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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This book presents a substantial description of the principles and applications of digital holography. The first part of the book deals with mathematical basics and the linear filtering theory necessary to approach the topic. The next part describes the fundamentals of diffraction theory and exhaustively details the numerical computation of diffracted fields using FFT algorithms. A thorough presentation of the principles of holography and digital holography, including digital color holography, is proposed in the third part. A special section is devoted to the algorithms and methods for the numerical reconstruction of holograms. There is also a chapter devoted to digital holographic interferometry with applications in holographic microscopy, quantitative phase contrast imaging, multidimensional deformation investigations, surface shape measurements, fluid mechanics, refractive index investigations, synthetic aperture imaging and information encrypting. Keys so as to understand the differences between digital holography and speckle interferometry and examples of software for hologram reconstructions are also treated in brief. Contents 1. Mathematical Prerequisites. 2. The Scalar Theory of Diffraction. 3. Calculating Diffraction by Fast Fourier Transform. 4. Fundamentals of Holography. 5. Digital Off-Axis Fresnel Holography. 6. Reconstructing Wavefronts Propagated through an Optical System. 7. Digital Holographic Interferometry and Its Applications. Appendix. Examples of Digital Hologram Reconstruction Programs
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Contents
Introduction
Chapter 1. Mathematical Prerequisites
1.1. Frequently used special functions
1.2. Two-dimensional Fourier transform
1.3. Linear systems
1.4. The sampling theorem
Chapter 2. The Scalar Theory of Diffraction
2.1. Representation of an optical wave by a complex function
2.2. Scalar theory of diffraction
2.3. Examples of Fraunhofer diffraction patterns
2.4. Some examples and uses of Fresnel diffraction
2.5. Collins’ formula
2.6. Conclusion
Chapter 3. Calculating Diffraction by Fast Fourier Transform
3.1. Relation between the discrete and analytical Fourier transforms
3.2. Calculating the Fresnel diffraction integral by FFT
3.3. Calculation of the classical diffraction formulae using FFT
3.4. Numerical calculation of Collins’ formula
3.5. Conclusion
Chapter 4. Fundamentals of Holography
4.1. Basics of holography
4.2. Partially coherent light and its use in holography
4.3. Study of the Fresnel hologram of point source
4.4. Different types of hologram
4.5. Conclusion
Chapter 5. Digital Off-Axis Fresnel Holography
5.1. Digital off-axis holography and wavefront reconstruction by S-FFT
5.2. Elimination of parasitic orders with the S-FFT method
5.3. Wavefront reconstruction with an adjustable magnification
5.4. Filtering in the image and reconstruction planes by the FIMG4FFT method
5.5. DBFT method and the use of filtering in the image plane
5.6. Digital color holography
5.7. Digital phase hologram
5.8. Depth of focus of the reconstructed image
5.9. Conclusion
Chapter 6. Reconstructing Wavefronts Propagated through an Optical System
6.1. Theoretical basis
6.2. Digital holography with a zoom
6.3. Reconstructing an image by Collins’ formula
6.4. Using the classical diffraction formulae to reconstruct the wavefront after propagation across an optical system
6.5. Conclusion
Chapter 7. Digital Holographic Interferometry and Its Applications
7.1. Basics of holographic interferometry
7.2. Digital holographic microscopy
7.3. Two-wavelength profilometry
7.4. Digital holographic photomechanics
7.5. Time-averaged digital holography
7.6. Tracking high-amplitude vibrations
7.7. Three-color digital holographic interferometry for fluid mechanics
7.8. Conclusion
Appendix. Examples of Digital Hologram Reconstruction Programs
A1.1. Diffraction calculation using the S-FFT algorithm
A1.2. Diffraction calculation by D-FFT
A1.3. Simulation of a digital hologram
A1.4. Reconstruction of a hologram by S-FFT
A1.5. Adjustable-magnification reconstruction by D-FFT
Bibliography
Index
First published 2012 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
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© ISTE Ltd 2012
The rights of Jun-chang Li & Pascal Picart to be identified as the author of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Cataloging-in-Publication Data
Picart, Pascal.
Digital holography / Pascal Picart, Junchang Li.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-84821-344-9 (hardback)
1. Holography--Mathematics. 2. Holography--Data processing. 3. Image processing--Digital techniques. I. Li, Junchang, 1945- II. Title.
QC449.P53 2012
621.36'750285--dc23
2011050552
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN: 978-1-84821-344-9
Introduction
Holography was invented in 1947 by the Hungarian physicist Dennis Gabor during his research into electronic microscopy [GAB 48, GAB 49]. This invention won him the Nobel Prize in physics in 1971. It took until 1962 [LEI 62, LEI 61, DEN 62, DEN 63] for the first lasers used for this technique to find concrete applications [POW 65, COL 71]. Holography is a productive mix of interference and diffraction. Interference encodes the amplitude and relief of a 3D object, and diffraction works as a decoder, reconstructing a wave that seems to come from the object that was originally illuminated [FRA 69]. This encoding contains all the information on a given scene: amplitude and phase – thus relief. Practically, the execution of a “Denisyuk” type [DEN 62, DEN 63] “analog” hologram is carried out in three steps: the first step concerns the recording of the interference pattern on a photosensitive support, typically a plate of silver gelatin; the second step involves the chemical process of development/treatment of the support (which typically lasts around a quarter of an hour with silver gelatin plates); and the last step is the process of the physical reconstruction of the object wave, where the laser is diffracted by the sinusoidal grating encoded in the photosensitive support, making the initial object “magically” appear. The magic of holography is explained by wave optics [GOO 72, BOR 99, COL 71, HAR 02, HAR 06]. Considering the constraints involved in the treatment of holograms (an essential stage of their development), which make their industrial use difficult for quality control in production lines, for example [SMI 94], the replacement of the silver support by a matrix of the discrete values of the hologram was envisaged in 1967 [GOO 67]. The idea was to replace the analog recording/decoding by a digital recording/decoding simulating diffraction by a digital grating. Holography thus became “digital” [HUA 71, KRO 72]. The attempts of the time suffered from a crucial lack of technological means permitting the recording of holograms while respecting sampling conditions and allowing the reconstruction of the diffracted field with a reasonable calculation time. From the 1970s up to the 1990s we witnessed a veritable boom of holography, as much from the point of view of applications [JON 89, RAS 94, SMI 94, KRE 96], as from the holography of art [GEN 00, GEN 08]. Some industrial systems based on dynamic holography are even currently commercialized [OPT 11]. The material used for the recording is a photoreactive crystal [TIZ 82]. Nevertheless, the difficulty of the treatment of the holograms and the relative complexity of the devices have impeded a real industrial penetration of the methods developed in the laboratory in the past 30 years. In parallel, at the same time, we witnessed the development of interferometry techniques (Twyman–Green and Fizeau interferometry) for the control of optical surfaces using phase-shifting methods [WYA 75, CRE 88, DOR 99]. The reader will notice that in the literature several terms have been used to describe “holography”, often among which is the term “interferometry”. With the advent of image sensors, the rapid development of “digital” interferometry [BRU 74] and of “TV holography” [DOV 00] has blurred the distinction between holography and interferometry. Holography was initially, and foremost, a non-conventional imaging technique permitting a true 3D parallax, whereas interferometry was a useful tool for the analysis and the measurement of wave fronts. With the development of laser sources and the increase in the resolution of image sensors, from which both disciplines have benefited, the frontier between Michelson interferometry, Mach–Zehnder interferometry, Twyman–Green interferometry, and holographic interferometry is henceforth much less marked than previously. The common objective of these methods is to record/reconstruct the smooth or speckled optical wave front. This means that these disciplines are intimately linked by the connectedness of their fundamentals. Thus, in this book, we can use the terms “holography” and “interferometry” interchangeably.
Even though the concepts of the implementation of digital holography had been known for some time, it took until the 1990s for “digital” holography based on array detectors to come about [SCH 94]. In effect, at the end of the 1980s, we witnessed important developments in two sectors of technology: microtechnological procedures have allowed the creation of image sensors with numerous miniaturized pixels, and the rapid computational treatment of images has become accessible with the appearance of powerful processors and an increase in storage capacities. These advances were made possible by the video games industry that boomed in the middle of the 1980s. From 1994, holography found new life in the considerable stimulation of research efforts. Figure I.1 shows graphs demonstrating the number of scientific publications in the domain of digital holography between 1993 and 2011 (keywords “digital holography”, source: ISI – Web of Sciences, 2011). The database lists more than 2,300 articles, of which 57 have been cited more than 57 times.
The most cited articles concern the methods of reconstruction, digital holographic microscopy, secure encoding, and metrological applications. The development of digital holographic microscopy from 1999 has led to commercial systems [LYN 11]. Figure I.1 shows that the explosion of digital holography dates from the start of the 2000s. This revival is explained in part by the appearance on the market of numerous laser sources (laser diodes or diode-pumped solid-state lasers), at moderate cost, giving the opportunity to develop compact and versatile systems. Thus, 10 years after this boom, it seems an opportune time to propose an introductory book on digital holography. This book describes the mathematical fundamentals, the numerical calculation of diffraction, and the reconstruction algorithms, and precisely explains a certain number of techniques and applications that use the phase of the reconstructed field.
Figure I.1.Graphs of the number of articles published and of citations since 1993
Analog or digital holography is closely related to the diffraction of light. However, in practice, it is often difficult to obtain analytical solutions to diffraction calculations, leading to the use of computational methods in order to obtain numerical results. The spectacular development of computational methods in recent years offers everybody the opportunity to calculate the Fourier transform of any image rapidly. Currently, very few works on optics are dedicated specifically to numerical calculations of diffraction, which is fundamental to digital holography. This is why we wish, with this book, to present the fundamentals of diffraction, summarize the different existing techniques of calculation, and give practical examples of applications. This book is for engineers, researchers, and science students at master’s level, and will supply them with the basics for entering the vast domain of digital holography.
This book is structured in seven chapters, an appendix, and a list of bibliographical references. The first chapter is a reminder of the mathematical prerequisites necessary for a good understanding of the book. In particular, it describes certain widely used mathematical functions, the theory of two-dimensional linear systems, and the Fourier transform as well as the calculation of the discrete Fourier transform. Chapter 2 introduces the scalar theory of diffraction and describes the propagation of an optical field in a homogeneous medium. The classical approaches are presented and the chapter is concluded with Collins’ formula that is used to treat problems of diffraction in waves propagating across an optical system. In Chapter 3, we develop the methods for calculating diffraction integrals using the fast Fourier transform, and we discuss sampling conditions that must be satisfied for the application of each formula. The fundamentals of holography are tackled in Chapter 4; we describe the different types of hologram and the diffraction process that leads to the magic of holography. The fundamentals being outlined, Chapter 5 presents digital Fresnel holography and the algorithms of reconstruction by Fresnel transform or by convolution. We also present methods of digital color holography. This part is illustrated by numerous examples. Chapter 6 is an extension of Chapter 5 in the case where the field propagates across an optical system. The seventh and last chapter considers digital holographic interferometry and its applications. The objective is to propose a synthesis of the methods that exploit the phase of the reconstructed hologram to provide quantitative information on the changes that any object (biological or material) is subjected to.
The Appendix proposes examples of programs for diffraction calculations and digital hologram reconstruction with the methods described in Chapter 5.
Chapter 1
Mathematical Prerequisites
Digital holography is a discipline that associates the techniques of traditional optical holography with current computational methods [GOO 67, HUA 71, KRO 72, LYO 04, SCH 05]. In the framework of the scalar theory of diffraction [BOR 99, GOO 72, GOO 05], digital holography tackles, based on diffraction formulae, the propagation of a light wave in an optical system, the study of interference between coherent light waves, and the reconstruction of surface waves diffracted by objects of various natures. In this context, the propagation of a light wave can be considered as the transformation of a two-dimensional signal by a linear system–the optical system. Various representations of the scalar amplitude of a light wave carrying information use special mathematical functions; the transformation of a light wave across a linear system uses a fundamental mathematical tool: two-dimensional Fourier analysis. The digital treatment of optical information leads us to treat the problems of sampling and discretization, under the restriction given by Shannon’s theorem. Thus, the mathematical prerequisites for a good understanding of this book concern the frequently used mathematical functions, the two-dimensional Fourier transform, and the notions of the sampling theorem [GOO 72].
1.1. Frequently used special functions
Many mathematical functions that we will present in this section are frequently used in this book. To understand their properties, we give a brief account of their physical meaning.
1.1.1. The “rectangle” function
The one-dimensional rectangle function is defined by:
[1.1]
This function is represented in Figure 1.1.
Figure 1.1.Rectangle function
Depending on the nature of the variable x, the rectangle function has various meanings. For example, if x is a spatial variable (a spatial coordinate in millimeters), we can use the function to represent the transmittance from a slit pierced in an opaque screen. In this book, we generally use the two-dimensional rectangle function that is obtained by the product of two one-dimensional functions. As an example, the following function is very useful:
[1.2]
This function is shown in Figure 1.2. It allows us to simply represent the transmittance from an aperture of a rectangular shape, centered on the point with coordinates (x0, y0) and of lengths a and b along the x- and y-axes, respectively. This binary function is very useful for considering the amplitude of an optical wave limited to a rectangular region, by eliminating the values outside the zone of interest.
Figure 1.2.Two-dimensional rectangle function centered on (x0,y0)
1.1.2. The “sinc” function
The one-dimensional sinc function is defined by:
[1.3]
Its curve is presented in Figure 1.3.
Figure 1.3.The sinc function
Also, the two-dimensional sinc function is formed by the product of two functions of independent variables:
[1.4]
Let us consider two positive values a and b; Figure 1.4 shows the curve of the function sinc2 (x/a, y/b). In Chapter 2, we will see that such a function represents the intensity distribution of Fraunhofer diffraction from a rectangular aperture illuminated by a coherent wave.
Figure 1.4.Two-dimensional sinc function
1.1.3. The “sign” function
The one-dimensional sign function is defined as:
[1.5]
The curve of this function is given in Figure 1.5.
If a function is multiplied by the function sgn(x–a), for a < 0, the sign of the function will be inverted. If a coherent optical field is multiplied by this function, the resulting change corresponds to a phase shift of π. We can also form a two-dimensional sign function by taking the product of two one-dimensional functions.
Figure 1.5.The sign function
1.1.4. The “triangle” function
The triangle function is defined as:
[1.6]
The curve of this function is given in Figure 1.6. Later, we will see that the Fourier transform of the function Λ(x) is sinc2(fx) (with the fx coordinate corresponding to the spatial frequency). This function will be very useful in the Fourier analysis of optical diffracting functions (e.g. diffraction grating). As noted earlier, we can form a two-dimensional triangle function by taking the product of two one-dimensional functions.
Figure 1.6.Triangle function
1.1.5. The “disk” function
In practice, an optical system is generally constructed with lenses whose mounts (cylinders) are circular in form. Their pupils are therefore circular and the disk function is often used to model the diffraction of circular elements (iris diaphragms, mounts, etc.). The definition of this function, in polar and Cartesian coordinates, is:
[1.7]
The surface of the disk function is given in Figure 1.7.
Figure 1.7.The disk function
1.1.6. The Dirac δ function
1.1.6.1. Definition
In the field of optical treatment of information, the Dirac δ distribution (henceforth called the δ “function”) in two dimensions is very widely used. Strictly speaking, δ is a distribution but for convenience we will hereafter call it a function. According to the Huygens–Fresnel principle of the propagation of light, a wave front can be considered as the sum of spherical “secondary” sources [BOR 99, GOO 72, GOO 05]. The two-dimensional δ function is often used to individually describe point sources. The fundamental property of the δ function is that, as for an infinitely narrow pulse of infinite height, the sum is equivalent to one (x and y being Cartesian coordinates). The δ function can be defined by various mathematical expressions, one of which is presented here.
[1.8]
Figure 1.8.Graph of fN (x) for N=1, 2, 4
Evidently, we can also define the two-dimensional δ function as:
[1.9]
To facilitate the use of the δ function, we give some equivalent definitions:
[1.10]
[1.11]
[1.12]
[1.13]
In the last expression, J1 is a first-order Bessel function of the first kind. Depending on the problem being studied, these definitions can be more or less appropriate and we can also choose which definition to apply in each case.
1.1.6.2. Fundamental properties
We will now consider some of the mathematical properties of the δ function. These properties will be used frequently in this book.
1.1.6.2.1. Contraction–dilation of coordinates
If a is any constant, we have:
[1.14]
1.1.6.2.2. Product
If the function φ(x) is continuous at the point x0, we have:
[1.15]
1.1.6.2.3. Convolution
Let us consider the convolution of two functions δ and φ:
[1.16]
Then we have:
[1.17]
The δ function is the unity of the convolution product.
1.1.6.2.4. Translation
The property of translation of the δ function is often used for theoretical analyses and proofs. Here we present this property and the corresponding proof. If φ(x) is continuous at the point x0, then we have:
[1.18]
[1.19]
If ε → 0, the first and third terms on the right will be zero, therefore:
[1.20]
In the same way, we can show that the two-dimensional δ function possesses the same property of translation.
[1.21]
1.1.7. The “comb” function
The comb function is a periodic series of δ functions. It is frequently used to model the sample of continuous functions. The definition of the one-dimensional comb function is:
[1.22]
Figure 1.9 shows the curves of δ(x) and comb(x). The two-dimensional comb function can be defined by the product of two one-dimensional comb functions:
[1.23]
Since the comb function is a periodic series of δ functions, it has analogous properties and is used in numerous analyses of optical signals.
Figure 1.9.The δ(x) and comb(x) functions
1.2. Two-dimensional Fourier transform
The Fourier transform is a very useful mathematical tool for the study of both linear and nonlinear phenomena. As the propagation of the optical field can be considered as a process of linear transformation of the “object” field to the “image” field, we are immediately interested in the two-dimensional Fourier transform [BOR 99, GOO 72].
1.2.1. Definition and existence conditions
The Fourier transform of a complex function g(x, y) of two independent variables, which we write here as F {g(x, y)}, is defined as :
[1.24]
Thus defined, the transform is itself a complex-valued function of the two independent variables G(fx, fy), called the spectral function, or spectrum, of the original function g(x, y). The two variables fx and fy are considered, without loss of generality, as frequencies. In optics, (x, y) are spatial variables and (fx, fy) are spatial frequencies (mm-1). Similarly, the inverse Fourier transform of the function G(fx, fy), which we write as F–1{G(fx, fy)}, is defined as:
[1.25]
We note that the direct and inverse transformations are completely analogous mathematical operations. They differ only by the sign of the exponent in the double integral. However, for some functions, these two integrals cannot exist in a mathematical sense. Therefore, we will briefly discuss the conditions of their existence. Among the various conditions, we concern ourselves with the following:
In general, one of these three conditions can be ignored if we can guarantee strict adherence to the other conditions, but this is beyond the scope of discussions in this book.
For the representation of real physical waves by ideal mathematical functions, in the analysis of tools, one or more of the existing conditions presented above may be more or less unsatisfied [GOO 72]. However, as Bracelet [BRA 65] remarked, “the physical possibility is a sufficient condition of validity to justify the existence of a transformation”. Furthermore, the functions of interest to us are included in the scope of Fourier analysis, and it is evidently necessary to generalize definition [1.24] somewhat. Thus, it is possible to find a transformation that has meaning for functions that do not strictly satisfy the existing conditions, provided that these functions can be defined as the limit of a sequence of transformable functions. In transforming each term of this sequence, we generate a new sequence whose limit is called the generalized Fourier transform of the original function. These generalized transforms can be handled in the same way as the ordinary transforms, and the distinction between the two is often ignored. For a more detailed discussion of this generalization, the reader may refer to the work of Lighthill [LIG 60].
To simplify the study of Fourier analysis, including this generalization, Table 1.1 shows the Fourier transforms of some functions expressed in Cartesian coordinates.
1.2.2. Theorems related to the Fourier transform
We now present some important mathematical theorems followed by a brief account of their physical meaning [GOO 72]. The theorems mentioned below will be used frequently as they constitute fundamental tools for the use of Fourier transforms; they allow us to simplify the calculation of solutions to problems in Fourier analysis.
Table 1.1.Fourier transforms of some functions expressed in Cartesian coordinates
1.2.2.1. Linearity
The transform of the sum of two functions is simply the sum of their respective transforms:
[1.26]
Where α and β are complex constants.
1.2.2.2. Similarity
[1.27]
This theorem is also known as the “contraction/dilation” theorem. It means that a “dilation” of the coordinates of the spatial domain (x, y) is expressed as a “contraction” of the coordinates in the frequency domain (fx, fy) and by a change in the amplitude and the width of the spectrum.
1.2.2.3. Translation
If F{g (x, y)}= G(fx, fy), then:
[1.28]
The translation of a function in the spatial domain introduces a linear phase variation in the frequency domain.
1.2.2.4. Parseval’s theorem
If F {g (x, y)}= G (fx, fy), then:
[1.29]
This theorem is generally interpreted as an expression of the conservation of energy between the spatial domain and the spatial frequency domain.
1.2.2.5. The convolution theorem
[1.30]
The Fourier transform of the convolution of two functions in the spatial domain is equivalent to the multiplication of their respective transformations. We will see in Chapter 3 that the Fourier transform can be calculated by the Fast Fourier Transform (FFT). This theorem offers the opportunity to calculate a convolution using FFT algorithms.
1.2.2.6. The autocorrelation theorem
If F{g (x, y)}= G(fx, fy), then:
[1.31]
[1.32]
This theorem can be considered as a particular case of the convolution theorem.
1.2.2.7. The duality theorem
Let us consider two functions f and g linked by the following integral development:
[1.33]
[1.34] F{f(x, y)}= g(fx, fy)
[1.35]
being equally:
[1.36]
giving
and by applying the Fourier transform operator to the left and right,
[1.38]
Hence the property of duality of the Fourier transforms:
if
then
[1.40] F {g(x, y)}= f(–fx,– fy)
1.2.3. Fourier transforms in polar coordinates
For a two-dimensional function with circular symmetry, it is more convenient to use polar coordinates. We consider a plane described by rectangular (x, y) and polar(r, θ) coordinates and the corresponding spectral coordinates are (fx, fy) and (ρ, φ), respectively. We then have:
[1.41]
[1.42]
Let f(x, y) be an original function with spectral function F(fx, fy). We can rewrite these as functions of polar coordinates:
[1.43]
By substituting these two relations into [1.23] and [1.25], we obtain direct and inverse Fourier transforms, respectively, in polar coordinates:
[1.45]
[1.46]
[1.47]
we can deduce the Fourier transform of gR(r) in polar coordinates:
[1.48]
[1.49]
We note that the mathematical forms of the direct and inverse transformations are the same.
1.3. Linear systems
An optical system allows the transformation of an input signal into an output signal. The device situated between the two planes (“input” and “output”) perpendicular to the direction of propagation will be henceforth called an “optical system”. An optical system may have linear or nonlinear properties. In most cases, considering the system to be linear as a first approximation, we are able to obtain sufficiently precise representations of the observed phenomena. Here we will consider only linear systems.
1.3.1. Definition
From a mathematical point of view, a linear system corresponds to a transformation operation. We conveniently represent such a system by an operator L{}, at whose output the two-dimensional function f(x, y) becomes a new function p(x′, y′). This is expressed as:
[1.50]
f(x, y) and p(x′, y′) are called the input function and the output function of the system, respectively.
Let us consider some input functions f1(x, y), f2(x, y),…, fn(x, y) and some output functions p1(x′, y′), p2(x′, y′),…, pn(x′, y′). We then have:
[1.51]
Assuming a1, a2,…, an to be complex constants, if the set of a system’s input and output functions satisfy:
[1.52]
and
[1.53]
then this system can be considered linear.
The linear approach presents a considerable advantage: it allows us to express the response of a system to any input function in the form of a response to “elementary” functions into which the input has been decomposed. In conclusion, if we can decompose, by a simple method, the input function into “elementary” functions for which the response of the system is well known, we will obtain the output function by the sum of these responses.
1.3.2. Impulse response and superposition integrals
Using the translation property of the two-dimensional δ function, we can express a function f(x, y) describing a light wave in the input plane as:
[1.54]
The physical meaning of this expression is that the distribution of the input optical signal f(x, y) can be considered as the linear combination of δ functions weighted by the value f(x0, y0) and shifted with respect to each other, the elementary functions of the decomposition being precisely these δ functions. Since the system is linear, its response to the input signal f(x, y) is determined by:
[1.55]
We notice that the number f(x0, y0) is a simple weighting factor applied to the elementary function δ(x–x0, y–y0). For any point with coordinates (x0, y0), f(x0, y0) is constant. According to its property of linearity, the operator L{} can move inside the summation (integral) sign, giving:
[1.56]
If we consider h(x, y; x0, y0) as the response of the system at the point (x, y) of the output space, when the input is a δ function situated at the point (x0, y0), we have:
[1.57]
The function h is called the impulse response of the system. The magnitude of the input and output of the system can then be related by the following equation:
[1.58]
This fundamental expression goes by the name of the “superposition integral”.
To completely determine the output signal, we note that we must know the responses to local impulses at every possible point of the input plane. In general, the determination of the impulse responses is very complex. However, we will see in the following section that for an important subclass of linear systems called invariant linear systems, which are invariant in the space, we can determine the impulse responses in a simple way. In most cases, an optical system can be approximated by a space-invariant linear system.
1.3.3. Definition of a two-dimensional linear shift-invariant system
Two-dimensional invariant linear systems are an important subclass of linear systems. If the impulse response of a system h(x, y; x0, y0) depends only on the distances (x–x0) and (y–y0), this system is considered as a linear shift-invariant system, that is:
[1.59]
Thus, the optical system is invariant in space if the output signal of a point in the input plane changes position only but not shape when the source point moves around the input plane. For a shift-invariant optical system, the superposition integral can be rewritten as:
[1.60]
This relation corresponds to the two-dimensional convolution of the input function with the impulse response of the system. Consequently, if an optical system is a linear shift-invariant system, on the condition that we are able to determine the impulse response of a point in the input plane (which is often considered on the axis of the system), whatever the optical input signal f(x0, y0), the output signal p(x, y) can be determined using expression [1.60].
1.3.4. Transfer functions
By taking the Fourier transform of both sides of [1.60] and using the convolution theorem, we obtain:
with:
[1.61a]
[1.61b]
[1.61c]
Expression [1.61] shows that the spectral function of the output signal is the product of the spectrum of the input signal with the function H(fx, fy). This product is the frequency response to one of the elementary functions of the input signal and the function H(fx, fy) is called the transfer function of the system. The transfer function of the system is determined by the Fourier transform of the impulse response [1.61c]. The output signal can be determined by the inverse Fourier transform of the spectrum of the output signal, that is:
[1.62]
The spectrum of the output signal P(fx, fy) can be calculated by the Fourier transform [1.61b]. If the transfer function of the system can be determined, the output signal can be obtained by [1.61b].
1.4. The sampling theorem
It is often convenient to represent a continuous function g(x, y) by a table of sampled values taken at a discrete set of points in the xy-plane. Current numerical methods allow the presentation, storage, and propagation of almost all information of a physical nature. It is intuitive that if the samples of the continuous function g(x, y) are taken at points sufficiently close together, the given samples are able to reliably represent the original function using a simple interpolation. However, for a given function, the question is to know the maximum sampling interval that we must respect. The answer is less evident. Yet, for a particular class of functions known as “bandwidth-limited functions”, the reconstruction can be carried out exactly, on the condition that the interval between two samples is not larger than a certain limit. A bandwidth-limited function is such that its Fourier transform is only non-zero on a finite region of the frequency space. The sampling theorem was initially proven by Whittaker [WHI 15] and was later revisited by Shannon [SHA 49] during his studies on information theory. This principle, which allows us to determine the maximum sampling interval, is called the Shannon–Whittaker sampling theorem.
The following section states the two-dimensional sampling theorem and refers to the work of Goodman [GOO 72].
1.4.1. Sampling a continuous function
Let us consider a set of samples of the function g(x, y), taken over a rectangular mesh. The sampled function gS(x, y) is defined as:
[1.63]
This function therefore consists of a set of δ functions separated by intervals of length X along the x-axis and of length Y along the y-axis, as shown in Figure 1.10, whose amplitude is the value of the function g(x, y) at the point being considered.
The volume enclosed by the δ function representation in the space and the xy-plane is proportional to the value of g(x, y) at each point of the sampling mesh. Applying the convolution theorem, we obtain the spectrum GS(fx, fy) of gs(x, y) by convoluting the transform of (x/X)comb(y/Y) with the transform of g(x, y), that is:
[1.64]
Figure 1.10.Two-dimensional sampling
Since:
[1.65]
then we have:
[1.66]
The spectrum of gs(x, y) can therefore be simply deduced by considering the spectrum of g(x, y) localized at each point with coordinates (n/X, m/Y) in the fxfy–plane, as shown in Figure 1.11.
Since we assumed that the function g(x, y) had a spectrum of limited scope, its spectrum G(fx, fy) is only non-zero in the corresponding frequency space domain. If X and Y are sufficiently small, in other words, if the samples are taken on points that are sufficiently close to each other, the intervals 1/X and 1/Y between the various spectral regions will be large enough to ascertain that the neighboring regions do not overlap. To determine the maximum interval between two sampled points, let us suppose 2BX and 2BY to be the dimensions following the respective directions of the fx- and fy-axes of the smallest rectangle containing the whole spectral domain of g(x, y). As shown in Figure 1.11, if the following two inequalities:
[1.67]
are satisfied, the different terms of the spectrum [1.66] of the sampled function are separated by the distances 1/X and 1/Y in the fx and fy directions, respectively. The maximum dimensions of the mesh of the sample network, which allow an exact restoration of the original function, are therefore 1/2BX and 1/2BY. Having determined the maximum allowed distances between samples, we now study how to obtain the spectrum of g(x, y) by a filter function, and how to reconstruct the original function g(x, y).
Figure 1.11.Spectrum of the sampled function
1.4.2. Reconstruction of the original function
Following Figure 1.11, we consider a two-dimensional rectangular function with sides 2BX and 2BY along the fx- and fy-axes, respectively. The filter function is:
[1.68]
We note that G(fx, fy) is obtained from GS(fx, fy) since:
[1.69]
This means that, if the sampled function gs(x, y) is considered as the input signal of a system, the function g(x, y) will be considered as the output signal. Thus, H(fx, fy) is the transfer function of the system. In this case, the identity [1.64] translates into the spatial domain by:
where
[1.71]
and h(x, y) is the impulse response of the filter, which is written as:
[1.72]
Consequently:
[1.73]
Finally, when we choose the maximum allowed values 1/2BX and 1/2BY for the sampling intervals X and Y, the identity becomes:
[1.74]
Expression [1.74] represents a fundamental result that we will henceforth call the Whittaker–Shannon sampling theorem. It states that the exact reconstruction of a bandwidth-limited function can be carried out from the sampled values of the function, taken after a suitable rectangular mesh sampling. The reconstruction is carried out by interpolating each sample point by an interpolation function constituted by the product of two sinc functions.
Note that this result is not the only possible sampling theorem. We chose two rather arbitrary sampling frames in the course of this study; with different assumptions we would have obtained a different sampling theorem. We first arbitrarily chose a rectangular sampling frame. Also, we chose the particular transfer function given by [1.68]. By making different choices we would establish other, equally valid theorems. For more detail, readers may refer to the articles by [BRA 56], [PET 62], and [LIN 59].
1.4.3. Space-bandwidth product
For a bandwidth-limited function g(x, y), which is mainly non-zero in a region of the xy-plane bounded by –LX ≤ x ≤ LX and–LY ≤ y ≤ LY and whose maximum sampling intervals along the fx- and fy-axes are 1/2BX and 1/2BY, respectively, to thus satisfy the sampling theorem, the minimum value of the number of sampling points able to represent the function g(x, y) is therefore:
This relation is called the space-bandwidth product of the function g(x, y) [GOO 05] and expresses the value of the product of the space and frequency surfaces in which the function g(x, y) and its spectrum F{g(x, y)} are bounded. As a result, for a two-dimensional bandwidth-limited function, the space-bandwidth product determines the minimum number of degrees of freedom, N, that correctly represent it. When g(x, y) is real, its number of degrees of freedom is N, since the samples are real; if g(x, y) is a complex function, its number of degrees of freedom becomes 2N as each sample must be represented by two real values. Given the theorems of similarity and translation relating to the Fourier transform, the dilation of the coordinates and the translation of the function in the spatial or spectral domain do not affect the space-bandwidth product of the considered function. This means that, for a given function, the number of degrees of freedom is constant. This number can therefore be considered as a significant piece of information expressing the complexity of the function, and as a criterion that allows us to verify whether there is any loss of information during the sampling process.
Chapter 2
The Scalar Theory of Diffraction
The calculation or expression of the propagation of coherent waves, the treatment of the interferences between these waves, and the reconstruction of images based on the different equations describing the diffraction of light are fundamental to the field of digital holography. Light being an electromagnetic wave, optical propagation is the process of diffraction of an electromagnetic wave in a four-dimensional (4D) space (three in space and one in time), which may be of a dielectric nature. In this chapter, we will address the description of the complex functions modeling an optical wave, then, from the electromagnetic theory based on Maxwell’s equations, we will introduce the wave equation describing its propagation. It turns out that even though the solution to the wave equation is in vector form, if both the size of diffracting objects and the distance across which the diffraction occurs are much greater than the wavelength, we can neglect the coupling between the electric and magnetic field vectors in the propagation process. Considering the electric field vector as a scalar, the solution to the wave equation can express the physical process of the propagation of light with high reliability. This approach is called the scalar theory of diffraction. In the framework of this theory, the propagation of light is rigorously described by the Kirchhoff and Rayleigh–Sommerfeld equations, and by the angular spectrum method [GOO 72, GOO 05]. The Fresnel diffraction integral, which comes from the Fresnel approximation, is a paraxial (small-angle) expression of these three formulations. These four methods constitute the classic formulation of diffraction, and on the condition that we know the optical field in a plane perpendicular to the direction of propagation, they can be applied to the calculation of optical fields in space, ahead or behind, with excellent reliability.
In this chapter, using the scalar theory of diffraction and the wave equation, we will deduce the angular spectrum and Fresnel diffraction integral methods. Experimentally, the propagation of light is often closely related to an optical system and the classical diffraction formulae are not always useful if we want to deal with a problem of the propagation of light across a structured system with various optical elements. Thus, by associating the matrix method with scalar diffraction, we will propose, at the end of the chapter, a generalized formulation of diffraction, the Collins approach [COL 70], which allows us to express the paraxial propagation of light across an optical system. The presentation of the different formulations of diffraction will be supplemented with examples so as to facilitate the understanding of their physical meaning.
2.1. Representation of an optical wave by a complex function
2.1.1. Representation of a monochromatic wave
In a space described by a Cartesian coordinate system xyz, a monochromatic wave, at the point P(x,y,z) at any instant t, can be represented by a trigonometric function of the form [GOO 72, COL 70, YAR 85]:
[2.1]
with amplitude U(x,y,z), frequency v, and phase φ(x, y, z). The expression is useful for describing a monochromatic optical wave of infinite length. In reality, a perfect monochromatic wave does not exist, as the process that generates it only exists for a finite duration, widening the spectrum of frequencies. However, we may consider the existence of a quasi-monochromatic wave, having a narrow frequency band centered on the average frequency of the wave. Such a wave is produced by a laser. Using Euler’s formula, with , expression [2.1] can be rewritten as:
[2.2]
[2.3]
This equation describes the complex amplitude of the optical wave. It depicts the amplitude and phase of the optical field at each point in space, determining the spatial distribution of the field. In the study of the propagation of light, the intensity distribution is an important physical quantity. The complex representation of a wave considerably simplifies the calculation of this intensity distribution. For example, if we consider the superposition of N waves U1(x,y,z), U2(x,y,z), …, UN(x,y,z), the resulting amplitude is given by the sum of the waves:
[2.4]
Given that the intensity distribution is proportional to the modulus squared of the amplitude, then:
[2.5]
thus, we have:
[2.6]
This expression is the product of the sum of complex exponents. The calculation is considerably simplified compared to the case where the waves are represented by trigonometric functions. Plane waves and spherical waves have pride of place in optics. We discuss their properties and representations in the following sections.
2.1.2. Complex amplitude of the optical field in space
2.1.2.1. Plane waves
The principal property of a plane wave is that its wavefront is a plane. In a homogeneous medium, the wavefront is perpendicular to the direction of propagation. For a plane wave propagating in a direction given by the direction cosines cos α, cos β, cos γ, the complex amplitude is expressed by:
[2.7]
[2.8]
This expression can be generalized to represent any wave.
2.1.2.2. Spherical waves
The wavefront of a spherical wave is spherical. From the generalization of expression [2.8], if the point source of the spherical wave is at the origin of a Cartesian coordinate system, the spherical wave of amplitude U0 is expressed by a trigonometric function:
[2.9]
with . We note that the amplitude is proportional to the inverse of the distance between the point source and r, the observation point. If the spherical wave is divergent, k and r have identical directions and [2.9] can be expressed in the form:
[2.10]
For a convergent spherical wave, the directions of k and r are opposed, and we then have:
[2.11]
The complex amplitude of a spherical wave can therefore be expressed by:
[2.12]
When the center of the spherical wave is at the point (xc, yc, zc) instead of the origin, the expressions are identical, with r substituted for:
[2.13]
2.1.3. Complex amplitudes of plane and spherical waves in a front plane
The above analysis conveys the complex amplitudes of plane and spherical waves in a 3D space. However, it is often useful to determine the amplitude in a front plane perpendicular to optical axis or the axis of propagation. For a given plane, the representation of the complex amplitude of different types of wave is therefore necessary. We will analyze two notable examples: plane and spherical waves.
2.1.3.1. Complex amplitude of a plane wave in a front plane
[2.14]
Since is constant and independent of (x,y), it can be rewritten as:
[2.15]
with
[2.16]
If the interference of this wave with other coherent waves is not to be considered, then the constant phase factor is generally neglected and [2.15] is the usual representation of the complex amplitude of a plane wave.
