reflecting surface to point B, forming the angle of incidence 0i and the angle of reflection Or, measured from the normal to the surface. The time required for the ray of light to travel the path AO + OB is given by t - (AO + OB)/v, where v is the velocity of light in the medium containing the points AOB. The medium is considered isotropic for convenience. From the geometry, we find 1 t(z) - ~([h~ + ( d - z)~] 1/2 + [h~ + z2l 1/2 ).

(2.2-1)

d

A

'~

z

B

hi h2

/ / / / / / / / / / / / / / / / / / / /

Figure 2.3 Incident (AO) and reflected (OB) rays.

2.2 Reflection a n d Refraction

13

According to the least time principle, light will find a path that extremizes t(z) with respect to variations in z. We thus set d t ( z ) / d z = 0 to get a-~

_

[h~-'F-(d-z)2] 1/2 - -

z

[h 2+z2]

1/2

(2.2-2)

or

sin Oi = sinO~

(2.2-3a)

0~ - 0r.

(2.2-3b)

so that

We can readily check that the second derivative of t(z) is positive so that the result obtained corresponds to the least time principle. Equation (2.2-3) states that the angle of incidence is equal to the angle of reflection. In addition, Fermat's principle also demands that the incident ray, the reflected ray and the normal all be in the same plane, called the plane of incidence. Let us now use Fermat's principle to analyze refraction as illustrated in Figure 2.4. Ot and 0i are the angles of transmission and incidence, respectively, measured once again from the normal to the interface. The time taken by the light to travel the distance AOB is t(z)-

AOw1 +

OB,/)2 -- [h12-jrz2]l/2~Vl+ [h~+(d-z)2]~/~~ ,

(2.2-4)

where vx and v2are the light velocities in media 1 and 2, respectively. In order to minimize t(z), we set dt __ --

z vl[h~+z211/2 --

(d-zl v2[h ~+(d_z)2ll/2 = O.

(2.2-5)

2 Geometrical Optics

14

hi 0

Medium 1 Medium 2 h2

B f

Figure 2.4 Incident (AO) and transmitted or refracted (OB) rays. Using the geometry of the problem, we conclude that

si,~o~= ,i,~o,. "V1

(2.2-6a)

V2

Now v~,~ - c/n~,~ where n~,~ are the refractive indices of media 1 and 2, respectively. Equation (2.2-6a) may be restated as

~i,~o~_ '2z sinOt

~

(2.2-6b)

n 1 '

where n ~ / n I is the relative refractive index of medium 2 with respect to medium 1. Equation (2.2-6) is called Snell's law o f refraction. Again, as in reflection, the incident ray, the refracted ray, and the normal all lie in the same plane of incidence. Snell's law shows that when a light ray passes obliquely from a medium of smaller refractive index into one that has a larger refractive index, it is bent toward the normal. Conversely, if the ray of light travels into a medium with a lower refractive index, it is bent away from the normal. For the latter case, it is possible to visualize a situation where the refracted ray is bent away from the normal by

2.3 Refraction in an Inhomogeneous Medium

15

exactly 90 ~. Under this situation, the angle of incidence is called the critical angle Oc, given by s i n O~ - n 2 / n 1

(2.2-7)

When the incident angle is greater than the critical angle, the ray originating in medium 1 is totally reflected back into medium 1. This phenomenon is called total internal reflection (TIR). The optical fiber uses this principle of total reflection to guide light, and the mirage on a hot summer day is a phenomenon due to the same principle.

2.3 Refraction in an Inhomogeneous M e d i u m In the last Section, we have discussed refraction between two media with different refractive indices, i.e., possessing a discrete inhomogeniety in the simplest case. Consider, now, a medium comprising a continuous set of thin slices of media of different refractive indices as shown in Figure 2.5. At every interface, the light ray satisfies Snell's law according to nl s i n

01

--

n 2 s i n 02 -- n a s i n 03 = ...

(2.3-1)

Thus, we may put n s i n 0 -- nl s i n O1 ,

(2.3-2)

where n(z) and O(z) stand for the refractive index and the angle in a general layer, respectively, at location z. In the limiting case of continuous variation of the refractive index, which defines an i n h o m o g e n e o u s m e d i u m , the piecewise linear trajectory of the ray becomes a continuous curve, as shown in Figure 2.6. If d s represents the infinitesimal arc length along the curve, then

16

2 Geometrical Optics

t3

n2

n3

I

n4

I

I

I04~

04,

I I

,

I

///5

Figure 2.5 Rays in a layered medium in which the refractive index is piecewise continuous.

(d~)~ - (dx)~ + (d~) ~,

(2.3-3)

where we restrict ourselves to two dimensions. Also, from Figure 2.6, dz/ds

-

(2.3-4)

s i n O.

&

~A

Z

Figure 2.6 The path of a ray in a medium with a continuous inhomogeniety.

2.3 Refraction in an Inhomogeneous Medium

17

Combining (2.3-2), (2.3-3) and (2.3-4), we obtain

~dx)

2

=

(~)

n2 T~21siTt20 1 - -

(2.3-5a)

1

or alternatively, by differentiating with respect to z, d2x __ dz 2 ~

1 dn2(x) n~ s i n 2 0 1 dx

(2.3-5b) "

The solution of Eq. (2.3-5) gives the direction a ray will take in a medium with a refractive index variation n(x), and is a special form of the ray or eikonal equation.

Example 2.1 Homogeneous Medium n ( z ) - constant. From (2.3-5a), we observe that the solution is that of a straight line, as expected.

Example 2.2 Square Law Medium -

(2.3-6)

-

Substituting for n 2 from Eq. (2.3-6) into Eq. (2.3-5b) yields d2x

n (2)

a~ = - ~o~i~o~X(z) '

(2.3-7)

where 01 is the angle the ray makes with the x (transverse) axis at x - 0. The solution of the above equation is of the form

n(2)

x(z) - A sin ( i ,~~20~i,~0----Sz + r

/

,

(2.3-8)

where the constants A and r can be determined from the initial position and slope of the ray. Note that rays with smaller launching angles c~ (-rc/2-01)have a larger period, however, in the paraxial

18

2 Geometrical Optics

approximation (i.e., for small launching angles), all the ray paths have approximately the same period. The case discussed above approximately explains the mechanism of light propagation through graded index opticalfibers, to be discussed in more detail in Chapter 4.

2.4 Matrix Methods in Paraxial Optics In this Section, we consider how matrices may be used to describe ray propagation through optical systems comprising, for instance, a succession of spherical refracting and/or reflecting surfaces all centered on the same axis, which is called the optical axis. Unless otherwise stated, we will take the optical axis to be along the z-axis. As we will see shortly, the "coordinates" of a ray at a certain plane perpendicular to the optical axis can be specified by a vector, which contains the information of the position of the ray and its direction. It would therefore be convenient if, given this information, we can find the "coordinates" of the ray at any other plane, again normal to the optical axis, by means of successive operators acting on the initial ray coordinate vector, with each operator characteristic of the optical element through which the ray travels. We can represent these operators by matrices. The advantage of this matrix formalism is that any ray can be tracked during its propagation through the optical system by successive matrix multiplications, which can be easily programmed on a digital computer. This representation of geometrical optics is elegant and powerful, and is widely used in optical element designs. We will only consider paraxial rays, implying rays that are close to the optical axis, and whose angular deviation from it is small enough such that the sine and tangent of the angles may be approximated by the angles themselves. The reason for this paraxial approximation is that all paraxial rays starting from a given object point intersect at another point after passage through the optical system. We call this point the image point. Nonparaxial rays may not give rise to a single image point. This phenomenon, which is called aberration, is outside the scope of this book. The optics of paraxial imaging is also sometimes called Gaussian

optics. In what follows, we will first develop the matrix formalism for paraxial ray propagation or ray transfer, and examine some of the properties of ray transfer matrices. We then consider several illustrative

19

2.4 Matrix Methods in Paraxial Optics

examples. For instance, we examine the imaging properties of lenses, and derive the rules for ray tracing through an optical system.

2.4.1 The ray transfer matrix Consider the propagation of a paraxial ray through an optical system as shown in Figure 2.7. Restricting ourselves to one transverse direction (z), a ray at a given cross-section or plane may be specified by its height z from the optical axis and by its angle or slope which it makes with the axis. Thus the quantities (z,O) represent the coordinates of the ray for a given z-constant plane. However, instead of specifying the angle the ray makes with the z-axis, it is customary to replace the corresponding by v = n0, where r~ is the refractive index at the z-constant plane.

/4 j

s

S"

~

,~

~

~

,~, ,~'

,,,B / /

t

/

X2

X1

z (Optical axis)

Input plane

Optical system

Output plane

Figure 2.7 Reference planes in an optical system.

In Figure 2.7, the ray passes through the input plane with coordinates (Zl, Vl - n101), then through the optical system, and finally through the output plane with coordinates (z2, v 2 - n202). In the paraxial approximation, the corresponding output quantities are linearly dependent on the input quantities. We can, therefore, represent the transformation from the input to the output in matrix form as

2 Geometrical Optics

20

(Z2)-- ( C DB)(yl),o1.

(2.4-1)

The A B C D matrix above is called the ray transfer matrix, and as we shall see later, it can be made up of many matrices to account for the effects of a ray passing through various optical elements. We can consider these matrices as operators successively acting on the input ray coordinate vector. We state here that the determinant of the ray transfer matrix equals unity, i.e., A D - BC = 1. This will become clear after we derive the translation, refraction and reflection matrices. Let us now investigate the general properties of an optical system from the A B C D matrix.

Property 1: If D = 0, we have from Eq. (2.4-1) that v2 = Czi. This means that all rays crossing the input plane at the same point viz. z i, emerge at the output plane making the same angle with the axis, no matter at what angle they enter the system. The input plane is called the frontfocalplane of the optical system [see Figure 2.8(a)]. Property 2 : I f / 3 = 0, z2 = Azl (from Eq. (2.4-1)). This means that all rayspassing through the input plane at the same point (z l) will pass through the same point (z2) in the output plane {see Figure 2.8(b)]. The input and output planes are called the object and image planes, respectively. In addition, A - z2/zi gives the magnification produced by the system. Furthermore, by inverting the A B C D matrix and the fact that A D - BC = 1, we note from Eq. (2.4-1) that zx = Dz2 since B = 0. The implication of this is that the point z2 is imaged at Zl with magnification 1/A. Hence, the two planes containing zland z2 are called conjugate planes. Moreover, if A - 1, i.e., the magnification between the two conjugate planes is unity, these planes are called the unit or principal planes. The points of intersection of the unit planes with the optical axis are the unit orprincipalpoints. The principal points constitute one set of cardinal points. Property 3: If C = 0, v2 = DVl. This means that all the rays entering the system parallel to one another will also emerge parallel, albeit in a new direction [see Figure 2.8(c)]. In addition, D(r~l/n2) - 02/01 gives the angular magnification produced by the system.

2.4 Matrix Methods in Paraxial Optics

21

If D = n2/nl, we have unity angular magnification, i.e., 02/01 = 1. In this case, the input and output planes are referred to as the nodal planes. The intersections of the nodal planes with the optical axis are called the nodal points[see Figure 2.8(d)]. The nodal points constitute the other set of cardinal points. Property 4: If A = 0, x2 = B y 1 . This means that all rays entering the system at the same angle will pass through the same point at the output plane. The output plane is the back focal plane of the system [see Figure 2.8(e)].

(a)

(b)

_ _ _ - -_: = = - Z T __ _ . 0 2

"

(c)

N odal

planes

"

(d)

/> (e)

F i g u r e 2.8 Rays at input and output planes for (a) D - 0, (b) B - 0, (c) C - 0, (d) the case when the planes are nodal planes, and (e) A - 0.

When a ray passes through an optical system, there two types of processes, translation and refraction (and, reflection; this is treated later), that we need to consider determine the ray's progress. As the rays propagate

are usually sometimes, in order to through a

2 Geometrical Optics

22

homogeneous medium, they undergo a translation process. In order to specify the translation, we need to know the thickness of the medium and its refractive index. However, when a ray strikes in interface between two regions of different refractive indices, it undergoes refraction. To determine how much bending the ray undergoes, we need to know the radius of curvature of the boundary and the values of the refractive indices of the two regions. We shall investigate the effect each of these two processes have on the coordinates of a ray between the input and the output planes. In fact, we will derive the ray transfer matrices for the two processes. Figure 2.9 shows a ray traveling a distance d in a homogeneous medium of refractive index n. Since the medium is homogeneous, the ray travels in a straight line. The set of equations of translation by a distance d is X2 -- Xl -t- d t a n O i ,

(2.4-2a)

nO2 - - n 0 1 o r v2 -

(2.4-2b)

Vl.

___~:~____02

X2

Input plane

Output plane

Z=Zi

Z=Z2

F i g u r e 2.9 A ray in a h o m o g e n e o u s m e d i u m o f refractive index n .

These equations relate the output coordinates of the ray with its input coordinates. We can express this transformation in a matrix form as

ix2/ iv2 ( 0 J )l/Xl/ l

2.4 Matrix Methods in Paraxial Optics

23

The 2 x 2 ray transfer matrix, called the translation matrix T , is defined

as T-

1

0

d/n ) 1

"

(2.4-4)

Note that its determinant is unity. We now adopt the following convention: when light rays travel across a distance d from the plane z - Zl to the plane z = z2(see Figure 2.9), z 2 - Zl will be taken to be positive for a ray traveling in the +z direction and negative for a ray traveling in the -z direction. Therefore, in the latter case, we take the refractive index of the medium to be negative so that the value of (z2 - Zl)/n in the translation matrix will remain positive. We next study the effect of a spherical surface separating two regions of refractive indices nl and n2 as shown in Figure 2.10. The center of the curved surface is at C and its radius of curvature is R. The ray strikes the surface at the point A and gets refracted. Note that the radius of curvature of the surface will be taken as positive (negative) if the center C of curvature lies to the right (left) of the surface. Let x be the distance from A to the axis. Then the angle r subtended at the center C becomes

sin r ,.~ x / R ~ r

nl / n2 Figure 2.10 Ray trajectory during refraction at a spherical surface.

(2.4-5)

2 Geometrical Optics

24

We see that in this case, the height of the ray at A, before and after the refraction, is the same, i.e., x2 = Xl. We therefore need to obtain the relationship for v2 in terms of Xl and Vl. Applying Snell's law [see Eq. (2.2-6)] and using the paraxial approximation, we have

(2.4-6)

n l O~ = n2Ot.

From geometry, we know from Figure 2.10 that 0i = 01 + r Ot = 02 + r Hence,

and

rtl 0 i -- Yl -~- n l X l / R ,

(2.4-7a)

n20t = v2 + n 2 x 2 / R .

(2.4-7b)

Using Eq. (2.4-6), (Eq. 2.4-7) and the fact that x z = x2, we obtain V2

---

R

hi--n2

+ Vl.

(2.4-8)

The matrix-vector equation relating the coordinates of the ray after refraction to those before refraction becomes ( ~12 )~- , 2( _ p

0)1 (~)~1 ,

(2.4-9a)

where the quantity p given as p

--

Tt2--721

\z-, -r

R

.lu][Q.A-O]'~'I

is termed the refracting power of the spherical surface. With R measured in meters, the unit of p is called diopters. If an incident ray is made to converge (diverge) by a surface, the power will be assumed to be positive (negative) in sign. The (2 x 2) transfer matrix is called the refraction matrix 7-r and it describes refraction at A for the spherical surface: 7 ~ _ ( l_p

0)1

Note that the determinant of 7~ is also unity.

(2.4-10)

2.4 MatrixMethodsinParaxialOptics

25

2.4.2 Illustrative examples Example 2.3 Plane Parallel Layers Consider a medium of thickness d, divided into two regions of thicknesses di and d2, and having the same refractive index n, as shown in Figure 2.11. The system of equations relating the output coordinates to the input coordinates are given by (x2)_v2 (10

d2/n)l (zO)_v~ (10 d2/n)(ll

0 di/n)l

(Z~)vx (2.4-11)

dl

r d

Input plane Common plan (x~, v~) ~D. (x,, v,)

d~

Output plane _~ (x2, v2)

Figure 2.11 Plane parallel layers of thicknesses dl and d2. The overall system transfer matrix T can be written as 1 7

-

T2%

-

0

d2/n

dl/n

1)(0 1)-(o 1

1

(dl + d2)/n "~ (2.4-12) 1

J

as expected. Note that the overall system matrix T is expressed in terms of the product of the two individual matrices Ti and T2 written in order from right to left. The order of the matrix multiplication is important, as matrix multiplication is not commutative in general.

2 Geometrical Optics

26

A similar situation applies when a region of thickness d comprises i layers, each having a thickness dj and refractive index nj. We can write the overall system matrix T as i

dj/nj)_ .

.

.

1

.

j=l

O

Edj/nr

1

j=l

0

. (2.4-13)

1

In writing (2.4-13), we have neglected the refraction matrices at each plane of separation between successive layers, since the refraction matrix reduces to the identity matrix in the paraxial regime.

Example 2.4 A Single Lens Consider a single lens as shown in Figure 2.12. It is evident that the system matrix for the lens consists of two refraction matrices and a translation matrix"

S -

(1 o)(1 n2-nl

1

R2 refraction at

0

1 translation

surface 2

1

/71

/"/2

1

(2.4-14)

R1 refraction at

surface 1

/71

Surface

o)

n2-nl

7

\ Rl

(x,, v,)

Surface

2

R2

(x2, v2)

Figure 2.12 A single lens. The radii of curvatures of surfaces 1 and 2 are R1,2, respectively.

2. 4 Matrix Methods in Paraxial @tics

27

For a thin lens in air, d ~ 0 and nl - 1. Writing n2 -- n for notational convenience, Equation (2.4-14) becomes $=(

I_N

0 ) ( 1 10

01 ) (1 - P l

0)1 '

(2.4-15)

where Pl -- ( n - 1)/R1 and I92 - (1 - n)/R2 are the refractive powers of surfaces 1 and 2, respectively. Note that the translation matrix degenerates into a unit matrix. Equation (2.4-15) can be rewritten as ,9=(

1 -P2

0)( 1

1 -Pl

0) ( 1 1 --1/f

0) 1 -Sf,

(2.4-16)

where ,S'y is called the lens matrix and f is the focal length and explicitly given by [ -- ( 7 2 f

~

1)( 1

R1

1 )

R2

"

(2.4-17)

We will clarify the implication of the focal length in the following example where we discuss ray tracing through a thin lens. For RI>(<)0 and R2<(>)0, f>(<)0. If a ray of light is incident on the left surface of the lens parallel to the axis, the angle at which it emerges on the right surface may be found by using Eq. (2.4-1) with Vl = 0, and the ray transfer matrix for the thin lens, as in Eq. (2.4-16). It follows that for f>(<)0, the ray bends towards (away from) the axis upon refraction through the lens. In the first case, the lens is called a converging (convex) lens, while in the second case, we have a diverging (concave) lens. Example 2.5 Ray Tracing through a Single Thin Lens

In the previous example, we mentioned how rays parallel to the axis of a thin lens in air was bent towards or away from the axis after passing through a thin lens, depending upon whether the lens was converging or diverging. To carry this idea on a little further, consider the following cases:

28

2 Geometrical Optics

(a) Ray traveling parallel to the axis: The input ray vector is (z l, 0) r, hence the output ray vector is given, using Eq. (2.4-1) and Eq. (2.4-16), as (Zl, - z ~ / f ) r. This ray now travels in a straight line at an angle - 1/f with the axis. This means that if Z l is positive (or negative), the ray after refraction through the lens intersects the optical axis at a point a distance f behind the lens if the lens is converging (f>0). This justifies why f is called the focal length of the lens. All rays parallel to the optical axis in front of the lens converge behind the lens to a point called the back focus [see Figure 2.13(a)]. In the case of a diverging lens (f<0), the ray after refraction diverges away from the axis as if it were coming from a point on the axis a distance f in front of the lens. This point is called the front focus. This is also shown in Figure 2.13(a).

(b) Ray traveling through the center of the lens: The input ray vector is ray (0, Vl) T, hence the output ray vector is given using Eq. (2.4-6), Eq. (2.4-16) as (0, vl)T. This means that a ray traveling through the center of the lens will pass undeviated as shown in F i gure 2.13 (b).

(c) Ray passing through the front focus of a converging lens" The input ray vector is given by (x~, x~/f) T, so that the output ray vector is (Zl, 0) T. This means that the output ray will be parallel to the axis, as shown in Figure 2.13(c). In a similar way, we can also show that for an input ray on a diverging lens appearing to travel toward its back focus, the output ray will be parallel to the axis.

2.4 Matrix Methods in Paraxial Optics

29

1 s

front focus focus

(a)

(b) front focus

(c) ~o

f
Figure 2.13 Ray tracing through thin converging and diverging lenses.

Example 2.6 Imaging by a Single Thin Lens Consider an object OO' located a distance do in front of a thin lens of focal length f, as shown in Figure 2.14. Assume that (Zo, vo)T represents the coordinates originally from point O', and traveling towards the lens. Then the output ray coordinates (z, v) at a distance z behind the lens can be written in terms of the input ray coordinates, two translation matrices and the transfer matrix for the thin lens as:

O

~

I

O

A W

A W

f

f do

~.~

di

Figure 2.14 Imaging by a single lens.

I

2 Geometrical Optics

30 (1

(;)

-

z)

0

(

1

1

-1/f

-

do)(xo)

0)(1 1

1-do/f

0

1

Vo

Vo

9

(2.4-18)

Assume for a moment that the object is a point source on the axis, i.e., Xo = 0. Consider two rays emanating from the point object, one along the axis, the other at an angle %. Upon refraction through the lens, the on-axis ray will emerge undeviated, while the other ray will emerge at an angle v. We define the image of the point object to be the point where these two rays meet. Clearly, in this case, the image will be a point on the axis. We can calculate the distance z - di along the axis behind the lens where the image will form by setting Xo and x equal to zero in Eq. (2.418). This yields the celebrated thin-lens formula: 1

1 _

1

(2 4-19)

do + ~ - 7"

The sign convention for do and di is as follows, do is positive (negative) if the object is to the left (right) of the lens. If di is positive (negative), the image is to the right (left) of the lens and it is real (virtual). Now, returning to Eq. (2.4-18), we have, corresponding to the image plane, the relation

(~i)_ v,

(1-dill -1If

0 1-do/f

)(Zo) ~o '

(2.4-20)

where we have written (xi, vi) for the coordinates of the ray at the image plane z - di. For Xo r 0, we obtain x_~ = :Co

M -

1

di f

_ --

f-di f

_ --

f f-do

_ --

a, [using Eq.(2.4-19)] (2.4-21)

do

where M is called the magnification of the system. If M >0 (<0), the image is erect (inverted).

2.4 Matrix Methods in Paraxial Optics

31

Example2.7 Two-lensCombination Consider two thin lenses of focal lengths fl = f2 = f, separated by a distance d - 4 f in air (see Figure 2.15). Assume an object a distance of 2 f units to the left of lens L1. The matrix chain from the object to the image becomes

fl

f2

L2 Figure 2.15 A two-lens system.

S'-

( 10

d1i ) 1s (

0

2f)1 '

(2.4-22)

where S is the overall transfer matrix for the two lens system as

S -

(

1

-1/f2

0)(1 4f)( 1

0

1

1

-1/fl

0)

1 "

(2.4-23)

Using Eq. (2.4-22) and Eq. (2.4-23), we can express S ~ as

S' - ( - 32/f + 2di/f

- 2f + d~'] 1 J

(2.4-24)

from which we can find the position of the image by setting B = - 2f + di = 0. Hence, di = 2 f and the magnification (A) is then equal to 1, implying a real erect image of the same size as the object.

2 Geometrical Optics

32

2.5 Ray Optics using MATLAB 1. Ray coordinates passing through a single lens system In the following program, we will find the ray coordinates an arbitrary distance z behind a lens of focal lens f when the input ray vector ro = (0, 1) for a ray starting from an object located a distance do in front of the lens is specified. To and T/denote the translation matrices for the ray in air before and after the lens (corresponding to object and image distances, respectively), while S f is the lens matrix. The product of the three S - TiSfTo gives the overall system matrix for the optical system. The program gives the output ray vector r~ . All dimensions have been written in centimeter.

clear %Input object distance do do=15; %Input focal length of lens f f=-lO; % Input distance behind lens z z=30; %Input object ray coordinates ro ro-[O;1]; To=[ 1,do;0,1 ];

sf:[1,0;-(1/f),l]; Ti=[ 1,z;0,1 ]; S=Ti*Sf*To; %Check for determinant of overall matrix S detS=det(S) %"image" ray coordinate is ri ri=S*ro Output from MATLAB detS = 1

ri = 0 -0.5000 Table 2.1 MATLAB code for ray traveling through a single lens, and the corresponding MATLAB output.

2.5 Ray Optics using MA TLAB

33

This means that if the input ray starts from the axis, the output ray meets the axis a distance of 30 cm behind the single lens. In other words, for an object distance do of 15 cm, the image distance z - di is 30 cm. Finding the determinant of the overall system matrix ,5' is a check of the computations. Note that the ray coordinates at any plane z behind the lens can be found by substituting a number for the value of z in the program. To find the magnification of the imaging system, we can put in input ray coordinates of say, (1,1)r. Using the same program as above, the output ray coordinates at z=30 cm (the image plane) works out to be -2.0. This means that the magnification of the system equals -2, which corresponds to an inverted real image of twice the size as the object, as expected.

2. Ray coordinates passing through a double lens system In the following program, we will find the ray coordinates an arbitrary distance z behind a two-lens combination of focal lengths fl,2 and separated by a distance d when the input ray vectors ro~,2 for rays starting from an object located a distance do in front of the lens are specified. To and T/denote the translation matrices for the ray in air before and after the lens (corresponding to object and image distances, respectively), T~ denotes the translation matrix for a ray traveling between the two lenses, while ~fl,2 are the lens matrices, respectively. The product of $ - ~iSfZT-dSfl~o gives the overall system matrix for the optical system. The program gives the output ray vectors ril,2. All dimensions have been written in centimeter.

clear %Input "object" distance do do=10; %Input focal length of lens fl,f2 fl-10; f2=10; %Input distance between lenses d d=20; % Input distance behind lens z z=10; %Input object ray coordinates ro 1,ro2

2 Geometrical @tics

34 rol=[0;1]; ro2=[ 1;0]; To=[ 1,do;0,1 ]; Sfl=[ 1,0;-( 1/fl ), 1]; Td-[1,d;O,1]; sf2:[1,0;-(1/f2), ~]; Ti=[1,z;0,1]; S=Ti*Sf2*Td*Sfl*To; %Check for determinant of overall matrix S detS=det(S) %"image" ray coordinate is ri ri l=S*ro 1 ri2=S*ro2 Output from MATLAB detS = 1

ril = 0 -1 ri2 = -1 0 Table 2.2 MATLAB code for ray traveling through a two lens system, and the corresponding MATLAB output.

This means that if the input ray starts on-axis a distance of 10 cm in front of the first lens (ray coordinate roi in the program above), the output ray meets the axis a distance of 10 cm behind the second lens. In other words, for an object distance do of 10 cm from the first lens, the image distance z - di is 10 cm behind the second lens. This is the classic example of a two-lens imaging system that will be also discussed in detail later in connection with image processing. Note also that if the input ray is parallel to the axis (ray coordinate %2 in the program above), the output ray is also parallel to the axis. The image is real and inverted, with a magnification equal to unity. In general, for a two lens system with unequal focal lengths fi and f2separated by a distance d - fi + f2, the magnification equals f 2 / f i .

2.5 Ray Optics using MATLAB

35

3. Finding the image location in a single lens system The previous two examples using MATLAB assumed a certain distance for z. The values of z were tacitly chosen to conform to the image plane corresponding to a given object location. The following program is an extension of Example 1 in which the distance z is varied so as to determine the image plane. To do this, the object is taken to be an on-axis point, and the ray coordinates monitored behind the lens. If the position of the ray is sufficiently close to the axis behind the lens, the corresponding value of z is the location of the image. The program output shows that indeed the image location works out to be a distance of 30 cm behind the lens for an object distance of 15 cm in front of the lens with focal length equal to 10 cm. clear %Input object distance do do=15; %Input focal length of lens f f=10; % Vary distance behind lens z to find image distance forj=l:100 z=j; %Input object ray coordinates ro ro=[0;1]; Yo=[1,do;0,1]; Sf=[ 1,O;-(1/f),1]; Ti=[1,z;O,1]; S=Ti*Sf*To; %ray coordinate after lens is ri ri=S*ro; if abs(ri(1,1))<0.001 %image distance is di di-z end end Output from MATLAB di = 30 Table 2.3 MATLAB code for locating image plane for single lens imaging, and the corresponding MATLAB output.

36

2 Geometrical Optics

The procedure above can be readily extended for any imaging system.

Problems 2.1

Derive the laws of reflection and refraction by considering the incident, reflected and refracted light to comprise a stream of photons characterized by a momentum p - h k , where h - h/27c, h being Planck's constant, and k is the wavevector in the direction of ray propagation. Employ the law of conservation of momentum, assuming that the interface, say, y = constant, only affects the y-component of the momentum. [This provides an alternative derivation of the laws of reflection and refraction.]

2.2

Commonly used construction for the refractive index variation in an optical fiber is given as: n ( x ) -- no(1 A ( x / a ) ~ / ) 1/2. -

Assume a ray as shown in Figure P2-2. Find the direction(s) of the ray at a distance x - a. Sketch the variation of the direction of the ray at x = a as a function of A.

Incident

ray ~.

/7o

Figure P2.2

2.3

A thin glass beaker of 5 cm diameter is filled up with tap water of refractive index 1.4. Find the focal length of the cylindrical lens thus formed. Use the matrix formalism.

2.4

An object 3 cm tall is 15 cm in front of a converging lens of focal length 20cm. Using the matrix formalism, find the location and

2.5 Ray Optics using MATLAB

37

magnification of the image. Draw a ray diagram from the object to the image. 2.5

Referring to Figure 2.15, show that the equivalem focal length f of the two-lens combination can be expressed as 1

7

_

1

1

fl -~ f2

--

d

flf2'

assuming d < fl + f2. Specify the conditions under which the equivalent focal length concept can be used. 2.6

Consider a system of two thin lenses as shown in Figure 2.15. Let fl - 4cm, f2 = - 6 c m , and d = 3cm. An object 2cm tall is placed 12cm in front of the positive lens. Calculate the location and magnification of the image. Also, draw the complete ray diagram from the object to the final image. In addition, write a MATLAB program to verify your calculations.

2.7

Develop a MATLAB code to find the location of the image for an object located l m in front of the surface of a thick lens having surfaces with radii of curvature l m each. Assume that the thickness of the lens is 0.25 m.

2.8

An object is placed 12cm in front of a lens-mirror combination as shown in Figure P2.8. Using ray transfer matrix concepts, find the position and magnification of the image. Plane mirror

f=

lOcm

.

.

.

.

.

.

.

.

.

.

0 ~

12cm

,

,~

.~

20cm

Figure P2.8

/ / / / / / / / / / -/ /

2 Geometrical Optics

38

2.9

Show that the ray reflection matrix for a ray reflected from a surface of radius of curvature R is given by

(1 0) 2n/R

1'

where n is the refractive index of the medium in which the reflecting surface is placed. A point object is placed a distance 2m away from a concave mirror of radius of curvature R = - 8 0 c m . Find the location of the image. 2.10

Develop a MATLAB code to determine the ray coordinates after one round trip in an optical resonator comprising two concave mirrors of radius of curvature -1 m each, and separated by a distance of 1 m. Use the ray reflection matrix you derived in Problem 2.8, along with the ray translation matrix. Assume that the region between the two mirrors is air. Raise the round trip matrix to arbitrary powers using MATLAB and examine the coefficients of these matrices. What can you infer from your results?

References 2.1 2.2

2.3 2.4 2.5 2.6 2.7 2.8

Banerjee, P.P. and T.-C. Poon (1991). Principles of Applied Optics. Irwin, Illinois. Feynman, R., R. B. Leighton and M. Sands (1963). The Feynmann Lectures on Physics. Addison-Wesley, Reading, Massachusetts. Gerard, A. and J. M. Burch (1975). Introduction to Matrix Methods in Optics. Wiley, New York. Ghatak, A. K. (1980). Optics. Tata McGraw-Hill, New Delhi. Goldstein, H.(1950). Classical Mechanics. Addison-Wesley, Reading, Massachusetts. Hecht, E. and A. Zajac (1975)" Optics. Addison-Wesley, Reading, Massachusetts. Klein, M. V.(1970). Optics. Wiley, New York. Nussbaum, A and R. A. Phillips(1976). Contemporary Optics for Scientists and Engineers, Prentice-Hall, New York.

39

Chapter 3 Propagation and Diffraction of Optical Waves

3.1 Maxwell's Equations: A Review 3.2 Linear Wave Propagation 3.2.1 Traveling-wave solutions 3.2.2 Intrinsic impedance, the Poynting vector, and polarization 3.3 Spatial Frequency Transfer Function for Propagation 3.3.1 Examples of Fresnel diffraction 3.3.2 MATLAB example: the Cornu Spiral 3.3.3 MATLAB example: Fresnel diffraction of a square aperture 3.3.4 Fraunhofer diffraction and examples 3.3.5 MATLAB example: Fraunhofer diffraction of a square aperture 3.4 Fourier Transforming Property of Ideal Lenses 3.5 Gaussian Beam Optics and MATLAB Example 3.5.1 q-transformation of Gaussian beams 3.5.2 Focusing of a Gaussian beam 3.5.3 MATLAB example: propagation of a Gaussian beam In Chapter 2, we introduced some of the concepts of geometrical optics. However, as stated there, geometrical optics cannot account for wave effects such as diffraction. In this Chapter, we introduce wave optics by starting from Maxwell's equations and deriving the wave equation. We thereafter discuss solutions of the wave equation and review power flow and polarization. We then discuss diffraction at length through use of the Fresnel diffraction formula, which is derived in

3 Propagation and Diffraction o f Optical Waves

40

a unique manner using Fourier transforms. In the process, we define the spatial transfer function and the impulse response of propagation. We also describe the distinguishing features of Fresnel and Fraunhofer diffraction and provide several illustrative examples. Specifically, we analyze diffraction from rectangular and circular apertures and analyze the diffraction of a Gaussian beam. We also discuss wavefront transformation by a lens and show the Fourier transforming properties of a lens. Gaussian beam lensing is also discussed. In all cases, we restrict ourselves to propagation in a medium with a constant refractive index.

3.1 Maxwell's Equations: A Review In the study of electromagnetics, we are concerned with four vector quantities called electromagnetic fields: the electric field strength E (V/m), the electric flux density D (C/m2), the magnetic field strength H (A/m), and the magnetic flux density B (Wb/m2). The fundamental theory of electromagnetic fields is based on Maxwell's equations. In differential form, these are expressed as V . D -- p,

(3.1-1)

V.B

(3.1-2)

= 0,

VxE

-

V•

--

oB 0t ~ J

--

(3 1-3) J~+O~

i)t ~

(3.1-4)

where J is the current density (A/m 2) and p denotes the electric charge density (C/m3). J~ and p are the sources generating the electromagnetic fields. We can summarize the physical interpretation of Maxwell's equations as follows: Equation (3.1-1) is the differential representation of Guass's law for electric fields. To convert this to an integral form, which is more physically transparent, we integrate Eq. (3.1-1) over a volume V bounded by a surface S and use the divergence theorem (or Gauss's

theorem), fv V . D dV = fs D. dS,

(3.1-5)

3.1 Maxwell's Equations: A Review

41

to get f s D . dS = fv p dV.

(3.1-6)

This states that the electric flux flsD. dS flowing out of a surface S

enclosing a volume V equals the total charge enclosed in the volume. Equation (3.1-2) is the magnetic analog of Eq. (3.1-1) and can be converted to an integral form similar to Eq. (3.1-6) by using the divergence theorem once again, f s B . dS - O.

(3.1-7)

The right-hand sides (RHSs) of Eqs. (3.1-2) and (3.1-7) are zero because, in the classical sense, magnetic monopoles do not exist. Thus, the magnetic flux is always conserved. Equation (3.1-3) enunciates Faraday's law of induction. To convert this to an integral form, we integrate over an open surface S bounded by a line C and use Stokes's theorem, fs (V • E ) - d S - f E. ds

(3.1-8)

~a ( E . d g =

(3.1-9)

and - f ~B .dS. "s

This states that the electromotive force (EMF) fc E . dg induced in a loop is equal to the time rate of change of the magnetic flux passing through the area of the loop. The EMF is induced in a sense such that it opposes the variation of the magnetic field, as indicated by the minus sign in Eq. (3.1-9); this is known as Lenz's law. Analogously, the integral form of Eq. (3.1-4) reads f e l l . r i g = f o P T/ " dS + fsJc . dS,

(3.1-10)

which states that the line integral of H around a closed loop C equals the total current (conduction and displacement)passing through the surface

3 Propagation and Diffraction of Optical Waves

42

of the loop. When first formulated by Ampere, Eqs. (3.1-4) and (3.1-10) only had the conduction current term J~ on the RHS. Maxwell proposed the addition of the displacement current term OD/Ot to include the effect of currents flowing through, for instance, a capacitor. For a given current and charge density distribution, note that there are four equations [Eqs. (3.1-1)- (3.1-4)] and, at first sight, four unknowns that need to be determined to solve a given electromagnetic problem. As such, the problem appears well-posed. However, a closer examination reveals that Eqs. (3.1-3) and (3.1-4), which are vector equations, are really equivalent to six scalar equations. Also, by virtue of the continuity equation, V.J~+~

= O,

(3.1-11)

Eq. (3.1-1) is not independent of Eq. (3.1-4) and, similarly, Eq. (3.1-2) is a consequence of Eq. (3.1-3). We can verify this by taking the divergence on both sides of Eqs. (3.1-3) and (3.1-4) and by using the continuity equation [Eq. (3.1-1 1)] and a vector relation, V.(V•

(3.1-12)

to simplify. The upshot of this discussion is that, strictly speaking, there are six independent scalar equations and twelve unknowns (viz., the x, y, and z components of E, D, H, and B) to solve for. The six more scalar equations required are provided by the constitutive relations, D = eE,

(3.1-13a)

B=#H,

(3.1-13b)

where e denotes the permittivity (F/m) and/z the permeability (H/m) of the medium. The e and /~ are tensors in general; however, in many cases, we can take e and # to be scalar constants. This is true for a linear, homogeneous, isotropic medium. A medium is linear if its properties do not depend on the amplitude of the fields in the medium. It is homogeneous if its properties are not functions of space. The medium is, furthermore, isotropic if its properties are the same in all direction from any given point. For most of this book, we will assume the medium to be linear, homogeneous, and isotropic. However, and the propagation

3.2 Linear Wave Propagation

43

of waves in inhomogeneous and nonlinear media will be examined in Chapter 4. Returning our focus to linear, homogeneous, isotropic media, constants worth remembering are the values of e and # for free space or vacuum" e0 - (1/3670 x 10-9F/m and #0- 47r x 10 . 7 H/m. For

dielectrics, the value of e is greater than e0, because the D field is composed of free-space part e0E and a material part characterized by a dipole moment density P (C/m2). P is related to the electric field E as P = XeoE,

(3.1-14)

where X: is the electric susceptibility and indicates the ability of the electric dipoles in the dielectric to align themselves with the electric field. The D field is the sum of e0E and P,

D=eoE+P=e

o(I+X)E

=

e oerE,

(3.1-15)

where er is the relative permittivity, so that e : (1 + X )Co.

(3.1-16)

Similarly, for magnetic materials # is greater than #0.

3.2 Linear Wave Propagation In this section, we first derive the wave equation and review some of the traveling-wave type solution of the equation in different coordinate systems. We define the concept of intrinsic impedance, Poynting vector and intensity, and introduce the subject of polarization.

3.2.1 Traveling-wave solutions In Section 3.1 we enunciated Maxwell's equations and the constitutive relations. For a given J~ and p, we remarked that we could, in fact, solve for the components of the electric field E. In this subsection, we see how this can be done. We derive the wave equation

44

3 Propagation and Diffraction of Optical Waves

describing the propagation of the electric and magnetic fields and find its general solutions in different coordinate systems. By taking the curl of both sides Eq. (3.1-3), we have V xV

xE = -Vx

~ ( V x B ) = / z ~ ( VDx H )

~R_ bt

bt

(3.2-1)

where we have used the second of the constitutive relations [Eq. (3.113b)] and assumed # to be space- and time-independent. Now, employing Eq. (3.1-4), Eq. (3.2-1) becomes V

x V

x E =

D2E

-#e~-#~

DJc

t,

(3.2-2)

where we have used the first of the constitutive relations [Eq.(3.1-13a)] and assumed e to be time-independent. Then, by using the vector relationship V x V xA=V(V.A)

- V 2A,

V 2 - V . V,

(3.2-3)

+ V ( V . E).

(3.2-4)

in Eq. (3.2-2), we get

V2E-

D2E

If we now assume the permittivity e to be space-independent as well, then we can recast the first of Maxwell's equations [Eq. (3.1-1)] in the form V-E-

p6

(3.2-5)

~

using the first of the constitutive relations [Eq. (3.1-13a)]. Incorporating Eq. (3.2-5) into Eq. (3.2-4), we finally obtain V2E

-- # e -~E ~ = #~

+17 V p ,

(3.2-6)

which is a wave equation having source terms on the RHS. In fact, Eq. (3.2-6), being a vector equation, is really equivalent to three scalar equations, one for every component of E. Expressions for the Laplacian (V 2) operator in Cartesian (x, y, z), cylindrical (r, r z), and spherical (R, 0, 4)) coordinates are given as follows"

3.2 Linear Wave Propagation ~7 2

-- 02

02

~,2

_

1 ~

~oot

45

32

~x2 + ~-~ + ~----~;

cyl ~7s2ph_

32

(3.2-7)

~2

32

Or2 + 7~7 + ~--~ + ~

;

02 ~R2 + ~2 ~~ +

+

1 32 R2sin20 3~b2

(3.2-8) ~_ff ~2 ~

~

cotO 0 R 2 30"

(3 2-9)

In space free of all sources (J~ = 0, p = 0), Eq. (3.2-6) reduces to the homogeneous wave equation "' "~v2~ =

#e

O2E Ot2 9

(3.2-10)

A similar equation may be derived for the magnetic field H, V2H =

# e ~21t

Ot2 9

(3.2-11)

We caution readers that the V'2operator, as written in Eqs. (3.27)-(3.2-9), must be applied only after decomposing Eqs. (3.2-10) and (3.2-11) into scalar equations for three orthogonal components. However, for the rectangular coordinate case only, these scalar equations may be recombined and interpreted as the Laplacian V '2r e c t acting on the total vector. Note that the quantity #c has the units of (1/velocity) 2. We call this velocity v and define it as v 2--

•

#e

(3.2-12)

For free space, # = #0, e = e0, and v = c. We can calculate the value of c from the values of e0 and #0 mentioned in Section 3.1. This works out to about 3 x 108 m/s. This theoretical value, first calculated by Maxwell, was in remarkable agreement with Fizeau's previously measured speed of light (315,300 km/s). This led Maxwell to conclude that light is an

electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.

3 Propagation and Diffraction of Optical Waves

46

Let us now examine the solutions of equations of the type of Eqs. (3.2-10) or (3.2-11) in different coordinate systems. For simplicity, we will analyze the homogeneous wave equation ~2~ Ot 2

_ v2V,2~ b _

O,

(3.2-13)

where ~b may represent a component of the electric field E or of the magnetic field H and where v is the velocity of the wave in the medium. In Cartesian coordinates, the general solution is ~ ( x , y, z, t) =

c l f (~ot - ko~X - ko~V - ko=Z)

+ c~g (COot+ k o x + ko~!t + k0z )

(3.2-14)

with the condition 2

2

%

--

~L+ %+ k~= -

%

k0~

=

v2

(3.2-15)

where c~and c2are some constants. In Eq. (3.2-15), coo is the angular f r e q u e n c y (rad/s) of the wave and k0 is the propagation constant (rad/m) in the medium. Since the ratio Coo/ko is a constant, the medium of propagation is said to be nondispersive. We can then reexpress Eq. (3.214) as ~b(x, y, z, t) = elf(COot- ko . R ) + c2g (Wol + k 0 9R),

(3.2-16)

where R = xax + yav + zaz,

(3.2-17a)

k ~ --

(3.2-17b)

koa ~ +ko~av +koa~.

k 0is called the propagation vector and I k 0 l - k0; a~, a v, and a~ denote the unit vectors in the x, y, and z directions, respectively. In one spatial dimension (viz., z), the wave equation [Eq. (3.213)] reads 02~b

~t2

V 2 ~%

= 0,

(3.2-18)

3.2 Linear Wave Propagation

47

and its general solution is ~(z,t)

-- c f ( w 0 t - k 0z)+c~g(W 0 t + k oz),

v - %k0.

(3 .2-19)

Note that Eq. (3.2-14) or (3.2-16) comprises the superposition of two waves, traveling in opposite directions. We can define a w a v e as a disturbance of some form characterized by a recognizable amplitude and a recognizable velocity or propagation. Let us now consider a special case: c I r O, c~ = O. Observe that if ~b is a constant, so is W o t - ko 9R . Hence, ko 9R -

Wot + constant.

(3.2-20)

But this is the equation of a plane perpendicular to k 0 with t as a parameter; hence the wave is called a p l a n e wave. With increasing t, k0. R must increase so that Eq. (3.2-20) always holds. For instance, if k0 - k 0a~ ( k o > 0) and R - za~, z must increase as t increases. This means that the wave propagates in the +z direction. For c~ - 0, c~ ~ 0, we have a plane wave traveling in the opposite direction. The w a v e fronts, defined as the surfaces joining all points of equal phase Wot + k0 9R, are planar. Consider now the cylindrical coordinate system. The simplest case is that of cylindrical symmetry, which requires that ~(r, qS, z, t) -- ~(r, z, t). The qS-independence means that a plane perpendicular to the z axis will intersect the wavefront in a circle. Even in this very simple case, no solutions in terms of arbitrary functions can be found as was done previously for plane waves. However, we can show that harmonic, z- and qS-independent solutions of the form c exp[ j(Wot + k o r)], r >> 0 and C constant,

(3.2-21)

approximately satisfy the wave equation [Eq. (3.2-13)]. We remark that the exact solution has a Bessel-function type dependence on r if we assume ~ to be t i m e - h a r m o n i c , that is, of the form ~ = Re[~bp(r)e j%t], where R e [ . ] means "the real part of."

48

3 Propagation and Diffraction of Optical Waves

Finally, we present solutions of the wave equation in a spherical coordinate system. For spherical symmetry ( i9/i9q5 = 0 = ig/i~0), the wave equation, Eq. (3.2-13), with Eq. (3.2-9) assumes the form R

(~ ~

2~

2 ~) -]- ~ ~

=

O2(R@) _ 1 O2(R~) OR2 v2 ot 2 .

(3 2-22)

Now, Eq. (3.2-22) is of the same form as Eq. (3.2-18). Hence, using Eq. (3.2-19), we can write down the solution of Eq. (3.2-22) as cI

= -~ f ( W o t -

c2

koR) + -~g(Wot + koR)

(3.2-23)

with wo/ko - v. The wavefronts are spherical, defined by koR = k o [X2 +

y2 +

z21% = C~ot + constant.

(3.2-24)

3.2.2 Intrinsic impedance, the Poynting vector, and polarization So far in our discussion of Maxwell's equations and the wave equation and its solutions, we made no comments on the components of E and H. The solutions of the wave equation [Eq. (3.2-13)] in different coordinate systems are valid for every component of E and H. We point out here that the solutions of the wave equation discussed previously hold, in general, only in an unbounded medium. We will not discuss the nature of the solutions in the presence of specific boundary conditions in this book. In this subsection, we first show that electromagnetic wave propagation is transverse in nature in an unbounded medium and derive the relationships between the existing electric and magnetic fields. In this connection, we define the intrinsic or characteristic impedance of a medium, which is similar in concept to the characteristic impedance of a transmission line. We also introduce the concept of power flow during electromagnetic propagation and define the Poynting vector and the irradiance. Also, we expose readers to the different types of polarization that the electric field might possess during propagation. In an tinbounded isotropic, linear, homogenous medium free of sources, electromagnetic wave propagation is t r a n s v e r s e in nature. This

3.2 Linear Wave Propagation

49

means that the only components of E and H are those that are transverse to the direction of propagation. To check this, we consider propagating electric and magnetic fields of the forms E --

E~+E v+E~

-- R e { E o , exp[j(w ~t - k0z)]a, + E0vexp[ j(Wot- k0z)]a v + E0~exp[ j ( w o t - k0z)]a, }, H

-

(3.2-25)

Hz+Hv+Hz

= Re{Ho~exp[ j(Wot- k0z)]a~ + H0vexp[ j(Wot - koz)]a v + Ho~exp[ j(COot- koz)]a~ },

(3.2-26)

where E0~, E0y, E0~ and Ho~, Hov, Hoz are (complex) constants in general. We put Eq. (3.2-25) in the first of Maxwell's equations [i.e., Eq. (3.1-1)], with p = 0, and invoke the constitutive relation, Eq. (3.1-13a), to derive ~-o-{Eozexp[ j(Wot - koz)] } - O, implying E0~ = 0.

(3.2-27a)

This means there is no component of the electric field in the direction of propagation. The only possible components of E then must be in a plane transverse to the direction of propagation. Similarly, using Eqs. (3.1-2) and (3.1-13b) we can show that H0~ - 0.

(3.2-27b)

Furthermore, substitution of Eqs. (3.2-25) and (3.2-26) with Eo~= 0 - Hoz into the third of Maxwell's equations, Eq. (3.1-3), yields

3 Propagation and Diffraction of Optical Waves

50

koEoyax

-

koEoxay

=

-

# o(Hoxax +

Hoyay).

We can then write [using Eqs. (3.2-12) and (3.2-15)] Hox-

1E0y,

H0y _

I Eox,

(3.2-28)

where ~7--

6do

-c-tt = v# -

( ~ ) ,/2

(3.2-29)

'~o

is called the intrinsic or characteristic impedance of the medium. The characteristic impedance has the units of V/A, or S?. Its value for free space is r/0 = 37792. Now using Eqs. (3.2-25)-(3.2-29), we see that E . H = 0,

(3.2-30)

meaning that the electric and magnetic fields are orthogonal to each other, and that E x H is along the direction of propagation (z) of the electromagnetic field. Similar relationships can be established in other coordinate systems. Note that E x H has the units of W/m 2, reminiscent of power per unit area. All electromagnetic waves carry energy, and for isotropic media the energy flow occurs in the direction of propagation of the wave. Note that this is not true for anisotropic media. The Poynting vector S, defined as S = E • H,

(3.2-31)

is a power density vector associated with an electromagnetic field. In linear, homogeneous, isotropic unbounded medium, we can choose the electric and magnetic fields to be of the form E(z, t ) - R e [ E o e x p [ j ( c % t - k0z)] ] a~, Eo

H(z, t ) - R e [ ~ e x p [j(c~0t- k0z)] ] a v,

(3.2-32a) (3.2-32b)

3.2 Linear Wave Propagation

51

where E 0 is, in general, a complex quantity. This choice is consistent with Eqs. (3.2-28)-(3.2-30). Note that S is a function of time. It is more convenient, therefore, to define the time-averaged power, or irradiance I, w~ 2~/~o IEol2 IEol2 I= ~fo ISl dt - 2~ - ev 2 9

(3.2-33)

In subsequent chapters, the irradiance will be referred to as the intensity. Unless otherwise stated, the intensity will always be taken to be proportional to the magnitude squared of the complex field. In the remainder of this subsection, we introduce readers to the concept of polarization of the electric field. The polarization describes the time-varying behavior of the electric field vector at a given point in space. A separate description of the magnitude field is not necessary, because the direction of H is definitely related to that of E. Assume, for instance, that in Eq. (3.2-25), Eoz = 0 and Eoz = IEoz[,

Eov = IEou[ e -jr176

where 4)0 is a constant. First, consider the case where 4)0= 0 or+Tr. components of E are in phase, and E--

(IEoxlax + IEo,lay)COS(~ot- koz).

(3.2-34)

Then, the two

(3.2-35)

The direction of E is fixed on a plane perpendicular to the direction of propagation (this plane is referred to as the plane o f polarization) and does not vary with time, and the electric field is said to be linearly polarized. As a second case, assume r = + 7r/2 and [E0x[ = [E0v[ = Eo. In this case, from Eq. (3.2-25), E = Eocos(a;ot - koz)a~ + Eosin (a;ot -/coz)ay.

(3.2-36)

When monitored at a certain point z - z o during propagation, the direction of E is not longer fixed along a line, but varies with time according to 0 - C~ot- ko Zo, where 0 represents the angle between E and the (transverse) x axis. The amplitude of E (which is equal to Eo) is, however, still a constant. This is an example of circular polarization of

52

3 Propagation and Diffraction of Optical Waves

the electric field. When r = -7r/2, Ey leads E~ by 7r/2 [see Eq. (3.236)]. Hence, as a function of time, E describes a clockwise circle in the x-y plane as seen head-on at z = z o. Similarly, for r = + 7r/2, E describes a counter-clockwise circle. In the general case, E -- E~ + Eu =

E~a~+ Eyay

= IEo~lcos(wot- koz)ax + IEoylcos(wot- k o z - r

(3.2-37)

o

As in the case of circularly polarized waves ( where IE0~[2 + IE0u[2 = E 2 = constant), the direction of E is no longer fixed because E~ and Ey vary with time. We can trace this variation on the E ~ - E y plane by eliminating the harmonic variations of (Wot- koz) in Eq. (3.2-37). To do this, we note that e~

IEoyJ

-

cos(w ot - k0z)cosr

+ sin(w ot - k0z)sinr

{ ( )2}1/2 ----

IE0~IE~ COSr ~ +

1--

~e~

sine ~

(3.2-38)

After a little algebra we can reexpress Eq. (3.2-38) as

E~

-2

( )( ) NL-] e~

~ev

cosr o+

(

~ev

-

sin2 r

(3.2-39)

which is the equation of an ellipse; hence, the wave is said to be elliptically polarized. Note that for values of r to 0 or + 7r and • 7r/2 (with IEoxJ = IE0 l = Eo), the polarization configurations reduce to the linearly and circularly polarized cases, respectively. Figure 3.1 illustrates various polarization configurations corresponding to different values of r to demonstrate clearly linear, circular, and elliptical polarizations. In this figure, we show the direction of rotation of the E-field vector with time, and its magnitude for various r When r 0 or + 7r, the E-field is linearly polarized and the E vector does not rotate. Unless otherwise states, we will, throughout the book, assume all electric fields to be linearly polarized and choose the transverse axes such that one of them coincides with the direction of the

53

3.3 S p a t i a l F r e q u e n c y Transfer F u n c t i o n f o r P r o p a g a t i o n

electric field. The magnetic field will, therefore, be along the other transverse axis and will be related to the electric field via the characteristic impedance r/.

~b0 = 0

0<~b o < z / 2

~b0 = z / 2

z / 2 < qko < z

qko = z

leo l -lEo l Figure 3.1

Various polarization configurations corresponding to different value o f r ([Eo~ [ # [Eov[), unless otherwise stated).

3.3 Spatial Frequency Transfer Function for Propagation In this Section, we derive the spatial transfer function for wave propagation. We then derive the important Fresnel diffraction formula and the Fraunhofer Diffraction formula commonly used in Fourier optics. We also provide some illustrative examples as well as MATLAB examples for the clarification of the formulas developed. We start from the wave equation, Eq. (3.2-13), expressed in Cartesian coordinates: 1 02r

v 2 0t 2

02r + ~2r

--- 0x2

~

~2r

(3 3 1)

+ 0z2.

9-

We now assume that the wavefunction r y, z, t)comprises a complex amplitude Cp(x, y, z) riding on a carrier of frequency wo" r

y, z, t) = Re{~bv(x, y, z)exp(jcoot)}.

(3.3-2)

Substituting Eq. (3.3-2) into Eq. (3.3-1), we get the Helmholtz equation for Cp, i~2r + ~2r

~2r

ko_ wo

(3.3-3)

3 Propagation and Diffraction of Optical Waves

54

By taking the 2-D Fourier transform, i.e., .7"~y, of Eq. (3.3-3) and upon some manipulations, we have d2~p

+ k (l

2

k?~

kg) ~ ; - O,

(3.3-4)

where ~p(kz, ky;z) is the Fourier transform of ggp(X, y, z). readily solve Eq. (3.3-4) to get

~p(kz, ky;Z) = ~pO(kx, ky) e x p [ - jkov/1 - k~/k~ - k2y/k~ z],

We now

(3.3-5)

where ~;o(k~, ky) = ~;(k~, ky;z - O) = 9r~u{r

y, z - O) } - 9r~u{r

y) }.

We can interpret Eq. (3.3-5) in the following way: Consider a linear system with ~p0(k~, ky) as its input spectrum (i.e., at z - 0) and where the output spectrum is ~p(k~, ky;z). Then, the spatial frequency response of the system is give by fflp(kx,ku.z) 9 = 7-t

ky;z)

= e x p [ - j k oi l

- k 2 / k~ - k 2 / k 2 z ] .

(3.3-6)

We will call 7-[ (kz, kv;z) the spatial frequency transfer function of propagation of light through a distance z in the medium. To find the field distribution at z in the spatial domain, we take the inverse Fourier transform of Eq. (3.3-5):

47r2

f f w,o(kxky)exp[- j k o l l

- k2z/k 2 - k2y/k 2 z ]

• exp [ - j k z x - jkvy] dkxdky. (3.3-7)

3.3 Spatial Frequency Transfer Function for Propagation

55

Now, by substituting ~po(k,, kv) = 9v~y{ ~bp0(X, y)} into Eq. (3.3-7) and defining G(x-

x', y -

y", z) -

4rr2Zf f e x p [ - j k 0 4 1 - kx2/k2 - k2/kg z]

• exp[ - j k ~ ( x - x') - j k y ( y - y')] d k x d k y ,

(3.3-8)

we can express Eq. (3.3-7) as Cp(X, y, z) = f f Cpo(x', y ' ) G ( x - x', y - y'; z) dx'dy'

= r

y) 9 G(~, y; z).

(3.3-9)

The result of Eq.(3.3-9) indicates that G(x, y; z) is the spatial impulse response o f propagation of the system. By changing of variables: x - rcosO, y - rsinO, k~ - pcosr and ky - p s i n r G(x, y; z) can be evaluated as [Stark (1982)] G(~, y; z) -

a(~o~O, ~~o;

j k o e x p ( - j k o v/r2+z 2 ) 2 7rv/r 2+ z 2

z) -

G(~; z)

z

1

v/r 2+ z 2 ( 1 -~- jk0 v/r2+z 2 )"

We now make the following observations: (1) For z >> A0 - 2rc/ko, (1 q-

1

jko x/r2 +z 2

)

(3.3-10)

~ 1 and the term

can be ignored. Z (2) = cos g), where cos g~ is called the obliquity factor v/r2+z 2

(3)

and ~ is the angle between the positive z-axis and the line passing through the origin of the coordinates. Using the binomial expansion, the factor v/r2+z 2 _ v / x 2 + y 2 + z 2 ~ z + x2+v2 2z , provided x 2 + y 2 < < z 2. This condition is called the paraxial approximation, which leads to cos q5 ~ 1 . If the condition is used in the more sensitive phase term and only used the first expansion term in the less sensitive denominators of the first

3 Propagation and Diffraction of Optical Waves

56

e',,a

and second terms of Eq. (3.3-10), G(r; z)becomes the socalled free-space spatial impulse response, h ( x , y ; z ) , in Fourier Optics [Banerjee and Poon (1991), Goodman (1996)]" h(x, y; z) - exp( - jkoz) ~y;Jk~exp . (3.3-11)

[-jk~(x2+y2)J2z

By taking the 2-D Fourier transform of h(x, y; z), we have

-- exp( - jkoz)exp[J(k~+k~)Z2ko )

(3.3-12)

H(kx, ky;z) is called the spatial frequency response in Fourier Optics. Indeed, we can derive Eq. (3.3-12) directly if we assume that k~ + k v2 << k02,meaning that the x and y components of the propagation vector of a wave are relatively small, we have, from Eq. (3.3-6) p( kx ,kv; z )

9~o(k~,k~) = 7-t (k~, ky; z) = exp[- jko(1

- (k2x + k~)/k 2 z ]

- exp(--Jk~ = H(kz, ky;Z).

(3.3-13)

If (3.3-11) is now used in (3.3-9), we obtain

r

v, z) -

r exp(-

y) 9 h(~, y; z)

jkoz)~J~~f fr

x exp [-jko ( x -

x') 2 + y - y' )2)]

dx'dy'.

(3.3-14)

3.3 Spatial Frequency Transfer Function for Propagation

57

Equation (3.3-14) is termed the Fresnel diffraction formula and describes the Fresnel diffraction of a beam during propagation and having an arbitrary initial complex profile ~bp0(x, y). Figure 3.2 shows a blockdiagram relating the input and output planes. To obtain the output field distribution, we need to convolve the input field distribution with the spatial impulse response h(x, y; z). R

_ [(x-

x') 2 .-I-(y-y'

)2 Jr-z~],~P(x,y)

h(x, y; z)

y

= e-lk~ jko e-lk"(x2 +y2)/2z

z=0

Z=Z

2 zz

Input plane

Output )lane

~0(x,y)

~,~ (x, y, z)

Figure 3.2 Block diagram of wave propagation in Fourier optics. The input and output planes have primed and unprimed coordinate systems, respectively.

3.3.1 E x a m p l e s of F r e s n e l d i f f r a c t i o n Example 1" Point Source

(3.3-~5) By Eq. (3.3-14),

r

y, ~) - [~(~)e(y)],h(~, y; z) jko e x p [ - jkoz - Jk~ ~z -_ 2~

]"

(3.3-16)

Consider the argument of the exponent in Eq. (3.3-16). Note that using the binomial expansion previously used, we have

3 Propagation and Diffraction of Optical Waves

58

jko exp( - jko [z 2 + x 2 + y2 ]1)

Cp(x, y, z) --

jko exp( -

27rR

jkoR)"

(3.3-17)

thus Eq. (3.3-16) represents the paraxial approximation to a diverging spherical wave, as expected. E x a m p l e 2: Plane W a v e

(3.3-18)

Cpo(X, y) -- 1. Then tFpo(kz, ky) - 47reS(kx)5(ky),

so that, using Eq. (3.3-13). 9 p(kx, ky; z ) -

47r2(5(kx)(5(kv)exp(- j k o z ) e x p [j(k~+k~)z2k0] -- 47c25(kx)5(ky)exp( - jkoz)exp[ j(k~+k~)~2ko ] k~=O=ky

= 47r2~5(kx)~5(kv)exp( - jkoz)

(3.3-19)

= ~poexp( - jkoz). Hence, ~p(X, y, z) - r

- jkoz),

that is, the plane wave travels undiffracted, as expected. E x a m p l e 3: Plane W a v e through a R e c t a n g u l a r A p e r t u r e

Cpo(X, y) -

rect(~)rect(~).

Using Eq.(3.3-14), Cp(x ' y, z) -- e x p ( - j k 0 z ) ~jko f-oo oo f ~~ rect(T)rect(T) x' y'

(3.3-20)

59

3.3 Spatial Frequency Transfer Function for Propagation

- 3 ~.ko {(x-x)

xexp - exp( -

t 2 + ( y - y ' ) 2 } ] dx'dy'

g(~)g(y), jkoz) ~jko

where

9(x) - <,/2 exp

g(Y) - <,/2 exp

[ [

- 3 ~9ko (X -

(3.3-21)

Xt

- J ~ko( Y -

] ]

)~ dx', (3.3-22)

y' )~ d y'.

These integrals are substantially simplified by the change of variables

r

ko 1/2 (~'- ~), (~)

-

kO 1/2

(~)

y/

(-y),

(3.3-23a)

yielding 1/2 "7r<2 f<<2exp( - 3~ ) de, 1 1/2

(3.3-23b)

f:2 e x p ( - j g 7r~2)dr/, 1

where

<: _ (~o)

1//2

(~~+ x)

~2 ( ;co (~_x), 1/2

-

-

,

rh -- -

(~o)

(lg + y ) , (3.3-23c)

~

(~0~)

1/2

(~-~)

We can recast Eqs. (3.3-23b) in the forms ~/2

~)- (~:)

[{C(r

-- C(
-- S(
(3.3-24a)

3

60

Propagation and Diffraction of Optical Waves

1/2

[{C(~72

)

--

C(~]1)}

--

j{S(~72 ) -- S(~]I )

(3.3-24b)

}],

where C (a)and S' (a) denote the Fresne! integrals, defined by -

f:

cos

7rt2 dr,

--

fo

(3.3-25)

slnz-dt"

"

We can evaluate the Fresnel integrals by using the Cornu spiral [see Figure 3.3], which is a simultaneous plot of C(a) and S(a) for different values of a. Now, we visualize a quantity C(a) + jS(a) to be a complex phasor joining the origin to the point a on the spiral. Thus, { C ( ~ ) - C ( ~ ) } + j { S ( ~ ) - S(~1) } is the phasor defined by the line joining the point ~ to the point ~on the spiral, so that 9(z), as defined by Eqs. (3.3-24), is the complex conjugate of this phasor multiplied by (TrZ/ko)1/z. We can compute 9(Y) similarly and can numerically evaluate the Fresnel diffraction pattern. 0.8 ........................................ :............................................................................................................................. "... : :

0.4

!

.....

. 9

0.2

. . . . . . . . . . . . . . . . . . . . . . . . . . . 9

-0.2

.............................................................................................................

-0.4

-0.6

-0.8

t

.........................................................................................................................................

-1

-0.5

0

0.5

1

Figure 3.3 The Cornu spiral, used to compute the Fresnel diffraction pattern of a rectangular aperture. The points 1,2,3 and -1 located on the graph correspond to the values of a.

3.3 Spatial Frequency Transfer Function for Propagation

61

Observe that at a very small distance from the aperture (kol ~,/z >> 1), (1,2 ----+ ~ - oo for z > T I/2,

t

+oo for z < T 1/2,

(3.3-26) ----+ ~ - c~ for y > q: l/2,

~71.2

(

+oo for y<~: l/2,

so that

'

~ +0.5 for z < m 1/2,

(3.3-27) C(T]12)_ '

S(T],,2)----+ { - 0 . 5 | b r y > m / / 2 , + 0.5 for y < q: 1/2.

In Eqs. (3.3-26) and (3.3-27), the subscripts 1 and 2 in ( and r/, respectively corresponds to t h e - and + signs on the limits on z and y. Substituting Eqs. (3.3-27) into Eqs. (3.3-23) and, thereafter, the results into Eq. (3.3-21), we finally obtain

~2p(z, y, z) ~_ rect(z/1)rect(y/1),

(3.3-28)

which implies that deep within the Fresnel region, the field distribution obeys the results predicted from geometrical optics, as expected.

3.3.2 Matlab example: the Cornu Spiral The Cornu spiral shown in Figure 3.3 can be generated using the following MATLAB code, shown in Table 3.1. In addition, the code can generate the Fresnel integrals as a function of c~, as shown in Figure 3.4.

3 Propagation and Diffraction of Optical Waves

62

%The Cornu Spiral clear dx=0.0 l; %dx implies the increment distance between the intervals t=0:dx: 10; %"t" represents alpha x=0; %"x" represents C as a function of alpha y=zeros(401); %"y" represents S as a function of alpha y-y(1,:); xl =0; y 1=zeros(401 ); yl=yl(1,:); for m = 1:401 %these "for" loops are used to evaluate the integrals for n = 1:m x(n)=cos((pi.*t(n).A2)./2).*dx; y(m) - y(m)+x(n); x l (n)=sin((pi.*t(n).A2)./2).*dx; y 1(m)=y 1(m)+x 1(n); end end Y=fliplr(y); Y 1-fliplr(y I );

%this flips the current original graph to the position %in the third quadrant

T=-4:0.01:4; BY(1:401)=-Y; BY(401:801)=y; BYI(1:401)=-Y1; BY 1(401:801 )--y 1; plot3(BY,BY 1,Y) view(0,90) grid on

%this combines the two existing graphs into one spiral

%plots the original spiral (FIGURE l) %rotates it for the viewer to view from a birds eye view

figure(2)%plots the second figure which is a side view of C and S in relation to alpha plot(T,BY,'r') %this plots the C graph in relation to alpha hold on plot(T,BY1) %this plots the S graph in relation to alpha grid on Table 3.1 MATLAB program to generate the Cornu spiral (cornu_spiral.m).

3.3 Spatial Frequency Transfer Functionfor Propagation 0.8

1

!

1

i.

:

l

?,

. . . . . . . . . .

,

;

!

o.~ ..................... ! ......................!........................ i............c(a-)..-. :

.

.

.

.

/'

9

-0.4

-

i)

.

i

'.

/r

.

.

.

.

I -3

-2

t

f,

........... ' ' ; +

#,

, i",.. J,

,

,,,,,,; -1 t'l

.............. : . . . .

\

/ >

.

.

.

"

.

.

.

.

i

, ,,

9

,J

-4

/

',

..........................................

/

~.,,

-0.8

(

9

'x ,. / !' "i, '

_o.o

:~

,

/z ......

.......................................................

-,,' S\

:.

/

-0.2

/

'

}

" i~

',\

'i

7 ............... ~ : " : ~ ;

:

r

0

i

,,

9

0.6

63

..... i -1

. . .

9

:

.....

t 0

i 1

J

,

2

J. . . . . . 3

4

Figure 3.4 Fresnel integrals as a function of c~

3.3.3 M A T L A B example: Fresnel diffraction of a square aperture The m-file, Fresnel.m, generates four figures as shown below. Figure 3.5 shows the square aperture, rect(z/a)rect(y/a), which is illuminated by a plane wave. Figure 3.6 shows the real part of h(z, y; z), with z defined by a parameter called sigma in the program below. If the units of Figure 3.5 and Figure 3.6 are in centimeters, the program then generates a - 0.4336 cmand z = 4.9033cm for the physical parameters used in the simulation. Figure 3.7 and Figure 3.8 show the cross-section of the square aperture and the Fresnel diffracted amplitude, respectively.

64

3 Propagation and Diffraction of Optical Waves

%Fresnel diffraction of a square aperture, rect(x/a,y/a). clear L=1 ; %L : length of display area N=256; %N : number of sampling points dx=L/(N - 1); % dx : step size %Create square image, M by M square, rect(x/a), M=odd number M=lll; a=M/256 R=zeros(256); %assign a matrix (256x256) of zeros r-ones(M); %assign a matrix (MxM) of ones n=(M - 1)/2; R(128-n: 128+n, 128-n: 128+n)=r; %End of creating input image R %Axis Scaling for k=1:256 X(k) = 1/255 *(k-1 )-L/2; Y(k) = 1/255*(k-1 )-L/2; %Kx=(2*pi*k)/((N-l)*dx) %in our case, N=256, dx = 1/255 Kx(k)=(2 *pi *(k- 1))/((N-1 )* dx)- ((2 *pi*(256-1 ))/((N- 1)* dx))/2; Ky(k)=(2 *pi *(k-1 ))/((N-1 )* d x)-((2 *pi* (256-1 ))/((N-1)* dx))/2; end %Fourier transformation of R FR=( 1/256)A2*fft2(R); FR=fftshifi(FR); %Free space impulse response function h(x,y;z), factor exp(-jkoz) suppressed %sigma=ko/(2*z) sigma = 1" 10* 1.5"337.5"2; z=pi/((0.6328* 10A(-4))*sigma) % wavelength used is 0.6328 #m for r=1:256, for c = 1:256, %compute h(x,y;z) with Gaussian apodization against aliasing h(r,c)=j*(sigma/pi)*exp(4 "200' (X(r).A2+Y(c).A2))*exp(j* sigma* (X(r).A2+Y(c).A2));

3.3 Spatial Frequency Transfer Function for Propagation

65

end end H=( 1/256)A2 *fftZ(h); H=fftshift(H); HR-FR.*H; H=( 1/256)A2" fft2(h); H=fftshift(H); HR-FR.*H; hr=ifft2(HR); hr=(256A2)*hr; hr=fftshift(hr); figure(! ) %Image of the square object image(X,Y,255*R); co lotto ap(gray( 256 )); axis square figure(3) % Plot of cross-section of the square plot(X +dx/2, R(:, 127)) grid axis([-0.5 0.5 -0.1 1.2]) figure(2) %Plot of real part ofh(x,y;z) image(X, Y,255 *real(h )/max (m ax(re a 1(h)))) co lorm ap(gray( 256 )) axis square %Plot of absolute value of Eq.(3.3-14) figure(4) %Plot of 1-D cross-section of Fresnel diffracted pattern plot(X +dx/2,abs(hr(:, 127))) grid axis([-0.5 0.5 -0.1 max(max(abs(hr)))* 1]) Table 3.2 MATLAB code to generate the Fresnel diffraction pattern of a rectangular aperture (fresnel.m).

66

3 Propagation and Diffraction of Optical Waves

Figure 3.5 Square aperture

Figure 3.6 Display the real part of h(x, y; z), z - 4.9 cm.

67

3.3 Spatial Frequency Transfer Functionfor Propagation

7 ....

1

08

4-

06

-

J

_

. . . . i

04

_

_

I

021 L /

-

I

~

I

t

I

I I

I I

~ I

I I

) )

i

T

(-

I

I

I

I

f

I

I

I

[

I

d

_

-I-

-I-

-I-

-I

_1-

-.

I

I

I

]

I

I

)

I

I

1

I

I

I ....

I -

-

-I-

-I

I

J

t

I

I

E

I _

I_

f

) --

-t

I

I

1 1 -

-

i ~

-

I T

I

I

I

I

I

I

J-

-

-I-

-

I-

-

4-

I

!

I

t

I

I

I

i

i

{

I

I

I

I

I

I

-r

1

I ....

I

I

1

I

I

I

I

-II

-I-

I 0 . . . . . .

L_. . . . . i______A____ -05

-04

,03

__ -02

l

I

I

t

k

I

-01

0

01

02

03

04

I

I

)

I

I

i --

-I

1

1 -

-

1

I--

-

I -

-I

-

q

-

I

-

I

I

I

I

1 f

I 1-

I

I

1

I

I I-

I

t -

I

)

I

-

-I

I

I

....

I

I

7

-I-

t

I

I

i .....

I

-I

I

Figure 3.7 Central cross section of Figure

3.5

~4

~ 3 ~ 2

~1

t-

I I

i

I

~

I

-

I

I F

I

I

I

I

I

I

I ___A____ 0

01

I

--

I

! ~5

05

I

r .....

t I

__

-

I / f I \~1

-

t-

I I

I

1

L___~___L 02

-

1 t

03

..... 04

05

Figure 3.8 Central cross section of diffracted amplitude at z - 4 . 9 c m

3.3.4 Fraunhofer diffraction and examples So far, we have studied the effect of propagation on the amplitude and phase distributions of various initial wave profiles. At all times, we examined the Fresnel diffraction pattern, which is determinable through the Fresnel diffraction formula Eq. (3.3-14). The range of applicability of this formula is from distances not too close to the source [see observation (1) following Eq. (3.3-10)], typically from about 10 times the wavelength. However, it is not always easy to determine the diffraction pattern, as was seen in the example of the rectangular aperture. In this section we examine a method of calculating the diffraction pattern at distances far away from the source or aperture. More precisely, observe that if our observation plane is in the far field, that is, /zt2 ko ~

t__

y t2x )~a~

2

= ZR ~

z,

(3.3-29)

where zn is the Rayleigh range, then the value of the exponent e x p [ - j k 0 ( x ' 2 + fl'2)]~a~/2z is approximately unity over the input plane (x~,y~). Under this assumption, which is commonly called the Fraunhofer approximation, Eq. (3.3-14) becomes

3 Propagation and Diffraction of Optical Waves

68

r

y, z) -

exp(-

jkoz) ~zJk~exp [-jk0_VTz(x2 +

x f fCpo(X', y')exp IJk~ -- exp( -

jkoz) ~jko exp [-jk0

y2)]

+ yy')] dx'dy'

(x

(3.3-30) kY=koy/:

Equation (3.3-30) is termed the Fraunhofer diffraction formula and is the limiting case of the Fresnel diffraction studied earlier. The first exponential in Eq. (3.3-30) is the result of the phase change due to propagation, whereas the second exponential indicates a phase curvature that is quadratic in nature. Note that if we are treating diffraction of red light (A0 = 0.6328 #m) and the maximum dimensions on the input plane are 1ram, then z >> 5m. In what follows, we consider various examples of Fraunhofer diffraction when a plane wave passes through different apertures. E x a m p l e 1" Slit o f F i n i t e W i d t h

~bp0(x, y) -- rect(~).

(3.3-31)

Note that because we are usually interested in diffracted intensities (i.e.,l~bpl 2, the exponentials in Eq. (3.3-30) drop out. Furthermore, the other term besides the Fourier transform, namely, (ko/2rcz), simply acts as a weighting factor. The intensity profile depends on the Fourier transform, and we will therefore concentrate only on this unless otherwise stated. In our problem, also note that there is no obstruction to the incident plane wave in the y dimension when it emerges from the aperture. Thus, there will not be any effect of diffraction in the y direction. We only need to take the Fourier transform in x to find the Fraunhofer diffraction pattern. Using Eq. (3.3-31), 5x {rect(v~)x } _ lxsinc(-~).Z~kx

(3.3-32)

3.3 Spatial Frequency Transfer Function for Propagation

69

Hence, using Eq. (3.3-30) and neglecting the y direction for simplicity, Cp(x, z) ~ lxsinc( ~).Zxko

(3.3-33)

........J......................... i....................... i........................................ i....................J........................ i........ 0.8

0.6

I(O)/I(O)

0.4

0.2

i

I

-3

-2

-1

o(

0

--

I

~

X

2 zz

/

2

3

Figure 3.9 Fraunhofer diffraction pattern of a rectangular slit

Observe

that

the

first

zero of the sinc function occurs at x• 27rz/lzko - -+-AoZ/lz, and it is between these points that most of the diffracted intensity (e([~p]2) falls (see Figure 3.9). We also observe that the angle of spread O~p ~_ x / z , during diffraction is of the order of A0/1 xIn fact, we can simply find the spread angle from a quantum mechanical point of view [Poon and Motamedi, 1987]. Consider light emanating from an aperture of width l x, as shown in Figure 3.10.

3 Propagation and Diffraction of Optical Waves

70

A

A

r

X

lx

,,- Z ......

Aperture B

!

Figure 3.10 Geometry for determination of the angle of spread Ospduring diffraction

Lines AA' and BB' represents rays emanating from the endpoints of the aperture. Quantum mechanics relates the minimum uncertainty in position Ax of a quantum to the uncertainty in its momentum Ap~ according to (3.3-34)

A x A p ~ h,

where h - h/27r [ h - 6.625 • 10 -34 J-s (Planck's constant)]. Now, in our problem Ax - l~, because the quantum of light can emerge from any point on the aperture. Hence, by Eq. (3.3-34), h zxp, ~ T2.

(3.3-35)

We define the angle of spread Osp, assumed small, as O~p ~ ~p,

P~

/Xp,

P0 '

(3 3-36)

where Pz and P0 represent z component of the momentum, and the momentum of the quantum, respectively. But P o - hko' where k o is the propagation constant; hence,

3.3 Spatial Frequency Transfer Function for Propagation

Osp

1

kol~

_

1

71

(3.3-37)

Ao

27r l~ '

where Ao denotes the wavelength in the medium of propagation of the light. Thus, the angle of spread is inversely proportional to the aperture width.

Example 2: Circular Aperture

~/po(x, y) - Cpo(r) - circ(~).

(3.3-38)

Note that r - (x 2 + y2) 89and that circ(r/ro)denotes a value 1 within a circle of radius ro and 0 otherwise. The first step is to compute the Fourier transform of the circ-function. Now, Sxy{r

-

tgpo(kx, ky) - f-2 f L Cv~

y)exp[j(kxX + kyy)]dxdy. (3.3-39)

Introducing new variables x - rcosO, y - rsinO,

k, - k~cosr ky - k~sinr

(3.3-40)

Eq. (3.3-39) transforms to A

= f~of:~ ~bpo(r)exp[jk~r(cosOcosr + sinOsinr - f ( r ~po (r) ~~exp[jkrrcos(O - r

(3.3-41) A

where we have employed the circular symmetry of ~ 0 ( x , y) -

~p0 (r).

But, 27r

J0._.(/3) = 2f02~ exp[jflcos(0 -

r

(3.3-42)

72

3 Propagation and Diffraction o f Optical Waves

where J0(fl)is the zeroth-order Bessel function. [Note that the LHS of Eq. (3.3-42) is independent of r Using Eq. (3.3-42) in Eq. (3.3-41), we obtain

B{~,o(~)} -

~,o(k~) - 2~ f o ~ o

(~)ao(k~)d~.

(3.3-43)

Equation (3.3-43) defines the Fourier-Bessel transform and arises in circularly symmetric problems. Now, substituting Eq. (3.3-38) in Eq. (3.3-43), we get (3.3-44) Finally, using the relationship

a,]~ ( a ) -- s fl Jo ( fl ) d fl,

(3.3-45)

Eq. (3.3-44) becomes ~pO(l~r) __

2rrro Jl(rOk, r)

(3.3-46)

kr

which is the desired Fourier transform. Hence, from Eq. (3.3-30), we have

kx =koz/z ky=koY/z A

= 13{ ~,0} kr=kor/z

or I"

%(~, z) - ~ ( ~ , y, ~) ~

2rrroz

roko

ko~ J1(-7- ~)

(3.3-47)

The intensity is proportional to ]~v(r, z)t 2 and is sketched in Figure 3.11. The plot is called an Airy pattern. The two arrows in the figure point to the first and second zero of the pattern, which are at ~=1.22 and r respectively.

3.3 Spatial Frequency Transfer Function for Propagation

1 0.9

L

0.8.__ 0.7

i

i

~

I . . . . ~-

- f ./ . !~. ~ . .

I

II-I

t

I

I

_ _~

I, iI

4 . . . .

0.6__

_~ I

.~

0.4

. . . . _~ . . . .

0.2 0.

t ....

i---

i

I

_ _j

.

[. . . . . .

'

I

.;

i

-1-I . . . .

I 1--

I I

/

I-- -- --i-- --I -- t

-4- . . . . . . ~ . .

I I

1 1

j

I 1

!1

/ I -I-

....

0.5

0.3

;

t

4. . . . .

73

....

I

"--

I

-'--t'I

I I

tI .... -I-

1

~ . . . . . . . .

I~1--~_~ ---I----i--tlf It . . . . . . 17. I I-I------'t- . . . . ~-" -- - -- -I-

I

I

I

I

t

I

I

I

t

t

T I I

rI I

I

1

I

I

\ I

I

I

I I. . . .

-~ T .l . . .

iV-l

II

i

.

'

T

-2

-1

0

1

-

I I

~

-

-

-

~

3

# Figure 3.11 Plot of [2Jl(Trf)/Tr(] 2 against (= rokor/Trz. This represents the Fraunhofer diffraction pattern of the circular aperture.

The circular aperture is of special importance in optics because lenses, which form an important part in any optical system, usually have a circular cross section. In what follows, we derive the Rayleigh criterion, which dictates the resolution of an optical imaging system. Resolution is a figure of merit that determines how close two object points can be such that they are clearly distinguished, or resolved, by the optical system. Note from Figure 3.11 that most of the intensity lies between }r < 1.227r. Consider, now, two Airy patterns superposed on each other with a certain distance of separation between their peaks. It is easy to understand that the peaks of the main lobes should be no closer than 1.22 units (in () for them to be discernible in the superposed picture. In the context of our problem, this translates to

3 Propagation and Diffraction of Optical Waves

74

~0k__~0r > 1.227r Z

m

or r OLmin __ ( 7 ) m i n

__ (1.22)rr ~

koro

Ao 0.61(7o)

(3.3-48)

where O~min is assumed small and represents the minimum angle between the beams contributing to the two main lobes to facilitate resolution. Note the omnipresence of the parameter )~o/ro. To see where the Rayleigh criterion, as enunciated in Eq. (3.348), plays an important role in imaging systems, consider the arrangement shown in Figure 3.12, where we have two point sources, P1 and P2, a distance x 0 apart on the front focal plane of a lens of focal length f. Two beams should propagate on the right-hand side of the lens, at an angle to each other. The beam diameter is, however, finite, due to the finite diameter do of the lens. In fact, the lens of finite diameter can be replaced by one with, hypothetically, an infinite diameter, followed by a circular aperture of diameter do. In our case, the circular aperture is illuminated by plane wavefronts, because a point source on the front focal plane of a lens of infinite aperture produces plane waves behind it. Thus, on the observation plane, we can see the diffraction patterns of the aperture. Looking from the observation plane, we can therefore distinguish P1 and P2 as long as the angle between the two beams is more than c~,~i~. From the geometry of Figure 3.12 it follows that the minimum separation (X0)min between P1 and P2 is determined from (XO)mi~ f ~

OLmin - -

Ao 1.22 (N).

(3.3-49)

or (XO)mi n ~

1.22Ao(~).

This criterion is often called the A bbe condition.

(3.3-50)

75

3.3 Spatial Frequency Transfer Function for Propagation

I _I_P, Lens-"--""--~

................. Observation plane

Figure 3.12 Optical arrangement to derive the Abbe condition

The problem discussed above can be switched around, by asking for the minimum angle between two beams starting from sources A and /3 at infinity and passing through the lens such that their images can be resolved or, equivalently, by inquiring about the minimum separation between their peaks. The answer, found conventionally in optics texts, is identical to Eq. (3.3-50). The parameter f/do is the f-number of the lens. For instance, a lens with a 25-mm aperture and a focal length of 50mm has an f-number equal to 2 and is designated as f/2. Cameras have a fixed lens but a variable aperture (diaphragm) with typical f-number markings of 2, 2.8, 4, 5.6, 8, 11, and so forth. Each consecutive diaphragm setting increases the f-number by x ~ ; hence the amount of light energy reaching the film for the same exposure time is cut in half because this is proportional to the area of the beam and, hence, to 1/(f-number) 2. Note also that for the same amount of light energy to reach the film, the product of the exposure time and irradiance must be constant; hence, the exposure time is proportional to (f-number) 2 in this case. Logically, the f-number is sometimes called the speed of the lens. Thus an f/1.4 lens is said to be twice as fast as an f / 2 lens.

76

3 Propagation and Diffraction of Optical Waves

3.3.5 M A T L A B example: Fraunhofer diffraction of a square aperture This m-file, shown in Table 3.3, generates the four figures below. Figure 3.13 is the square aperture ~ b p o ( x , y ) - rect(~)rect(-v) " a Figure 3.14 is the real part of h(x, y; z), with z defined by a parameter called sigma in the program below. If the units of figure 3.13 and figure 3.14 are in centimeters, the program then generates a - 0.0431 c m and z--- 98 cm. Figure 3.15 and figure 3.16 show the Fraunhofer diffracted amplitude ]r = 2ko- - a~ 2 sinc(akox ' y,z)l ~ ) s i n c ( ~ ) ,akoy and its central cross-section of the diffracted amplitude, according to Eq.(3.3-39), respectively. %Simulation of Fraunhofer diffraction of square aperture clear L=I; N-256; dx=L/(N - 1); % Input square image, M by M square, rect(x/a)rect(y/a), M=odd number M=I1; a=M/255 R=zeros(256); r=ones(M); n=(M -1)/2; R(128-n: 128+n, 128-n: ! 28+n)=r; %End of input image %Axis Scaling for k=1:256 X(k) = 1/255 *(k- 1)-L/Z; Y(k)= 1/255 *(k- 1)-L/2; %Kx=(2 *pi *k)/((N- 1)*dx) %in our case, N-256, dx=1/255 Kx(k)=(2 *pi *(k-1))/((N-1 )*dx)-((2 *pi* (256-1 ))/((N-1 )*dx))/2; Ky(k)= (2 *pi*(k-1))/((N- 1)*dx)-((2 *pi *(256-1 ))/((N- 1)*dx))/2; end %Fourier transformation of R FR-(1/256)A2*fft2(R); FR=fflshifl(FR);

3.3 Spatial Frequency Transfer Function for Propagation %Free space impulse response function h(x,y;z), factor exp(-jkoz) suppressed %sigma-ko/(2*z) sigma = 1.5"33 7.5; z=pi/((0.6328* lO/'(-4))*sigma) % wavelength used is 0.6328 microns %compute h(x,y;z) without Gaussian Apodization for r=1:256, for c= 1:256, h(r,c)=j*(sigma/pi)*exp(0*(X(r).A2+Y(c).A2))*exp(j*sigma*(X(r).A2+Y(c).A2)); end end H=( 1/256)/\2*fftZ(h); H=fftshift(H); HR=FR.*H; hr=ifft2(HR); hr=(256A2)*hr; hr=fftshift(hr); %Image of the rectangle object figure( 1) image(X,Y,255*R); co lorm ap(gray(256 )); axis square %Plotting the real part of h figure(2) image(X,Y,255 *real(h)/max(max(real(h)))) co lotto ap(gray(256 )) axis square % Fraunhofer diffraction pattern of a square figure(3) image(X,Y, 10000*(abs(hr)).A2/(max(max(abs(hr))))A2 ) co lormap(gray( 256)); axis square axis([-0.43 0.43 -0.43 0.43]) %Cross-section of fig. 3 figure(4) plot(X,abs(hr(:, 127))) grid axis([-0.43 0.43 0 max(max(abs(hr)))* 1.1 ]) Table 3.3 MATLAB program for generating the Fraunhofer diffraction pattern of rectangular aperture (fraunhofer.m).

77

3 Propagation and Diffraction of Optical Waves

78

F i g u r e 3.13 Square aperture.

F i g u r e 3.14 Real part of

h(x, y; z)

at z - 98 c m with wavelength - 0.6 #m.

3.3 Spatial Frequency Transfer Functionfor Propagation

79

Figure 3.15 Fraunhofer diffracted amplitude of the square aperture shown in Figure 3.13. ....

i I

-

t

i

t

0.3

--

7

-

-

l

-

J .

I

I

t

I

I

.

.

.

.

.

I I

0.25

1

"

~

-

--

.

.

.

.

.

.

,_

_

t I

I

r

I I

0.2

--1

. . . . . . .

1

. . . . .

f I

I

I

I

t

f

~ . . . . . 4- . . . .

+-;

I

I

tJ

1

I

I

I

!J

I

. . . .

4

I

-~

. . . .

/4-

t

I ~l

.

.

.

.

~

--~

--

----

I

t

/

0

_~

I

...............

-0.4

,

LI t

-0.3

,

, ",

:

I

'

t

l

L

,

+, .

1

t ,

.

.

.

.

.

.

.

I

' . . . .

I '

t. . . . . . . . .

I. . . .

I

~-

1

t

~ I

I

I

'

. . . . .

'1

I . . . .

'I ,

I

t

I

J

4--

.

I

'

I

,q

I

. . . .

I

t

t

I

I

I

. . . . .

t.

,

r

I

I

. . . .

I

+

I

0.1--~

0.05-

t

f

I I

0.15

-''

I

I . . . .

4-

t

~,l . . . .

'~

I

I . . . .

I.

I .

.

.

.

I-

~

l"

1

1

I

I'

I

I

I

I

t

I

l . . . . . .

t_'

I

I

I

~

I

I

1

;

I

1

L

L

I

, ,'

, ',

}.

I .

.

.

.

.

ll_

I I l

"

-

t

[ _.

-

b t

,, ,,

I,

t,

,,, ;

-0.2

-0.1

0

0.1

[

0.2

"____1. . . . . . . .

0.3

Figure 3.16 Central cross- section of Figure 3.15.

0.4

3 Propagation and Diffraction of Optical Waves

80

3.4 Fourier Transforming Property of Ideal Lenses The transformation of a plane wavefront to a spherical wavefront is essential for applications such as focusing. In this subsection, we study how the lens, which is a phase object, can perform this transformation. For an ideal lens of focal length f, its phase transformation function, tf(x, y), is given by

tf(x, y) -- exp [ j ~ko (x 2 + y2 )].

(3.4-1)

The reason for this is that for a uniform plane wave incident upon the lens, the wavefront behind the lens is a converging spherical wave (for f>0) that converges ideally to a point source a distance z = f behind the lens. Upon comparing with the paraxial approximation for a diverging spherical wave, as given by Eq. (3.3-16), Eq. (3.4-1) readily follows. Let us now investigate the effect of placing a transparency t(x, y) immediately against the lens, whether in front or behind, as shown in Figure 3.17.

/)

t(x,y)

t(x, y)

f

Plane wave

Plane wave

Figure 3.17

A transparency immediately before and after an ideal lens under plane-wave illumination

In general, t(x, y) is a complex function such that if a complex field Cp(x,y) is incident on it, the field immediately behind the transparency-lens combination is given by Cp(x, y)t(x, y)tf(x, y) ~2p(X,y)t(x,y)exp[j~f(x2+ y2)], where we have assumed that the

3.4 Fourier Transforming Property of Ideal Lenses

81

transparency is infinitely thin, as is the case for the ideal lens. Then, under illuminated by a unit-amplitude plane wave, i.e, r y) - 1, the field immediately behind the combination is given by ko (x 2 + y2 t(x,y)exp[j~ )]. We then find the field distribution at a distance z - f by using the Fresnel diffraction formula, Eq. (3.3-14), as

~bp(X, y, z - f) - exp( - jkof) ~J k 0 e x p [ @ ( x 2 + y2 )] ~o x f ft(x', y')exp [jT(xx' + yy')]dx'dy'

-- exp( -

jkof) ~zJk~e x p [ ~ ( x 2 + y2)] (3.4-2) kz =kO:r/ f , kv=koy/f

where x and y denote the transverse coordinates at z = f. Hence, the complex field on the focal plane (z = f) is proportional to the Fourier transform of t(x,y) but with a phase curvature [depicted by the exponential on the RHS of Eq. (3.4-2)]. Note that if t(x, y)=1, i.e., the transparency is totally clear, we have ~p(X, y, z - f ) c v ~5(x, y), which corresponds to the focusing of a plane wave by a lens. For an ideal divergent lens, its phase transformation function is given by ko X 2

y2

exp[-ji-ff( + )]. All physical lenses have finite apertures and we can model this physical situation as a lens with an infinite aperture followed immediately by a transparency described by what is called the pupil function Pl (x, y) of the lens. Typical pupil functions are rect(x/X)rect(y/X) or circ(r/R), where X , Y , and R are some constants. Hence, if we have a transparency t(x, y) against a lens with a finite aperture, the field at the back focal plane of the lens is given by

r

y, z - f) o( ~xv{t(x, y)p~(x, y)}

(3.4-3) kx = k o z / f , ky=koY/f

under plane wave illumination.

3 Propagation and Diffraction of Optical Waves

82

Example 1 Transparency in Front of a Lens

t(x, y)

Suppose that a transparency is located at a distance do in front of a convex lens with an infinitely large aperture and is illuminated by a plane wave (see Figure 3.18).

~(x,y;f)

Plane wavefronts Figure 3.18 Plane-wave illumination of a transparency t(x, y) located a distance d0in front of a converging lens of focal length f.

t(x, y) t(x, y).

Assuming the field to the left of to be of unit strength, the field to the right of the transparency is then This travels a distance do to the lens; hence, using the transfer function approach to wave propagation, and Eq. (3.3-13), we have

front of lens

--exp(-jkodo)T(kx

ky)exp[j(k~+k~)d~ 2k0 '

where

T(kx, and

- f

V) }

(3.4-4)

3.4 Fourier TransformingProperty of ldeal Lenses

front o f lens

-

83

v)

front o f lens

}

"

In what follows, we will state the method that we use to find the field in the back focal plane of the lens. We then write down the final expression in terms of Fourier transform operators, in order to avoid repeating lengthy but similar mathematical expressions. First, note that be taking the inverse Fourier transform of Eq. (3.4-4) we get Cp(x, y) immediately in front of the lens. Hence, the complex amplitude in the back focal plane of the lens can be found from Eq. (3.4-2) by replacing t(x, y) by the field immediately in front of the lens. This gives jk0 exp( - jkodo)exp( - jko f) Cp(x, y, 9f) - 2-Y] x exp

.ko

x 2 + y2)]

• flZ'xy{fl2zy-l{T(kx ky)exp[ j(kg+k~)d~ '

_ 27rf jk0 e x p [ - jko(do + f)]exp --

x T(k~, Icy)

2k0

ko

}}

kx =kox/f kY=koY/f

do X2 + y2)]

- jg-ff(1 - T) (

(3.4-5) kz = k o z / f ky-=koy/f "

Note that, as in Eq. (3.4-2), a phase curvature factor again precedes the Fourier transform, but vanishes for the special case do = f. Thus, when

the object (transparency) is placed in the front focal plane oft he convex lens the phase curvature disappears, and we recover the exact Fourier transform on the back focal plane. Fourier processing on an "input" transparency located on the front focal plane may now be performed on the back focal plane, as will be seen later on in next chapter. Thus, in essence, a lens brings the Fraunhofer diffraction pattern (usually seen in the far field) to its back focal plane through the quadratic phase transformation.

3 Propagation and Diffraction of Optical Waves

84

3.5 Gaussian Beam Optics and MATLAB Examples In this Section, we will study the propagation or Fresnel diffraction of a Gaussian beam. Consider a Gaussian beam in two transverse dimensions with initially plane wavefronts. It can be expressed in the form 2

Cpo(x, y) -- e x p [ - (x 2 + y2)/w o ], where w0 is called the waist of the Gaussian beam. transform of this is

(3.5-1) The Fourier

2 2 ~vo(kz, kv) - rCwo2exp[- (k~ + kv)wo/4 ].

(3.5-2)

Using (3.3-13), the spectrum after propagation by a distance z is given by

9 p(k.. kv;z ) - '.~vo(kx.

kv)exp(

-

jkoz)exp[j(k~ + kv)z /2ko]

2 2 2 = rcw2exp[- (k 2 + ky)wo/4]exp(jkoz)exp[j(k 2 + ky)z/2ko] 2 = 7rwgexp(- jkoz)exp[j(k2x + ky)q/2ko],

(3.5-3)

where q is called the q-parameter of the Gaussian beam, defined as

q = z + jzR,

(3.5-4)

with zR defined as the Rayleigh range of the Gaussian beam" zR - k0~0~/2.

(3.5-5)

The shape of the beam after propagation through a distance z can be found by taking the inverse Fourier transform of Eq. (3.5-3): k0~0~ Cv(x, y, z) - exp( - jkoz)j--~qexp[jko(x 2 + y2 )/2q]

which reduces, after considerable but straightforward algebra, to

(3.5-6)

3.5 Gaussian Beam Optics and MA TLAB Example

Cp(X, y, z) -- w(z)W~e-(x2+y2)/w2(z)e-jko(x2+y2)/2R(z)e-Jr

85

e_,-Jk~ (3.5-7)

where

w:(z) - ~[1 + (~):]; R ( z ) - (z ~ + z ~ ) / z ; r - - ta~-l(z/zR).

(3.5-8)

Note, from (3.5-8), that (1) The width

w(z) of the Gaussian beam is a monotonically increasing

function of propagation z, and reaches x/~ times its original width or waist w0 at z - zR, the Rayleigh range; (2) The radius of curvature R ( z ) o f the phase fronts is initially infinite, corresponding to an initially plane wavefronts, as defined by (3.5-1), but reaches a minimum value of 2zR at z = zR, before starting to increase again. This makes sense, since far from the source z - 0, and well past the Rayleigh range, the Gaussian beam resembles a spherical wavefront, with the radius of curvature approaching z, the distance of propagation; (3) The slowly varying phase r monotonically varied from 0 at z - 0 to - ~-/2 as z-4oc, with a value of - ~-/2 at z - zR.

3.5.1 q-transformation of Gaussian beams The q-parameter of a Gaussian beam makes it real convenient to track an arbitrary Gaussian beam during its propagation through an optical system. Consider, for instance, the propagation of a Gaussian beam through a distance d. From (3.5-3) it is easy to see in the spatial frequency domain, propagation by a distance d amounts to multiplying the spectrum by an exponential term exp[j(kx2 + k2)d/2ko], besides a constant factor exp( - jkoz),

9 ~(kx, k~;z + d) - ~O(kx,

k~;z)~J(k~+k~)d/2k~

-- 7rw~eJ(kZx+k2)q/2koej(k2+k2)d/2ko

3 Propagation and Diffraction o f Optical Waves

86

= 7rw2ej(k~+k~ )%/2k~ .

(3.5-9)

Thus, the new spectrum is characterized by a new qe given by translation law: qe = q + d.

(3.5-10)

An optical system would usually comprise lenses and/or mirrors spaced apart from each other. While Gaussian beam propagation in between lenses and mirrors can be tracked using the translation law above, we need to develop the law of q-transformation by a lens. Note that the transparency function for a lens is of the form exp[jko(x 2 + y2)/2f]. The optical field immediately behind the lens is therefore the product of the optical field immediately in front of the lens and the transparency function, and can be expressed as

j__5~q

kow~e- jko(X 2+y2)/2ql -jko(x2 +y2) /2q ejko(x2 +y2) /2 f ____j--~q

where qz, the transformed q, is given by 2-

1

qt -- q

i

f"

(35-11)

For simplicity we do not discuss the q-transformation of a Gaussian beam when it is reflected from a concave or convex mirror. However, note that the effect of the mirror is also to change the phase front of the incident light, as in the case of a lens, and hence the law of qtransformation is similar to that derived above, with f denoting the focal length of the mirror. The laws of q-transformation due to translation and lensing can be incorporated into a single relation using the A B C D parameters introduced in Chapter 2. The q-parameter transforms in general according to the bilinear transformation

q,

_

Aq+B

Cq+D

(3 5-12)

once the A B C D parameters of the optical system are known. For instance, the A B C D matrix for translation is (01 d1), while that for a lens

3.5 Gaussian Beam Optics and MATLAB Example

87

i s ( i _ l / f o)~ . Substitution of the requisite values for A B , C , D for of the cases of translation and lensing gives the relations derived in Eq.(3.5.10) and (3.5.11) above.

3.5.2 Focusing of a Gaussian beam Based on the laws of q-transformation of a Gaussian beam developed above, let us analyze the focusing of a Gaussian beam by a lens of focal length f. Assume that a Gaussian beam of initial waist w0, and correspondingly initial q - q o - j z R - jkow2/2, is incident on a lens with a focal length f( > 0). Notice that the initial q is purely imaginary corresponding to a Gaussian beamwidth initially plane wavefronts. After propagation through a distance z behind the lens, the q of the beam is transformed as

q(z)-

(3.5-13)

fqo _qt_Z

The Gaussian beam is said to be focused at the point z - z f where the q(z)becomes purely imaginary again. Thus, setting q ( z f ) - j p ( z f ) = jkow2f/2 in (3.5.13), we obtain jp(zf )

-

j

fzR R -+- (Zf f2+z2

f,+z~

9

(3.5-14)

Equating the imaginary and real parts and simplifying,

Z4

f2+z2,

(3.5-15)

w} = f2+z~. f~g

(3.5-16)

zf-

and

Note that the Gaussian beam does not exactly focus at the geometrical back focus of the lens. Instead, the focus is shifted closer to the lens, and tends to the geometrical focus at f as w0~oc. Also, for large w0,

3 Propagation and Diffraction of Optical Waves

88

(3.5-17)

w f ~ w o f / z R -- A f /TrWo,

where A0 is the optical wavelength. As an example, for w0 = 3 mm, A0 = 633 nm, and f = 10 cm, the focal spot size w I ~ 20#m.

3.5.3 M A T L A B Gaussian beam

example:

propagation

of

a

The following example shows propagation of a Gaussian beam with initially plane phasefronts, using the paraxial transfer function for propagation discussed earlier in the Chapter. The MATLAB code is given below, and the plots from the output appear in Figures 19(a) and (b). The three dimensional plots of the magnitude of the optical field are viewed along the x l (or x) axis. The width of the Gaussian beam monotonically increases, as evidenced by the plots. Note that it is convenient, as done here, to normalize all lengths, e.g., by multiplying them with the propagation constant /Co. In the example, we show the propagation of a two-dimensional Gaussian beam (y, in the program below) having a normalized initial waist equal to 3. This normalized Gaussian beam is Fourier transformed (z0, in the program below) and multiplied by the normalized transfer function for propagation (v, in the program below) to yield the spectrum of the diffracted beam (zp in the program below). Upon inverse Fourier transforming zp, we get the diffracted field (yp in the program below) after propagation. Note that the normalized Rayleigh range zR works out to be equal to 4.5, as evident by comparing Figures 19(a) and (b). As clear from Figure 19(b), the peak amplitude drops to 1 / ~ times its initial value of unity, after propagation through z -- 4.5. "Energy" and "Energy_p" as the program's outputs serve to confirm that, before and after propagation, the total energy of the beam reminds constant.

clear % The following two lines define the 2-d grid in space. x 1=[ - 10:20./64.:- 10+63 .*20/64.]; x2=[ - 10:20./64.:- 10+63.*20/64.]; % delz is the step size, and N is the number of steps. z=4.5; w0=3;

3.5 Gaussian Beam Optics and MA TLAB Example

89

% y is the initial 2-d Gaussian beam. y - 1.0*(exp(-x 1.*x 1/W0A2))'*(exp(-x2.*X2/W0A2)); z0=fft2(y); % The energy statements below and in the last line is to check for numerical accuracy. energy=sum( sum(abs(y.A2 ))) figure (1), mesh (xl,x2,abs(y)), view (90,0); % The following two lines define the 2-d grid in spatial frequency. u 1--[- 10:20./64.:- 10+63.*20/64.]; u2=[ - 10:20./64.:- 10+63.*20/64.]; % v is the transfer function for propagation. v=(exp(i*u 1.*ul *z/2.))'*(exp(i*u2.*u2*z/2.)); w=fftshift(v); % Shifting is required to properly align the FFT of y, and w for multiplication. zp=z0.*w; yp=ifft2(zp); figure (2), mesh (xl,x2,abs(yp)), view (90,0); energy_p=sum(sum(abs(yp.A2)))

Table 3.4 MATLAB program for the propagation of Gaussian beam (gaussianprop.m).

0.9 0.8 0.7

'.'.'.'.'.'.'.'i'.'.'.'.'.'.'.'.'i'.'.'.'.'.'.'.'.i.'.'.'.'.'.'.'." t

.; ........

"

! ........

. . . . . . . .

: .........

. . . . . . . . .

: ........

. . . . . . . .

,.

. . . . . . . .

0.6

........ i ........ ! ........ i 0.5

t~

~ ~-,

........ ~........ i ........ ~ ....... ::

.

~ ...... :......... i........ i ........ i

.......

i ........

!........

i ......

.......

! ....... i

! ........ i

i ...... !

i /

1'4 t t ,,"

'\

........

::. . . . . . . . .

::. . . . . . . .

7

i

,,~,

",

0.4 0.3

~

....... i / i '

,-

,

:

..... 7........

:

:

:

: ........

! ........

!

:

:

:

:

4

6

0.2 0.1

:.'.'.'.'.'.'.'.i.'.'.'.'.'.'.'." ii

0 -10

-8

-6

-4

-2

0

(a)

2

................. 8

10

3 Propagation and Diffraction of Optical Waves

90

.8 . . . . . . . .

0.7

0.6

-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

........

i ........

i ........

~........

i ....

.........

~.........

~ ........

! ........

i~,

.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

!': i!

~,~

"" i ..... i........ ! ........ i......... :........ ! ~k ~,:~

..............

; .......

:

/

/:

0.5

\

0.4

/

0.3

!

/

0.2

,, ~"\\. ."\\ j--

,, ',,

,,\ 0.1

i

:

,,x,

S

9

:

i

:

"

:

',~

0

-10

!_~ -8

-6

-4

-2

0

2

4

__ 6

8

10

(b) F i g u r e 3 . 1 9 C r o s s - s e c t i o n a l plots ( l o o k i n g d o w n the x - a x i s ) o f (a) initial G a u s s i a n and (b) after travel by a distance e q u a l to the R a y l e i g h r a n g e o f the n o r m a l i z e d G a u s s i a n beam.

Problems 3.1

For a time-harmonic uniform plane wave in a linear isotropic homogenous medium, the E and H fields vary according to exp ( ~ 0 t - k0. R ) . Show that in this case, Maxwell's equations in a source-free region can be expressed as ko.E--O, ko-H=O, ko • ko • H -- -- woeE.

3.5 Gaussian Beam Optics and MA TLAB Example

91

Verify Eq. (3.2-23) by direct substitution into the wave equation,

3.2

02r 2 0r _ oR 2 + R OR --

()2 ko ~

02r Ot2"

Consider three plane waves traveling in three different directions, as shown in Figure P3-3. (a) Assuming that they all start in phase at the point O and have the same frequency, calculate the intensity I ( y ) a t the point P on the observation plane in terms of the relative field strength a, the angle r and the distances y0 and z0 where one can observe maximum and minimum fringe contrasts. ( F r i n g e c o n t r a s t is

3.3

[I(y)]max/[I(y)]min.) Observation plane

@

@+l:a<<1

P(Yo , Zo)

small

Z-O Z=Z o

Figure P3.3

3.4

Find a paraxial approximation to a wavefront, in the plane z - 0, that converges to the point P as shown in Figure P3.4. A transparency t ( x , #) is now placed at z = 0 and illuminated by the converging wavefront. Assuming Fresnel diffraction from z = 0 to the plane z = z0, find the intensity pattern on the observation plane. Comment on the usefulness of illuminating an aperture or transparency using a spherical wavefront instead of a plane wavefront.

3 Propagation and Diffraction of Optical Waves

92 t(x,y)

(x,y)

z=O

Z = Z

o

Figure P3.4 3.5

Find the

Fresnel diffraction v) -

pattern of an initial amplitude profile

- ?) +

+ ?)]6(y).

3.6

Verify that Eq. (3.3-28) is the Fresnel diffraction pattern of a square aperture of side l, at a distance z << k012away, by first confirming Eqs. (3.3-26) and (3.3-27).

3.7

Determine the Fresnel diffraction pattern of a straight edge u(x), where u(x) denotes the unit step function, illuminated by a plane wavefront. Plot the intensity distribution. Reconcile your answer with what you would expect in the very near and far fields, respectively.

3.8

Examine the MATLAB code given in Table 3.2 and re-program the code to verify that S(ec) - C(oc). Also find the value of

3.9

Examine the MATLAB code given in Table 3.2 and find the range of z that you think will give reasonable results on the Fresnel diffraction pattern of a square aperture. Give reasons why, for some range of z used, that you will not get reasonable answers.

3.5 Gaussian B e a m Optics and MA TLAB Example

93

3.10

In Eq. (3.3-33), take the limit as lx --+ oc. Do your results make physical sense? Give reasons for any discrepancy.

3.11

Find the Fraunhofer diffraction pattern of two slits described by the transparency function,

ix xj2] [x+xj2]

t ( x , y) - rect .... xo

+ rect

Z0

,

in two ways" (a) by directly taking the Fourier transform; (b) by using the notion of convolution and then using the Fourier transform. Suppose that the slits are now illuminated by light containing two frequencies Cal and cJ2. Suppose also that in the Fraunhofer diffraction pattern, the second off-axis maximum of color a21 coincides with the third off-axis maximum of color c~2. Find the ratio of the two frequencies (wl/c~2) constituting the illuminating light. 3.12

Find the Fraunhofer diffraction pattern of a sinusoidal amplitude

grating described by

3.13

1

+ -m cos

(kxo x

)}rect( -[)rect( ) , where

m < 1. Plot the intensity distribution for Fraunhofer diffraction along the x-axis and label all essential coordinates along the axis. A plane wave of amplitude A propagating in the + z direction is incident on an infinite series of slits, spaced 5' apart and a wide at z = 0, as shown in Figure P3.13 (Such a grating is called a Ronchi grating.) Find an expression for the amplitude distribution in the Fraunhofer diffraction of the aperture. Sketch the intensity distribution on the observation plane, labeling the coordinates of any important points.

3 Propagation and Diffraction of Optical Waves

94

. y

•

/

)Z

a.....a, ////A ////A ~ILIA ////A ////A ////A ////A //I/

Z

Figure P3.13

3.14

Find the Fraunhofer diffraction pattern of a sinusoidal phase grating described by exp[j(m/2)sin(k,ox)]rect(x/1)rect(y/1). Describe qualitatively what you would expect if the phase grating is moving along x with a velocity V - ~/k~o.

3.15

Examine the MATLAB code given in Table 3.3 used for Fraunhofer diffraction simulations. Verify that (3.3-29) is satisfied for the parameters used in the simulations. Find the range of z that you think will give reasonable results on the Fraunhofer diffraction pattern of a square aperture. Give reasons why, for some range of z used, that you will not get reasonable answers.

3.16 Using the A B C D formalism and the q-transformation, find how the width and radius of curvature of a Gaussian beam transform as the beam goes across a plane interface. Reconcile your results with a simple ray diagram and/or a physical explanation. 3.17

Consider the propagation of an elliptical Gaussian beam profile described by y)

-

exp[

-

-

WOy

95

3.5 Gaussian Beam Optics and MATLAB Example

Show that Eq. (3.5-8) describing the waist and the radius of curvature may be applied separately to the x and y dependences of the elliptic Gaussian.

3.18

Assume that a Gaussian beam with a waist radius Cpo (x, y) - A exp (

Wo,

x2+y2/ Wg

travels a distance d to a lens with a focal length f. Calculate the radius of the beam at the back focal plane of the lens. 3.19

Often, the beam emerging from a laser diode has an elliptical Gaussian cross section,

Cpo(x,y) - Aexp [- ( instead of a circular Gaussian cross section. If a lens of focal length f ( > 0) is placed right next to the exit aperture of the laser diode, calculate the distance z0 where the on-axis amplitude of the beam will be maximum. 3.20 A medium with a quadratic index profile n2(r) an ABC D matrix coskz -ksinkz

(1/k)sinkz) coskz

-n~) -n(Z)r 2 has

k2 __ n(2)/iZO"

A Gaussian beam with initially plane wavefronts and waist w0 is launched at z = 0. How far does it need to travel to transform back to its original shape? Also, for what value of w0 is the width of the Gaussian beam unchanged during propagation through the medium? Use the laws of q-transformation of Gaussian beams. 3.21

Assume that a Gaussian beam with initially plane wavefronts incident on a convex lens with a given focal length. Write a

3 Propagationand Diffraction of Optical Waves

96

MATLAB code to track the propagation of the Gaussian beam an arbitrary distance behind the lens. Show plots of the beam profile (a) immediately behind the lens, and (b) at the back focal plane of the lens. Change the initial waist size of the beam and plot the beam size at the back focal plane as a function of the input beam waist size.

References 3.1 3.2

3.3 3.4 3.5 3.6 3.7 3.8

3.9 3.10 3.11 3.12 3.13

Banerjee, P.P.(1985). Proc. IEEE 73 1859-1860. Banerjee, P.P. and T.-C. Poon (1991), Principles of Applied Optics, Irwin., Born, M. and E. Wolf (1983). Principles of Optics. Pergamon, New York. Cheng, D.K. (1983). Field and Wave Electromagnetics. AddisonWesley, Reading, Massachusetts. Ghatak, A.K. and K. Thygarajan (1978). Contemporary Optics. Plenum, New York. Goodman, J.W. (1996). Introduction to Fourier Optics. McGrawHill, New York. Hecht, E and A. Zajac (1975). Optics. Addison-Wesley, Reading, Massachusetts. Korpel, A. and P. P. Banerjee (1984). Proc IEEE 72 1109. Lee, D. L. (1986). Electromagnetic Principles of Integrated Optics. Wiley, New York. Poon, T.-C and M. Motamedi (1987). Applied Optics 26 46124615. Schiff, L. I. (1968). Quantum Mechanics. McGraw-Hill, New York. Stark, H., ed. (1982). Applications of Optical Transforms. Academic Press, Florida. Yu, F. T. S. (1983). Optical Information Processing. Wiley, New York.

97

Chapter 4 Optical Propagation in Inhomogeneous Media

4.1 Introduction: The Paraxial Wave Equation 4.2 The Split-step Beam Propagation Method 4.3 Wave Propagation in a Linear Inhomogeneous Medium 4.3.1 Optical Propagation through Graded Index Fiber 4.3.2 Optical Propagation through Step Index Fiber 4.3.3 Acousto-optic Diffraction 4.4 Wave Propagation in a Nonlinear Inhomogeneous Medium 4.4.1 Kerr Media 4.4.2 Photorefractive Media

4.1 Introduction: The Paraxial Wave Equation In Chapter 3, we derived the Helmholtz equation for the phasor part of the optical field propagating through a material of uniform refractive index no = c/v, where c is the velocity of light in vacuum, and v is the velocity of light in the medium" i)2r

ox~

_.[_

022/3p

i)2'Op

o-p- + ~

+ kgr

0,

lc0_ ~o. V

(4.1-1)

If we write r

y, z) - r

y, z)exp( -

jkoz)

(4.1-2)

98

4 Optical Propagation in Inhomogeneous Media

and assume that r

is a slowly varying function of z in the sense that

tOVe/OZl ~ ko ~ Ce

(4 ~ 1-3)

we can derive the paraxial wave equation for ~br as" Off)e/O z

where Vt2

denotes

the

__

1

2jko

~72//)e

transverse

(4 1-4)

'

Laplacian

02

D2

~V +oy~"

Fourier

transforming Eq. (4.1-4) with respect to the variables x and y leads to the ordinary differential equation d%

_

dz

--

j (k~+k~) 2 2 ~ 2k o

e

(4.1-5)

which when solved yields the paraxial transfer function similar to Eq.

(3.3-12)"

H~(kx , k v ; z ) -

exp[J(k~+k~)z 2k ~ ) 1.

(4.1-6)

If we wish to consider propagation in a material where the propagation constant or equivalently the refractive index is a function of position, either due to profiling of the material itself (such as a graded index fiber or a grating) or due to induced effects such as third order nonlinearities, the paraxial wave equation changes to Off)e/OZ--

1 Vt2 ~ - j A n k o ~ 2jk0

(4.1-7)

The quantity A n is the change in the refractive index over the ambient refractive index no. Eq. (4.1-7) is a modification of (4.1.4) and can be derived from the scalar wave equation when the propagation constant or equivalently the velocity of the wave is a function of (x, y, z) explicitly as in gratings or fibers, or implicitly such as through the intensitydependent refractive index. The paraxial propagation equation (4.1.7) is a partial differential equation that does not always lend itself to analytical solutions, except for some very special cases involving special spatial variations of A n or when as in nonlinear optics, one looks for particular soliton solution of

4.2 The Split-step Beam Propagation Method

99

the resulting nonlinear partial differential equation (PDE) using exact integration or inverse scattering methods. We will discuss some of the exact and analytical solutions for these special cases first. Numerical approaches are often sought for to analyze beam (and pulse) propagation in complex systems such as optical fibers, volume diffraction gratings, Kerr and photorefractive (PR) media etc. A large number of numerical methods can be used for this purpose. The pseudospectral methods are often favored over finite difference methods due to their speed advantage. The split-step beam propagation method is an example of a pseudospectral method. We will discuss this numerical technique first, and results using the technique later on in this Chapter.

4.2 The Split-step Beam Propagation Method To understand the philosophy behind the split-step beam propagation method, also called simply the beam propagation method (BPM), it is useful to rewrite Eq. (4.1.7) in the form [Agrawal, 1989] A

A

O~b~/Oz - (D + S)r where / ~ -

1 Vt2 is the 2jko

(4.2-1)

linear differential operator representing A

diffraction of the beam, and S - - j A n k o is a space-dependent or nonlinear operator (see, for instance, the structure of (4.1-7)). Thus, in general, the solution of (4.2-1) can be symbolically written as A

A

~b~(x, y, z + zXz) - exp[(D + S)Az]~b~(x, y, z) A

A

if D and S are assumed to be z-independem. noncommuting operators D and S, A

.

(4.2-2)

.

.

Now for two

A

.

1

I.

A

e x p ( D A z ) e x p ( S A z ) -- exp[DAz + S A z + ~[D, S](Az) A

A

2

+ ...] (4.2-3) A

A

A

A

according to the Baker-Hausdorff formula, where [D, S ] - D oe - S D represents the commutation of D and S . Thus up to second order in /kz, exp((D + S ) A z ) ~_ e x p ( D A z ) e x p ( S A z ) (4.2-4) A

A

A

A

100

4 Optical Propagation in Inhomogeneous Media

which implies that in (4.2-1) the diffraction and the inhomogeneous operators can be treated independent of each other. The action of the first operator on the RHS of (4.2-4) is better understood in the spectral domain. Note that this is the propagation operator that takes into account the effect of diffraction between planes z and z + Az. Propagation is readily handled in the spectral or spatial frequency domain using the transfer function for propagation written in (4.1-6) with z replaced by Az. The second operator describes the effect of propagation in the absence of diffraction and in the presence of medium inhomogenieties, either intrinsic or induced and is incorporated in the spatial domain. A schematic block diagram of the BPM method in its simplest form is shown in Figure 4.1. There are other modifications to the simple scheme, viz., the symmetrized split-step Fourier method, and the leap-frog techniques, these are discussed in detail elsewhere [Banerjee and Misra (1993)]. Initial Profile q/, (x, y; z - O)

J % (L, L , z) - Fx,y { ~ (x, y, z) }

~e(kx , ky ; z + A z ) - ~ e ( k

x , ky , z)exp l/(kx "

2

2

l ~,; (~, y, z + ~ ) - F~-){'e (k~ , k, ;z + ~ ) }

~(x,y,z

+ Az) - ~; ( x , y , z + Az) exp{SAz}

Figure 4.1 Flow diagram for the split step Beam Propagation Method

101

4.3 Wave Propagation in a Linear Inhomogeneous Medium

4.3 Wave Propagation in a Linear lnhomogeneous Medium Thus far, we have only considered wave propagation in a homogeneous medium, characterized by a constant permittivity e. In inhomogeneous materials, the permittivity can be a function of the spatial coordinates x, y and z. To study wave propagation in inhomogeneous materials, we need to return to Maxwell's equations (3.1-1)-(3.1-4) and rederive the wave equation. Our starting point is Eq. (3.2-4) which we rewrite here for a source-free medium (J~ = 0):

V2E-#e

(4.3-1)

o2E - - V ( V ' E ) Ot 2

~

Now, from (3.1-1) with p = 0, and (3.1-13a), V-(eE)

(4.3-2)

= e V - E + E - V e = 0.

With (4.3-2), we can rewrite (4.3-1) as 02E

(4.3-3)

If the spatial variation of e is small over a wavelength of the propagating field, we can neglect the last term on the LHS of (4.3-3) to write V2E-

02E

# e - ~ - 0,

(4.3-4)

where e = e(x, y, z). Note that Eq. (4.3-4) is similar to the homogeneous wave equation for the electric field [see Eq. (3.2-10)] derived earlier. For notational convenience, we return to our generic dependent variable ~(x, y, z, t) and adopt V 2 ~ - # e - b -~V - 0 ,

e-e

(x , y, z)

(4.3-5)

as our model equation, where we have assumed # = #0 for simplicity.

4 Optical Propagation in Inhomogeneous Media

102

4.3.1 Optical propagation through graded index fiber Recall from Sec.2.3 that the refractive index profile in such a fiber can be modeled as in Eq. (2.3-6). Equivalently, we can incorporate the inhomogeniety through a permittivity profile of the form [Haus (1984)] e(x, y, z) - e(x, y) - e(O)(1

x~h2 +y~) ,

(4.3-6)

where we have assumed two transverse dimensions instead of one as in Eq. (2.3-6). We wish to study the propagation of arbitrary beam profiles through the inhomogeneous medium modeled by Eq. (4.3-6). However, analytical solutions of Eq. (4.3-5) with Eq. (4.3-6) is difficult for arbitrary initial conditions. We will, therefore, first look for a propagating plane wave solution which can have an arbitrary crosssectional amplitude and/or phase profile. Thus, we set r

y, z , t ) - ReiCh(x, y ) e x p ( j ( c ~ o t

- kz)],

(4.3-7)

where k is as yet arbitrary, and substitute into the wave equation (4.3-5). This gives

v~r

+ [~,o~(~, y ) - k~]r - o.

(4.3-8)

We denote by k0 the propagation constant of an infinite plane wave propagating in a medium of uniform dielectric constant c(0), i.e., /Co - ~o[#oe(O)] 1/2.

(4.3-9)

When we introduce (4.3-6) into (4.3-8) with (4.3-9) and use the normalized variables --

we get

-

x,

r/--(~)

fl,

(4.3-10)

4.3 Wave Propagation in a Linear Inhomogeneous Medium

oo~ ~

+ ~o ~

103

+ [ a - (~ + rfi ) ] ~ - 0; ~ ( ~ , ~ ) - r

u),

(4.3-11)

where A-

(k~

]Co

(4.3-12)

"

We solve Eq. (4.3-11) using the commonly used separation of variables technique. To this end, we assume r162 X(~)Y(~7), substitute in Eq. (4.3-11) and derive two decoupled ordinary differential equations (ODEs) for X and Y. These are A

o~

-

)X

-

O,

oo~~v+(Av

- r/2 ) Y - 0,

(4.3-13a) (4.3-13b)

with Ax + Ay - A. Each equation in (4.3-13) is of the same form as that arising in the analysis of the harmonic oscillator problem in quantum mechanics [Schiff (1968)]. The solution to Eq. (4.3-13a) is Xm(~c) - Hm(~C)exp( - ~2/2), A~ - 2m + 1, rn - 0, 1, 2, ... (4.3-14) where the H,~s are called the Hermite polynomials. Hermite polynomials are H0(~) - 1; HI(~) - 2~; H2(~) - 4~ 2 - 2; ...

The first few

(4.3-15)

The solutions to X(~r are called Hermite-Gaussians. The first few are plotted in Figure 4.2. Similar solutions hold for Y(r/).

104

4 Optical Propagation in Inhomogeneous Media

Hm(~)e-#-" 12 2.5

F

r

"

/

2

//

/

\

/~\

m=2 ~",

i

,

1.5

l 0.5

-

m=l

-

\'

/i/

-0.5

-~/i -1.5

[

-3

J !

/

'\ f

b

-2 -4

, .

-2

-1

0

E

1

2

3

Figure 4.2 The three lowest-order Hermite-Gaussian functions. Eq.(4.3-1 1) thus has the general solution r

r

H m ( ~ ) H . ~ ( q ) e x p ( - (~2 + q2)/2) '

(4.3-16)

with A -- Amn - 2(m -+-n -+- 1), m , n - O, 1,2,...

(4.3-17)

r is called the m o d e p a t t e r n or m o d e profile of the m n - t h mode. The fundamental mode, with m - n - 0 (Aoo - 2)is given by ~be00(X, y) -- (2/70:/2 exp( -- (x 2 + y 2 ) / w 2) W

(4.3-18)

where w-

(2h ~0) 1/2 "

(4.3-19)

4. 3 Wave Propagation in a Linear Inhomogeneous Medium

105

The propagation constant k - k ~ of the mn-th mode is obtained from Eq. (4.3-12) with Eq. (4.3-17)" k2mn_ k02(1_ 2(~+~+l))k0h,

(4.3-20)

indicating that the propagation constant decreases with increasing mode number. Thus, higher order modes have higher phase velocities. The above analysis is indicative of the mode patterns that are characteristic of a multimode optical fiber. An arbitrary excitation at the input of the fiber can be tracked by decomposing it into the characteristic modes discussed above. The Hermite-Gaussian functions form an orthogonal basis, enabling such a decomposition to be made easily. We also comment that multimode fibers possess a distinct disadvantage in the sense that different modes travel with different velocities, leading to m o d a l d i s p e r s i o n . Optical pulses traveling through multimode fibers are more easily dispersed or spread in time than single mode fibers, which are more commonly used for optical communication. Single mode fibers are usually constructed with a step-index geometry, where the core of the fiber has a constant refractive index which is higher than that of the cladding. The refractive indices are chosen in such a way that outside the core, only evanescent solutions exist, and that only one mode, viz., the zeroth order, can propagate inside the core. As stated earlier, propagation of arbitrary initial profiles are much more difficult to analyze. An efficient way of doing this is numerically, using the BPM method. We assume that the graded index medium had a refractive index variation of the form (see Eq. (2.3-6)): -

_

+

y:)

(4.3-21a)

or

n ( x ) ~_ n o - (n (21/2no)(X 2 + y2),

(4.3-21b)

where no denotes the intrinsic refractive index of the medium and n t2/is a measure of the gradation in the refractive index. In this case, the operator S becomes _ j k o ( n ( 2 1 / 2 n o ) ( X 2 + y2).

(4.3-22)

4 Optical Propagation in Inhomogeneous Media

106

Propagation of a Gaussian beam in a medium with a graded index profile is shown in Figure 4.3. The contour plots show the initial (Gaussian) beam profile, the beam profile where the initial Gaussian attains its minimum waist during propagation before returning back to its original shape again, due to periodic focusing by the graded index distribution. As discussed above, there exists a specific eigenmode (a Gaussian of a specific width, related to the refractive index gradient) for which the beam propagates through the material without a change in shape as a result of a balance between the diffraction of the beam and the guiding due to the parabolic gradient index profile. The contour plot of such a beam is shown in Figure 4.4. Note that in this case, the contour plot remains unchanged during propagation.

4Oi

I

30

i

10 10

20

30

40

50

83

10

3D

(a)

30

40

5:)

eO

(b)

10

20

30

40

50

80

(c) Figure 4.3 Contour plots showing typical periodic focusing of an initial Gaussian profile. The initial profile in (a) focuses to a minimum width in (b) and regains its original shape in (c), demonstrating periodic focusing.

4.3 Wave Propagation in a Linear Inhomogeneous Medium

107

~

1

~ f

~,

3o t

'i',;

'L.-L>.

' '

. . o

2O 10 I'0

20

~0

40

50

~0

Figure 4.4 Sketch of the fundamental mode in a graded index fiber. Note that the width of the beam is smaller than the initial profile in Figure 4.3(a) and larger than the profile in Figure 4.3('o).

The M A T L A B program to generate the Figs. 4.3 and 4.4 is shown in Table 4.1 below. Note that the program uses the split step beam propagation method described in Section 4.2 above. The effect of diffraction is handled in the spatial frequency domain and the effect of the material inhomogeniety in the spatial domain. The material inhomogeniety causes an incremental phase change at every step of the split step technique. The induced phase change is proportional to

ko(n(21/2no)(X 2 + y2), which then defines the operator S as in Eq. (4.322). clear x 1=[-4. :8./64. :-4+63. *8/64. ]; x2= [-4. :8./64. :-4+63. *8/64. ]; del=l.0; %Input the Gaussian beam y= 1.0*(exp(-2.*x 1.*x 1))'* (exp(-2.*x2.*x2)); energy=sum(sum(abs(y. ^2))) figure (1),contour (abs(y)) ul =[- 1. :2./64. :- 1.+63 .*2./64. ]; u2 =[- 1. :2 ./64. :- 1.+63. *2 ./64. ]; %Input the transfer function for propagation v=(exp(i*ul.*ul *del))'* (exp (i'u2. *u2*del));

108

4 Optical Propagation in Inhomogeneous Media

w=fftshift(v); forj=1:125; z=fft2(y); zp-z.*w; yp=ifft2(zp); %Input the phase change due to quadratic refractive index p--(exp( 1.0*i*x 1.*xl *del))'* (exp( 1.O*i*x2.* x2* de 1)); yp=yp.*p; zp=fft2(yp); zp=zp.*w; yp=ifft2(zp); y=yp; end figure (2), contour(abs(y)); energy_p=sum(sum(abs(yp. A2))) Table 4.1 MATLAB code used to study propagation of Gaussian beam through a graded index fiber.

4.3.2 Optical propagation through Step Index Fiber Contrary to the graded index fiber, the step index fiber has a refractive index profile of the form [Ghatak and Yhyagarajan (1989)]"

~(~)_

{.co,

r
(4.3-23)

rid, r>a (cladding),

where r is the transverse radial coordinate. To analyze optical propagation through this fiber, we start from Eq. (4.3-8), rewriting it in the form

VtG + [ko2n 2 ( ~ ) -

k 2 ]r

(4.3-24)

- 0, W~ - V~(~, r

Since the medium has cylindrical symmetry, we write (4.3-25) Substituting Eq. (4.3-25) straightforward algebra,

into

Eq.

(4.3-24),

we

obtain

after

4.3 Wave Propagation in a Linear Inhomogeneous Medium R1 ( r2

-

d2R +

dR)

+

r2

--

109

ld2~ -2 1 ,

(4.3-26)

where 1 is a constant. The C-dependence will be of the form cos(lr or sin(lr The function 9 should also satisfy the condition q~(r + 27r) = ~ ( r hence admissible values of 1 are 1 = 0,+ 1,-+-2..... The r-dependent part of the equation in (4.3-26) above can now be rewritten as r

d2R + r-y7~ dR ) + (r2[kgn2(r) - k 2] - 1 2 ) R - O.

(4.3-27)

Using Eq. (4.3-23), the above equation can be written as the following set of two equations" r -yT~ d2R +r-yr-~ dR) + (r 2 [konco 2 2 - k 2] - 12) R - 0 ; r < a

(4.3-28a)

r2 -d2R ~ v + r ~dR) - ( r2 [k 2 - k.2oncl2 ] + /2 ) R - 0 ;

(4.3-28b)

r

>a.

Both equations are standard forms of the Bessel differential equation. Physical solutions, finite at r = 0 for Eq. (4.3-28a), are the Bessel functions of the form AJl([ k onto 2 2 - k 21/2r) ] . Similarly, physical solutions for Eq. (4.3-28b), which tend to zero as r --+ oc are modified Bessel functions of the form B K l ( [ k 2 - k 0rid] 2 2 1/2r) . The values of A and t3 can be determined from the boundary conditions, namely the continuity of r and Or at r = a. Note that waves will be guided down the fiber as long as 2 ],c2n2o>],c2:>k27Zcl.

(4.3-29)

Furthermore, by examining the boundary conditions in detail (we do not do this here for simplicity), it turns out that there can be only a discrete number of admissible solutions for k for a given l, and these are usually labeled with the index m = 1,2,3, .... The modes in a fiber are usually called the LPl~ modes, LP standing for linearly polarized. The mode structures of the modes for a step index fiber are plotted in Figure 4.5 [also see Ghatak and Thyagarajan (1989)]. Typically these modes are

110

4 Optical Propagation in Inhomogeneous Media

plotted for fibers with a certain V number. The V number of a fiber is defined as 2 V - koa(n~o

-

2

ncl

)

1/2

(4.3-30)

.

Suffice to state here that the lowest mode, LP0~, can be often approximated as a Gaussian. Fibers with V-numbers between 0 and 2.4 can only support one guided mode, namely the LP01 mode. Such a fiber is called a single mode fiber and of great importance in communication systems.

~'/'e

\ - \

LPo2

t

Xp22 Px'~t

/

\/

LPo, 21

LPll

.................. ~ . . . . . . .

1.0

J

r/a

2.0

Figure 4.5 Radial intensity distribution of some lowest order modes in a step index fiber of V - 8.

4.3 Wave Propagation in a Linear Inhomogeneous Medium

111

4.3.3 A c o u s t o - o p t i c Diffraction As a final example of propagation of light through an inhomogeneous medium, we will consider a c o u s t o - o p t i c (AO) d i f f r a c t i o n , or diffraction of light by acoustic waves. The interaction of light and sound is called a c o u s t o - o p t i c i n t e r a c t i o n . An AO modulator comprises an acoustic medium (such as glass or water) to which a piezoelectric transducer is bonded. Through the action of the piezoelectric transducer, the electrical signal is converted to ultrasonic waves propagating in the acoustic medium with a frequency that matches that of the electrical excitation to the transducer. The pressure in the sound wave creates a traveling wave of compression and rarefaction, which in turn causes analogous perturbations of the index of refraction. Thus the AO device such as shown in Figure 4.6 may be thought of as a phase grating with an effective spatial period equal to the wavelength A of the sound in the acoustic medium. ~/'m

~2 ~ / " l/r/c .

i

I

A~

2~B P'o

I I/]. m

X

t

v

Z

Figure 4.6 Interaction of light and sound fields in an AO medium. The sound field propagating in the medium is S - R e { S e ( x , z)exp[j(f~t - Kx)]}.

4 Optical Propagation in Inhomogeneous Media

112

It is well known that a grating splits light into several diffracted orders (see Problems 3.13 -15). It can be shown that the directions of the scattered or diffracted light inside the AO cell are governed by the

grating equation." sinr

= sinr

+ mA/A,

(4.3-31)

where r is the angle of the m-th order diffracted light, r is the angle of incidence, and A is the wavelength of light in the medium. The angle between adjacent diffracted orders is twice the Bragg angle CB:

s i n C B - A / 2 A - K/2ko,

(4.3-32)

where k0and K are the wavenumbers of the incident light and the sound, respectively. A more accurate approach considers the interaction of light and sound as the collision of photons and phonons. For these particles to have well-defined momenta and energies, we must assume that classically, we have interaction of monochromatic plane waves of light and sound, that is we assume that the width L of the transducer is sufficiently large in order to produce plane wavefronts of sound at a single frequency. In the process of collision, two conservation laws have to be obeyed, namely, the conservation of energy and the conservation of momentum. If we denote the wavevectors of the incident light, scattered light and the sound as ko, k+l and K , respectively, as shown in Figure 4.7(a), we can write the conservation of momentum as

ltko + h K

= hk+l

or, equivalently, k+a = ko + K .

(4.3-33)

The corresponding conservation of energy leads to cu+l = cJ0 + f2,

(4.3-34)

113

4.3 Wave Propagation in a Linear Inhomogeneous Medium

where w0, f~ and W+l are the angular frequencies of the incident light, sound and the scattered light respectively. The interaction described above is called the upshified interaction. Since IK I << ]kol, or equivalently,/( <
(4.3-35)

w-1 = w0 - f~.

(4.3-36)

and

CB

K

K

"A

(a)

(b)

Figure 4.7 Wave-vector diagram for (a) upshifted diffraction, and (b) down-shifted diffraction.

In a practical situation, the incident light is not always a plane wave but is a beam, e.g., Gaussian. In this case, the incident beam or profile must be decomposed into its angular plane wave components to analyze its interaction with the traveling sound field or refractive index profile. Clearly, not all of the angular plane wave components of the incident optical field will be at Bragg incidence, i.e., exactly at the Bragg angle. For off-Bragg incidence, upshifted or downshifted scattering still takes place, although with reduced diffraction efficiency. In order to quantitatively analyze diffraction efficiencies, as well as the effect of incident beams on a traveling sound column, we need to resort to the

114

4 Optical Propagation in Inhomogeneous Media

wave equation with added inhomogeniety, as in the case of the optical fiber. A further degree of complexity is introduced due to the traveling nature of the inhomogeniety, but this can be handled rather elegantly knowing that its effect is the Doppler shift of the optical wave frequency, such as given by relations (4.3-34) and (4.3-36) above. Our starting point is Eq. (4.3-4), which we rewrite in the form 02r

-

Ot 2

02r

-

r

-

z, t),

(4.3-37)

where v is the light velocity in the AO medium, c is the intrinsic permittivity in the AO medium and e~ represents the acoustically driven perturbation in the permittivity. We have assumed all optical fields to be polarized in the y direction in Figure 4.6). The optical field in the AO cell can be expressed as

r

z, t) - ~ R e { r

z) exp[j(wmt

-

k,~xX -

k,~z)]}.

(4.3-38)

m

Figure 4.6 shows the scattering of an arbitrary light field, ~bi,~c, incident, for instance, nominally at the Bragg angle CB, on the AO cell. In Eq. (4.3-38), Cm represents the complex amplitude or envelope of the m-th diffracted order, f~m is the temporal frequency of the m-th order light, and km~, kmz represent the x and z components of the propagation constants of the m-th scattered order. By extending the simple heuristic treatment of acousto-optics described earlier, it can be readily shown that w.~

-

wo + mf~.

(4.3-39)

Also, note that k . ~ - kosinr and k . ~ - kocosr Let us assume a sound field propagating along the positive xdirection with amplitude Sr z) in the AO medium: e' - eC R e { S r

z)exp[j(f~t - Kx)]},

(4.3-40)

where C is the AO interaction constant. We shall now assume S~(x, z) - A, a complex constant for simplicity. Upon substituting Eqs. (4.3-38) and (4.3-40) into Eq. (4.3-37), and gathering the coefficients of exp[j(w + mf~)t], we obtain, after some algebra [Banerjee and Tam (1991)],

115

4.3 Wave Propagation in a Linear Inhomogeneous Medium

o~r

Ox 2

2j(kmxOr

Ox

-k- k m z

ocm) Oz

- + - ( k g C / 2 ) { A * ~ b m _ l e x p [ - j(k(m-1)z -- kmz)Z]

-Jr- A r

j(k(m+l)z

-- k m z ) Z ] }

-- O,

(4.3-41)

where we have assumed slow variation of G~ with respect to z. Finally, upon Fourier transforming Eq. (4.3-41) with respect to x, we find that the angular plane wave spectra of the various diffracted orders ~ evolve according to Oq~mOz -- J (kx2+2Gk~

~ m -- j D ~ m + l - jEff2m-1

(4.3-42)

where D = ( k o C A * / 4 ) e x p { - jko[COSCm+l E - (koCA/4)exp{

--

cO8r

- jko[cOSCm-1 -- CO8r

(4.3-43)

In Eq. (4.3-42), the first term within the brackets on the right hand side represents the propagational diffraction effect [compare with Eq. (4.15)]. The second and third terms on the right hand side model the AO interaction. For nominal Bragg incidence (r - + CB), only the 0-th and -lst diffracted orders exist, and exact analytical expressions for the socalled A O spatial interaction transfer functions can be determined. For instance, from Eq. (4.3-42) and limiting ourselves to two orders, we obtain O%o~= j (k~+2k~ko~)2ko:~0 -- j e t - 1 O~zl - - j (k2x+2k~k-lx) ~ - 1 2k-1~

-- j D ~ o ,

(4.3-44) (4.3-45)

where ko~ = kosinCB = - k - i x , koz = k o C O S C B = k - l z . The constants D and E are simplified to D - k o C A * / 4 , E - k o C A / 4 .

116

4 Optical Propagation in Inhomogeneous Media

From Eqs. (4.3-44) and (4.3-45), we can obtain the acousto-optic spatial transfer functions analytically for 0-th and -1 st diffracted orders as [Banerjee and Misra (1993)]"

Ho(kx, Z - L ) -

exp(j k~L-~0)(cos v/(kxkoxL/ko) 2 + (o~/2)2

(4.3-46)

sin ~/(k~kozL/ko) 2+ (c~/2) 2

and

H_~ (kx, z -- L) -

f .k~L) - j ( a / 2 ) e x p ~ 3-~o

•

sin ~/( kxko~L / ko)2+ (c~/2) 2 i(kxkoxL/ko) 2+(c~/2) 2

,

(4.3-47)

where c ~ - k o C I A [ L / 2 represents the peak phase delay the light encounters during propagation through the AO cell of length L. Also note that k o ~ L / k o - Q A / 4 7 r , where Q is called the Klein-Cook parameter. Equations (4.3-46) and (4.3-47) define the interaction transfer functions for AO Bragg diffraction of incident optical beams or profiles. The propagational diffraction effect through the length of AO cell is /,

represented

k2L

by the exp (j -~0 ] phase term.

The remaining

terms

characterize the AO interaction process. Equations (4.3-46) and (4.3-47) show that, in essence, the propagational diffraction effects are decoupled from the AO interaction process. Figure 4.8 shows typical variations of the AO transfer functions for the zeroth and -1 st order versus spatial frequency kx for two different values of c~. The undiffracted or 0 order shows a highpass spatial frequency response, while the diffracted or-1 order shows lowpass characteristics. Note that by changing c~ or the sound pressure, we can achieve programmable highpass filtering in the zeroth order. In Chapter 7, we will show how this highpass characteristic can be used as a novel technique for programmable edge enhancement of images using acoustooptics.

4. 4 Wave Propagation in a Nonlinear Inhomogeneous Medium

" ""

I

"

-'-1 9

",'\,

I

/f

", ",..,,..

Magnitude

/' ',. . . . . .

.,'

, ,

, ,

9

0.5

I

"

."

~

,

-6.104

l" ....

117

,

I

I

I

I

I

-4.10 4

- 2 . 1 04

0

2"104

4"104

6"104

k~(1/m)

(a) [

Magnitude

.].

9

0.5

,

I

\

/' e

I

,

'\ ',,

,

/ ._

if" 0 -6"10 4

~, \,

,j

,' .

". 9 9

--,

,:,, ......

-4.10 4

-2'10 4

0

2'10 4

4'10 4

6"104

kx(1/m)

(b) Figure 4.8 Bragg regime transfer functions. The solid line is for c~ - 0.47r, the dotted line is for c~-0.87r. (a) magnitude of the zeroth order transfer function IH0(kx, Z - L)l; (b) magnitude of the -1 order diffracted order transfer function

[H-I (kx, z - L)I.

4.4 Wave Propagation in a Nonlinear Inhomogeneous Medium Thus far, all our discussions on optical propagation and optical processing were limited to linear media. In this Section, we take a simplified look at the vast area of nonlinear optics. This deals with optical propagation through nonlinear media and its effects. We first introduce the concept of nonlinearity qualitatively in the simplest possible way. Some of the effects of nonlinearity are second harmonic

118

4 Optical Propagation in Inhomogeneous Media

(and subharmonic) generation, self-refraction, two- and multi-wave coupling, bistability, phase conjugation and soliton propagation in optical fibers. While some materials have an instantaneous response to the optical field as in Kerr materials, others have a nonlinear response through the slow redistribution of space charges as in photorefractive

(PR) media.

4.4.1 Kerr Media Conventionally, the description of nonlinearities is given in terms of the nonlinear induced polarization in the medium or an amplitudedependent refractive index. In the former, the induced polarization P, having components Pi, i - 1, 2, 3 is expressed in terms of a Taylor series expansion of the electric field component E) according to [Yariv (1997)] :

Pi - r

j

+ Y~XijkEjEk + Y~ XidkIEjEkEl + ...)

j,k

=

oGx jEj + J

=

0(P? + PY ),

j,k..l

oP/L (4.4-1)

In Eq. (4.4-1), pNC refers to the nonlinear part of the polarization Pi. Also, Xij is the linear susceptibility tensor, Xijk and Xijkl are the nonlinear susceptibility tensors. Xijk, the second order nonlinearity coefficient, is generally responsible for the linear electro-optic effect and second harmonic generation or three wave mixing. X~ijkl,the third order nonlinearity coefficient, is responsible for third harmonic generation, and more importantly, the nonlinear refractive index n2 in Kerr-type materials, to be defined below. We remark that the tensor representation is not usually necessary to explain the physics of the nonlinear effects mentioned above. However, readers should be familiar with it in case of cross- referencing with existing literature. We now derive the wave equation for the vector electric field E in terms of the nonlinear polarization pNL, whose components are written as pNr. From Maxwell's equations (3.1-1)-(3.1-4) and using

4.4 Wave Propagation in a Nonlinear Inhomogeneous Medium

119

D = e0E + P,

(4.4-2)

we have 02E

V x V • E = V(V. E)- V2E-

--lzoeo-~ - I z o

02p

ate,

(4.4-3)

similar to the way we derived the wave equation for homogeneous isotropic media in Chapter 3. There, for a source-free region, we argued that V . E = 0 , since V . D - - 0 . However, this is only true in the general case if the E field is transverse to the direction of propagation. In this case, 02E Ot 2

1 V2E_

1

#oeo

o2p

(4.4-4)

eo Ot 2 "

Now, on the basis of Eq. (4.4-1), p_

(4.4-5)

pL + p N L .

If the medium is isotropic in the linear regime, i.e., X~ij- ~ where XJ denotes the linear susceptibility, P = e0xE +

pNL,

(4.4-6)

and Eq. (4.3-4) reduces to

02E or2

v2V2E

m

1 02P NL

~

or2

,

(4.4-7)

where e-e0(l+~)

v2 = ~ #06

(4.4-8) "

Equation (4.4-7) is often called the n o n l i n e a r o p t i c s e q u a t i o n . Note that in a simple case where X~ijk- 0, X~ijkl- ~ 1 1 1 1 - ~(3)and Ej,k,1 -- E 1 - - ~b, the nonlinear optics equation becomes 02~b Ot 2

V2V2

cox (3) 02~) 3 ~

m

e

Ot 2 "

(4.4-9)

4 Optical Propagation in Inhomogeneous Media

120

Assume now that ~2(x, y, z,t) -- Re{r

y , z ) e x p [ j ( w o t - koz)]}

(4.4-10)

as in Eq. (4.3-7), where r y , z ) is a slowly varying function of z. If this is substituted in Eq. (4.4-9) and coefficients of exp[j(c~0t - k0z)] are collected, we get after some algebra, 1 V2tCe _ jA~r~kOr e Off)e / OZ -- 2jko

(4.4-11)

Z~n

(4 4-12)

where --

n2 I~)e [2

, n2--

3 (3) g~n0X 9

Note that Eq. (4.4-11) with (4.4-12) is of the same form as Eq. (4.1-7), showing that the effect of induced inhomogeniety due to the cubic nonlinearity can be handled numerically by the split-step beam propagation method. In some cases, as we will discuss below, Eq. (4.411) with (4.4-12) has solutions which do not change its shape with propagation, called spatial solitons. While spatial solitons in one transverse dimension are stable and can be expressed analytically, those in two transverse dimensions with circular symmetry may be unstable. Eq. (4.4-11) with (4.4-12) is called the nonlinear S c h r o d i n g e r equation. The nonlinear Schr6dinger equation with renamed variables is also used to analyze pulse propagation in nonlinear optical fibers. In our quest for the expression for 1~ I that does not depend on z, we substitute r

y,z) -- a ( x , y)exp[ - j ~ z ]

(4.4-13)

in Eq. (4.4-11) to get V2t a -- - 2t~koa - 2k~a 3.

(4.4-14)

Consider first the case when we have one transverse dimension, namely, x. In this case, Eq. (4.4-14) becomes

121

4. 4 Wave Propagation in a Nonlinear Inhomogeneous Medium

d2a ----

dx2

-- 2 n k o a -

(4.4-15)

2 k g a "3

It can be readily verified that a solution of a ( x ) is of the form [Banerjee and Poon (1991)] (4.4-16)

a ( x ) -- A s e c h ( K x ) ,

where A-(

2~ )1/2 n2ko

K--(

'

1 )1/2 "

(4.4-17)

2~k0

We note from above that n<0 and n2>O for a physical solution. This makes sense since for n2>0, the refractive index is greater where the beam amplitude is higher. For a s e c h beam shape, this means that the center of the beam sees the highest refractive index, which then decays off-axis, much like a graded index fiber. Thus, the nonlinearly induced inhomogeniety causes the beam to maintain its shape by balancing the effects of spreading due to diffraction. A typical plot of this profile is shown in Figure 4.9.

/\

0.9 0.8

/' l

0.7 0.6 0.5

!

0.4 0.3 0.2 0.1 0 -8

-6

-4

-2

4

;

6

8

Figure 4.9 Typical plot of a normalized sech function, showing the shape of a spatial soliton in one transverse dimension. For two transverse directions, viz., x and y, we only consider the case where there is radial symmetry, and express the transverse Laplacian in polar coordinates as ~t 2_

0~

Oz2 d-

02 c3y2

--

d2 ~

-~

1 d r dr"

(4.4-18)

4 Optical Propagation in Inhomogeneous Media

122

Using the definitions

a

-

2n2k0

-

~~ r -

1 2~k0

.-r~

(4.4-19)

we can rewrite (4.4-14) in the form d2~ _4 1 d~ d'~2 "? a7

a

+ ~3=0

(4.4-20)

This ordinary differential equation (ODE) has no analytic solutions; the solutions that tend to zero as ~ -+ oo are obtained by numerical methods [Chiao et al. (1964), Haus (1966)]. These are shown Figure 4.10. Note that the solutions are multimodal in nature, with the mode number m depending on the initial condition a(0). In a way, these solutions are remeniscent of the modes in a graded index fiber, discussed in Sec. 4.3.

2~

h

2"21

rn =

Mode number

-2

Figure 4.10 Numerical solutions to Eq. (4.4-20) showing the amplitude profiles for lowest order modes with radial symmetry.

4.4 Wave Propagation in a Nonlinear Inhomogeneous Medium

123

In general, it is not possible to find analytical solutions for the v a r i a t i o n o f a b e a m shape with p r o p a g a t i o n t h r o u g h a cubically n o n l i n e a r m e d i u m , and one has to resort to n u m e r i c a l t e c h n i q u e s to a n a l y z e the propagation. A n e x a m p l e o f such a p r o g r a m written in M A T L A B follows below. W e use the split-step b e a m p r o p a g a t i o n m e t h o d o u t l i n e d in Section 4.2. The plots in Figure 4.1 1 (a) and (b) are c r o s s - s e c t i o n a l cuts o f the b e a m shapes in two transverse d i m e n s i o n s . The p r o g r a m can be readily m o d i f i e d to analyze p r o p a g a t i o n o f b e a m s w i t h a oned i m e n s i o n a l transverse profile. W h e n the b e a m profile b e c o m e s too steep, i.e., the w i d t h a p p r o a c h e s the w a v e l e n g t h , one has to resort to s p e c i a l i z e d m e t h o d s to simulate the propagation. One such t e c h n i q u e i n v o l v e s using an adaptive f u l l - w a v e l e t t r a n s f o r m technique, w h i c h is currently u n d e r d e v e l o p m e n t .

clear % The following two lines define the 2-d grid in space. x 1=[-4. :8 ./64. :-4 +63. *8/64. ]; x2 = [-4. :8 ./64. :-4 +63. *8/64. ]; % delz is the step size, and N is the number of steps. delz=l.0; N=35; % y is the initial 2-d Gaussian beam. y-2.01 *(exp(-2.*x 1.*x 1))'*(exp(-2.*x2.* x2))+0.000* rand(64); % The energy statements below and in the last line is to check for %numerical accuracy. energy=sum(sum(abs(y.A2))) figure (1), mesh (xl,x2,abs(y)), view (90,0) % The following two lines define the 2-d grid in spatial frequency. u 1=[ - 1.:2./64.:- 1.+63.'2./64.]; u2=[ - 1. :2./64. :- 1.+63.* 2./64. ]; % v is the transfer function for propagation. v=(exp(i*u 1.*u 1*delz*0.05))'*(exp(i*u2.*u2*delz*0.05)); % Shifting is required to properly align the FFT of y, and w for multiplication. w=fftshift(v); %The part from for to end in the "DO" loop is a split-step % beam propagation method where medium nonlinearities can be % incorporated in the spatial domain in the variable p below. forj=l:N; z=fft2(y); zp=z.*w; yp=ifft2(zp); yint=(yp. *conj (yp)+conj (yp). *yp)/2.0; p=exp(-i*yint*0.05);

4 Optical Propagation in Inhomogeneous Media

124

%p=l; yp=yp.*p; zp=fft2(yp); zp=zp.*w; yp=ifft2(zp); y=yp; end figure (2), mesh (xl,x2,abs(y)), view (90,0) energy_p=sum(sum(abs(yp. A2))) Table 4.2 MATLAB code used to study propagation of Gaussian beam through a Kerr medium.

MATLAB OUTPUT: energy = 203.0776 energy_p 203.0776

0-4

;

-~

"I

0

I

I

2

~

;

L-- ~

~

~

o

1 "~

3

4

Figure 4.11 (a) Initial Gaussian beam shape and (b) Beam shape after propagation in the Kerr medium. Notice that self-focusing increases the peak value. Energy is conserved in the process as shown by the MATLAB output above.

4.4.2 Photorefractive Media The photorefractive (PR) effect has been used in a wide variety of applications, viz., image processing, optical interconnections, optical data storage, optical limiters and self-pumped phase conjugators. When a PR material is illuminated by a light beam or by a fringe pattern generated by the interference of two light beams, photoexcited carriers are redistributed in the volume of the crystal through diffusion and recombination. This sets up a space-charge field which, through the linear electro-optic effect, gives rise to a intensity-dependent refractive index profile.

125

4. 4 Wave Propagation in a Nonlinear Inhomogeneous Medium

Charge transport in PR materials can be modeled using the Kukhtarev equations [Yeh (1992)]. We will not discuss them in detail, but mention that for a diffusion dominated PR material, these equations can be decoupled in the steady state to yield an ordinary differential equation for the space charge electric field E~ (x, y, z). An approximate solution of this differential equation is

E~ ~

(4.4-21)

kBT V I e ~/s+I '

where e is the electronic charge, kB is the Boltzmann constant, T is the temperature, s is the ionization cross-section per unit photon energy and /3 is the thermal generation rate. The term ~ / s is important if the beam intensity profile I decays to zero for large x , y , which represent directions transverse to propagation (z) of the beam in the PR material. I ( x , y;z) denotes the intensity distribution along x, y at a position z in the PRmaterial. Now, this electrostatic field induces a refractive index change An for polarized (say along x, see Figure 4.12) plane waves of light polarized in the PR material, assumed BaTiO3 from now on, through the linear electro-optic effect, given by

An(X, y, z, O) ~ -- naOrecosO sin20 Esx,

(4.4-22)

where E~x is the x-component of E~, no is the refractive index, and r~ is the effective linear electro-optic coefficient. The angle 0 in Eq. (4.4-22) is defined in Figure 4.12.

Fan

I_...___._---

f

x 3 ~ z

Incident beam Photorefractive crystal Figure 4.12 Geometryused to study beam fanning in PR BaTiO3. The c-axis is in the x - z plane.

126

4 Optical Propagation in Inhomogeneous Media

It can be readily shown that, in general, propagation through the PR material under the slowly varying envelope approximation may be modeled by means of the PDE as in Eq. (4.1-7): Offye/O z __ 2jko 1 V2t ~.Ye _ j / ~ n ~ O C e

(4.4-23)

where An is defined through Eqs. (4.4-22) with (4.4-21). For values of 0 around 40 degrees, a symmetric beam could induce an asymmetric refractive index profile, leading to beam bending and distortion (also called deterministic beam f a n n i n g [Banerjee and Misra (1993)] in the far field. However, for some other value of 0, for instance 90 degrees, Banerjee and Misra's theory predicts symmetric beam shaping. In this respect, the nature of the optical nonlinearity in a PR material is more involved as compared to that in a nonlinear Kerrtype material. We point out that in a Kerr-type material for instance, only an asymmetric beam profile can cause beam bending, as reported in Swartzlander et al. (1988), while a symmetric beam undergoes selff o c u s i n g or defocusing, as discussed earlier. In the remainder of this section, we will present the results for far-field beam profiles, assuming that a Gaussian beam of peak intensity I0 and waist W is incident on a PR material such as BaTiO3 material of thickness L. We assume that the waist forms at the center of the sample z = L / 2 in the absence of any PR effect. Numerical simulations for propagation have been performed on the basis of Eqs. (4.4-21)-(4.4-23) by employing the split-step beam propagation technique. Figures 4.13(a) and (b) show the normalized far field intensity patterns with W and power P - 7rio W2/2 a s parameters. The results show, for instance, that beam distortion and fanning is seen to reduce at very low (high) and very high (low) values of P (W). Quantitatively, for a fixed power P (viz., 1.5 mW), we see the negligible fanning for sufficiently large values for W (viz., 70 microns). This is because beams of sufficiently large waist size induce a smaller induced refractive index change An, since, according to Eqs. (4.4-21) and (4.4-22), the gradient of the intensity is smaller for large W. On the other hand, the reason for the low fanning for a sufficiently small value of W is that effectively, the beam width, if monitored over most of the sample is large (due to a large diffraction angle), again implying a reduced PR effect. This, in turn, implies that propagation through the crystal is predominantly diffraction limited. The

127

4. 4 Wave Propagation in a Nonlinear Inhomogeneous Medium

variation of beam distortion and fanning with beam power P is not as easy to explain. Suffice to mention that beam fanning is reduced for high and low beam powers. For low beam powers, the induced refractive index change An is small, since the gradient of the intensity is smaller. VI For large powers, 3/s+z "~ -77L, and hence independent of the peak intensity or power. Thus, while there is a shift in the peak of the beam in the far field (due to the V operator), the distortion in the beam is negligible. The PR response modeled by relations (4.4-21)-(4.4-23) above can also be used to analyze two-beam coupling, phase conjugation, holographic recording and reconstruction, and to explain image processing such as edge enhancement and correlation in such materials.

w=O.O2 5 rnm

>.I--

--.-------

W=O.04 mm W-O,07 mm

Ze ~6 I-Z

t 1 ii,

I

i

w

i k

m

'

9 Z

t

| L I i

I h

,1

g -20,0

--12.0

-.i,o

4,.0 KX

(l/m)

(a)

12.0

20.0

"10'

4 Optical Propagation in Inhomogeneous Media

128

P.O,2 mW

r t

P - I , S mW P=25 mW

I

t'

ii 0 t i

>t-.

l

uad I-Z

[

S

r 'i

I

,

I

rv,

t'

]

..

2~

! t b

', , t

p

,

r29 Z

0

l

t

!

d r

e,,J

,

o

,

, \

;

q

', b

t 0 g~

-,

-20,0

-~2,0

-4,o

s

KX (l/m)

4.0

12,0

~0.o

"10"

(b) F i g u r e 4.13 Normalized far-field profiles for Gaussian input to PR barium titanate. (a) P = 1.5 mW, and W = 25 microns (dotted line), W = 40 microns (continuous line), W = 70 microns (dashed line); (b) W = 40 microns, and P = 0.2 m W (dotted line), P = 1.5 mW (continuous line), P = 25 m W (dashed line).

Problems 4.1

Show the orthogonality properties of the Hermite-Gaussian modes propagating through a graded index fiber by proving that

where X~(~) - H ~ ( ( ) e x p ( - ~2/2).

4.4 WavePropagation in a Nonlinear Inhomogeneous Medium

129

4.2

Consider a step index fiber with n~o=1.46; nd=1.457, and core radius of 5 microns. Find the wavelength above which the fiber can sustain only one mode. For red light at 633 nm, how many modes will be supported by the fiber?

4.3

Develop a MATLAB program using the beam propagation method to study propagation of a Gaussian beam through a step index fiber. You can write the refractive index variation in the form of a radially symmetric super-Gaussian, viz.,

Art--ncl ~ (rico-

nc/)exp[--

(X2+a2Y2)P]

where p is an integer. Start with p = 1, and take the input Gaussian beam width to be that of the fundamental mode of a graded index fiber admissible by the refractive index profile under a parabolic approximation. You can then increase p and make the profile resemble more and more like a step function. 4.4

For near phase synchronous downshifted AO Bragg diffraction, i.e.,r CB + Ar the 0th a n d - l s t order diffracted light fields evolve according to

dr dz

d~)-i dz

_

_

-

--

-

jar

exp( -- jKzAr ;

ja~0exp( +

jKzAr

where a is an interaction constant and / ( denotes the sound wavenumber, z denotes the nominal direction of propagation of the light. Solve for ~0 and ~-1 using the boundary conditions ~0(z - 0) - ~i~; ~ - l ( z - 0) - 0. Show how your results can be reconciled with the transfer functions for AO interaction derived in the Chapter. 4.5

Develop a MATLAB code to study the propagation of the sech profile in a cubically nonlinear medium, assuming one transverse dimension. Your results should show that the beam with the right amplitude and width does not change its shape during propagation. Now perturb the amplitude and the width, one at a

4 Optical Propagation in Inhomogeneous Media

130

time and study the behavior of the profiles during propagation. Do you observe periodic focusing and defocusing? 4.6

Starting from the paraxial wave equation in the presence of a cubic nonlinearity, show that during the propagation of a complex envelope r there is conservation of power. To do this, assume r z) - a(x, z)exp[ - jr z)] and show that

f ~ a2(x, z) dx - constant. 4.7

Use the split-step method to analyze propagation of a onedimensional Gaussian beam of waist size W0=100A through a material of thickness 100A with a refractive index profile + ~ ( ~ / w 0 ) , Ixl < 5w0. Let no=l.5 and c~ - 0.015. Determine the far-field profile. Compare your results with the physical inferences from Gaussian beam propagation through a diffusion dominated PR material, discussed in the Chapter. Point out similarities and differences. -

4.8

Derive the induced nonlinear focal length of a length Az of a Kerr type material in terms of the nonlinear refractive index coefficient n2, the beam (assumed Gaussian) amplitude and width. Using the laws of q-transformation for translation and lensing (see Chapter 3), derive the differential equation for the variation of the width of the beam during propagation through such a nonlinear material.

References 4.1 4.2

4.3 4.4 4.5

Agrawal, G.P. (1989). Nonlinear Fiber Optics, Academic. Banerjee, P.P. and R.M. Misra (1993). Optics Communications, 100 166. Banerjee, P.P. and T.-C. Poon (1991). Principles of Applied Optics, Irwin Banerjee, P.P. and C-W Tam (1991). Acustica 74 181. Chiao, R., E. Garmire and C.H. Townes (1964). Phys. Rev. Lett. 13 479.

4.4 Wave Propagation in a Nonlinear Inhomogeneous Medium

4.6 4.7 4.8

4.9 4.10 4.11

131

Ghatak A. and K. Thyagarajan (1989). Optical Electronics, Cambridge. Haus, H. (1966). Appl. Phys. Lett. 8 128. Haus, H. (1984). Waves and Fields in Optoelectronics, PrenticeHall. Swartzlander, G.A., H. Yin and A.E. Kaplan (1988). Optics Letters 13 1011. Yariv, A. (1997). Optical Electronics in Modern Communications, Oxford. Yeh, P. (1993). Introduction to Photorefractive Nonlinear Optics, Wiley.

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133

Chapter 5 Single and Double Lens Image Processing Systems

5.1 5.2 5.3 5.4 5.5

Impulse Response and Single Lens Imaging System Two-Lens Image Processing System Examples of Coherent Image Processing Incoherent Image Processing and Optical Transfer Function MATAB Examples of Optical Image Processing 5.5.1 Coherent lowpass filtering 5.5.2 Coherent bandpass filtering 5.5.3 Incoherent spatial filtering

5.1 Impulse Response and Single Lens Imaging System In this Section, we examine the field distribution at an arbitrary distance behind the lens having a pupil function p~(x,y) when a transparency t(x, y), upon plane wave illumination, is placed a distance do in front of the lens. We derive the impulse response of the system first and then find out the output distribution due to the transparency t(x, y). Considering a point source as our object or input to the optical system as shown in Figure 5.1, we represent the input as ~5(x - x0, y - Y0) mathematically. Now we can track the propagation of the complex field from the input (or object) plane to the plane of the pupil function by using either the Fresnel diffraction formula or the transfer function approach to wave propagation. Using the former, we have, from Eq. (3.3-14),

5 Single and Double Lens Image Processing Systems

134

jko

exp( - jkodo) 2+do

Cp(X, y, z -- do) -

X

(5.1-1)

exp]-jk~ ( x - Xo)2+ y - yo)2)]-'_, I -+Vo

The field immediately behind the lens, Cp[behindlens, is given by multiplying Eq. (5.1-1) by the pupil function and the phase transformation introduced by the lens of focal length f" ~)p[behind lens --

exp( - jkodo) ~Jk~ exp [-J%w~0( x - x 0)2 + y _ yo)2)] ko 2 + y2 )]. x pz(x,y)exp [jvf(x

p f(x,y)

gtpI Behind lens

#

6(X-Xo,y- y o)

~p(x,y,'z) ! |

,,i | <

d o

~

Z

!

Figure 5.1 Single-lens imaging system with a point object. Finally, we can express the field at an arbitrary distance z to the right of the lens by the Fresnel diffraction formula again as C p ( X , y, Z) -- Cptbehind lens * h ( x , y; z)

135

5.1 Impulse Response and Single Lens Imaging System

jko exp( - jkodo) 2~do jko = exp( - jkoz) ~z x f fexp [-jko ~o(X' _

]

Xo)2 +( y, _ Vo)2

ko x pi(x', y')exp[jTf(x'

2

+

yt2

x exp -jko ( x - Xt)2 + ( y _

)]

yt )2] dx'dy'

= C exp [ -5~o -jk~ (Xo2 + Yo2)] exp[ -jk~ •

X2

f fv,(~', y')exp @ [ E 1 + 7

1

_

+ y2)] ~](x'

2 + y,2)]

x x exp jko[(NXo + z)X' + (EYo + zy )Y' I dx~ dye'

(5.1-2)

jko 2~do" jko Let z - di -+-/kz where di where C - exp( - jko(z + do)) 2~z satisfies the thin-lens formula E1 + N1 - 71 and assumeAz is small Eq. (5.1-2) becomes _

_

-jk o Cp(x, y, z) - C exp _g~o(Xo2+y2)]exp[-jko 2(<+Az)(x 2 + y2 )]

•

ffpr

-jk~

(x' + 2 y,2)l

9 + (N yo + zv )Y' I dx~ dy~ x exp jko[(N~o + z)X' -

-~-o(Xo~ + yo~

x {exp[ o2~Z

)] exp [~(~+~zl( -3"

(z2 +y)]pr

+

Ix, Y/} (5.1-3)

Defining the so-called defocused pupil function as

5 Single and Double Lens Image Processing Systems

136

Paz(X,y)-exp -jkoAz 2d~ ( x2 + y2 ) ] Pl ( x , y ) ,

(5.1-4)

Eq. (5.1.3) becomes

-jko X 2 y2 ] + Yo2)l exp 2(e~+Az)( + )

C p ( x , y , z ) - C e x p [ -jk~ xo

~o

~

~

• P~:(ko(~o + d:+zx~),kO(~o + ei+Az

))

'

(5.1-5)

where P~is the Fourier transform of PAz" Equation (5.1-5) is the impulse response of the single-lens system at the observation or output plane z = d i + Az away from the lens due to an off-axis point source at the input. Having determined the impulse response of a single-lens system where the lens has a pupil function p~ (x, y), we now find the response of the system for an arbitrary input t(x,y). To do this, note that ~5(x - x ~, y - y~) on the object plane yields a response according to [see Eq. (5.1-5]

o Xt2 + y~2]) exp [ - j k o ( X 2 + y2 ) 6 ( x - x' ' y - y') ~ Cexp [-jk -53;-0( 2(d~+Az) x'

z

y'

y ))

• P~:(kO(~o + d~+ZXz),kO(~o + ~,+zx~ on the output plane; hence,

t(x, y) -

f f t(x', y')5(x - x', y - y')dx'dy' Cexp

-jko ( X2 y2 ] 2(d~+Az) + ) f f t(x', yt )exp

-~~ j ko X 12 + yt 2 ) 1

x~ z Y~ Y x PA~(ko(E + d~+Az),kO(E + d~+Az))dx'dy'.

(5.1-6)

If we now assume that do >> ~lko(X,2 + y,2 )]max and di-+-/kz >2> l k0(x 2 + y2)[ma~ the condition reminiscent of Eq (3.3-29), i.e. in this case, we consider a small region of the field amplitude on the output plane from a small region of object space, Eq. (5.1-6) becomes

5.1 ImpulseResponse and SingleLens ImagingSystem

137

t(x, y) - c f f t(x', y') x x PA= (k0(~x 0/ + d~+zx~) k0(Eyr + ~

Y ))dx'dy'

d~ + A z

"

(5.1-7)

We shall now consider a special case when Az - 0, i.e., the imaging is in focus. Equation (5.1-7) reduces to, after some manipulations, within some constant: t ( ~ , y) -

t(x/M,

y/M)

kox

koy )

9P~( d~ , ~

,

(5.1-8)

Pf is the Fourier transform of Pl, M - - di/doand t ( x / M , y / M ) is called the geometrical image of the object. The result indicates that a transparency t(x, y) placed on the object plane produces where

a field on the image plane proportional to the geometrical image t ( x / M , y / M ) convolved with the impulse response of the imaging system [also known as the coherent point spread function (PSF)] defined as

h~(~ y ) - P i ( ~~ ko~) di ~ di "

(5.1-9)

The Fourier transform of hc(x, y) will be called the coherent transfer function H~(k~, ky), given by, to within some constant,

H~ (kx , k~) - p~ ( -~,k~ k0 ' -d~k~ k0 )"

(5.1-10)

We remark that, in retrospect, the impulse response in Eq. (5.1-9) can be obtained directly by setting x0 = y0 = 0in Eq. (5.1-5) with Az = 0. We end this Section with a reminder that the pupil function pz(x,y) can represent, in effect, an arbitrary transparency placed immediately against the lens, and different image processing operations can be accomplished by properly designing the pupil. The field on the image plane is

~/pi(x, y) o( t ( x / M , y / M ) 9 h~(x, y) and hence the corresponding image intensity is

(5.1-11)

5 Single and Double Lens Image Processing Systems

138

Z~(x, y) - Ir

y)l: oc l t ( x / M , y / M ) 9 h~(x, y)l ~. (5.1-12)

5.2 Two-Lens Image Processing System While on the topic of impulse response and the coherent transfer function, it is instructive to discuss these with reference to a two-lens system as shown in Figure 5.2.

Object(input) plane

Fourier plane (x,y) sf (x,y )

Image(output) plane

L2

Figure 5.2 A two-lens imaging processing system.

The two-lens system is traditionally attractive for image processing because, in the configuration shown in the Figure, the Fourier transform of the input transparency appears on the common focal plane, or Fourier plane. In order to perform Fourier-plane processing on the input transparency, we can insert a transparency on the Fourier plane that will suitably modify the Fourier transform of the input transparency. The Fourier plane transparency is commonly called a spatial filter si(x , y). Example of image processing or spatial filtering using a two-lens system appear in a section later.

5.2 Two-Lens Image Processing System

139

We shall use some of the results of Chapter 3 to facilitate our derivation of the impulse response of the two-lens system. According to Eq. (3.4-5), when a transparency t(x, y) is placed on the front focal plane of lens L1 as shown in Figure 5.2, the field distribution on the common focal plane is given by, ignoring some inessential constants, T(kox/fl,koy/fl), where we have assumed that the transparency is illuminated by a plane wave. After this field distribution is modified by the spatial filter, we finally can find the field distribution on the back of lens L2, using Eq. (3.4-5) again and neglecting some constant, as

7x {T(kox/fl,

y)}

~x =k0z/f2

ky=koY/f2

kox'

- f ft(x', = t(~/M, y/M)

+

kox koy'

+

koy

kox koy ) , 9 Sj( ~ , ~

(5.2-1)

where M - - f 2 / f l is the magnification factor and Sf is the Fourier transform of s I. Comparing Eq.(5.2-1) with Eqs. (5.1-10) and (5.1-9), which describes the impulse response of a one-lens system, we see the striking similarities: the spatial filter s~ in the two-lens system can be compared with the pupil function p~ of the single-lens system, and fl and f~ take on the roles of do and di, respectively. Hence, the Fourier plane in the two-lens system is also known as the pupil plane, and the spatial filter is also called the pupil of the system. In addition, we can define the impulse response of the two-lens system or the coherent PSF as h~(.,

kox koy )

y ) - S ~ ( ~ , i~

9

(5.2-2)

The Fourier transform of the pupil is the coherent PSF of the system. The corresponding coherent transfer function is

H~(kx, k v) -

s:(7:~kXko, -:~k,ko )"

The image intensity can be found by Eq.(5.1-12), defined in Eq.(5.2-2) and with M - - f2 / fl.

(5.2-3)

where h~(x,y)

is

140

5 Single and Double Lens Image Processing Systems

5.3 Examples of Coherent Image Processing Example 1 Contrast Reversal Consider a periodic amplitude transmittance in one dimension, for simplicity, as shown in Figure 5.3(a). Suppose this is placed in the object plane of our two-lens system of Figure 5.2 and a spatial filter in the shape of a tiny opaque stop is introduced on the axis as a processing element. Avoiding all the complicated mathematics, we can argue that the spatial filter essentially removes the DC component A X 1 / ( X 1 + X2) from the object field. Assuming fl = f2 = f, the field on the image plane will have the distribution shown in Figure 5.3(b). The intensity distribution is plotted in Figure 5.3(c). For XI>X2, it is easy to see that the image intensity shows a contrast reversal over the object intensity: regions that were previously brighter now appear darker after the image processing. Of course, there is a certain amount of background illumination, and the method fails if XI<X2.

t(x)

(a)

A

9

9

9

O

9

9

9

X

5.3 Examples of Coherent Image Processing

141

~p

(b) MY 2

X: + X 2

X

AX 1 X~+X2

Intensity

(c)

(AX,) ~ ( X 1 + X2

)2

(AXe) ~ ( X 1 -at- X 2

)2

Figure 5.3 Contrast Reversal. (a) Input transparency; (b) optical field on image plane with the DC component removed; (c) intensity distribution corresponding

to (b).

142

5 Single and Double Lens Image Processing Systems

Example 2 Seeing a Phase Object It is easy to understand why it is not generally possible to see a phase object of the type t ( x ) - e j~(x), (5.3-1a) where we have assumed a 1-D object for simplicity. This is because the intensity transmittance, proportional to is equal to a constant. One way of seeing the structure of phase objects is to employ the twolens system of Figure 5.2 with a small stop on the axis on the Fourier plane as in Example 1 in this Section. Assuming r << 1,

It(x)l 2,

t ( x ) _~ 1 + j r

(5.3-1b)

so that, on the Fourier plane, the field immediately in front of the spatial filter (the small stop) is

+je(kox -7-) } , where (I)( kx ) - ,7-~{ r that

(5.3-2)

}. The opaque stop now removes the ~i(x), so

~(~) o( ~(k~ T)"

(5.3-3)

Hence, on the image plane the intensity becomes Ii o< r

x),

(5.3-4)

and the intensity variation shows the nature of the phase structure. It is also possible to monitor different derivatives of the phase, if desired. For instance, by inserting a Fourier-plane filter with the characteristic

s~(x) - ax 2,

(5.3-5)

143

5.3 Examples of Coherent Image Processing

where a is some constant, it is easy to show that the intensity variation on the image plane is 2

d2r

9

dx 2

(5.3-6)

Example 3 Removing Horizontal Lines from a 2-D Grating The input transparency is a grating represented as

t(x,y)-

(() (() rect }

x

9 E 5(x-mXo) m=--oo

rect ~

9 ~'~ (5(y- nYo) n=--

) )

,

(5.3-7)

(:x)

as shown th Figure 5.4(a). The field on the common focal plane of the image processing system in Figure 5.2 (assume f l - f 2 - f for simplicity) is proportional to the Fourier transform of the input transparency t(x, y), k~u Cp (x, y) ~ 47r2 ~xy sinc ( k~~ x ) ( sinc

x

~5

kx

2Xo ~

m=--oo

= 47c2 Xoyoko~sinc

•

~5 m=--oo

5 kv

2~,~ Yo

kz =kOX/ f

n=--oo

(

x

27rf

ky=koY/f

sinc

2~mf k0x0

5 y - ~k0y0 n=--oo

)

.

(5.3-8)

The Fourier-plane field distribution is shown in Figure 5.4(b). Suppose, now, we desire to see only the vertical lines on the image plane. As we show below, this can be achieved by placing a spatial filter on the Fourier plane of the type

5 Single and Double Lens Image Processing Systems

144

sf(x,y)-

4~f

rect ( ~ )

(5.3-9)

Yf < koYo"

This spatial filter is essentially a rectangular slit in one dimension parallel to the x-axis and is drawn in Figure 5.4(c). The field to the right of this spatial filter is thus proportional to 47r2 XoYoko ~sinc

27rf

x m--~--

koXo

,

(x)

hence, on the back focal plane of the second lens,

@i(x, y) oc rect

(x)

9 ~ (5 x - m X o

,

(5.3-10)

m~-oo

this is, we retrieve only the vertical lines, as shown in Figure 5.4(d). Horizontal lines may be retrieved similarly, by reorienting the slit along the y-axis. Before ending this example, we want to point out that the spatial filter used here had the amplitude characteristics of a 1-D low-pass filter. However, we mentioned nothing about the phase characteristics. A discussion on the realization of complex spatial filters having both amplitude and phase characteristics is presented in the next Chapter.

5.3 Examples of Coherent Image Processing

145

kY

T 0

0

0

0

0

0

0

0

9

0

0

0

0

0

0

0

0

0

0

0

0

0

0

9

0

0

0

0

0

9

0

0

0

0

0

9

9

0

9

0

9

0

9

9

0

0

0

0

9

kx

(b)

l

y

~ l ~ / j j j jjJ~

~"~~

X

Yf

(c)

146

5 Single and Double Lens Image Processing Systems

Figure 5.4 Image processing of the transparency function in Eq.(5.3-7). (a) Original transparency; (b) intensity pattern on Fourier plane; (c) spatial filter; (d) processed image with only vertical lines remaining.

5.4 Incoherent Image Processing and Optical Transfer Function Our discussion on diffraction and the Fourier-transforming and imaging properties of lenses has so far been based on the fact that the illumination of the object is spatially coherent. This means that the complex amplitudes of light falling on all parts of the object vary in unison, that is, any two points on the object receive light that has a fixed relative phase that does not vary with time. Before the advent of lasers, which form excellent coherent sources, spatially coherent illumination could be simulated by starting from an intense point source of light and collimating the light by a lens. Conversely, an object may be illuminated with light having the property that the complex amplitudes on all parts of the object vary randomly, so that any two points on the object receive light of illumination is termed spatially incoherent. Light from extended sources, such as fluorescent tube lights, is incoherent. How does this affect the results on the impulse response and transfer function of imaging systems? Many books have discussions and theoretical development on this question and in essence we summarize the results as follows [Banerjee and Poon (1991), Goodman (1996), Yu (1983)]" A

5.4 Incoherent Image Processing and Optical Transfer Function

147

coherent system is linear with respect to the complex fields and hence Equations (5.1-11) and (5.1-12) hold for coherent optical systems. On the other hand, an incoherent system is linear with respect to the intensities and to find the image intensity, we perform convolution with intensity quantities as given below:

Ii(x, y) oc It(x/M, y/M)12 9 Ibm(x, y)l 2.

(5.4-1)

]h~(x,y)l 2 is often called the intensity point spread function (IPSF) of the optical system. Whereas the coherent transfer function H~(k~, kv) relates the Fourier transform of the object field under coherent illumination to that of the image field, according to [see Eq. (5.1-11)] (5.4-2) the Fourier transform of the image intensity is related to the object intensity under incoherent illumination according to

U~y{I{(x, y) } o( G y { l t ( x / M , y/M)12} OTF(k~, ky), (5.4-3) where

OTF(k~, kv) - U~v{ [h~(x, y)[2} =

G) |

G)

(5.4-4)

is the optical transfer function of the incoherent imaging system and | denotes correlation. The O T F can be expressed explicitly as

OTF(k~, kv) - f f H~ (k'~, tc'~) H~(k" + k~, k'y + kv) dx'dy'. (5.4-5) Some general properties of the O T F follow the properties of correlation: IOTF(G, ky)l _< IOTF(0, 0)1,

(5.4-6)

OTF(-

(5.4-7)

and - G) - OTE*(kx,

5 Single and Double Lens Image Processing Systems

148

Example 4 0 T F of a Two-Lens System

Consider a two-lens system as shown in Figure 5.2 with fl - f2 - f and sf(x, y) - rect (x/X)rect(y/Y). Using Eq. (5.2-3), the coherent transfer function is (5.4-8)

H~(k~,ky)-rect Xko/f rect Vko/I "

We plot Hc(kx, kv) versus k~ with ky constant in Figure 5.5 (a); the plot versus kv can be drawn similarly. The OTF is the autocorrelation of H~ and is plotted in Figure 5.5(b). Observe that the passband of the OTF is twice as much as that of H~. This in turn implies that, using incoherent illumination, it is possible to transmit twice the range of spatial frequencies of the object than under coherent illumination; however, the spectrum of the object transmitted through the passband is modified by the shape of the OTF.

H

OTF C

(a)

kx Xk o

o

k~ Xko

2f

2f

0

Xk,,

j

f

F i g u r e 5.5 The coherent transfer function and the OTF of a two-lens system with s/(x, y) - rect (x/X)rect(y/Y)

Next, consider

sf(x y)

[ ()(x

rect xxx----~~ + rect

x~

x

Xo~

X .

5.4 Incoherent Image Processing and Optical Transfer Function

149

We plot H~(k~,ky) versus kx with kv constant in Figure 5.6(a) along with the OTF in Figure 5.6(b). Note that even though it may be positive to achieve band-pass characteristics with coherent illumination, incoherent processing always gives rise to inherently low-pass characteristics as its point spread function is real and positive [see Eq. (5.4-1)]. Much attention has been focused on devising methods to realize band-pass characteristics using novel incoherent image processing techniques [see, e.g., Lohmann and Rhodes (1978), Stoner (1978), Poon and Korpel (1979), Glaser (1987), Indebetouw and Poon (1992)]. In Chapter 7, we will discuss a technique for the so-called bipolar incoherent image processing whereby the synthesis of bipolar point spread functions in incoherent optical systems is possible.

H

(a)

I

[

xoko

xoko

f

f

Xko

Xk o

f

f

kX

5 Single and Double Lens Image Processing Systems

150

OTF

(b)

M~o f

_ 2xoko

f

0

_~

f

2xoko

f

Figure 5.6 (a) The coherent transfer function and (b) the OTF of a two-lens system /

with sf(x,

y)

--

%.

{rect[(x - xo)/X] + rect[(x + xo)/X]}rect(y/Y)

5.5 MATLAB Examples of Optical Image Processing 5.5.1 Coherent lowpass filtering The m-file, lowpass_filtering.m, simulates coherent image processing. The file first generates the "Spoke Pattern" as input image, shown in Figure 5.7. For the parameters chosen in the file and a circular spatial filter being used [Figure 5.8 (a)] in a two-lens system, Figure 5.8 (b) shows the lowpass filtered image of the input. The program also calculates the radius of the spatial filter, R[cm], in the spatial domain and its corresponding radius, kr0[radian/cm], in the spatial frequency domain.

5.5 MA TLAB Examples of Optical Image Processing

Figure 5.7 Spoke Pattern.

Figure 5.8 (a) Lowpass spatial filter.

151

152

5 Single and Double Lens Image Processing Systems

Figure 5.8 (b) Lowpass-filtered Spoke pattern.

clear %L : length of back ground %N : sampling number L=I; N=256; % dx : step size dx-L/(N-1 );

%@@@@@@@@@@@@@@@@@@@@@@ %Input a spoke pattern %spoke pattem s=256; n=40; %n is the number of spokes for x=l:s % xt=(2*(x- 1)/(s- 1))- 1; % for y=l:s yt=(2*(y-1)/(s-1))-1; % r=xtA2+ytA2; t=floor(atan2(yt,xt)/pi*n); % Spo(x,y)=(r<-0.90)*(mod(t,2)>0.5); end % end %

@ @ @ @ @ @

5.5 MATLAB Examples of Optical Image Processing

%Input image Input-Spo; % % End of input image

153

@

%@@@@@@@@@@@@@@@@@@@@@@ %Axis Scaling for k =1"256 X(k) = 1/255*(k- 1)-L/2; Y(k) = 1/255*(k- 1)-L/2; %Kx=(2*pi*k)/((N - 1)*dx) %in our case, N=256, dx=1/255 and maximum radian freq is 2*pi/dx--- 800 rad./unit length Kx(k)=(2 *pi *(k- 1))/((N- 1)* dx)- ((2 *pi* (256-1 ))/((N- 1)* dx))/2; Ky(k)=(2*pi*(k-1 ))/((N-1)* dx)-((2*pi* (256-1 ))/((N-1)* dx))/2; end %Display the input in the space domain figure( 1) image(X,Y,256 *Input) xlabel ('cm'); ylabel ('cm'); colormap(gray(256)) title('Spoke- original object') axis square %Rescaling the input image % (M =-l) %Flip the input image Input=fl ip lr(Input); Input=flipud(Input); S_Input=zeros(256); S_Input=Input; %Display image without filtering figure(2) image(X,Y,256*S_Input); co lorm ap(gray( 256)) title('image without filtering'); xlabel ('cm'); ylabel ('cm'); colormap(gray(256)) axis square %Fourier transform of the input image F_Input=( 1/256)A2" fft2(S_Input); F_Input=fftshift(F_Input);

154

5 Single and Double Lens Image Processing Systems

O~********************************************** ~*******

%circular opening at the Fourier plane corresponds to lowpass filtering * %Low pass filter * pu=zeros(256); % * Ro=60; %M determimes the size of the low pass filter * for n = 1:256 % * for m = 1:256 % * if (n-127)A2+(m-127)A2
%Display spatial filter, sf(x),in the space domain figure(4) image(X,Y,256*pu) co lormap(gray( 256 )) title('spatial filter in spatial domain') xlabel ('cm'); ylabel ('cm'); axis square %Spatial filter, sf(-f*kr/Ko), in frequency domain; %kr is spatial frequency in the r-direction and f is the focal length of the two-lens system %Hence, spatial filter is circ(-f*kr/Ko*R)=circ(-f*kr*KX/Ko*R*KX), where KX=2*pi/dx as L = 1 % Define sigma =f'*KX/Ko, we have circ(-sigma*kr/2R*KX). %Therefore, radius of circ(.) is kr0=R*KX/sigma [radian/unit length]. kr0=R*2*pi/dx % if sigma is chosen to be unity<<<<<<<. %Flip the spatial filter pu=fliplr(pu); pu--flipud(pu); S_pu=zeros(256); S_pu=pu; %Display the flter in frequency domain figure(5) im ag e( Kx,Ky ,256 *S_p u) colormap(gray(256)) title('Spatial filter in frequency domain') xlabel ('radian/cm'); ylabel ('radian/cm'); axis square

5.5 MATLABExamples of OpticalImage Processing

155

%Filtering in Frequency domain F_Output=F_Input.*S_pu; Output=ifft2(F_Output); Output=(256A2)*Output; %Output=fftshift(Output); %Display the filtered output figure(6) image(X,Y,256*abs(Output)/max(max(abs(Output)))) colormap(gray(256)) title('Filtered Image') xlabel ('cm'); ylabel ('cm'); axis square Table 5.1 MATLAB code to simulate coherent lowpass filtering: the spatial filter is a circular opening at the Fourier plane (lowpass_filtering.m).

5.5.2 Coherent bandpass filtering We simulate coherent bandpass filtering through the use of an annular filter in the Fourier plane. The MATLAB code in Table 5.1 can be revised to effect this type of filtering. Specifically, the following subprogram (Table 5.2) can be used to replace the portion of the program outlined by "***..." in Table 5.1. In addition, the line marked by " <<<<<<<" in Table 5.1 should be commented out. Figure 5.9 shows the annular spatial filter used for bandpass filtering. The width of the annular opening is controlled by parameters a and b in Table 5.2. Figure 5.10 shows the bandpassed filtered image of the Spoke pattern shown in Figure 5.7. In Table 5.3, we show a sub-program which can take bitmap file as an input image. The bitmap image file should be 256 • 256 in size in order for the program to work. The portion of the program in Table 5.1 outlined by "@@@..." should be replaced by this sub-program. The bitmap file used for the next simulation is "sq_circ.bmp" and it is shown in Figure 5.11. Figure 5.12 shows the image with M - - 1. Note that the image is inverted and flipped from left to right. Figure 5.13 shows the annular filter in the spatial domain with a - 50 and b - 1 0 and Figure 5.14 presents the bandpass-filtered image. It is interesting to point out the well-known edge detection capability of bandpass filtering as demonstrated in the Figure. To be precise, the edge of the image is at the location of zero-crossing.

5 Single and Double Lens Image Processing Systems

156

%Annular opening at the Fourier plane corresponds to bandpass filtering %Bandpass filter pu=zeros(256); % outside radius = a/255 (unit length), inside radius = b/255 a=30; b=20; for n = 1:256 for m=1:256 if (n- 127)A2+(m- 127)A2
%Input a bitmap file image Input=imread( 'sq_circ.bmp','bmp'); Input=double(Input(:,:,l)); % taking one color % End of input image Table 5.3 MATLAB sub-program to take bitmap-file as input.

5. 5 MA TLAB Examples of Optical Image Processing

Figure 5.9 Annular filter.

Figure 5.10 Bandpass filtered output of input image shown in Figure 5.7.

157

158

5 Single and Double Lens Image Processing Systems

Figure 5.11 Bitmap input image.

Figure 5.12 Imaging without filtering but with M -

- 1.

5.5 MATLAB Examples of Optical Image Processing

Figure 5.13 Annular spatial filter used for processing Figure 5.11.

Figure 5.14 Banpass filtered output of Figure 5.11.

159

5 Single and Double Lens Image Processing Systems

160

5.5.3 Incoherent spatial filtering In the last simulation, we want to show incoherent spatial filtering and demonstrate that even with an annular spatial filter in the Fourier plane, edge detection is not possible with incoherent illumination in the standard two-lens image processing system. Table 5.4 lists the MATLAB code for incoherent filtering. Using the exact annular spatial filter used for coherent filtering, as shown in Figure 5.13, the O T F is shown in Figure 5.15 and the filtered output is given in Figure 5.16. Note that the filtered output is a lowpass version of the input image, i.e., the edges of the image have been smoothed instead of emphasized as in coherent filtering.

%Simulation of incoherent filtering with annular filter in the Fourier plane %L : length of back ground %N : sampling number L=I; N=256; % dx : step size dx=L/(N - 1); .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

%Input a bitmap filte image Input-imread('sq_circ.bmp','bmp'); Input=double(Input(:,:,l)); % taking one color %End of input image .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

%Axis Scaling

for k - 1:256 X(k)- 1/255 *(k- 1)-L/Z; Y(k) = 1/255 *(k- 1)-L/Z; %Kx=(2 *p i*k)/((N- 1)* dx) %in our case, N=256, dx=1/255 Kx(k)-(2 *pi*(k- 1))/((N-1 )* dx)- ((2 *pi *(256-1 ))/((N-1 )* dx))/2; Ky(k)=(2 *pi *(k- 1))/((N-1 )* dx)-((2 *pi*(256-1 ))/((N- 1)* dx))/2; end %Display the input in the space domain figure( 1)

5.5 MA TLAB Examples of Optical Image Processing

image(X,Y,256*Input) co lormap(gray( 256 ) ) title('Input image') xlabel ('cm'); ylabel ('cm'); axis square %Rescaling the input image % (M =- l) %Flip the input image Input=fliplr(Input); Input=fl ipud(Input); S_Input=zeros( 256); S_Input=Input; %Display input image without filtering figure(2) image(X,Y,256*S_Input); colormap(gray(256)) title('Image without filtering, M =-1'); xlabel ('cm'); ylabel ('cm'); axis square %Fourier transform of the input image F_Input=( 1/256)A2" fft2(S_Input); F_Input=fftsh ift(F_Input); %Annular filter pu=zeros(256); %outside radius = a/255 (unit length), inside radius = b/255 a=50; b=10; for n = 1:256 for m=1:256 if (n- 127)/`2+(m- 127)/`2
161

162

5 Single and Double Lens Image Processing Systems

%Display filter in the space domain figure(3) image(X,Y,256*pu) colormap(gray(256)) title('Filter in the spatial domain') xlabel ('cm'); ylabel ('cm'); axis square %Flip the filter pu-fliplr(pu); pu=flipud(pu); S_pu=zeros(256); S_pu=pu; %F S pu is the Fourier transform of H(kx,ky) F S _pu=fft2(S_pu); F_OTF=F S pu.*conj(F_S_pu); S_pu=fftshift(fft2(F_OTF)); S_pu=S_pu/(max(max(abs(S_pu)))); %S_pu is the OTF of the system %Display the filter in Frequency domain figure(4) image(Kx,Ky,256*abs(S_pu)) colormap(gray(256)) title('OYF of the system') xlabel ('radian/cm'); ylabel ('radian/cm'); axis square %Filtering in Frequency domain F_Output=F_Input.*S_pu; Output=i fft2 (F_Output); Output=(256/'2)*Output; %Display the filtered output figure(5) image(X,Y,256*abs(Output)/max(max(abs(Output)))) co lormap(gray(256 )) title('Incoherent filtered Image by annular spatial filter') xlabel ('cm'); ylabel ('cm'); axis square Table 5.4 MATLAB code for incoherent fltering (Incoherent_filtering.m).

5.5 MA TLAB Examples of Optical Image Processing

Figure 5.15 OTF of the system with spatial filter shown in Figure 5.13.

Figure 5.16 Incoherent filtered image.

163

164

5 Single and Double Lens Image Processing Systems

Problems 5.1

An input transparency t(z,y) is placed at a distance d < f behind a lens of focal length f. Find the field distribution at the back focal plane if a plane wave of unit amplitude is incident on the lens.

5.2

A monochromatic plane wave is obliquely incident on a transparency t(z,y) placed on the front focal plane of the converging lens L1 of the optical system shown in Figure 5.2. Assume fl = f2 = f. (a) Calculate the complex field at the back focal plane of Li (b) If a small pinhole is placed on this placed on the lens axis, predict the shape of the intensity pattern on the back focal plane of the second lens L2 as 0 varies.

5.3

Consider an optical system like that shown in Figure 5.2, with fl = f2 - f. An object with the amplitude transmission function

f (x) - cos(27rx/a) + cos(27rx/b),

aCb,

is placed in the front focal plane of LI. In the back focal plane of the lens L2, we want to obtain a field pattern proportional to cos(27rz/a). How do we accomplish such a task? Perform the mathematical analysis in one transverse dimension. 5.4

Consider the image processing scheme of Figure 5.2. (a) What is the intensity distribution I(x,y) on the observation plane in terms of the input transparency function t ( z , y ) , assuming that there is no spatial filter placed on the common plane? Give simple reasons for your answer. (b) With f i - f 2 f, suppose that t(x, y) - e x p [ - jr y)]. Write down the expression for I(x,y). (e) With a transparency function as in (b), suppose now that a spatial filter 8 f ( X , y) -- a(x 2 _qt_y2)is introduced on the common focal plane between the two lenses. How is the resulting intensity

5.5 MA TLAB Examples of Optical Image Processing

165

related to the object phase? Do not make any approximations on

y) 5.5

Given a periodic transparency function OO

E

,

Tt------(:X:)

where T~ denotes the Fourier coefficients and K = 27r/X, where X is the repetition period, write down, using the transfer function concept of propagation, the spectrum of the complex field a distance z behind the transparency if it is illuminated with a plane wave. Hence, find at what distance z = zi the complex field becomes exactly the same as the input transparency function. (This is called a case of lensless imaging. ) 5.6

Derive the two properties of the OTF, as given in Eqs. (5.4-6) and (5.4-7) by starting from the definition of the OTF, given in Eq. (5.4-5).

5.7

For a circular pupil function pz(x, V) =circ(r/ro), find and plot the following for a single-lens imaging system" (a) the coherent transfer function; (b) the OTF

5.8

Referring to the two-lens coherent image processing system of Figure 5.2 and assume fl - f2 - f, suppose that a transparency tl(x, y) is placed on the object plane. For the spatial filter SI(X, y) of the form (a) sf(x, y) - T 2 ( k o x / f , koy/f), (b) s f (x, y) - T~ ( - kox/ f , - koy/ f ), where 7'2 (kx, k~) - 3c~v{t2 (x, y) }, find the expressions for the image intensity on the image plane in terms of tl(x,y) and t2(x, y). What type of mathematical operations does this system represent?

5.9

Consider the single-lens image processing system shown in Figure P.5.11, where F denotes the focal point of the lens with

5 Single and Double Lens Image Processing Systems

166

focal length f. Assuming a spatial filter sf(x, y) is placed on the back focal plane of the lens, find, from first principles, the response of the imaging system to a shifted delta function input,

5(x - xo, y - Yo). Object plane

s, (x,y)

F

lmagc plal~

#~i F i

5.10 Examine the MATLAB code given in Table 5.1 used for coherent lowpass filtering. Increase the radius of the circular filter by a factor of two and observe the change in the output. Explain the observed change. 5.11 Examine the MATLAB code given in Table 5.2. Change the value of b from 20 to 10 and explain the resulting filtered output.

References 5.1 5.2 5.3 5.4 5.5 5.6

Banerjee, P.P. and T.-C. Poon (1991). Principles of Applied Optics. Irwin, Illinois. Ghatak, A. K. and K. Thyagarajan (1978). Contemporary Optics. Plenum, New York. Glaser, I. (1987), In Progress in Optics, Vol. 24 ( E. Wolf, ed.) Chapter V North-Holland, Amsterdam. Goodman, J.W. (1996). Introduction to Fourier Optics. McGrawHill, New York. Indebetouw, G. and T.-C. Poon (1992), Optical Engineering 31 2159. Lohmann, A.W. and W.T. Rhodes (1978). Applied Optics 17 1411.

5.5 MATLAB Examples of OpticalImage Processing 5.7 5.8 5.9

167

Poon, T.-C. and A. Korpel (1979). Optics Letters 10 317. Stoner, W. (1978). Applied Optics 16 265. Yu, F. T. S. (1983). Optical Information Processing. Wiley, New York.

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169

Chapter 6 Holography and Complex Spatial Filtering

6.1 6.2 6.3 6.4 6.5 6.6

Characteristics of Recording Devices The Principle of Holography Construction of Practical Holograms Reconstruction of Practical Holograms and Complex Filtering Holographic Magnification Ray Theory of Holograms: Construction and Reconstruction

In the previous Chapter, we discussed the Fourier-transforming and imaging properties of lenses, the frequency response of imaging system under coherent and incoherent illumination, and simple image processing systems. Recall that image processing involves placing a suitable Fourier-plane spatial filter. For instance, in Example 1 of Section 5.3, we employed a spatial lowpass filter (at least in amplitude) to achieve the required processing. Those familiar with liner systems will remember that an ideal lowpass filter has not only a specified amplitude characteristic, but a phase characteristic as well. This leads to the question of how we can realized complex spatial filters in general for image processing purposes. In this Chapter we will address this topic along with a related area called holography. In that connection, we will first briefly introduce the properties of photographic films and other recording devices that are usually used in the construction of spatial filters and holograms.

6 Holography and Complex Spatial Filtering

170

6.1 Characteristics of Recording Devices We will briefly discuss the properties of three kinds of recording devices: (a) photographic film, (b) photorefractive materials, and (c) photopolymers. Every photographic film comprises a transparent base with a coating of a photosensitive emulsion of silver halide on top. When the film is exposed to light, the silver halide changes to metallic silver, which is opaque at optical frequencies. The process of fixing removes the unexposed halide by dissolving it away; hence, those parts of the film that receive no light become transparency after the fixing process. The entire process is much more gradual than it sounds, and parts of the film receiving more light appear darker than the parts that receive less light. This analog nature comes from the fact that the halides are deposited as micrograins on the transparent base, so that a certain amount of light energy can only enable the chemical reaction to occur in some of these grains. The transparency produced after fixing is called the negative, for reasons that are obvious from the preceding paragraph. When the negative is exposed to light, more light is transmitted through those regions on the negative that are more transparent. A figure of merit for this is the intensity transmittance: 7-,~ (x, y) -

average{

intensity transmitted at (x,y) intensity incident at (x,y)

}'

(6.1-1)

where the subscript n signifies the fact that we have a negative transparency and the average is taken over several film micrograins. During the exposure process, the greater the amount of light incident on the photographic film, the more silver is formed, and hence, Tn is comparatively lower. That is, T~ decreases with increasing exposure H defined as the product of the exposure time T and incident light intensity I(x, y). A standard procedure for quantifying all this is to define the photographic density D,

D ( x , y ) - lOgl0(1/7"n).

(6.1-2)

The variation of D with lOgl0H is usually specified for every film. Typical variations, called the Hurter-Driffield (H &D) curves, are shown

6.1 Characteristics of Recording Devices

171

in Figure 6.1. Note that the solid curve has a linear region with a slope ")In, which is most useful for recording purposes. The 7~ depends on the following: a) the type of film emulsion (i.e., the silver halide); b) the type of developer used; c) the developing time. Based on the type of film emulsion, we can have negatives of different c o n t r a s t s . A higher-contrast film has a larger value of ")in, as shown by the dashed curve in Figure 6.1.

D

7 Higher-contrast film

...." ..-

..:..'./"".:..:." -

lOglo H

D o

Figure 6.1 The H and Dcurves of typical films.

In the linear region (see Figure 6.1), D = 7 ~ l o g l o H - Do = 7 ~ l o g l o l T - Do,

(6.1-3)

6 Holography and Complex Spatial Filtering

172

so that, from Eqs. (6.1-2) and (6.1-3), V) -- I(n[Xn(*,

/G

-- 1 0 + D ~

(6.1-4)

Often, it is desirable to obtain a positive (transparency), which is a replica of the object. To achieve this, we photograph the first negative transparency (say, %1), producing a second negative transparency. This resulting transparency is now a positive copy of the original object and is characterized by an intensity transmittance 7p, where

7"p(X, y) --

Kn2(Knl [I(x, y)]--"Ynl)--"~n2

= Kp I(x, y)]-7~ ,

(6.1-5)

where % = ")/nl")/n2,/(p is a constant and % represents the overall gamma (7) for the two-step process. In Eq.(6.1-5) the subscripts 1 and 2 refer to the first and second stages involved in producing the final transparency. For most image processing or holographic applications, an overall 7 (= %) of 2 is desired. This means that the amplitude transmittance t(z, y) of the (positive) transparency is

tp(X, y) c~ I(x, y),

(6.1-6)

where I(x, y) is proportional to the intensity of the light incident on the original film from which %1 was constructed. In this sense, photographic films record amplitude holograms from incident intensity patterns. Photorefractive materials have been introduced in Chapter 4 in connection with wave propagation through these materials. In diffusion dominated PR materials, the induced change in refractive index A n is vz where I is the intensity distribution approximately proportional to ~/~+I, on the material. An examples of such a material is BaTiO3. In some other materials, such as LiNbO3, the photovoltaic effect predominates. In this case, A n is approximately proportional to I. In both cases, note that the material acts as an induced phase grating when an interference pattern falls on the material. Thus, PR materials can be effectively used to store phase holograms. Holograms can be written and read from some of these materials simultaneously. In some cases, holograms can be erased from the material by shining uniform illumination (coherent or

6.2 The Principle of Holography

173

incoherent) on the material. In other cases, the hologram written is semipermanent, and can be removed by heating the material beyond a critical temperature. Other examples of PR materials are KNbO3, BSO, SBN, and polymers. The last three require a bias voltage across them to aid in the recording process. For more information on PR materials, the reader is referred to Yeh (1993). Photopolymers provide a convenient recording material which can store phase holograms. Large-scale manufacture of these photopolymer films comprising photopolymerizable monomers in a safe, easy, dry process by DuPont has made commercial applications of holography possible. The recording from an interference pattern changes the thickness as well as the refractive index of the material. When exposed to an intensity pattern, polymerization of the monomers occurs more in the regions of higher intensity, with subsequent diffusion to regions of lower concentration. This leads to the phase modulation of the ensuing hologram. Commercial uses for the technology are in the areas of 3-D or holographic imaging, electronic displays fabrication of holographic optical elements (HOEs), etc.

6.2 The Principle of Holography Holography is like 3-D photography to most people. However, the use of holography to store 3-D optical information is probably one of the most important aspects from the scientific point of view. Holography has been an important tool for scientific and engineering studies, and it has found a wide range of applications. A greater and greater number of biologists, chemists, crystallographers, electrical engineers, mechanical engineers, physicists, and others use holography. A photograph is a 2-D recording of a 3-D scene. What is recorded is the light intensity at the plane of the photographic recording film, the film being light sensitive only to the intensity variations. Hence, according to Eq. (6.1-6),the developed film's amplitude transparency tp(x, y) o( I(x, y) -I~)p] 2, where r is the complex field on the film. As a result of this intensity recording, all information about the relative phases of the light waves from the original 3-D scene is lost. This loss in the phase information of the light field in fact destroys the 3D character of the scene, i.e., we cannot change the perspective of the image in the photograph by viewing it from a different angle ( i.e.,

174

6 Holography and Complex Spatial Filtering

parallax) or we cannot interpret the depth of the original 3-D scene. Holography is a method invented by Gabor in 1948, in which not only the amplitude but also the phase of the light field can be recorded. The word "holography" combines parts of two Greek words: holos, meaning "complete," and graphein, meaning "to write or to record." Thus, holography means the recording of complete information. Hence, in the holographic process, the film records both the amplitude and phase of the light field. The recorded film is now called a "hologram". When a hologram is properly illuminated, an exact replica of the original 3-D wave field is reconstructed. The principle of holography can be explained by using a point object since any object can be considered as a collection of points. We shall discuss holographic recording of a point object as an example. Figure 6.2 shows a collimated laser split into two plane waves and recombined through the use of two mirrors (M) and two beam splitters (BS). One plane wave is used to illuminate the pinhole aperture (our point object), and the other illuminates the recording film directly. The plane wave that is scattered by the point object generates a diverging spherical wave. This diverging wave is known as an object wave in holography. The plane wave that directly illuminates the photographic plate is known as a reference wave. Let ~bo represent the field distribution of the object wave on the plane of the recording film, and similarly let ~ represent the field distribution of the reference wave on the plane of the recording film. The film now records the interference of the reference wave and the object wave, i.e., what is recorded is given by [~r + ~o[ 2, provided the reference wave and the object wave are mutually coherent over the film. The coherency of the light waves is guaranteed by the use of a laser source and ensuring that the path difference between the two paths is less than the coherent length of the laser. This kind of recording is commonly known as holographic recording, distinct from a photographic recording in which the reference wave does not exist and hence only the object wave is recorded.

6.2 The Principle of Holography

175

laser source

<---

--> lens

BS

reference wave

ii .....

..,..

recording film p,-

pinhole aperture

object wave

Figure 6.2 Holographic recording of point object.

Now, referring back to our point object example, let us model the point object at a distance z0 from the recording film by an offset delta function, i.e., r y ) - 6 ( x - xo, y - yo). According to (3.3-14), the object wave arises from the point object on the film is given by r

-

~ ( ~ - xo, y - yo) 9 h ( ~ , y; z)

= exp( -

jk0 exp{ jkozo)2-~o

-

jko[(x

-

x0) 2 _Jr_

(y

yo)2]/2zo}.(6.2-1)

This object wave is a spherical wave. For the reference plane wave, we assume that the plane wave has the same phase with the point object at a distance z0 away from the film. Its field distribution on the film is, therefore, r - aexp( - jkozo), where a is the amplitude of the plane wave. Hence the intensity distribution

6 Holography and Complex Spatial Filtering

176

being recorded on the film, or the transmittance of the hologram, given by

tp(X, y) 0(. ]ff)r nt- r = la + -

where A -

is

2

jk0 exp{ 2--~0

jko[(x

-

Xo) 2 + (y - y0)2]/2z0}12

(6.2-2)

A + Bsin{ ~k0 [ ( x - Xo )2 + ( y - y o ) 2 ] } , q2+

(2k0z0)2and B -

~z0k-~ This expression is called a

Fresnel zone plate which is the hologram of a point object and we shall call it a point-object hologram. Note that the center of the zone plate specifies the location x0 and y0 of the point object and the spatial variation of the zone plate is governed by a sine function with a quadratic spatial dependence. Hence, the spatial rate of change of the phase of the zone plate, i.e., the fringe frequency, increases linearly with the spatial coordinates, x and y. The fringe frequency also depends on the depth parameter, z0. In communications and radar, a 1-D version of this signal is known as a chirp signal. If the reference wave were not there as in conventional photography, what is recorded on the film is k0 )2 ' which is a uniform intensity pattern. The 3t~(x, y ) ~ 1r 2 - (~--~o D information, i.e., x0, Y0 and z0, is mostly lost. Figure 6.3 shows some typical holograms of an off-axis point object for some positive values of x0, Y0 and z. Indeed we see that the fringe frequency contains the depth information, z, whereas the center of the zone, x0 and Y0, defines the transverse location of the point object. Table 6.1 presents the MATLAB code to generate Figure 6.3. For an arbitrary 3-D object, we can think of the object as a collection of points, and therefore, we can envision that we have a collection of zones on the hologram, with each zone carrying the transverse location as well as the depth information of each individual point. In fact, a hologram has been considered as a type of Fresnel zone plate and the holographic imaging process has been discussed in terms of zone plates [Rogers (1950)].

6.2 The Principle of Holography

177

Figure 6.3 Off-axis Fresnel Zone Plate (hologram due to an off-axis point object) : (a) for z = zo; (b) for z = zo/2.

%Fresnel zone plate FZP.m %display function is 1+sin(sigma*((x-x0)A2+(y-y0)A2)). All scales are arbitrary. % x0=y0=5.0, sigma-pi/(wavelength*z)

178

6 Holography and Complex Spatial Filtering

clear; ROWS=256; COLS=256; colormap(gray(255)) z=8 sigma= 1/z; y=-12.8; for r=l :COLS, x=-12.8; for c= 1:ROWS, %compute Fresnel zone plate FZP (r, c)=exp(j *sigma*(x-5 .)*(x- 5 .)+j *sigma*(y+ 5 .)*(y+5 .)); x=x+.l; end y=y+. 1; end %normalization max 1-max(FZP); max2-max(max 1); scale=l.0/max2; FZP=FZP.*scale; image( 127"( 1+imag(FZP))); axis square on axis off Table 6.1 MATLAB code to generate the Fresnel zone plate. So far, we have discussed the transformation of a point object to a zone plate on the hologram and this corresponds to a recording or coding process. In order to obtain the point object back from the hologram, we need a reconstruction or decoding process. This can be done by simply illuminating the hologram with a so-called reconstruction wave. Figure 6.4(b) corresponds to the reconstruction of a hologram of the point object located on-axis, i.e., for the simple case where x0 - Y0 - 0. Note that in practice, the reconstruction wave usually is identical to the reference wave, therefore we can assume the reconstruction wave to have a field distribution on the plane of the hologram given by Crc(Z,y)=a. Hence, the field distribution of the transmitted wave immediately after the hologram is Crctp(x, y) = atv(x, y)and the field at arbitrary distance of z away is, according to Eq. (3.3-14), given by the evaluation of atp(X, y) 9h(x, y; z). For the point-object hologram given by (6.2-2), we have, after expanding the sine term of the hologram

6. 2 The Principle of Holography

tp(x, y)-A

+ ~ {exp{j2~0 [ ( x -

179

xo) 2 + ( y - yo)2]}

} exp{ - j ~ k0 z o [ ( x - xo) 2 + ( y - yo)2]}

(6.2-3)

Therefore, we have three terms resulting from the illumination of the hologram by the reconstruction wave. These contributions, according to the convolution operation, a t ; ( x , y) 9 h ( x , y; z ) , are as follows: First term 9 a A 9 h ( x , y; z = zo) = a A , ( z e r o - o r d e r b e a m ) .

(6.2-4a)

Second term : exp{j2-~o [ ( x - xo) 2 + ( y - yo)2]}, h ( x , y; z - zo) ~5(x - xo, y - Yo), (real i m a g e ) .

(6.2-4b)

Third term" exp{ - J2--~o [(x - xo) 2 + (y - yo)2]}, h ( x , y; z 8( x - xo , y - yo) , ( v i r t u a l i m a g e ) .

- zo)

(6.2-4c)

In writing Eq. (6.2-4c), we have back-propagated the field immediately behind the hologram by a distance z0 to demonstrate that a virtual image forms in front of the hologram. Optical fields from this virtual image correspond to a diverging wave behind the hologram as illustrated in Figure 6.4(b). In the terminology of holography, the first term is the zero-order beam due to the bias in the hologram. The result of the second term is called the real image and the third term is the virtual image. Note that the virtual image and the real image are located at a distance z0 in front and back of the hologram, respectively. Figure 6.4 serves to summarize the holographic recording and reconstruction situation.

180

6 Holography and Complex Spatial Filtering

Plane reference wave

" Film

~

object Point

-_ i i i i i i i i -~z22-]~:""

Zo

Figure 6.4(a) Holographic recording.

point-object hologram reconstruction wave -

w,-

b,~ v

v

"

A v

observer

--

v

virtual image

real image

Figure 6.4(b) Holographic reconstruction.

6.2 The Principle of Holography

181

reference plane wave

/

recording film

object wave (a) hologram of three-point object

reconstruction pane wave

9

9

O

virtual image

real image (b)

Figure 6.5 Holographic recording and reconstruction of three-point object. Figure 6.5 shows the holographic recording of a 3-point object and its reconstruction. Note that the virtual image appears at the correct 3-D location of the original object, while the real image is the mirror-image of the original object, with the axis of reflection on the plane of the hologram. Indeed, the original 3-D wavefront has been completely stored and now reconstructed. This is further explained by inspecting Eq. (6.2-2). By expanding Eq. (6.2-2), we have t , ( x , v) -

ICr + r 2

= I~r 12 + 1~o12-+- ~r~o -+- ~o~r"

(6.2-5)

182

6 Holography and Complex Spatial Filtering

Note that the original object wavefront (the fourth term) ~bo has been successfully recorded (amplitude and phase) on the hologram. When the hologram is reconstructed, i.e., the hologram is illuminated by ~Pr~, the transmitted wave immediately behind the hologram is Crctp(x,y) = Crc[l~r [2 + [r + ~ o + ~o~br] 9 Assume for simplicity that ~c = ~b~ = constant- c on the film, such as the use of plane waves for recording and reconstruction, we therefore have ~br~tp(x,y)C[[~r ]2 + ]r + C2~o + C2~o]. The last term is identical to, within a constant multiplier, the object wave which was present on the plane of the hologram when the hologram was recorded. If we view the reconstructed object wave, we would see a virtual image of the object precisely at the location where the object was placed during recording with all the effects of parallax and depth. The term that is proportional to the complex conjugate of ~bo is responsible for the generation of the real image (also known as the twin image). Physically, it is the mirror image of the 3-D object as seen previously. Finally, the first two terms, i.e., the zero-order beam, is a space-variant bias term as ~o is a function of x and y in general. This would produce a space-variant background (noise) on the observation plane. Up to this point, we have discussed the recording and reconstruction of a point object using the principle of the so-called onaxis holography. The term "on-axis" refers to the use of a reference wave coaxially illuminating the hologram with the object wave. Although this technique can record 3-D information of the object, it has an annoying effect when viewing the reconstructed virtual image, the real image is also reconstructed along the viewing direction [see Figure 6.4(b)]. This is the so-called "twin-image problem" in holography. In addition, the zero-order beam, which is space-variant, produces its annoying diffraction along the same viewing direction. Off-axis holography is a method devised by Leith and Upatnieks in 1964 to separate the twin-image as well as the zero order beam from the desired image. To achieve off-axis recording, the reference plane wave is now incident on the recording film off-axis. Conceptually, referring back to Figure 6.2, this can be done by simply, for example, rotating beamsplitter between the pinhole aperture and the film in the clockwise direction such that the reference plane wave is incident on the film at an angle. The situation is shown in Figure 6.6(a).

6.2 The Principle of Holography

183

ence wave x

0

-

recording film

object wave

Figure 6.6(a) Recording with off-axis reference plane wave. The point object is z0 away from the film.

reconstruction wave ,..

hologram

g--

v

observer

virtual image

/

zero-order lzezm

real image

Figure 6.6(b) Holographic reconstruction of off-axis hologram.

6 Holography and Complex Spatial Filtering

184

Mathematically, the recording now reads tp(X, y) - ICr-4- ~ol 2, where ~r is now an off-axis plane wave given by a exp(jkoxsinO), and ~o is the spherical wave generated by an on-axis delta function. Similarly to Eq.(6.2-2), we now have, with x0 - Y0 - 0, jko exp{ tv(x , y) - [ a exp(jkoxsinO) + 2-~zo

-

j k o [ X 2 -+-

y2]/2zo} 12

(6.2-6)

ko [X2 _+_y2 ] + koxsinO} = A + B sin{ ~zo :

A + ~ {exp[j(~k~ [ x 2 + y2] +koxsin0)] exp[k -- j( /Co [X2 _t__y2] "-t- koxsin0)l }

k0 )2 and B - ~z--g" ~ko tp(x , y) given by Eq.(6.2-6) is where A - a 2 + (2-77o called an @ a x i s hologram. Again we will have three terms resulting from the illumination of the hologram by the reconstruction wave. By illuminating the hologram with a reconstruction wave identical to the reference wave, we have Crctp(x, y) immediately after the hologram, where ~ c - a e x p ( j k o x s i n O ) - Cr. Performing the convolution operation, Crctv(x , y ) . h(x, y; z), as in the case of on-axis holography, the three contributions are as follows:

First term" A a exp(jkoxsinO) 9 h(x, y; z = zo) exp(jkoxsinO), (zero-order beam)

(6.2-7a)

Second term: a exp(jkoxsine) exp [j(2@o Ix2 -+- y21 q_ koxsinO)] 9 h(x, y; z - zo)

6.2 The Principle of Holography

185

5( x + 2zosinO, y), (real image)

(6.2-7b)

Third term"

aexp(jkoxsinO)exp[- j( ~o ko [x2+ y2 ] + 6( x, y), (virtual image)

ko xsinO)]. h(x, y ; z

=

-

zo)

(6.2-7c)

The situation is shown in Figure 6.6(b). For sufficiently large 0, the three beams propagate along different directions, thus the virtual image can be viewed without the disturbances from the zero-order beam and the real image. This technique is also known as carrier-frequency holography as the holographic information is now riding on a spatial carrier at frequency kosinO/2rc = sin0/A0 [see the argument of the sine term in Eq. (6.2-6)]. For realistic parameter values, 0 = 45 ~ and Ao = 0.6pro for red laser light, sinO/Xo ~ 1,000 cycle/mm, this technique translates to a film resolution of about 1000 lines/mm in order to be able to employ this technique for holographic recording - a major drawback if the carrierfrequency hologram is to be displayed on some sort of spatial light modulator for real-time coherent reconstruction. A 2-D spatial light modulator is a device with which one can imprint a 2-D pattern on a laser beam by passing the laser beam through it (or by reflecting the laser beam off the device). A liquid crystal television (LCTV) (upon suitably modification) is a good example of spatial light modulators. In fact, we can think of a spatial light modulator as a real-time transparency because one can update 2-D images or holograms upon the spatial light modulator in real time without developing films into transparencies. Figure 6.7 illustrates the holograms obtained with on-axis and off-axis holographic techniques for a single on-axis point object. By comparing the two holograms, we see that off-axis recording places a stringent resolution requirement on recording medium. Indeed, on-axis holography seems to be prevalent nowadays, as it presents advantages such as the lower resolution requirement of spatial light modulators than off-axis alternatives for real time applications [Piesun et. al. (1997), Poon et al. (2000)].

186

6 Holography and Complex Spatial Filtering

Figure 6.7 Hologram of on-axis point object by (a) on-axis recording; (b) off-axis recording.

6.3 Construction of Practical Holograms In this Section, we consider two examples of the construction of off-axis holograms. Figure 6.8 shows the first example. The lens L forms a coherent beam from the point source, part of the which illuminates the object t(x, y). A prism bends the other part of the

187

6. 3 Construction of Practical Holograms

coherent beam through an angle 0 and helps provide the off-axis plane wave reference. The two fields, one from the object and assumed equal to ~2o(X,y) at the plane of the film [which is the Fresnel diffraction of t(x,y)] and the other forming the reference and equal to ~ ( x , y ) ~b~oexp(jkoysinO) (~b~oreal for simplicity), interfere on the plane of the film. Any relative phase difference between these two fields, due to varying propagation distances, can be included in a complex constant K multiplying ~o(X, y). We also assume that the amplitudes of the optical field incident on the object t(x, y) and the prism are equal. If a positive transparency tv(x, y) with an overall 3' of 2 is once again constructed, then

y)

+ I :t: + KCror

y)l: y)exp( -

jkoysinO)

+ K~o~o (x, y)exp(jkoysinO). (6.3-1)

Film

i t(x,y)

Z

Figure 6.8 Recording of Fresnel-hologram.

Equation (6.3-1) therefore gives the amplitude transmittance of the

Fresnel hologram constructed from the object t(x, y). As a variation of the setup of the off-axis hologram, consider the arrangement shown in Figure 6.9, with which we now intend to record

6 Holographyand Complex Spatial Filtering

188

the Fourier transform of the object. This is achieved by the lens L2, which projects the Fourier transform of a transparency h(x, y) on the plane of the film. The hologram will now have a transparency function

tp(x, y) oc r

+ IK~I 2 IHI 2 + Klr

- jkoysinO)

+ K;2/,,.oH*exp(jkoysinO), where H = H ( k o x / f , koy/f) -

.T'~y{h(x, y)}

(6.3-2) (6.3-3) kz =kox/f,

ky=koY/f

and where K1 is once again some complex constant. Thus, the hologram contains the magnitude and phase information of the Fourier transform of the transparency h(x, y) and is called the Fourier-transform hologram of h(x, y), or a complex spatial filter if intended for use in coherent image processing applications.

Y

0

(

Figure 6.9 Recording of Fourier-transform hologram.

_

Film

6. 4 Reconstruction of Practical Holograms and Complex Filtering

189

6.4 Reconstruction of Practical Holograms and Complex Filtering In this Section, we investigate the effect of illuminating the hologram just discussed with different complex optical fields. First, consider the Fresnel hologram as in Eq. (6.3-1) to be illuminated by a collimated beam, as shown in Figure 6.10. Relevant terms in the expression will be proportional to the third and fourth terms in Eq. (6.3-

1). tp(X, y)

i ' - ,

Plane-wave v--J~ illumination j i

I

I

i /

image i

~

11

1%

.,(

I

I I \

"~ %

~ ..,. , ~ " ~

Real irrage

1

Figure 6.10 Reconstruction of Fresnel hologram using a plane wave.

Recall that r is a complex field, due to the Fresnel diffraction of the light after it passes through the original transparency t(x, y), on the plane of the film. Hence, using Eqs. (3.3-14) and (6.3-1), the first relevant term, that is, the third term in Eq. (6.3-1), is, explicitly,

6 Holography and Complex Spatial Filtering

190

e-jko(z+sinyO) jkoK 27rz if)to

x f f t ( x ' , y')exp [-jk0~(x - x') 2 + y -

y')2)l dx'dy',

(6.4-1)

and this complex field represents the field distribution at distance z after the hologram. In the context of reconstruction, this term gives rise to a virtual image, as shown in Figure 6.10, because diverging spherical waves appear to emanate from the location of the virtual image, with the nominal direction of propagation at an angle 0 to the axis. Correspondingly, the fourth term in Eq. (6.3-1) is e-jko (z-sinOy) -jkoK.______~* Cro 27rz

• fft*(x',y')exp[

(x-

y,)2)ld ,dy,"

(6.4-2)

Note that this expression represents a collection of converging spherical waves with nominal direction of propagation at an angle - 0 to the axis. We now show that this complex field, after propagating a distance z ~, can give rise to a real image as shown in Figure 6.10, by employing the transfer function concept of propagation, as enunciated in Eq. (3.3-13). The Fourier transform of Eq. (6.4-2) is proportional to 2 T*( - kx, - kv)exp[- j (k 2 + ky)z/2ko], where we have disregarded the exponential before the integral for simplicity; hence, after propagation through a distance z ~, the Fourier transform of the complex field is 2 T * ( - k ~ , - k y ) e x p [ - j ( k 2 + kv)(z z')/2ko]. When z ' - z, the complex field, therefore, will be t*(x,y), a real image as discussed previously in connection with Eq. (6.2-5). Next, let us look at an application of the hologram described by Eq.(6.3-2). Remember that in chapter 5, we perform spatial filtering in the two-lens system by placing a spatial filter in the Fourier plane. The notion of complex spatial filtering becomes clear as we use holograms as spatial filters, and hence, the filter becomes the so-called complex filter. Assume, for instance, that the hologram given by Eq.(6.3-2), is inserted on Fourier plane of the two-lens system shown in Figure 6.11.

6. 4 Reconstruction of Practical Holograms and Complex Filtering

I

I

r . ,....~ i

o 0

. ,.....~

o

= 0

. ,....~

o

=

0 tJ o 0

.mq

o,,.~

0 ~J m

|

o o o,,,~

i

N o,..~

<

191

6 Holography and Complex Spatial Filtering

192

Now let a transparency g ( x , y) be placed in the front focal plane of lens L1 and illuminated with a plane wave. Then the field incident on the complex filter is K2G(kox/f, koy/f), where /(2 is a complex constant. This gets multiplied with the transparency t p ( x , y ) o f the complex filter as in Eq. (6.3-2). Hence, on the back focal plane of lens L2, the complex field is

y) o< 7x

{at (x, v)}

7xy{G[r

+ IKll 2

IH[ 2 +

+ K;r

KlCroHexp( - jkoysinO) (6.4-3)

} kz =koz / f , ky=kOV/f

Once again, the relevant terms are the third and fourth terms in Eq. (6.43). For instance, we can express the third term as Cpl (x, y) oc. .7"xy { G ( k o x / f

, koy/f)H(kox/f

, koy/f)

x exp( - j k o s i n O y ) } kz = k o z / f

ky=koy/f

cx 9( - x, - y) 9 h( - x, - y) 9 5 ( x , y - fsin0)

=9(-x,

-y),h(-x,

-(y-fsin0)),

(6.4-4)

whereas the fourth term yields r

(x, y)

oc h( - x, - y) |

g( - x, - (y + f sinO) ).

(6.4-5)

Notice that we have implemented two important mathematical operations optically, convolution and correlation, and with these two results centered around y = + fsin0, respectively, on the observation plane. Also, note that the first and second terms in Eq. (6.4-3) contribute to complex fields of the forms g( - x, - y) and h ( - x, - y ) , h * ( x , Y)*g( - x, - y), both of which are centered around

6.5 HolographicMagnification

193

y = 0. This is shown in Figure 6.11. For spatial separation of the undeviated light and the convolution and correlation, it is imperative to make 0 sufficiently large.

6.5 H o l o g r a p h i c

Magnification

Wavefront reconstruction is, in general, three-dimensional in nature. In what follows, we study the lateral and longitudinal magnifications of the holographic image. It is sufficient to use point sources for reference and reconstruction and to restrict ourselves to point objects, because a three-dimensional object can be represented as an ensemble of points.

-~

R

x=O

l

a

(a)

~1 Film

194

6 Holography and Complex Spatial Filtering

tp(X, y) X

l

Zr2

@

6

x=oT ~ Z0

(b) Figure 6.12 Geometry to find lateral and longitudinal holographic magnifications: (a) recording; (b) reconstruction. Consider the geometry for recording shown in Figure 6.12(a). The two point objects 1 and 2 and the reference source 3 emit spherical waves that, on the plane of the film, contribute to complex fields ~;1, ~p2, and ~;3, respectively, given by

~pl

--

Alexp ( - jkl

ff)p2 - -

Azexp

if)p3

A3exp

-

-

( - j k l { ( R + d) + 2(R+d) 1 [(x + ~

(

- jkl

{ 1

ll + ~l[(X -+- a) 2 -t- y2]

))

+ y2

,

1})

, (6.5-1b)

(6.5-1c)

where A1, A2, and A3 are complex constants, and where we have assumed kl ( - 27r/A1) to represent the wave number of the three waves. These three waves interfere on the plane of the film, which if recorded and converted into a positive transparency with an overall "7 of 2, yields the hologram

6.5 HolographicMagnification tp(X, y) o( (r

+r

195

+ Cp3)(r

(6.5-2)

+ Cp~ + Cp~).

Rather than write down the expression of tp(x, y) explicitly, we will, on the basis of our previous experience, pick out some relevant terms. The terms of interest are

tprell -- ~)p*lCp3 R-f1)+ 2~R[(x-- h)2+y 2] 2/1 [(X +

+

(6.5-3a)

]

1[( h) 2]

- - A ~ A 3 ( + j k l { (R + d - / 1 ) + 2(R+d)

+ y2

x+g

l [(x + a)2 + y2] }) 21~ tpr~13 --

(6.5-3b)

ff)plff)p3 (tprell ) ,

(6.5-3C)

tp~d4 -- ~p2~p*a -- (tp~d~)*.

(6.5-3d)

Suppose, now, that the hologram just constructed is illuminated with a reconstruction wave from a point source 4, as shown in Figure 6.12(b), and having a wave number k2 = (27r/A2). Then the complex field ~hp4 illuminating hologram is ~)p4

--

A4exp

(

- jk2

{12 + 1

~ [ ( x - b) + y2]

})

,

(6.5-4)

where A4 is a complex constant. We find the total field immediately behind the hologram by multiplying Eq. (6.5-4) with Eq. (6.5-2), relevant terms which are

196

6 Holography and Complex Spatial Filtering

i=

ff)pR rel~ -- ~)p4 tp reli ,

1,2,3,4,

(6.5-5)

where the tp~d~s are defined in Eqs. (6.5-3a)-(6.5-3d). Consider, first, the contribution from r ~d, and CpR ~d~. After propagation through a distance z behind the hologram, these fields are transformed according to the Fresnel diffraction formula. Note also that because both these fields are converging [see Eqs. (6.5-3a) and (6.5-3b)]. they will contribute to real images. Explicitly, the fields can be written [using Eq. (3.3-14)] as

[ (

Cp~l~ oc exp jkl

21~[(~ +

{

(R-11)+~-~

a) ~ + y~]

9e x p [ - j k 2 ( z +

1 [(

x-

}) ( { exp

- jk~ Z~ + ~ [ ( ~ - V)~+y ~]

})] (6.5-6a)

~fv2) ] z

I ( {

1 [(

)2 +y21

1 [(X + a)2 + y2}) ] exp ( -- jk2 { 12 + ~1[ ( x

2/1

9exp - j k 2

z+

2~

- b) 2 + y2]

})]

9

(6.5-65) Recall now that the convolutions above formally demand that we perform integrations by rewriting the functions in convolution with new independent variables x', y' and (x - x'), ( y - y'), respectively. It may be possible to equate the coefficients of x~2, y ~2, appearing in the exponentials, to zero, thus leaving only linear terms in x ~, y~. In fact, doing this for Eq. (6.5-6a) gives k__!_~ k~ k2 k~ = 2R 211 2/2 2z~ 1

0,

(6.5-7a)

6. 5 Holographic Magnification

197

where we have replaced z by z~. From Eq. (6.5-7a), we can solve for zr~to get

[( ) ] kl

z,~= ~

1

R

n

1

~

AiRlll2

-V~

A21112-(A212-+Alll)R

(6.5-7b)

'

where we have replaced the ks by the corresponding Is. At this distance, from Eq (6.5-6a),

@p rell (X ~,~, exp j

kl

2R

l~ -+- k2 F2 -+- ~

x

--00

+

O( ~

k2----TYt})Zr~ dxtdy'

X -31- Zrl

12

2k2 R

k2 11

~/

'

which is a (5 function shifted in the lateral direction and is the real image of the point object 1. A similar analysis of Eq. (6.5-6b) reveals that this is also responsible for a real image, expressible as

ff)p rel2 O( (~ Z + Zr2

~2 ~ 2k2 R+d

k2 ll

'y

'

with

[( k]

z~-

1

~ R+e

---

1

)1 1

ii -V~

)kl(ll~nLd)lll2 A21112-(A212+)~lll ) ( R + d ) .

(6.5-10)

Alternatively, we could find ~)p rel2and z~by comparing Eq. (6.5-6b) with Eq. (6.5-6a) and noting that we only need to change R to R + d and h t o - h to derive the former from the latter; hence, the same changes in Eqs. (6.5-7b) and (6.5-8) would readily yield Eqs. (6.5-10) and (6.5-9),

6 Holography and Complex Spatial Filtering

198

respectively. This gives the real image of the point object 2. These two real images are shown in Figure 6.12(b). We are now in a position to evaluate lateral and longitudinal magnifications. For instance, the longitudinal distance (along z) between the two images is zr~-z~,, so that the longitudinal magnification is M ~ long

=

(6 5 - 1 1 a )

z~2-z~ d

,.,.,

AIA2(/1/2) 2

-- (A21112_A2R12_A2Rll)2

(6.5-11 b)

using Eqs. (6.5-7b) and (6.5-10) and assuming R >> d. We evaluate the lateral distance (along x) between the two images by taking the difference between the locations of the two (5 functions in Eqs. (6.5-8) and (6.5-9), so that the lateral magnification is

Zr2 (b-nt---~--h 12 2k2 R+d-- kla)--Zrl(b k2 --ll ~2- 2k2 kl Rh -- kla) k2 ~'1 Ml'~t

-

--

h

[(b

.~a) ~2--A1 ~-1

1 )A2h] (Zr2-Zrl)+2(Zrl+Zr2 A1

(6.5-12)

In order to make this magnification independent of h (the lateral separation between the objects), we must set b

A2a_

b _

~2 t2

a --

A1/1"

0

implying (6 5 - 1 3 )

Then, from Eq. (6.5-12), ' "" // r t

--

(zrl+Zr2))~....~2 2R

A1

A21112 A21112-(A212+/~211)R

(6.5-14)

199

6. 6 Ray Theory of Holograms: Construction and Reconstruction

assuming R >> d. that

Note , upon comparing Eq.(6.5-1 l b) and (6.5-14),

Mrlong ~ A2 A1(Mlat) 2"

(6.5-15)

The locations of the virtual images, as well as lateral and longitudinal magnifications, can be similarly calculated starting from Eqs. (6.5-3c) and (6.5-3d). The preceding analysis demonstrates clearly how threedimensional images are formed from a hologram. As seen, image distortion (due to the lateral magnification not being equal to the longitudinal magnification) may, in general, result unless special care is taken to alleviate the problems. The preceding analysis also shows that, in principle, a hologram can be recorded using a particular color and read by some other color. In fact, sometimes this proves to be an advantage, because, as can be readily checked from Eq.(6.5-15), the choice

Mlat = ~A-~

(6.5-16a)

Mitt-

(6.5-16b)

ensures

Mlrng,

thus eliminating any image distortion for 3-D display.

6.6 Ray Theory of Holograms: Construction and Reconstruction Thus far, we have examined hologram recording and reconstruction from a wave optics standpoint. We would like to present in this section the ray theory of holograms which essentially can be derived from physical optical principles. The reason it is possible to present a ray theory of construction and reconstruction of holograms is that during the reconstruction process, we simply find the location and characteristics of the image, be it real or virtual. As we know from Chapter 2, imaging can be well explained using ray optical principles. Thus it is no surprise that a ray optical approach to hologram recording and reconstruction is feasible. In many cases, it provides a simple

200

6 Holography and Complex Spatial Filtering

alternative to the more formal but mathematically extensive physical optics approach. The discussion on ray theory of holograms follows closely the analysis called the Abramson ray tracing method [Olson (1989)]. As for an optical system comprising lenses, we first enunciate the laws of ray tracing for holograms. Assume the setup shown in Fig. 6.13 where we have a point object O and a point source R as a reference. The point C where the straight line RO intersects the film (which is later developed into a hologram, and assumed to be reinserted back at the same position as the original film for reconstruction) is called the optical center of the hologram. Rays passing through the optical center of the hologram are undeviated, much like the rays passing through the center of a lens on axis. The reason for this, which is clear from a simple wave optic picture, is that the local spatial frequency at C of the intensity interference pattern on the plane of the film is zero, since plane waves traveling in the same direction (along ROC) cannot produce transverse interference patterns. Assume now that we draw a perpendicular RP to the plane of the hologram. If R is also the position of the reconstruction beam (assumed to be a point source), the wave theory of holograms suggests that a virtual image will be formed at the same position as the original object O. In other words, one of the diffracted rays behind the hologram should pass through the point O, if extended in front of the hologram (see dashed line in the figure). The reason that the ray is diffracted is that at the point P, the local spatial frequency is nonzero. Physical optics dictates that the local interference pattern at P is of the form c o s / ~ x - [ej~3x + e-J~x]. Hence at P, one should expect two diffracted rays, PQ and PQ', making equal and opposite angles with respect to the axis of propagation z, as shown in the figure. The intersection of this second ray with the ray ROC (passing through the optical center of the hologram) determines the location of the second image, be it real or virtual.

201

6. 6 Ray Theory o f Holograms: Construction and Reconstruction

Q

"=

~ z "'-......~ Q,

c(o,o)

film or

hologram Fig. 6.13 Ray tracing through a hologram. A ray ROC passing through the center C is undeviated. A ray starting from R (the position of the reference and reconstruction "beam") and normally incident on the hologram is diffracted into two rays behind the hologram, making equal and opposite angles ft. One of the rays appears to originate from the location of the original object O. Assume that the center of the hologram C is taken to be the origin of the (z, x) coordinate system, and that the coordinates of R and O are respectively (zR, xR) and ( z o , x o ) . Then the coordinate of P is (0, xR). Hence, t a r i f f ~ f l -(xRxo)/zo. The slope of the line PQ' is ( x R - x o ) / z o , and the equation of the line can be written as x -

~ Rz o- X ~

+ xn

(6.6-1)

Now the slope of the line ROC is ( x o - x R ) / ( z o - z R ) , and since this passes through the point (0, 0), the equation of the line is x -

(6.6-2)

x o - z R z. zO - ZR

Solving for z and x from Eqs. (6.6-1) and (6.6-2), we can find the position of the second image (zz, xz) as x R

ZO -- ZR

2;1 --- ZO xo-xR 2zo-zn ~

6 Holography and Complex Spatial Filtering

202

XI

~

Zo

xR

(6.6-3)

2ZO--ZR

As evident from the ray diagram above, for a real image to form, zz>0. Since zo and zR are both negative, this implies that 21zo I < Izn I, as also apparent from the ray diagram above. When the object is closer to the plane of the film, the angle/3 is larger, so that the magnitude of the slope of PQ' increases. The line PQ' may assume a larger negative slope than the line ROC, facilitating intersection of the two lines for positive values ofz.

Problems 6.1

Two transparencies t~ (x, y) and t2(x, y) are placed a distance f in front of a converging lens of focal length f, as shown in Figure P6.1. If a film is placed on the back focal plane and a positive transparency with a "7 of 2 is constructed, find an expression for the amplitude transmittance of the transparency. This transparency is reinserted on the front focal plane of the same system. Find the intensity distribution on the back focal plane. Throughout the problem, assume plane-wave illumination of the input transparencies. This method of convolving two functions optically is known as joint-transform correlation.

t,(x,y)

Film

/

t (x,y)

J

Figure P6.1

6. 6 Ray Theory of Holograms: Construction and Reconstruction

203

6.2

Assume that a Fresnel hologram [as given by Eq.(6.3-1)] is placed on the front focal plane of a converging lens of focal length f. Find the intensity distribution at the back focal plane in terms of the original object t(x, y) from which the hologram was constructed. Assume plane-wave illumination of the hologram.

6.3

With reference to Figure 6.8, assume that the object t(x, y) = 5(x-xo,g-go). Find an expression for the positive transparency tp(x, y) constructed with an overall "7 of 2. This transparency tp is then illuminated by a plane wave traveling in the +z direction as shown in Figure 6.10. Find the locations and nature of the real and virtual images of the object. Suppose now the reconstruction beam is, instead, a plane wave traveling at an angle 00to the z-axis. Predict the new location of the real and virtual images.

6.4

A hologram constructed from Young's double-slit experiment, with an overall -y of 2, is illuminated with a reconstruction wave as shown in Figure P6.2. Find the location of the real images of the two slits. Hologram

Figure P6.4

204

6 Holography and Complex Spatial Filtering

6.5

Develop expressions for the longitudinal and lateral magnifications for the virtual images due to the recording and reconstruction geometry as shown in Figure 6.12. Start by examining Eqs.(6.5-3c) and (6.5-3d).

6.6

A hologram of a point source using a plane wave reference is first constructed with an overall -,/ of 2, as shown in Figure 6.4(a). Now a point reconstruction source is located at a distance do in front of the hologram. Find the location of the real image of the object.

6.7

Assume that an object is located at the coordinate (xo = 2 cm, zo = - 3 cm), and the reference is located at the coordinate (xR = 2 cm, ZR = - - 5 cm), with the center of the hologram being at (0,0). If the reconstruction beam is a point source at the same position as the reference, find the locations and nature of the two images. What happens if the object is now repositioned closer to the film so that :co = 2 cm, zo = - 2 cm?

6.8

Write a MATLAB program to plot Figure 6.7. What is the effect by increasing the angle of the off-axis reference plane wave? Explain your MATLAB results.

References 6.1 6.2 6.3

6.4 6.5 6.6 6.7

Banerjee, P.P. and T.-C. Pooh (1991). Principles of Applied Optics. Irwin, Illinois. Gabor, D. (1948). Nature 161 777. Ghatak, A. K. and K. Thyagarajan (1978). Contemporary Optics. Plenum, New York. Goodman, J.W. (1996). Introduction to Fourier Optics. McGrawHill, New York. Leith, E.N. and J. Upatneiks (1962). Journal of the Optical Society of America 52 1123. Olson, D.W. (1989). American Journal of Physics 57 439. Piestun, R., L. Shamir, B. Wesskamp, and O. Bryngdahl (1997). Optics Letters 22 922.

6.6 Ray Theory of Holograms." Construction and Reconstruction 6.8

6.9 6.10

6.11 6.12

6.13

205

Poon, T.-C., T. Kim, G. Indebetouw, B.W. Schilling, M. H. Wu, K. Shinoda, and Y. Suzuki (2000). Optics Letters 25 215. Poon, T.-C., M. Wu, K. Shinoda, and Y. Suzuki (1996). Proceedings of the IEEE 84 753. Rogers, G.L. (1950)Nature 166 237. Stroke, G. W. (1966). An Introduction to Coherent Optics and Holography. Academic Press, New York. Yeh, P. (1993). Introduction to Photorefractive Nonlinear Optics. Wiley. Yu, F. T. S. (1983). Optical Information Processing. Wiley, New York.

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207

Chapter 7 Contemporary Topics in Optical Image Processing

7.1 Theory of Optical Heterodyne Scanning 7.1.1 Bipolar incoherent image processing 7.1.2 Optical scanning holography 7.2 Acousto-optic Image Processing 7.2.1 Experimental and numerical simulations of 1-D image processing using one acousto-optic cell 7.2.2 Improvement with two cascaded acousto-optic cells 7.2.3 Two-dimensional processing and four-corner edge enhancement 7.3 Photorefractive Image Processing 7.3.1 Edge enhancement 7.3.2 Image broadcasting 7.3.3 All-optical joint transform edge-enhanced correlation 7.4 Dynamic Holography for Phase Distortion Correction of Images

In this Chapter, we utilize many of the concepts studied in the previous Chapters for achieving unique image processing operations such as bipolar incoherent image processing, scanning holography, edge enhancement, edge enhanced correlation etc. Acousto-optics is used to realize heterodyne scanning, leading to bipolar processing. It is also used to perform image processing such as edge enhancement and corner detection. Nonlinear materials such as photorefractives are also used for

208

7 Contemporary Topics in Optical Image Processing

edge enhancement and for edge enhanced correlation in a joint-transform correlation geometry. A novel technique using dynamic holography is described which corrects images corrupted by large and small scale phase distortions. MATLAB is a useful tool once again to analyze these optical systems, and relevant MATLAB programs and outputs are discussed as required.

7.1 Theory of Optical Heterodyne Scanning Coherent optical image processing as discussed in Chapter 5 is conceptually simple, and for this reason has received much attention in the last several decades. In a coherent system, complex spatial filtering is possible as the amplitude and phase of specific Fourier components can be directly manipulated in the Fourier plane. Synthesizing the transfer function is straightforward as it can be implemented by modifying the pupil function directly. However, since coherent systems are sensitive to phase, they are extremely susceptible to coherent noise corruption [Chavel and Lowenthal (1978)]. Incoherent systems have better signal-to-noise ratio than their coherent counterpart, but synthesizing the transfer function of an incoherent system is less straightforward than doing the same for a coherent system as the optical transfer function (OTF) is the autocorrelation of the pupil function [see Eq.(5.4-4)], or equivalently the point spread function (PSF) is real nonnegative [see Eq.(5.4-1)]. This means in particular that it is not possible to implement directly even the simplest enhancement and restoration algorithms such as highpass filtering and derivative operations, which requires a bipolar PSF. The past three decades have witnessed an impressive re-emergence of incoherent optical processing techniques [Indebetouw and Poon (1992)]. Instrumental in this evolution was the inclusion of a number of novel techniques, such as acousto-optic frequency shifting and heterodyne detection [Poon and Korpel (1979)], that made it possible to design practical incoherent systems with bipolar point spread functions. In what follows, we describe a practical optical system of the acousto-optic frequency shifting and heterodyne detection technique and we call the optical system as the acousto-optic heterodyning image processor (AOHIP). Figure 7.1 shows the AOHIP. The processor is a two-pupil

system.

209

7.1 Theoryof Optical Heterodyne Scanning AOFS BS LASER @ ~

f~t

coo+f~ object

3-D

~(~Y)

"~, ~~i ..... i/ Fo(x, y; z)

I

II'/I %

L2

t~(x,y)

I1

I

I &(x,Y) -

L1

BSl

2-D Scanning mirrors

Figure 7.1 Acousto-optic heterodyning image processor (AOHIP)" The bandpass filter is tuned at heterodyne frequency f~, and PMT is a photomultiplier for the detection of scattered light from the 3-D object.

Beamsplitters BS and BS1, and mirrors M and M1 form the Mach-Zenhner interferometer. A collimated laser at temporal frequency w0 is used to illuminate the pupil, Pl (x, y). The other pupil, p2(x, y), is illuminated by the laser of temporal frequency coo + f~. The laser's temporal frequency offset of f~ is introduced by an acousto-optic frequency shifter (AOFS) as shown in the Figure. The two pupils are located at the front focal planes of lens L1 and L2, both with focal length f. The two pupils are then combined by beamsplitter BS1 to focus the light onto the 2-D x - y scanning mirrors, which are located on the back focal plane of lenses L1 and L2. The combined optical scanning field at z away from the focal plane of the two lenses are then given by

P l z ( k o x / f , koy/ f ) exp(jcJ0t) + P 2 z ( k o x / f , k o y / f ) exp[j(aJ0+f~)t],

(7.1-1)

where P~z(kox/f, k o y / f ) is the field distribution z away from the scanning mirrors and is given by, through Fresnel diffraction,

P~(kox/f , koy/f) -

P i ( k o x / f , k o y / f ) 9 h(x, y; z), i - 1,2.

(7.1-2)

7 Contemporary Topics in Optical Image Processing

210

In Eq. (7.1-2), Pi(kox/f, koy/f) is the field distribution in the back focal plane of lenses L1 and L2 and is given by, aside from some inessential constant and a phase factor,

P~(kox/ f , koy/ f ) - ~{p~(x, y)}

kx = k o x / f , k y =k o Y/f

.

(7.1-3) h(x, y; z), aside from a constant phase shift, is the free-space impulse response and is given by

h(x, y; z) -

~zJk~exp[ -3Z'k~(x 2 + y2)]

(7.1-4)

with/c0 denoting the wavenumber of the light. Also the 9 in Eq. (7.1-2) denotes the 2-D convolution operation. Returning to Eq. (7.1-1), the combined optical field or the scanning pattern is used to perform a twodimensional scan of a 3-D object using the 2-D scanning mirrors. We now model the 3-D object as a collection of planar amplitude distribution ro(X, y;z) along its depth direction, z. z is a depth parameter which is measured away from the scanning mirrors as shown in Figure 7.1. The photomultiplier (PMT), which responds to the incident intensity of the optical transmitted field or scattered field (as illustrated in Figure 7.1), generates a current that is given by i ( x , y; z) -

f fA [P~z(kox'/f , koy'/f)eJ~ot + P2z(kOx'/f , koy'/f)e j(~~ 2

x ro(X + x', y + y';z) dx'dy'.

(7.1-5)

Note that the integration is over the area A of the photodetector, and x - x(t)and y - y(t) represent the instantaneous position of the scanning pattern. For uniform scan at speed V, x(t) - y(t) - Vt. The shifted coordinates of 1-'o represent the action of scanning. The heterodyne current at temporal frequency f~ of Eq. (7.1-5), after a bandpass filter tuned at frequency f~, becomes

7.1 Theory of Optical Heterodyne Scanning

211

ia(x, y; z) = Re [ f fA Pl*z(kox'/f , koy'/f) P2z(kox'/f , koy'/f) x [Fo(X § x', y § y';z)l~dx'dy'exp(jf~t)

,

(7.1-6)

where we have adopted the convention for phasor ~pas ~(x, y, t ) Re[g2p(X,y,t)exp(jf~t)}, and where Re[.] denotes the real part of its argument. Eq. (7.1-6) can be written as

ia(x, y) = Re[ia~(x, y; z)exp(jf~t)],

(7.1-7)

where kox~ ~ ( x , v; z) - f fA P~z( j , k~i ) P2z ( k~J , k~J J 9

,

• Iro(~ + ~', y + y ' ; z ) 1 2 d x ' d y ' is the output phasor which denotes the amplitude and the phase information of the heterodyne current which constitute the scanned and processed version of the object Iro(Z, y; z)] 2. Writing ia~(x, y; z) with the help of correlation operation, we have ia~(~ v" z) -

Plz( k~

ko,)pL(kox f f

,

ko,) | Ir o (x,y ;~)12 . f

(7.1-8)

Note that the optical scanning system is incoherent as only intensity distribution, i.e., II'ol 2, is processed. We shall now define the optical transfer function (OTF) of the system as

OTF~(kx, kv; z)= 9c{i~p(X, y; z ) } / f { I r o ( x , y;z)[2}.

(7.1-9)

Substituting Eq. (7.1-8) into Eq. (7.1-9), we have

o~~(~x,

~ , ~)- ~. { ~z (~0, ,9, ~0~) ~z.(~0~, , ~~0~)}

(7.1-10)

In terms of the pupils pl andp2, we substitute Eqs. (7.1-2) and (7.1-3) into Eq. (7.1-10) to obtain

7 Contemporary Topics in Optical Image Processing

212

O T F a ( k x , ky, 9z) -

21 exp [zJ~oo ( k2 + ky)

• f f p ~ ( x ' , Y')P2( x'+ fk~,y'+ fky)exp[j}(x'k~ +y'ky)]dx'dy'. (7.1-11) This equation states that OTFa of the system can be modified according to the selection of the two pupils. Now using Eq. (7.1-9) and re-writing Eq. (7.1-7) in terms of OTFa, we have

ia(x, y; z) = Re [ia~(x, y; z)exp(jf~t)]

(7.1-12)

= Re [9r-1 {$-{ [Fo (x, y; z)[2} OTFa(kx, kv; z)} exp(jf~t)]. Defining the point spread function (PSF) of the optical heterodyne scanning system as ha(x. v; z) -

(7.1-13)

7 -~ { O T F a }.

we can now re-write Eq. (7.1-12) in the spatial domain as

i~(x, y; z) = Re[lFo(X, y; z)] 2 | h~(x, y; z)exp(jf~t)]. (7.1-14) Equation (7.1-12) or (7.1-14) represents the scanned and processed output current modulated by a temporal carrier at frequency f~. We can demodulate and extract the in-phase and the quadrature components of the output current by mixing it with cos(f~t) and sin(f~t), and hence the idea of parallel processing. The demodulation system is shown in Figure 7.2, and the two outputs can be shown and given as

it(x, y; z) - Re[~-~{5{lFo(X, y; z)12}OTFa(kx,

ku; z)}]

= n~[Iro(X, y; z)l ~ | ha(~, y; z)]

(7.1-15a)

and

i~(x, y; z) - I m [ 5 - ~ { 7 { I r o ( X , y; z)12}OTF~(kx, k~; z)}] = I.~[Iro(x. y; z)l ~ | h~(x. y; z)].

(7.1-15b)

7.1 Theoryof Optical HeterodyneScanning

213

where Ira[.] denotes the imaginary part of the argument, and the subscripts "c" and "s" represent the use of cosf~t and sinf~t for mixing, respectively, to extract the information from if~.

cos f2t

0

LPF

i

PC ) Monitor

LPF ~ )

is sin f2t

Figure 7.2 Electronic demodulation of in(x, y): LPFs are lowpass filters. In Eq. (7.1-15), the input object, [1-'o(X, y; z)[ 2, has been assumed to be an infinitely thin 2-D object located at z away from the focal plane of lens L1 and L2, which is on the 2-D scanning mirrors as shown in Figure 7.1. Hence, to generalize Eq. (7.1-15) for 3-D objects, we need to integrate the equation over the depth, i.e., over z, of the 3-D object. Equation (7.1 - 15) becomes

ic(X, y) -

Re [ f 5 -l{7{Iro(x, y; z)12}OTFa(kx, ky; z)Idz]

- R [fJro( ,y; )f

| h (x,y;

(7.1-16a)

and

(7.1-16b) Note that we have left the z dependence out of the left-hand side of (7.116) to emphasize that the recorded and processed information is strictly 2-D even for 3-D objects, i~(x, y) or i~(x, y) represents the scanned and processed information and can be stored as a 2-D record in a computer if

7 Contemporary Topics in Optical Image Processing

214

these currents are stored in synchronization with the signals used to drive the x - y scanning mirrors.

7.1.1 Bipolar incoherent image processing In the beginning of this Section and Section 5.4, we mentioned about the severe limitations of incoherent processing with standard incoherent systems in that the optical transfer function (OTF) achievable is the autocorrelation of the pupil function, or equivalently the point spread function (PSF) is real nonnegative. Among the acousto-optic heterodyning image processor (AOHIP) discussed in the last section, a number of novel techniques have been devised to implement bipolar point spread functions in incoherent systems. These techniques are usually referred to as bipolar incoherent image processing in the literature [lndebetouw and Poon (1986), Mait (1986)]. In this Section, we demonstrate and simulate bipolar incoherent image processing achievable with the heterodyning image processor. For 2-D image processing, we assume that the object is now a 2D image and placed at the focal plane of lenses Lland L2 in Figure 7.1. Hence, ro(X, y;z) becomes Fo(x, y;0) - ro(X, y) with the 2-D scanning mirrors moved toward the lenses for scanning purposes. We now, as an example, modify Eq. (7.1-15a) to clarify the idea of bipolar image processing. By setting z = 0 in Eq. (7.1-15a), we have

y) -

y)l

}OTr

(7.1-17)

(kx,

= n [Iro( , y)l | ha( , y)], where OTFa(kx, ky) becomes, according to Eq. (7.1-11), ,

f kx

yt

f

ky)dx'dy ~, (7.1-18)

and ha(x, y) : f - ~ {OTFa(kx, kv)}. Note that the OTF achievable is no longer the autocorrelation of the pupil function, but the crosscorrelation of the two pupils in the incoherent optical system, and hence the point spread function (PSF) becomes bipolar- the very concept of bipolar incoherent image processing. For simulation purposes, we take Pl (x, y) = e x p [ - a l ( x 2 + y2)] _ e x p [ - a 2 ( x 2 + y2)], a Difference-ofGaussian aperture function (which is an annular-type pupil), and

7.1 Theoryof Optical Heterodyne Scanning

215

P2 (x, y) -- 5 (x, y), a small pin-hole aperture. The OTF then becomes, using Eq. (7.1 - 18), OTFa(k~, k v ) -

2

e-Crl(k~+kv

2

) --

2

2

e-Crz(k~+kv ) ,

(7.1-19)

where cr~ - a~ ( f ) 2 and cr2 - a~ ( f ) 2 . This OTF is a bandpass filtering function and according to Eq.(7.1-17), the spectrum of the intensity object, [lPo(Z,y)] 2, is modified by this bandpass filtering action. Table 7.1 is the MATLAB code used to simulate bipolar incoherent filtering and the results are shown in Figure 7.3. Figure 7.3 (a) is the input image and Figure 7.3(b) is its Fourier transform. Figures 7.3(c) shows the Difference-of-Gaussian OTF of Eq.(7.1-19) with crland cr2chosen in Table 7.1. Finally, Figure 7.3(d) shows the processed image, which clearly demonstrates edge extraction on the incoherent image. For readers who are interested in experimental results, we refer them to the literature [Inbebetouw and Poon (1984 and 1986) and Poon (1985)]. The use of the acousto-optic heterodyning image processor also has been applied to the applications of tunable incoherent spatial filtering and textural edge extraction [Poon, Park, and Indebetouw (1990), and Park and Poon (1988)].

%Reading input bitmap file clear,

I=imread('vatech256.bmp','bmp'); l=I(:,:, 1); figure(1)%displaying input colormap(gray(255));image(I) title('Input image') axis off %%Creating difference of-Gaussian OTF ROWS=256; COLS=256; sigma1=0.01, sigma2=0.03, %kx,ky are spatial frequencies ky=-12.8; for r=l :COLS, kx--12.8; for c-- 1:ROWS,

7 Contemporary Topics in Optical Image Processing

216

OTF(r,c)=exp(-sigma 1*kx* kx-sigma 1*ky* ky)-exp(-sigma2* kx* kx-sigma2* ky* ky); kx=kx+. 1; end

ky=ky+.l;

endmax 1=max(OTF); max2=max(max 1); scale= 1.0/max2; OTF=OTF.*scale; % Taking Fourier transform of I FI=fft2(I); FI=fftshift(FI); maxl =max(FI); max2=max(max 1); scale = 1.0/max2; FI=FI.*scale; figure(2)%Displaying Fourier transform of input, I colormap(gray(255)); image(10*abs(256* FI)); title('Fourier transform of input image') axis off figure(3)%Displaying OTF colormap(gray(255)); image(256*OYF); title('Difference-of-Gaussian OTF') axis off %FHI is the processed input image in the Fourier domain FHI=FI.*OTF; HI=ifft2(FHI); max 1-max(HI); max2-max(maxl); scale=l.0/max2; HI=HI.*scale; figure(4) colormap(gray(255)) image(abs(real(256*HI))); %Displaying the absolute value of processed image title('Processed image') axis off Table 7.1 MATLAB code for bipolar incoherent filtering (Bipolar_filtering.m).

7.1 Theory of Optical Heterodyne Scanning

Figure 7.3(a) Incoherent Image.

Figure 7.3(b) Magnitude spectrum of Figure 7.3(a).

217

218

7 Contemporary Topics in Optical Image Processing

Figure 7.3(e) Difference-of-Gaussian OTF for bandpass filtering of incoherent image.

Figure 7.3(d) Bandpass filtered image of input incoherent image shown in Figure 7.3(a).

7.1 Theory of Optical Heterodyne Scanning

219

7.1.2 Optical scanning holography Optical scanning holography (OSH) is a novel technique first suggested by Poon and Korpel in 1979, in which holographic information of an object can be recorded using heterodyne optical scanning. The 3-D object is two-dimensionally scanned by a timedependent Fresnel zone plate (TDFZP) to generate a hologram. The TDFZP is created by the superposition of a plane wave and a spherical wave of different temporal frequencies. Since an electronic processing technique is used in the context of holographic recording, the technique is real-time, bypassing the use of films for recording. Such holographic recording technique nowadays is commonly known as electronic holography [Kuo (1996)]. Let us see how the TDFZP can be generated using the acoustooptic heterodyning image processor shown in Figure 7.1. We choose the two pupils such that p~ (x, y) - i and p2(x, y) - 5(x, y) as illustrated in Figure 7.1. With this choice of the pupils, the OTF of the optical scanning system, according to Eq. (7.1-11), becomes z ( k2 + ky) 2 OTF~(k~, ky;z) - exp I - Jsvi0

= OTFo~h(kx, ky;z),

(7.1-20)

where the subscript "osh" denotes that the particular achieved OTF is for holographic recording. What is being recorded in two dimensions is a hologram, t(x,y). We shall use Eq.(Y.l-16a) as an example, i~(x,y) now becomes a hologram and is given by t(x, y) - Re [f9c-1 { f'{IFo(x, y; z)12}OTFo~h(kx,k,;z)}dz]

= R e [ f U - l { f ' { I F o ( X , y; z)12}exp [ - j2-~o(k~+ k2)] }dz]. (7.1-21) From the above equation, it is clear that holographic recording process in the frequency domain can be interpreted as the object's spectrum along its

7 Contemporary Topics in Optical Image Processing

220

depth (z) is being processed by the OTF of the form given by Eq. (7.120). To clearly see why this corresponds to holographic recording, we rewrite Eq. (7.1-21) in terms of convolution and we have t(x, y) -- Re[flFol2, h*(x, y; z) dz],

(7.1-22)

where interestingly Eq. (7.1-22) can be written in terms of h(x, y; z) which is the free-space impulse response defined in Eq. (7.1-4). Equation (7.1-22) can be rewritten in terms of correlation tsin(x, y)

--

Re[ f h(x, y; z) | [ro[2dz]

= f ko sin[kO x 2

2

y2

(7.1-23)

In writing the last step of (7.1-23), since IFo[2 represents intensity distribution which is strictly positive, the Re-operation has been distributed to the function h. The subscript "sin" in the left side of the equation denotes a sine function is involved in the calculation of the correlation integral. If we let [Fol 2 - ( 5 ( x - xo, y - yo, z - zo), tsin(x, y) k0 [(x - x0) 2 + (y - y0) 2 ]}, which is the hologram of an offset delta sin{ ~0 function as discussed in Chapter 6. Hence, in optical scanning holography, 3-D holographic recording process can be thought of as a 2D transverse correlation between the real part of the free-space impulse and the 3-D object's planar intensity response, ~ko sin[ko(x2+y2)] ~ distribution at z, [Fo(x,y;z)l 2. The resulting correlation is then integrated along the depth of the object to obtain the hologram of the whole 3-D object. To put Eq. (7.1-23) into a wider context, the _Reoperation can be replaced by/m-operation. This corresponds to the use of Eq.(7.1-16b) as an output:

tcos(X, y) -- I m [ f h(x, y; z) | Iro(X, y; z)[2dz] ~o ~o ( ~ + y~ )] -_ f ~~oS[Tz = ~o

os{ ~ [ ( x -

| rro(~,y; z)l~d z

xo) + ( y -

vo) I

(7.1-24)

7.1 Theoryof Optical HeterodyneScanning

221

for [Fo[2= ~5(x- x o , y - yo,z- zo). We shall call tsin(X, y) and t~o,(X, y) the sine-Fresnel zone plate (sine-FZP) hologram and cosine-FZP hologram of the object 1['ol2, respectively. Indeed, due to the fact that holographic information is available in electronic form as given by ia(x, y) as shown in Figure 7. l, we can have a sine-FZP hologram and a cosine-FZP hologram simultaneously through parallel processing as shown in Figure 7.2. In addition, once the two electronic holograms have been stored, say, in a computer, we can perform complex addition to form a complex hologram of the object. Specifically, we add Eqs. (7.1-23) and (7.1-24) to give a complex hologram tp(x, y) as follows:

tp(X, y) -- tcos(X, y) =kjtsin(x, y) +Jz0 = fexp[-5;-

(7.1-25)

)] | Iro( , y; z)l dz,

where we have neglected some constant for simplicity. In contrast to the sine-FZP and cosine-FZP hologram, the complex hologram has an advantage in that it will not give any twin-image noise upon reconstruction [see Eqs. (6.2-4)]. In addition, the complex hologram does not require any higher spatial resolution of the recording medium as in off-axis hologram for twin-image elimination [Bryngdahl and Lohmann (1968), Doh et al. (1996), Kim et al. (1997), Mishina et al. (1999), Poon et al. (2000), Takaki et al. (1999), Yamaguchi et al. (1997)]. Table 7.2 shows the MATLAB code for the simulation of optical scanning holography. The input object is a 2-D image as shown in Figure 7.3(a) with a distance specified by "sigma" in Table 7.2, where "sigma ''= z/2ko as defined in Eq. (7.1-20). Again, z is the distance of the object measured away from the focal plane of lenses Lland L2 as shown in Figure 7.1. Figure 7.4(a) shows the real part of the impulse response for optical scanning holographic recording at the distance z, i.e., Re[U-l{OTFo~h(kx, kv;z)}]. Figure 7.4(b) shows the sine-FZP hologram obtained according to Eq. (7.1-21) and Figure 7.4(c) is the cosine-FZP hologram with the operation Re[.] replaced by Ira[.] in Eq. (7.1-21). Figures 7.4(d) and (e) display the reconstruction of the sineand cosine-hologram, respectively. Note that twin image noise exists in these reconstructions. For no twin image noise upon reconstruction, complex holograms given by Eq.(7.1-25) are needed. This is left as an exercise for students.

7 Contemporary Topics in Optical Image Processing

222

The use of optical scanning holography has been applied to the applications of 3-D holographic fluorescence microscopy (1997), 3-D location of fluorescent inhomogenieties in turbid media (1998), and most recently 3-D optical image recognition (1999). Spatial light modulatorbased scanning holography for real-time reconstruction and 3-D display also have been explored [Duncun et al. (1992), Poon et al., 1997)].

%Reading input bitmap file clear, I=imread('vatech256. bmp','bmp '); I=I(:,:, 1); figure(1 )%displaying input colormap(gray(255)); image(I) axis off pause %%Creating OTF with SIGMA=z/2*k0 for optical scanning holography (Eq.(7.1-20)} ROWS=256; COLS=256; sigma=0.51 ; %kx,ky are spatial frequencies ky=-12.8; for r=l :COLS, kx---12.8; for c = 1:ROWS, O TF (r, c)= e xp (-j *s igm a* kx* kx-j *s igma* ky* ky); kx=kx+. 1;

end

ky=ky+.l; endmax 1=max(OTF); max2 =max(max 1); scale = 1.0/max2; OTF=OTF.*scale; % .... displaying the real part of h(x,y): h(x,y) is inverse transform of OTF

h=ifft2(OTF); h=iffishifi(h); max 1=max(h); max2=max(max 1); scale = 1.0/max2; h=h.*scale; figure(l) image(real(256*h)); title('Impulse response of optical scanning holography') axis off .

.

.

.

7.1 Theoryof Optical Heterodyne Scanning %Recording hologram % Taking Fourier transform of I FI=fft2(I); FI=fftshift(FI); max 1=max(FI); max2=max(maxl); scale = 1.0/max2; FI=FI.*scale;% FHI is the recorded hologram in Fourier domain FHI=FI.*OTF; HI=ifft2(FHI); max 1=max(HI); max2=max(maxl); scale=l.0/max2; HI=HI.*scale; figure(3) colormap(gray(255)); image((real(256*HI))); %Displaying the real part of hologram, sine-FZP hologram title('Sine-FZP hologram') axis off figure(4) co lormap (gray(255 )); image(5*imag(256*HI)); %Displaying the imaginary part of hologram, cosine-FZP hologram title('Cosine-FZP hologram') axis off %Reconstructing holograms %Reconstruction of sine-hologram,twin-image noise exists figure(5) colormap(gray(255)) FH=FHI; H=ifft2(FH); FRCH=fft2(real(H)).*conj (OTF); RCH=ifft2(FRCH); image(256 *abs(RCH)/max(max(abs(RCH)))) title('Reconstruction of sine-FZP hologram') axis off FH=FHI; %Reconstruction with cosine-hologram, twin-image noise exists figure(6) colormap(gray(255)) FRCH=fft2(imag(H)).* conj (OTF); RCH=ifft2(FRCH); image(256*abs(RCH)/max(max(abs(RCH)))) title('Reconstruction of cosine-FZP hologram') axis off Table 7.2 MATLAB code for optical scanning holography (osh.m).

223

224

7 Contemporary Topics in Optical Image Processing

Figure 7.4(a) Impulse response of optical scanning holography.

Figure 7.4(b) Sine-FZP hologram.

7.1 Theoryof Optical Heterodyne Scanning

Figure 7.4(e) Cosine-FZP hologram.

Figure 7.4(d) Reconstruction of sine-FZP hologram.

225

226

7 Contemporary Topics in Optical Image Processing

Figure 7.4(e) Reconstruction of cosine-FZP hologram.

7.2 Acousto-Optic Image Processing In Chapter 4, we have derived the transfer functions for acoustootpic diffraction for the zeroth and the diffracted orders, and shown from plots of the transfer function that the zeroth order works as a high-pass spatial filter, while the diffracted order has low-pass characteristics. We can more physically explain the phenomenon of high-pass spatial filtering in the zeroth order and consequent edge enhancement by approximating the transfer function (4.3-46) derived in Chapter 4. As mentioned earlier, the first exponential term is the propagational diffraction term. It can be excluded by imaging the object onto the final observation plane, in the absence of the sound. Under our experimental conditions, Ik~,~lcoxL/kol- Ik~axQA/47rl << Ic~/2l, hence the zeroth order transfer function can be expressed as [Cao et al. (1998)]

Ho(kx) ~ A-+-jBkx,

(7.2-1)

7.2 Acousto-Optic Image Processing

227

where the plus sign represents the case when r = + CB, and the minus sign is for r = - C B , A - cos(c~/2), and B 4~(,/2) 9 From the Fourier

transform

(see Table 1.1) the amplitude of the zeroth order ~0 after the AO cell can be approximately written as

5x{Of(x,y)/Ox}-

property

-jk~U~{f(x,y)},

r

y) = (AT B O / O x ) r

y),

(7.2-2)

where r is the incident optical field on the AO cell. Experimentally, it is noted that for the objects used and when c~ is near 7r, the second part of the right hand side of Eq. (7.2-2) is dominant. Therefore, the effect of the AO cell on the zeroth order is identical to edge enhancement by a 1-D gradient operator in digital image processing such as the Sobel mask. But, unlike digital image processing, this edge enhancement is real-time and the amount of enhancement is programmable since the c~ of the AO cell can be easily changed by varying the sound pressure.

7.2.1 Experimental and numerical simulations of 1-D image processing using one acousto-optic cell The experimental set up for the 1-D edge enhancement and edge smearing using one AO cell is shown in Figure 7.5 [Banerjee et al. (1997), Cao et al. (1998)]. A He-Ne laser (A = 633nm) is used as the light source. The object used is a circular aperture with transmission function r y) - ~)inc(r, O) circ(r/a) where a ~ 1 mm. Lenses 1 and 2 are used to decrease the effective object size by a factor of 2. The AO cell is put immediately behind the object plane. The AO medium is flint glass with a refractive index of 1.68, and the sound velocity in the medium is about 3.96 km/s. The AO interaction length is about 62.5 mm. The iris in the back focal plane of lens 3 is used to select the zeroth and first diffracted order images for display. Lenses 3 and 4 are used to exactly image the object on a CCD detector array of a laser beam analyzer. Two convex lenses with focal lengths of 150 mm and 100 mm are used as lenses 3 and 4 respectively. Therefore, the propagational diffraction effect between the AO cell and the CCD camera (which as explained above includes diffraction through the AO -

-

7 Contemporary Topics in Optical Image Processing

228

cell but is decoupled from the interaction process) has been taken into account.

HeNe

object Lens i

Lens 2

sound ~

AO

Lens 3

Ih s

Lens 4

CCD

~T[a.nsdvcer

Figure 7.5 Experimental set up for 1-D image processing using one AO cell.

Based on the above experimental conditions, and using a sound frequency of 55 MHz, we obtain Q ~ 28. The value of c~ is calculated from the relation tar~2(c~/2)- I1/Io, where I~ and I0 are the first diffracted and zeroth optical intensities respectively which are experimentally measured without the circular aperture. The iris at the back focal plane of lens 3 is used to first select the zeroth order image for display. At c ~ - 0.857r, the experimental result showing 1D image edge enhancement of the circular aperture is presented in Figure 7.6(a). Note that the right side is brighter than the left, signifying asymmetric edge enhancement. At a - 0.857r, One of the two edges is hardly distinguishable. The processed circular aperture is shown in Figure 7.6(b) If the first diffracted order image is selected by the iris at the back focal plane of lens 3, edge smearing occurs. This indicates the low-pass filtering characteristic of the first order.

7.2 A cousto-Optic Image Processing

229

Figure 7.6 Experimental results of 1-D edge enhancement using one AO cell and with a 2 mm diameter circular object (a) The edges along the 1-D AO interaction direction at c~ - 0.857r; (b) One of the two edges is hardly distinguishable at c~ - 0.67r. [Banerjee et al. (1997)].

A square transparency with 2 mm side is also used. Two vertical edges appear at c~=0.857r, for the undiffracted or zeroth order, with the right edge brighter than the left edge [Banerjee et al. (1997)]. Note in each case that the edge enhancement is one dimensional and occurs in the direction of propagation of the sound beam. Based on the experimental conditions described above and Q = 28, a Discrete Fourier Transform (DFT) method is used to simulate the experiment. The zeroth order transfer function for Bragg incidence is multiplied with the angular plane wave spectrum of the input circular or square objects to derive the spectrum of the processed image at the exit of the AO cell. The inverse Fourier transform of the exit spectrum is equivalent in shape to the field pattern at the CCD detector. For c~ - 0.857r, the numerical simulation results of the brightness profile for the zeroth order is shown in Figure 7.7.

230

7 Contemporary Topics in Optical Image Processing

Figure 7.7 Numerical simulations for one dimensional edge enhancement of a circular aperture for c~ = 0.857r. [Banerjee et al. (1997)].

Comparison of Figures 7.6(a) and 7.7 show that the experimental results and the numerical simulations match very well. Note that when c~ is near 7r under our experimental conditions, the edges are clearly evident, but when c~ is near or smaller than 0.57r, at least part of the edge is indistinguishable. As stated above, the optical intensities of the two edges are not exactly the same. Techniques to overcome this asymmetry will be discussed later below. From our numerical simulations, it is found that the asymmetry is due to the non-zero first term in the RHS of Eq. (7.2-2). This also can be explained using the schematic diagrams shown in Figure 7.8. With a square object (Figure 7.8(a)) and zero constant term, the optical field after first derivative operation (similar to the second term in RHS of Eq. (7.2-2)) can be represented as in Figure 7.8(b). The corresponding optical intensity diagram is shown in Figure 7.8(c); and there are two symmetric edges. Unfortunately, the constant term (cos(c~/2)) cannot reach zero in many experimental conditions, hence, with a nonzero value of the constant term, the asymmetric edges appear in Figure 7.8(d).

7.2 A cousto-Optic Image Processing

,:_-:,ptical fiel,:t

231

Opti,:al field 0

-2

1

rl

I

2

"1

"2

-i

0

I

2

0 -2

-I

0

i

2

!I,

l'l

i i .............................

,'

,

II, ~.................

,

-1

J,,l jJ"

*'1 jl

!:'r.,ti,:cA intensity

,'i -

,iJ

9 .............

0

-1

-

(1

i

" .......

"2

Ot:,tical Intensity

|

0

(c)

1

2

(a)

F i g u r e 7.8 Schematic diagrams to explain asymmetric first order derivative operation using one AO cell. (a) input (b) optical field after first derivative operation and zero constant term, (c) optical intensity after first derivative operation and with zero constant term, showing symmetric edge intensities; (d) optical intensity after first derivative operation and with non-zero constant term, showing asymmetric edge intensities [Banerjee et al. (1997), Cao et al. (1998)].

7.2.2 Improvement with two cascaded acousto-optic cells A way to remove the asymmetry is to employ two AO cells which are put in tandem and with contra-propagating sound waves. If an object

232

7 Contemporary Topics in Optical Image Processing

is incident on the first AO cell at +r the zeroth beam after the first AO cell is made to be incident on the second AO cell at - CB. It is assumed that the second cell has a sound wave counter-propagating to the sound in the first cell [Cao et al. (1998)]. Using Eq. (7.2-1), the complex optical envelope of the zeroth order after the first AO cell can be approximated as

~ ) (x, y) - (A~ - BiO/Ox)~;{~c(X, y)

(7.2-3)

and the zeroth optical envelope after the second AO cell can be approximately expressed as r

(x, y) - (A2 + ~ 2 0 / 0 x ) ~ ) ~ 1) (x, y)

(7.2-4)

where Ax, A2 and Bi, B2 are the same functions as A and B respectively, but the c~, Q, and A0 in the relations for A and B are the parameters of the first and second AO cells, respectively. From Eqs. (7.2-3) and (7.2-4), the envelope of the zeroth optical field after the two AO cells can be written as

r

y) - (AiA2 + A i B 2 O / O x - A2BIO/Ox + B1B2O2/Ox2)gZi~c(X, y).

(7.2-5)

When the parameters of the two AO cells are the same, A1 = A2, Bi = B2, then A iB2 = A2Bi, the second and third terms in parenthesis on the RHS of Eq. (7.2-5) which contains the first order derivative terms are canceled. If peak phase delay parameters c~1 and c~2 of both AO cells are near 7r (e.g., C[1 = OZ2 --0.8571" ), the AiA2 term is also very small, therefore, the second order derivative term is dominant, and works as a 1D Laplacian operator. The edge detection can be accomplished by locating zero crossings of the double peaks in both sides of edges. Furthermore, the zeroth order beam profile can be changed by adjusting the value of the constant term which varies with the sound intensity of both AO cells. This phenomenon can be explained using the 1D schematic diagrams in Figure 7.9. With zero constant term, and for a square object as shown in Figure 7.9(a), the optical field responsible for second derivative operation (see last term in RHS of Eq. (7.2-5) is shown in Figure 7.9(b). The zero-crossing between the double peaks around the

7.2 Acousto-Optic Image Processing

233

edges appears in the intensity profile (see Figure 7.9(c)), and the edge detection may be performed by locating the zero-crossing between the double peaks. Unfortunately, the AiA2 term cannot be always equal to zero in practice. To simulate the effect of the non-zero constant term, we add various positive values of constant to the last term in Eq. (7.2-5). Figures 7.9(d) and (e) are two examples of the intensity profiles with two different values of the constant respectively. For some range of values of the constant term, we see that the well-defined double peaks are deformed. As the value of the constant term increases from zero to the maximum peak value of the optical field profile in Figure 7.9(b), the outer peaks in the intensity profile which indicate the dark side of each edge become increasingly lower, and two zeroes appear around each edge (Figure 7.9(d)). If the value of the constant term exceeds the maximum peak value of the field in Figure 7.9(b), the outer peaks disappear and the valleys appear instead (Figure 7.9(e)). If the value of the constant is small compared to the maximum peak value of the field in figure 7.9(b), the edges can still be identified by locating the zerocrossing between the double peaks around each edge (e.g., Figure 7.9(d)). However, when the constant term is near to or larger than the maximum peak value (e.g., Figure 7.9(e)), only the symmetric inner peaks are evident which represents the inner or "bright" side of the edge. The experimental set up is shown in Figure 7.10, where a second AO cell is put immediately behind the first one and with contrapropagating sound wave. The sound frequencies of the two AO cells are selected to be slightly different in practice; the reason for this is detailed in Banerjee and Cao (1998). The sound frequency of the first and the second AO cells are taken as 55 MHz and 68 MHz, respectively. The experimental results for a square aperture of dimension 1 mm showing the edge location at the zero-crossing of the double peaks at OZl=OL2=0.8571 " is shown in Figure 7.11(a). For C~l - ~ 2 - 0.67r, the zero crossings of well-defined double peaks is no longer visible (Figure 7.1 l(b)), the symmetric bright side of each vertical edges are evident. The results agree well with the theoretical explanation advanced earlier, and with the numerical simulations based on the theory as shown in Figures 7.12(a) and 7.12(b). As clearly seen, the edge enhancement results are symmetric when two AO cells are used.

7 Contemporary Topics in Optical Image Processing

234

]r

,2,pti,_-.~ fielcl o

ii,

I

C:lt:,ti,::al field

11,.........

I,

,.,..,._,_._,'1'

'ltll

..........],, "2

~!,.

I1I ,I I

-1

1

-2

2

-1

(~)

'[Lll""-0

1

2

(b)

"1 i"r]

Opti,_-.N Intensity

{-:Ii:.,ti c:al

Intes~sit7

0

-2

, ...............!II -1

0

tl..........

2

1

(c)

'1

.......... I ol .........}j'I,............. 11' i 2 -2

-1

...........

0

(d) m

I

'1

II",,.

Optic~

i

Intensity

......... 7 1 1

,,,'~11

..............

'~1jD -2

-1

IJr .....

tlJ'l 0

1

2

(e)

Figure 7.9 Schematic diagrams to explain second order derivative effect in the case of cascaded AO cells with counterpropagating sound. (a) input (b) optical field after 2nd derivative operation with zero constant term, (c) optical intensity corresponding to (b); (d) optical intensity after second derivative, and with small constant term; (e) optical intensity after second derivative, and with small constant term [Cao et al. (1998)].

235

7.2 Acousto-Optic Image Processing

H eNe

object ,i

AO I AO Lens 2 cell 11~ cell 2

Lens 1

r

Lens 3

Iris

Lens 4

I

"i"

CCD

.....

I

I

I

I

Figure 7.10 Experimental set up for 1D edge enhancement using two cascaded AO cells with contra-propagating sound. [Cao et al (1998)].

Figure 7.11 Experimentalresults of edge detection using two AO cells with contrapropagating sound with a's equal to (a) 0.857r, (b) 0.67r.

7.2.3 Two-dimensional processing and four-corner edge enhancement In this case, the two AO cells are placed so that they are orthogonal to each other. In one, the sound propagates along z, while in the other the sound propagates along y. The first cell performs processing along the z direction, while the second performs the processing along y. Conceptually, the two-dimensional processing can be understood by examining the approximated transfer functions similar to Eq. (7.2-2):

7 Contemporary Topics in Optical Image Processing

236

Figure 7.12 Numerical simulations for edge detection using two cascaded AO cells with contra-propagating sound. (a) zero-crossing location of edges with both c~'s -- 0.857c, (b) symmetric inner edges with both a's=0.67r [Cao et al. (1998)]. r

y) - (A1 -

BlO/Ox)r

y),

(7.2-6)

and the zeroth order field after the second AO cell is, approximately, r

(x, y) - (A2 -

B20/Oy)r

1) (x, y),

(7.2-7)

where A1, A2 and B1, /32 are the same functions as A and /3 respectively, but the c~, Q, and A0 in the relations for A a n d / 3 are the

7.2 Acousto-Optic Image Processing

237

parameters of the first and second AO cells respectively. From Eqs. (7.26) and (7.2-7), and assuming A1 - A2 = 0, the envelope of the zeroth optical field after the two AO cells can be written as ff)~2)(X, Y) -- B1B2(O2/OxOy)~)inc(X, Y).

(7.2-8)

Experimentally, a 2 mm • 2 mm square transparency (the size is demagnified by a factor of 2 using lenses 1 and 2 as shown in Figure 7.13) is used as the incident object. [Banerjee et al. (1997), Cao et al. (1998)]. The sound frequencies of the two AO cells are set at 55 MHz. By adjusting the sound pressure of the two AO ceils, the values of OL1 and Oz2 can be equal to 0.857r. The results, imaged on and recorded by a CCD camera, are shown in Figure 7.14. Figure 7.14(a) shows the vertical edge enhancement, as stated before, when only the first AO cell is active. With only the second AO cell in operation, the two horizontal edges appear in Figure 7.14(b). With both AO cells in operation, the second order partial derivative operation is achieved and is evident by the fourcorner enhancement of the square object, as shown in Figure 7.14(c). Numerical simulations, not shown here, are in agreement with the experimental results. Therefore, it is feasible to perform the symmetric four-corner detection in real-time using AO cells. AO cell 1

I

He-Nelaser object lens-I

,

fl

fl

~a'~sducer

/

lms-2 ] /

f2.,f<.

/

AO cell2

lens-3

/-\- f~

k

,

iris lms4

f~

, f4

CCD

f4,

lrm~sducer

Figure 7.13 Experimental set up for 2D image processing with two orthogonally oriented AO cells, where fl - 6 are the focal lengths of lens-1- lens-4 respectively. The object is put in the front focal plane of lens 1. Iris is put in the back focal plane of lens 3 to select the zeroth order image. Lenses 3, 4 are used to exactly image the result onto the CCD camera. [Cao et al (1998)].

238

7 Contemporary Topics in Optical Image Processing

Figure 7.14 Experimental results of the second order partial derivative operation of a square object with 2 mm side. (a) 1D edge enhancement along the transverse direction x with only the first AO cell in operation at OzI --0.8571"; (b) 1D edge enhancement along the transverse direction y with only the second AO cell in operation at c~2 = 0.857r; (c) second partial derivative operation result which is evident by the 4comer enhancement of the square object [Banerjee et al. (1997)]. B e f o r e closing, we r e m a r k that the results p r e s e n t e d also s h o w the a s y m m e t r y p h e n o m e n o n d i s c u s s e d earlier. The intensities o f the f o u r c o r n e r s are not the same. This p r o b l e m can be o v e r c o m e by u s i n g t w o

7.3 Photorefractive Image Processing

239

cascaded AO cells with contra-propagating sound waves in orthogonal orientations, respectively.

7.3 Photorefractive Image Processing Beam propagation through photorefractive materials has been analyzed in Chapter 4. In this Section, we will discuss some novel image processing applications of PR materials. Recently, we have observed scattering of an optical beam of wavelength 514 nm and a diameter of approximately 0.5 mm in a crystal of PR KNbO3 into a cone of semiangle about 1 degree and eventual reorganization into six symmetrical spaced spots on the cone [Banerjee et al. (1995)]. The near field pattern exhibits a surprisingly symmetric periodic hexagonal array [Banerjee et al. (1995)]. We have proposed a theory for the onset of transverse instabilities based on local and nonlocal nonlinearities and on the generation of both transmission and reflection gratings [Banerjee et al. (1999b)]. First we will discuss novel image processing applications of the setup when a second optical field, for instance, the spatial Fourier transform of an object transparency is coincident in the crystal along with the pump beam described above. As discussed below, we can achieve phase conjugation in a two-beam coupling geometry. We can also perform edge detection without any externally placed or previously recorded high-pass spatial filter, through a simple variation in the setup geometry. Furthermore, through yet another simple change in the experimental set-up and procedures, we can broadcast the information of the transparency at every location of the far-field hexagonal spots [Banerjee et al. (1996)]. Referring to Figure 7.15, we employ the pump beam and the spatial Fourier transform of the object transparency t(x, y) formed by a lens L of focal length f = 8.83 cm (also called the signal beam) to write a hologram in the KNbO3:Fe crystal of dimensions 5 • 5 x 5mm [Banerjee et al. (1996). The c-axis is oriented along the direction of propagation of the pump beam and the polarization of all incident beams are horizontal. The pump beam power is 7.5 mW, and the power of the beam illuminating the transparency is 3.0 mW. In traditional phase conjugation, a reading beam counterpropagating to the pump beam (also called the counterpropagating pump beam) gives an output

240

7 Contemporary Topics in Optical Image Processing

counterpropagating to the signal beam. The phase of this resulting beam is reversed with respect to the signal beam, justifying the name phase conjugate. In the setup shown in Figure 7.15, no counterpropagating pump is used. Instead, the incident pump beam simultaneously reads out the hologram; this generates the conventional 0 and + 1 orders, along with a higher (-1) order which is the phase conjugate of the object. This is shown in Figure 7.16b (monitored at a distance of 1 m behind the crystal), which depicts the phase conjugate of the picture in Figure 7.16(a), which is our test object. The phase conjugate (as well as the edge enhanced image, to be discussed later) can be monitored using a laser beam analyzer. For brevity, we have not shown the virtual image of the object (which appears in the +1 direction), and the hexagonal spots produced due to self-diffraction of the pump beam. All optical fields behind the crystal are also horizontally polarized. We have taken care in the experiment to ensure that the object t is placed exactly at the front focal plane of the lens L. We remark that in practice, if the size of the object is large one may insert a beam expander before the transparency, and employ a confocal two-lens arrangement after the transparency to reduce the effective size of the object before Fourier transforming with the lens L.

Xr

0

1

,I"

____---

4_i-t --____ -1

J J mirror

Kl~O3:Fe iar~L ~a-am~ ~'ar~:y t

A

X

Figure 7.15 Experimental setup for observing forward phase conjugation and edge enhancement. Both appear in the -1 order [Banerjee et al. (1996)].

7.3 Photorefractive Image Processing

241

First-order diffraction from thin crystals of PR BSO and BTO has been observed in connection with applications to real-time holographic interferometry. We believe that the reason for the generation of the -1 order in our experiment may be also due to the fact that a thin hologram is stored in the KNbO3 sample. This is made possible by the fact that the angle between the nominal directions of propagation of the two participating beams is rather large in our case (about 10 degrees), and furthermore, the finite extent of the Fourier transform T of t has a small area of overlap with the pump beam. The hologram is stored as a weak phase grating in the material, with the induced change in the refractive index being a function of the intensity of the interference pattern between the pump and the spatial Fourier transform of the object, facilitated through the nonlinear material response of the material. We note that our experiment shows that it is possible to simultaneously record and read a hologram using the same pump beam. We note that an additional phase conjugated wave, generated by diffraction of the back-reflected pump beam, propagates back along the direction of the signal wave. This additional wave can also be used for image formation, as well as performing correlation with phase objects[ Banerjee et al. (1996)].

7.3.1 Edge enhancement Edge enhancement of the object due to differential spatial filtering, as described below, is achieved through a minor change in the experimental setup. The object t is slightly displaced from the front focal plane of the lens L. Figure 7.16c shows edge enhancement of our test object, monitored at a distance at a distance of about 40 cm behind the crystal. Once again, the optical field comprising the edge is horizontally polarized. The presence of the edge enhanced output suggests the presence of an induced differential spatial filter in the crystal.

242

7 Contemporary Topics in Optical Image Processing

Figure 7.16 (a) picture of the transparency t, (b) phase conjugate of the input transparency, (c) edge detection of the input transparency [Banerjee et al. (1996)].

A simple explanation for the generation of the phase conjugate and edge detection in a PR material with nonlocal diffusive nonlinearity may be advanced as follows. As stated in Chapter 4, the induced change in the refractive index A n is proportional to VI, where I is the intensity distribution of the interference pattern in the PR material. In general, I = I ( x ) , and assuming one transverse dimension, I(x) - 1 + T(kox/f)exp(

- jkoxsinO)exp( - j/3x 2) + c.c.;

(7.2-9)

where/3 = ( k o / 2 f ) ( 1 d o / f ) , T represents the Fourier transform o f t suitably scaled to spatial coordinate x on the back focal plane of the lens L, which is the Fourier transform plane. Also,/c0 denotes the propagation constant of light, f is the focal length of the lens, do is the object distance of the transparency in front of the lens, and 0 is the angle between the two beams. The first exponential represents the effect of the angle between the two incident fields, while the second exponential denotes the effect of phase curvature on the back focal plane of L due to the offset ( d o - f) of the object from the front focal plane. Hence, An contains terms such as j k o ( s i n O ) T * ( k o x / f ) e x p ( j k o x s i n O ) e x p ( j / 3 x 2) and j/3xT*(kox/f)exp(jkoxsinO)exp(j/3x2). If we assume a weak phase grating, it is easy to show that a phase transparency function exp( - jr ~_ 1 - j r r - k o A n L ~ f f can approximately represent the action of the crystal, where/c0 is the propagation constant of the light in the material and Lef f denotes the effective interaction length of the two participating incident optical fields. To understand how the phase conjugate or the edge enhanced object is formed, note that this phase grating is

7.3 PhotorefractiveImage Processing

243

simultaneously read out by the pump beam (assumed to be a plane wave ). If/3 = 0 (i.e., t is placed at the front focal plane of L), the first term is responsible for the creation of the phase conjugate of the object. Theoretically, this phase conjugate should appear in the far-field of the hologram. On the other hand when t is sufficiently offset from the front focal plane of L, the second term dominates, and is responsible for the edge enhanced output. It can be readily shown that the edge enhanced image forms at z - f 2 / ( d o - f ) b e h i n d the Photorefractive crystal. The presence of the multiplier z in the second term above essentially simulates a differential spatial filter characteristic which operates on the Fourier transform of the object. Thus no external spatial filter or previously recorded filter as discussed in Chapter 6 is required in our case to accentuate the high spatial frequencies that is essential for edge detection. Note that, in general, some of the phase conjugated output will appear in the same general direction as that of the edge enhanced output. A final note about the experimental results: the object was moved through different offsets from the front focal plane of the lens L. Edge detection is clearly observed when do>f, with little "contamination" from the phase conjugate, in agreement with the above theory. In our experiment, the edge enhanced image was in focus at a distance of 40 cm behind the PR crystal, which implies a misfocus of the object by 2 cm from the front focal plane of the lens, which is in reasonable agreement with our experimental conditions. With different amounts of the object misfocus ( d o - f), the locations behind the crystal where the edge enhanced image comes to focus change. In some cases, double edges were observed, possibly from multiple reflections from the crystal faces; however, by fine tuning the position and orientation of the crystal, these could be eliminated, yielding a single edge-enhanced image. When the conjugate was observed, we still noted a slight amount of edge enhancement, possibly due to a small, though nonnegligible, misfocus (i.e., do not exactly equal to 0, or due to the fact that the phase conjugate, according to our theory, ideally forms in the far field behind the crystal. In the experiment, the phase conjugate was monitored 1 m away so that intensities are still detectable using our beam analyzer.

7.3.2 Image broadcasting Finally we will describe a simple variation of the experimental procedure described above to achieve broadcasting of an input object

244

7 Contemporary Topics in Optical Image Processing

into more than one location in space. Traditionally, the decomposition of optical beams into several directions can be achieved using gratings, or diffractive optics in the general case. To achieve image broadcasting, we reduce the angle between the two participating optical fields in our case to zero. Furthermore, we first expose the crystal to the pump beam for a few seconds to set up the transmission and reflection gratings which are responsible for directing the pump into a hexagonal pattern in the far field. As reported in Banerjee et al. (1999b), spatially and temporally stable hexagonal patterns can be achieved when the incident pump beam is exactly normal to the PR interface. We remark in this connection that the hexagonal pattern may be rotated by having the pump beam make a small angle (about 0.05 degrees) to the PR interface [Banerjee et al. (1995)]. Thereafter, with the pump switched off, the crystal is illuminated with the Fourier transform of the object t. The far-field pattern is shown in Figure 7.17 and shows the broadcasting of the letter "T" to the locations of the hexagonal pattern. It is also possible to change the orientation of the pattern by exposing the crystal to the pump beam sufficiently long and so that the pump makes a small angle (approximately 0.04 degrees) with the normal to the incident surface of the crystal. The prolonged exposure reorients the far field hexagonal pattern. Reillumination of the induced gratings created by the pump with the Fourier transform of the object broadcasts the images at this new orientation.

Figure 7.17 Image Broadcasting. The image of the letter T is broadcast to the location of each of the six spots in the hexagonal pattern [Banerjee et al. (1996)].

7.3 PhotorefractiveImage Processing

245

7.3.3 All-optical joint transform edge-enhanced correlation Correlation, using a joint transform correlation scheme (see Problem 6.1), can also be achieved using PR materials. In many cases, edge-enhanced joint transform correlation is possible because of the added advantage of using PR materials having properties similar to that described above. We have performed edge enhanced correlation using PR polymers. Polymer materials are attractive for their high gain property, low light level response and ease of fabrication [Matsushita et al. (1999)]. However, in the material that we have used, a bias voltage, typically of the order of 25 V/micron is required. A schematic of the experimental setup is shown in Figure 7.18. It is the typical two-beam coupling configuration with a reading beam coming on the bisector axis [Banerjee et al. 2000a]. The proposed configuration provides simplicity of the setup since all beams are traveling nominally in the same direction. The two objects tl and t_l are placed at the front focal plane of the Fourier transform lens L1. A grating is formed in the material due to the interference between the optical fields that carry the Fourier transforms of the objects. A reading beam in the bisector direction reads the grating, and finally L2 and L3 are used to perform the Fourier transform of the diffracted beams to obtain the far field distributions. We would like to point out that the reading beam is turned on for a short time after the grating is formed from the two object beams, and the correlation output decays with time, due to grating erasure. Recording of data for the correlation output is done at the beginning of the readout process. Alternatively a beam from another laser source could be used as the reading beam.

246

7 Contemporary Topics in Optical Image Processing

Figure 7.18 Joint transform correlation schematic using a PR material [Banerjee et al. (2000a)].

It turns out that in a typical experiment, one observes higher order diffraction resulting from, amongst others, the fact that the material is thin, and hence acts as a thin phase hologram or grating. Details of distinguishing the correlation results from the higher order diffraction is outlined in [Banerjee et al. (2000a)]. The two objects used in this experiment are shown in Figure 7.19. We have set P1 - 3.3 mW, P-1 5.4 mW, reading beam power = 17.6 mW. Beam powers are monitored immediately after beam splitting to generate the three beams. The object Fourier transform lens has a focal length 75 cm and the output Fourier transfrom lenses have focal length of 15 cm each.

7.3 Photorefractive Image Processing

247

Figure 7.19 Objects used for autocorrelation and cross-correlation [Banerjee et al. (2000a)].

The results for autocorrelation and cross-correlation for an applied bias voltages of V=1000 V and 1300 V, recorded using a CCD camera and MacPhase software, are shown in Figure 7.20a-d. Note that the autocorrelation results are much sharper than the cross-correlation results, as expected, and in agreement with our simulations. The presence of three peaks, one primary and two secondary, which we notice from Figure 7.20 a,b when examined carefully, is characteristic of edge enhanced correlation, and is described in detail in [Banerjee et al. (2000a)] .The correlation intensity is proportional to the applied voltage, which agrees with simulation results. Before closing, we would like to point out that by eliminating the reading beam, it is possible to obtain triple correlation between the two input transparencies. This concept has been recently used to identify biological specimens, e.g. algae, and their concentrations [Banerjee et al. (2000b)].

248

7 Contemporary Topics in Optical Image Processing

Figure 7.20 (a) Autocorrelation result for 1000V, (b) autocorrelation result for 1300V, (c) cross-correlation for 1000V, (d) cross-correlation for 1300V[Banerjee et al.

(2000a)].

7.4 Dynamic Holography for Phase Distortion Correction of Images Holographic correction for aberrations in telescopic systems has been the subject of extensive recent studies [Gonglewski and Vorontsov (1998)]. In almost all schemes, the technique involves writing the hologram of the phase distortion in a holographic recording medium and reading out this hologram with light from the object that passes through the same phase distortion. In many cases, this is experimentally demonstrated using a point source as the object, or using coherent light as an illumination for the object. In some cases, the chromatism of the holographic corrector is compensated for by using an auxiliary diffraction grating, providing the possibility of operation over a wide spectral range. Recording media such as an optically addressed spatial

7. 4 Dynamic Holography for Phase Distortion Correction of Images

249

light modulator (SLM) have been used to implement the phase correction. Reading of the hologram with a different wavelength or with "white" light has shown deterioration in the image quality. To date, optically addressed SLMs used for dynamic holographic recording and readout are composed of a ferroelectric liquid crystal (FLC) - photoconducting layer sandwich [Beresnev et al. (1998)]. The photoconducting layer is made from amorphous hydrogenated silicon carbide (a-SiC:H). The sandwich is biased in the 5 - 40 V range using transparent indium tin oxide electrodes, producing a field across the FLC region, which depends on the bias, as well as on the local photoconductor response to the incident optical intensity. Diffraction efficiencies as high as 20 - 35% for spatial frequencies in the range of 100 lines/mm can be obtained. Refresh rates in excess of 1 kHz with write beam energy densities of less than 100nJ/cm 2 can be achieved. The concept behind removal of phase distortions is straightforward and simple. A hologram of the phase distortion is recorded and then read out by the contaminated image passing through the same phase distortion. It is easy to show (see Problem 7.15) that the readout contains the corrected image in one of the diffracted orders. One of the other diffracted orders has twice the phase distortion. We now describe a setup used to compensate for both large and small scale phase distortion of images in the laboratory using optically addressed FLC SLMs [Banerjee et al. (1999a)]. The setup is shown in Figure 7.21. The hologram of the phase distortion is recorded in real time on the optically addressed FLC-SLM using green light from an Ar laser. Large scale and severe distortions within the modulation transfer function passband of the SLM are introduced using the plexiglass that was heated to the point where it developed large random changes in thickness, providing phase distortions over 207v. Small scale distortions are synthesized by shining a speckle pattern from a diffuser onto a fiberoptic bundle connected to a liquid crystal light valve. We find that amongst other contributing factors, it is most important to image the phase distortion onto the recording medium for phase correction to be effective. The typical angle between the nominal directions of propagation between the two writing beams is about 3 degrees. The light from the Ar laser is passed through a spatial filter and collimated to ensure plane wavefronts. Care is taken to avoid any additional phase distortions in the paths of the two writing beams by using a series of confocal two-lens imaging systems. The frequency of the bias voltage

250

7 Contemporary Topics in Optical Image Processing

across the SLM is optimized to achieve m a x i m u m diffraction efficiency from the SLM. The readout of the hologram is first performed using the object illuminated by coherent red light from a He-Ne source. This light, which is also passed through a spatial filter and collimated, is carefully introduced into the system upon illuminating the object so that it overlaps with the Ar laser light and with identical wavefronts. The FLCSLM is opaque to green light, hence, the light behind the S L M only comprises the undiffracted and diffracted red light. The undiffracted order and the diffracted order that contains the phase correction are individually imaged with single lens systems onto CCD cameras to display the distorted and corrected versions of the input image.

Figure 7.21 Schematic setup of wavefront distortion correction using dynamic holography recorded in ferroelectric SLMs. The pertinent diffracted order is imaged to get the compensated image of an arbitrary object. The plexiglass provides large scale aberrations and the speckle field that provides the small scale aberrations is generated by placing ground glass near the back focal plane of a collimated Ar laser light. This is transported to the LCLV using a fiber-optic bundle. (a) undistorted object. This is imaged through the system in the absence of any phase aberrations. Note the coherent noise during imaging. (b) object distorted through small and large scale aberrations. This was recorded in the undiffracted order behind the FLC-SLM. (c) compensated image in the first order. Note again the presence of speckle or coherent noise in the system [Banerjee et al. (1999a)].

7. 4 Dynamic Holography for Phase Distortion Correction of lmages

251

Results are shown in Figures 7.21 (a)-(c). Note that large scale and amplitude phase distortions as well as small scale aberrations are significantly compensated for using the holographic system in one of the diffracted orders. In other diffracted orders, for instance the one on the other side of the undiffracted beam, the phase distortion can be more severe than that of the uncompensated image. This is due to the fact that the phase of this diffracted order is twice that of the uncompensated image. Figure 7.21(c) also shows the residual aberrations. Possible causes for these residual aberrations are phase distortions in the reference beam from propagation and imaging through optical hardware, coherent noise, nonuniformity in optically addressable SLM layer thickness, wavelength mismatch between recording and reading beams etc. We have also performed the experiment on distortion correction when the object is illuminated with an incoherent white light source which simulates a spatio-temporally incoherent illumination of the object. This is an interesting practical problem, since it involves incoherent imaging and is therefore devoid of coherent noise. It was easy to do this in our experiment since nowhere in the system is any processing or compensation done on the Fourier plane. Although the effect of phase distortion on incoherent imaging is different from the coherent case, we have observed very good quality phase distortion suppression for images illuminated by incoherent light, as shown in Figures 7.22 (a)-(c). The effect of small scale distortion is less pronounced in this case, along with the lack of coherent noise, as shown in Figure 7.22(b). Figure 7.22(c) shows the corrected image. We point out that the intensity is weaker than for coherent illumination, and is a result of the fact that the light energy diffracted from the grating in the SLM is small since the illumination is incoherent and there is a much weaker component of the red that is transmitted through the FLC-SLM sandwich. We have also repeated the experiment using a narrow band red color filter in between the white light source and the object to simulate primarily spatially incoherent illumination, and obtained similar results as well.

7 Contemporary Topics in Optical Image Processing

252

Figure 7.22 Phase distortion correction using incoherent illumination of the object. (a) undistorted object imaged through the system without any small or large scale distortion. Note the absence of coherent noise. (b) distorted image monitored in the undiffracted order. The effect of small scale distortions is less than in the coherent case. (c) compensated image in the diffracted order. The intensity is much weaker than the compensated image using coherent illumination; however, coherent noise is absent [Banerjee et al. (1999a)].

The experiments have been repeated for various amounts of phase distortion by changing the position of the plexiglass plate and the focusing beam incident on the diffuser for generating the speckle pattern. For instance, Figures 7.21 (b),(c) above show the limiting case of the maximum compensatible phase distortion using coherent light. We have demonstrated phase distortion correction of coherent and incoherent images using dynamic holography with optically addressed SLMs over a large range of distortions. The advantages of the method are its simplicity, speed (depending on the speed of the SLM) and the ability to compensate for large scale distortions which is not possible using conventional adaptive optics technology. The remaining distortions may be removed using adaptive optics techniques using micro-electromechanical devices (MEMS), and is the subject of a lot of current research.

Problems 7.1

With reference to Eq.(7.1-18), find OTFa(tc,, kv) for Px (x, y) rect(x/X) and p2(x, y) - rect[(x - xo)/X]. What kind of spatial filtering the pupils are synthesizing?

7.4 Dynamic Holographyfor Phase Distortion Correction of Images

253

7.2

Find the point spread function if the optical transfer function of the acousto-optic heterodyning image processor is given by Eq. (7.1-19). For ~1 = 0.01 and c71 = 0.02, plot the point spread function and its optical transfer function for a fixed # and ky, respectively. Is the resulting point spread function bipolar?

7.3

Find the sine-FZP, cosine-FZP, and complex hologram of a threepoint object given by 5(x, y; z - zo) + 5 ( x - xo,y; z - zo) + (~(x,y; z - (zo+Azo))

7.4

Hologram scaling: After we have recorded the three holograms of the three-point object, obtained in problem 7.3, we want to scale the hologram to t36(Mx, My) if t36(x, y) is the original hologram, where M is the scale factor. Find the location of the real image reconstruction of the three-point object if the hologram is illuminated by a plane wave. Draw a figure to illustrate the reconstruction locations of the three points.

7.5

With reference to Problems 7.3 and 7.4 and defining the lateral magnification Mt~ as the ratio of the reconstructed lateral distance to the original lateral distance :co, express Mlat in terms of M.

7.6

Defining the longitudinal magnification Mlo~9 as the ratio of the reconstructed longitudinal distance to the original longitudinal distance Az0 for hologram t36(Mx, M y ) given in problem 7.4, express Mlo~g in terms of M.

7.7

With reference to Table 7.2 and for "sigma" equal to 0.5 l, write a MATLAB code for the reconstruction of a complex hologram. Compare your results with Figures 7.4 (d) and (e) and make some comments.

7.8

With reference to Problem 7.7, write a MATLAB code to reconstruct the complex hologram at "sigma" equal to 0.61. Interpret your results.

7.9

Determine the transfer function of the zeroth order if light is incident nominally normal to the direction of sound propagation

7 ContemporaryTopics in OpticalImage Processing

254

in a cell with a finite Q. Argue that one can obtain notch filtering in the zeroth order. 7.10

Draw the schematic diagram of the experimental setup using AO cells to get symmetric two-dimensional corner enhancement. Validate your answer by programming the transfer functions using MATLAB and using a square aperture as the object.

7.11

Using the one-dimensional symmetric edge enhancement setup, write a MATLAB program to examine the edge enhancement of a circular aperture.

7.12

How would you obtain two-dimensional edge enhancement of a square aperture using AO image processing? Give reasons for your answer.

7.13

Outline a four-wave mixing scheme that yields edge enhancement of an object using a PR material. Hint: Make use of the fact that using linearized theory, the modulation depth of the hologram written in a PR material is a function of the amplitude ratio of the two interacting waves writing the hologram.

7.14

Show that using the setup for correlation using PR materials, and by eliminating the reading beam, one can obtain triple correlation between the input objects. Point out any advantage(s) of using triple correlation to identify objects.

7.15

Assume that a thin phase hologram of a phase object r y) is recorded using a plane wave as a reference. If now the hologram is read out by an optical field t(x, y)r y), show that one of the diffracted orders contains t(x, y). Describe what you would expect in the other diffracted orders.

References 7.1 7.2

Banerjee, P.P., D. Cao and T-C Poon (1997). Applied Optics 36 3086. Banerjee, P.P., E. Gad, T. Hudson, D. McMillen, H. Abdeldayem, D. Frazier and K. Matsushita (2000a). Applied Optics 39 5337.

7.4 Dynamic Holography for Phase Distortion Correction of Images

7.3 7.4

255

Banerjee, P.P., L.A. Beresnev and M~A. Vorontsov (1999a). Proc. SPIE 3760 83. Banerjee, P.P., N. Kukhtarev and J.O. Dimmock (1999b).

"Nonlinear self-organization in photorefractive materials", invited book chapter in Photorefractive Materials and Applications, F.T.S. Yu ed., Academic, N.Y. 7.5 7.6

7.7 7.8 7.9 7.10

7.11 7.12

7.13 7.14

7.15 7.16

7.17 7.18 7.19 7.20 7.21

Banerjee, P.P., N. Kukhtarev, T. Kukhtareva, J. Jones and E. Ward (2000b). Proc. SPIE 4110-07. Banerjee, P.P., H-L Yu, D. Gregory, N. Kukhtarev and H.J. Caulfield (1995). Optics Letters 20 10. Banerjee, P.P., H-L Yu, D. Gregory and N. Kukhtarev (1996). Optics and Photonics Technology Letters 28 89. L. A. Beresnev, L.A., P. Onokhov, W. Dultz (1998). SPIE Proc. 3432 151. Bryngdahl, O and A. Lohmann (1968). Journal of the Optical Society of America 58 620. Cao, D., P.P. Banerjee and T.-C. Poon (1998). Applied Optics 37 3007. Chavel, P. and S. Lowenthal (1978). Journal of the Optical Society of America 68 559. Doh, K., T.-C. Poon, M. Wu, K. Shinoda, and Y. Suzuki (1996). Laser & Optics Technology 28 135. Duncan, B.D, T.-C. Poon, M.H. Wu, K. Shinoda and Y. Suzuki (1992). Journal of Modern Optics 39 63. Gonglewski, J.D. and M. A. Vorontsov (1998). Artificial turbulence for Imaging and Wave Propagation, Editors, SPIE Proc. 3432. Kim, S.-G., B. Lee, and E.-S. Kim (1997). Applied Optics 36 4784. Kuo, C.J., Guest editor (1996). Optical Engineering 35, June Issue. Indebetouw, G. and T.-C. Poon (1984). Applied Optics 23 4571. Indebetouw, G. and T.-C. Poon (1986). Optica Acta 33 827. Indebetouw, G., T. Kim, T.-C. Poon, and B. Schilling (1998). Optics Letters 23 135. Indebetouw, G. and T.-C. Poon (1992). Optical Engineering 31 2159. Mait, J (1986). Journal of the Optical Society of America A 3 1826.

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7.22 7.23 7.24 7.25 7.26 7.27 7.28 7.29 7.30

7.31 7.32

7.33

7 Contemporary Topics in Optical Image Processing

Mishina, T., F. Okano, and I. Yuyama (1999). Applied Optics 39, 3703. Matsushita, K., P.P. Banerjee, S. Ozaki and D. Miyazaki (1999). Optics Letters 24 593. Park, J. and T.-C. Poon (1988). IEEE Transactions on Geoscience and Remote Sensing 26 839. Poon, T.-C. and A. Korpel (1979). Optics Letters 10 317. Poon, T.-C. (1985) Optics Letters 10 197. Poon, T.-C, J. Park, and G. Indebetouw (1990). Optical Engineering 29 1507. Poon, T.-C., K. Doh, B. Schilling, K. Shinoda, Y. Suzuki, and M. Wu (1997). Optical Review 4 567. Poon, T.-C. and T. Kim (1999). Applied Optics 38 370. Poon, T.-C., T. Kim, G. Indebetouw, B.W. Schilling, H.Wu, K. Shinoda, and Y. Suzuki (2000). Optics letters 25 215. Schilling, B.W., T.-C. Poon, G. Indebetouw, B. Storrie, K. Shinoda, and M. Wu (1997). Optics Letters 22 1506. Takaki, Y., H. Kawai, and H. Ohzu (1999). Applied Optics 23 4990. Xia, J., D. Dunn, T-C. Pooh and P.P.Banerjee (1996). Optics Communication 128 1.

257

SU BJEC T I N D E X

1-D gradient operator, 227 asymmetric, 231 1-D Laplacian operator, 232 1-D second order derivative, 234 2-D image processing, 235 2-D mixed derivative, 237 3-D display, 222 Abbe condition, 74 ABCD formalism, 94 ABCD matrix, 20 ABCD parameters, 86 aberration, 18 Abramson ray tracing method, 200 acoustic medium, 111 acousto-optic cell, 114 acousto-optic diffraction, 111 acousto-optic heterodyne image processing, 215 acousto-optic heterodyne image processor, 208, 211 acousto-optic image processing, 226 acousto-optic interaction downshifted, 113 near phase synchronous, 129 upshifted, 113 acousto-optic interaction length, 227 acousto-optic material, 253 acousto-optic spatial frequency response highpass filter, 116, 226 lowpass filter, 116 acousto-optics, 111,207 contrapropagating sound waves, 231,233, 235,236 spatial interaction transfer functions, 115, 116, 226, 252 Airy pattern, 72 algae, 247 amplitude transmittance, 172 angular frequency, 46 angular magnification, 20 annular filter, 157, 159 aperture circular, 71,227 difference of Gaussian, 214 rectangular, 58 square, 233,237 autocorrelation, 147, 214, 247 back focal plane, 21 background illumination, 140 Baker-Hausdorff formula, 99 bandpass filter, 155,209, 215,218

barium titanate, 172 beam distortion, 126 beam fanning, 125, 126 deterministic, 126 beam propagation method, 99, 105 flow diagram, 100 beam shaping, 126 Bessel differential equation, 109 Bessel function, 72 bilinear transformation, 86 biological specimen, 247 bipolar incoherent image processing, 149, 207, 214,216 bistability, 118 Boltzmann constant, 125 Bragg angle, 114 Bragg incidence, 113, 115 cardinal points, 20 carrier frequency holography, 185 cascaded acousto-optic cells, 231,235,236 c-axis, 125 characteristic impedance, 50 charge density, 40 chirp signal, 176 coherent image processing, 140, 208 coherent noise, 251 coherent transfer function, 137, 139, 148, 165 comb function, 3 complex amplitude, 53 complex spatial filter, 144, 169, 188, 189, 190, 191,208 conjugate planes, 20 conservation of energy, 112 conservation of momentum, 112 conservation of power, 130 constitutive relations, 42, 44 continuity equation, 42 contrast, 171 contrast reversal, 140, 141 convolution, 1, 6, 178, 184, 191,192, 196 corner detection, 207 Cornu spiral, 60, 61 correlation, 1, 6, 127, 147, 191,192, 207, 211, 220, 245,253 edge enhanced, 207, 208 properties of, 147 critical angle, 15 cross-correlation, 214, 247 current density, 40 cylindrical wave, 47

258

DC component of image, 140 delta function, 2, 5, 57, 175, 197, 215 developer, 171 diaphragm, 75 dielectric, 43 diffraction efficiency, 113,249 diffraction orders, 253 diffractive optical elements, 173 diffrative optics, 244 diffusion, 124 digital image processing, 227 dipole moment density, 43 discrete Fourier transform, 3,229 divergence theorem, 40 Doppler shift, 114 double lens system, 33 edge detection, 239, 242 edge enhanced correlation, 247 edge enhancement, 127, 207, 208,226, 227, 240, 241,243,253 I-D, 227, 228, 229 asymmetric, 231 corner detection, 235,238, 253 horizontal, 237, 238 of circular aperture, 228, 230 symmetric, 232, 233,234 vertical, 237, 238 edge extraction of incoherent image, 215 textural, 215 edge smearing, 228 effective interaction length, 242 electric field strength, 40 electric flux density, 40 electromagnetic wave, 48 electromotive force, 41 electronic charge, 125 electronic demodulation, 213 electro-optic effect, 118, 125 emulsion, 171 exposure, 170 exposure time, 170 far field, 67 Faraday's law of induction, 41 Fast Fourier transform, 4 Fermat's principle, 9, 10, 11 ferroelectric liquid crystal, 249, 251 fiber-optic bundle, 250 film, 169, 172 first-order diffraction, 241 fixing, 170 f-number, 75 focal length, 27 focal spot size, 88

Subject Index focus back, 28 front, 28 Fourier plane, 138 Fourier series, 3 Fourier transform, 1, 3 continuous, 4 inverse, 1 two-dimensional, 2, 54 Fourier transform hologram, 188 recording, 188 Fourier transforming property of lens, 80 Fourier-Bessel transform, 72 four-wave mixing, 253 Fraunhofer approximation, 67 Fraunhofer diffraction slit of finite width, 68 Fraunhofer diffraction, 67 circular aperture, 73 rectangular aperture, 79 slits, 93 Fraunhofer diffraction formula, 68 Fresnel diffraction, 91, 92,209 square aperture, 63, 92 straight edge, 92 Fresnel diffraction formula, 53, 57, 81,133,134 Fresnel diffraction pattern, 60 Fresnel hologram, 187, 203 Fresnel integral, 61 Fresnel integrals, 60 Fresnel zone plate, 176 cosine, 252 off-axis, 177 sine, 252 time dependent, 219 Fresnel zone plate hologram cosine, 221,225 sine, 221,225 fringe contrast, 91 front focal plane, 20 gamma, 194, 202, 203,204 gamma of film, 172 Gauss's law, 40 Gauss's theorem, 40 Gaussian, 3 Gaussian beam, 84, 85, 86, 113, 126 elliptic, 92, 95 focusing, 87 fundamental mode, 107 in graded index fiber, 106 propagation, 88 self-focusing, 124 waist, 84 Gaussian optics, 18 geometrical image, 137

Subject Index geometrical optics, 9 grating 2-D, 143 induced, 244 reflection, 239 sinusoidal amplitude, 94 sinusoidal phase, 94 transmission, 239 grating equation, 112 gratings, 244 harmonic oscillator, 103 H-D curves, 171 Helmholtz equation, 53, 97 Hermite polynomials, 103 Hermite-Gaussian functions, 103, 104 Hermite-Gaussian modes, 128 heterodyne current, 210, 211 heterodyne detection, 208 hexagonal array, 239 hexagonal pattern, 244 higher order diffraction, 246 highpass filter, 208 hologram, 169, 249 complex, 221 decoding, 178 dynamic, 249 off-axis, 221 optical center, 200 phase, 172,253 ray theory, 199 recording, 245 scaling, 252 thin, 241,246, 253 transmittance, 176 transparency function, 188 hologram readout, 245 hologram transparency function, 195, 219 holographic readout, 249 holographic fluorescence microscopy, 222 holographic imaging, 173 holographic interferometry, 241 holographic magnification, 193 holographic reconstruction, 127, 178, 180, 189 holographic recording, 127, 173, 174, 178, 180, 219 holographic recording medium, 248 holography, 169, 173 dynamic, 208, 248 electronic, 219 off-axis, 183, 184 on-axis, 182 homogeneous medium, 11, 17, 42 Hurter-Driffield curves, 170 image broadcasting, 243

259

image distance, 33 image edge enhancement, 116 image location, 35 image plane, 20 image processing, 207 imaging by single lens, 29 ~maging system single lens, 132 imaging system two lens, 138 impulse response, 5 of imaging system, 137 of propagation, 55, 56, 57, 210 incoherent filtered image, 163, 218 incoherent illumination, 251 incoherent image processing, 146, 208 indium tin oxide, 249 induced inhomogeniety, 120 induced polarization, 118 inhomogeneous medium, 15, 101, 102 intensity, 43, 51 intensity transmittance, 142, 170 intrinsic impedance, 43, 48, 50 ~onization cross-section, 125 irradiance, 51 lsotropic medium, 42 joint transform correlation, 202, 245,247 edge enhanced, 245 Kerr media, 99 Kerr medium, 118 Klein-Cook parameter, 116 leap frog technique, 1O0 lens concave, 27 converging, 27 convex, 27 diverging, 27 front focal plane, 83 matrix, 27, 87 phase transformation, 83, 134 phase transformation of, 80 speed of, 75 thick, 37 thin, 27 transparency in front of, 83 lensless imaging, 165 lens-mirror combination, 37 Lenz' s law, 41 linear medium, 42, 101 linear system, 4 space invariant, 5 linear wave propagation, 43 linearity, 3 liquid crystal light valve, 250

260

liquid crystal television, 185 lithium niobate, 172 lowpass filter, 144, 150, 166 LP modes, 109 magnetic field strength, 40 magnetic flux density, 40 magnification, 20, 30, 31, 33, 34, 139 lateral, 193, 198, 199, 204, 252 longitudinal, 193, 198, 199, 204, 252 MATLAB example, 32, 33, 35, 61, 63, 76, 88, 107, 123,150, 155, 160, 177, 215,221 matrix methods, 18 Maxwell's equations, 40, 90, 101 micro-electromechanical devices, 251 mirror scanning 2_D, 209 modal dispersion, 105 mode number, 105 mode pattern, 104 mode profile, 104 mode structure in step index fiber, 110 multimodal solutions, 122 negative, 170 nodal planes, 21 nondispersive medium, 46 nonlinear focal length, 130 nonlinear inhomogeneous medium, 117 nonlinear optics, 117 nonlinear optics equation, 119 nonlinear partial differential equation, 99 nonlinear polarization, 118 nonlinear refractive index, 118, 120 nonlinear Schrodinger equation, 120 object, 200 object distance, 33 object plane, 20 object wave, 174 obliquity factor, 55 off-axis holography, 185 construction, 186 off-Brag incidence, 113 on-axis holography, 185 operator commutation property, 99 diffraction, 100 inhomogeneous, 100 linear, 99 nonlinear, 99 space-dependent, 99 optical 3-D image recognition, 222 optical fiber, 36, 99 graded index, 18, 102 nonlinear, 120

Subject Index single mode, 110 step index, 108, 129 optical heterodyne scanning, 207 point spread function, 212 optical path length, 10 optical scanning holography, 219, 220, 221,222 impulse response, 224 optical transfer function, 147, 163, 165,208, 211,212,214,215 difference of Gaussian, 218 low pass characteristics, 149 passband, 148 two lens system, 148 orthogonal acousto-optic cells, 235,237 parallel processing, 212 paraxial approximation, 24, 55, 58 paraxial ray, 18, 19 paraxial transfer function for propagation, 98 paraxial wave equation, 97, 98 modified, 98, 120, 125, 130 Parseval's theorem, 7 periodic focusing, 106 periodicity, 3 phase conjugate, 240, 243 phase conjugation, 118, 127, 239, 242 phase distortion, 249 large scale, 249, 250, 251 residual, 251 small scale, 249, 250, 251 phase distortion correction of images, 208, 248, 249, 250, 251 phase modulation, 173 phase object visualization, 142 phase transparency, 242 phasor, 2 phonon, 112 photoconducting layer, 249, 251 photographic density, 170 photomultiplier, 210 photon, 36, 112 momentum of, 36, 70 photopolymers, 173 photorefractive effect, 124 photorefractive image processing, 239 photorefractive material, 99, 172, 207, 253 barium titanate, 128 BTO, 241 diffusion dominated, 124 polymer, 245 potassium niobate, 239 photorefractive medium, 124 photovoltaic effect, 172 physical optics, 9 piezoelectric transducer, 111 pinhole, 215

Subject Index Planck's constant, 70 plane of incidence, 13 plane of polarization, 51 plane parallel layers, 25 plane wave, 47, 58, 90 point source, 57 point spread function, 208, 212, 214 bipolar, 208 coherent, 137, 139 intensity, 147 polarization, 43, 48, 51 circular, 51 elliptical, 52 linear, 51 positive, 172, 194, 202, 203 Poynting vector, 43, 48, 50 principal planes, 20 principal points, 20 principle of least time, 11, 13 propagation constant of mode, 105 propagation vector, 46 pump beam, 239 counterpropagating, 239 pupil function, 81,133, 139, 165, 211,212, 214, 219 annular, 214 defocused, 135 pupil plane, 139 q-parameter, 84, 85 q-transformation, 85, 86, 94 quadratic index profile, 95 ABCD matrix, 95 quantum of light, 70 radius of curvature, 26, 85 ray coordinate, 20 ray coordinates, 32, 33 ray transfer matrix, 19, 20 Rayleigh range, 67, 84, 85, 88 Rayleigh's criterion, 73, 74 rays, 9 reading beam, 239 real image, 179, 185, 190, 196, 197, 200 recombination, 124 reconstruction, 201 reconstruction beam, 204 reconstruction wave, 178 rectangle function, 3 rectangular slit, 144 reference, 200, 201,204 reference wave, 174 reflection, 11 laws of, 36 reflection matrix, 38 refracting power, 24

261

refraction, 11 laws of, 36 refraction from spherical surface, 23 refraction matrix, 24 refractive index, 9 amplitude dependent, 118 change, 98, 125 relative, 14 refractive index change, 172 refractive power, 27 relative permittivity, 43 residual phase aberration, 251 resolution, 73 Ronchi grating, 93 scan speed, 210 sech profile, 121, 129 second harmonic generation, 117 self-focusing, 124, 126 self-refraction, 118 separation of variables, 103 sifting property, 5 sign convention, 30 signal beam, 239 signum function, 7 silver halide, 171 sinc function, 68 single lens, 26 single lens system, 32, 35 Snell's law, 24 Snell's law of refraction, 14 Sobel mask, 227 soliton, 118, 120 spatial, 120 spatial, l-D, 121 spatial, 2-D, 122 sound amplitude, 114 sound wave, 111 space charge field, 124, 125 spatial filter, 138, 142, 143, 150, 164 differential, 241,243 highpass, 239 spatial filtering, 138 incoherent, 160 tunable incoherent, 215 spatial light modulator, 185,248, 249, 250, 251 spatially coherent light, 146 spatially incoherent light, 146 speckle field, 250 spectrum, 54, 84 spherical wave, 48 converging, 190 diverging, 58, 190 split step beam propagation method, 99, 107, 129 square-law medium, 17 Stokes' theorem, 41

Subject Index

262

subharmonic generation, 118 susceptibility, 43 symmetrized split step technique, 100 symmetry, 3

two-beam coupling, 127, 245 two-lens combination, 31, 37 equivalent focal length, 37 two-wave coupling, 118,239

tensor, 118 thermal generation rate, 125 thin lens formula, 30, 135 third harmonic generation, 118 time harmonic fields, 47 total internal reflection, 15 transfer function, 6 of propagation, 53, 54, 100, 190 transfer function approach, 82, 133 translation matrix, 23, 86 traveling wave solutions, 43 triangle function, 7 triple correlation, 247 twin image, 182 twin image elimination, 221 twin image problem, 182 two pupil system, 208

unit planes, 20 unit points, 20 virtual image, 179, 185, 190, 200 volume diffraction gratings, 99 wave equation, 43, 44 homogeneous, 45 in inhomogeneous medium, l 01 wave optics, 9 wavefront, 47 wavelet transform, 123 Young's double slit experiment, 203 zero-order beam, 179, 184 z-transform, 3