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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley Series in Pure and Applied Optics
- Sprache: Englisch
This book discusses statistical methods that are useful for treating problems in modern optics, and the application of these methods to solving a variety of such problems This book covers a variety of statistical problems in optics, including both theory and applications. The text covers the necessary background in statistics, statistical properties of light waves of various types, the theory of partial coherence and its applications, imaging with partially coherent light, atmospheric degradations of images, and noise limitations in the detection of light. New topics have been introduced in the second edition, including: * Analysis of the Vander Pol oscillator model of laser light * Coverage on coherence tomography and coherence multiplexing of fiber sensors * An expansion of the chapter on imaging with partially coherent light, including several new examples * An expanded section on speckle and its properties * New sections on the cross-spectrum and bispectrum techniques for obtaining images free from atmospheric distortions * A new section on imaging through atmospheric turbulence using coherent light * The addition of the effects of "read noise" to the discussions of limitations encountered in detecting very weak optical signals * A number of new problems and many new references have been added Statistical Optics, Second Edition is written for researchers and engineering students interested in optics, physicists and chemists, as well as graduate level courses in a University Engineering or Physics Department.
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Table of Contents
Cover
Series
Title Page
Copyright
Dedication
Preface– Second Edition
Preface– First Edition
Chapter 1: Introduction
1.1 Deterministic Versus Statistical Phenomena and Models
1.2 Statistical Phenomena in Optics
1.3 An Outline of the Book
Chapter 2: Random Variables
2.1 Definitions of Probability and Random Variables
2.2 Distribution Functions and Density Functions
2.3 Extension to Two or More Joint Random Variables
2.4 Statistical Averages
2.5 Transformations of Random Variables
2.6 Sums of Real Random Variables
2.7 Gaussian Random Variables
2.8 Complex-Valued Random Variables
2.9 Random Phasor Sums
2.10 Poisson Random Variables
Chapter 3: Random Processes
3.1 Definition and Description of a Random Process
3.2 Stationarity and Ergodicity
3.3 Spectral Analysis of Random Processes
3.4 Autocorrelation Functions and the Wiener–Khinchin Theorem
3.5 Cross-Correlation Functions and Cross-Spectral Densities
3.6 Gaussian Random Processes
3.7 Poisson Impulse Processes
3.8 Random Processes Derived from Analytic Signals
3.9 The Circular Complex Gaussian Random Process
3.10 The Karhunen–Loève Expansion
Chapter 4: Some First-Order Statistical Properties of Light
4.1 Propagation of Light
4.2 Thermal Light
4.3 Partially Polarized Thermal Light
4.4 Single-Mode Laser Light
4.5 Multimode Laser Light
4.6 Pseudothermal Light Produced by Passing Laser Light Through a Changing Diffuser
Chapter 5: Temporal and Spatial Coherence of Optical Waves
5.1 Temporal Coherence
5.2 Spatial Coherence
5.3 Separability of Spatial and Temporal Coherence Effects
5.4 Propagation of Mutual Coherence
5.5 Special Forms of the Mutual Coherence Function
5.6 Diffraction of Partially Coherent Light by a Transmitting Structure
5.7 The Van Cittert–Zernike Theorem
5.8 A Generalized Van Cittert–Zernike Theorem
5.9 Ensemble-Average Coherence
Chapter 6: Some Problems Involving Higher-Order Coherence
6.1 Statistical Properties of the Integrated Intensity of Thermal or Pseudothermal Light
6.2 Statistical Properties of Mutual Intensity with Finite Measurement Time
6.3 Classical Analysis of the Intensity Interferometer
Chapter 7: Effects of Partial Coherence in Imaging Systems
7.1 Preliminaries
7.2 Space-Domain Calculation of Image Intensity
7.3 Frequency Domain Calculation of the Image Intensity Spectrum
7.4 The Incoherent and Coherent Limits
7.5 Some Examples
7.6 Image Formation as an Interferometric Process
7.7 The Speckle Effect in Imaging
Chapter 8: Imaging Through Randomly Inhomogeneous Media
8.1 Effects of Thin Random Screens on Image Quality
8.2 Random-Phase Screens
8.3 The Earth's Atmosphere as a Thick Phase Screen
8.4 Electromagnetic Wave Propagation Through the Inhomogeneous Atmosphere
8.5 The Long-Exposure OTF
8.6 The Short-Exposure OTF
8.7 Stellar Speckle Interferometry
8.8 The Cross-Spectrum or Knox–Thompson Technique
8.9 The Bispectrum Technique
8.10 Adaptive Optics
8.11 Generality of the Theoretical Results
8.12 Imaging Laser-Illuminated Objects through a Turbulent Atmosphere
Chapter 9: Fundamental Limits in Photoelectric Detection of Light
9.1 The Semiclassical Model for Photoelectric Detection
9.2 Effects of Random Fluctuations of the Classical Intensity
9.3 The Degeneracy Parameter
9.4 Noise Limitations of the Amplitude Interferometer at Low Light Levels
9.5 Noise Limitations of the Intensity Interferometer at Low Light Levels
9.6 Noise Limitations in Stellar Speckle Interferometry
Appendix A: The Fourier Transform
A.1 Fourier Transform Definitions
A.2 Basic Properties of the Fourier Transform
A.3 Tables of Fourier Transforms
Appendix B: Random Phasor Sum
Appendix C: The Atmospheric Filter Functions
Appendix D: Analysis of Stellar Speckle Interferometry
Appendix E: Fourth-Order Moment of the Spectrum of a Detected Speckle Image
Bibliography
Index
Wiley Series
End User License Agreement
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Guide
Cover
Table of Contents
Preface– Second Edition
Begin Reading
List of Illustrations
Chapter 1: Introduction
Figure 1.1 An optical imaging system.
Chapter 2: Random Variables
Figure 2.1 Examples of distribution functions. (a) Discrete random variable; (b) continuous random variable; and (c) mixed random variable.
Figure 2.2 Typical probability density functions for (a) discrete, (b) continuous, and (c) mixed random variables. For the discrete and mixed cases, the labels and heights of the delta functions are meant to represent their areas.
Figure 2.3 The shaded sections on the
u
-axis are the regions for which the random variable
Z
will be less than or equal to the particular value of
z
shown.
Figure 2.4 The transformation . The gray horizontal line on the
u
-axis bounded by is the region .
Figure 2.5 Example of a one-to-one probability transformation.
Figure 2.6 Plots of (a) the probability density before transformation, (b) the transformation law, and (c) the probability density after transformation.
Figure 2.7 The shaded and cross-hatched region represents the area within which .
Figure 2.8 The Gaussian probability density function.
Figure 2.9 Contours of constant probability density for a joint Gaussian density with , and (a) , (b) , and (c) . The density functions have been normalized to have value unity at the origin.
2.8 Complex-Valued Random Variables
Figure 2.10 Contours of constant probability density in the complex plane for a circular complex Gaussian random variable.
Figure 2.11 Random phasor sum.
Figure 2.12 Rayleigh probability density function.
Figure 2.13 Sum of a constant phasor of length
s
and a random phasor sum.
Figure 2.14 Rician density function. The parameter
k
represents the ratio .
Figure 2.15 Probability density function of the phase of the sum of a constant phasor and a random phasor sum, as a function of the parameter .
Figure 2.16 Large constant phasor with length
s
plus a small random phasor sum.
Chapter 3: Random Processes
Figure 3.1 An ensemble of sample functions, where and are the parameter values for which the joint density function is specified.
Figure 3.2 Sample functions of a nonstationary process.
Figure 3.3 A stationary process that is nonergodic.
Figure 3.4 The hierarchy of classes of random processes.
Figure 3.5 Sample function of a random telegraph wave.
Figure 3.6 Autocorrelation function and power spectral density of a random telegraph wave.
Figure 3.7 Transformation of cross-spectral density under linear filtering.
3.7 Poisson Impulse Processes
Figure 3.8 (a) A sample function of a Poisson impulse process, together with (b) the corresponding rate function.
Figure 3.9 Filtered Poisson impulse process. (a) Filter response when a single impulse lies in the time interval . (b) Typical sample function when . (c) Typical response when .
Figure 3.10 Region of integration.
Figure 3.11 Fourier spectra of (a) a monochromatic real-valued signal and (b) its complex representation.
Figure 3.12 Construction of an analytic signal from a real signal.
Figure 3.13 Power spectrum of a narrowband signal.
Chapter 4: Some First-Order Statistical Properties of Light
Figure 4.1 Propagation geometry.
Figure 4.2 Probability density function of the instantaneous intensity of polarized thermal light.
Figure 4.3 Probability density function of the instantaneous intensity of unpolarized thermal light.
Figure 4.4 Old and new coordinate systems after rotation by angle .
Figure 4.5 Probability density function of the instantaneous intensity of a thermal source with degree of polarization .
4.4 Single-Mode Laser Light
Figure 4.6 Probability density functions of (a) amplitude and (b) intensity for a perfectly monochromatic wave of unknown phase.
Figure 4.7 Normalized average output intensity of a single-mode laser as a function of the pump parameter
w
.
Figure 4.8 Normalized standard deviation of output intensity of a single-mode laser as a function of pump parameter
w
.
Figure 4.9 Risken's solution for the probability density function of the output intensity of a single-mode laser oscillator for various pump parameters .
Figure 4.10 Probability density function of the amplitude of a wave consisting of equal-strength, independent sinusoidal modes. The total average intensity is held constant and equal to unity. A Gaussian curve is indistinguishable from the curve on this plot.
Figure 4.11 Probability density functions of intensity when independent modes of identical strength are added. The total average intensity is assumed to be unity in all cases. When , the probability density function is a negative-exponential density.
Figure 4.12 Ratio of standard deviation to mean for the intensity of light emitted by a laser oscillating in independent, equal-strength modes.
Figure 4.13 Pseudothermal light produced by a laser and a moving diffuser.
Chapter 5: Temporal and Spatial Coherence of Optical Waves
Figure 5.1 The Michelson interferometer, including the source , the lenses and , mirrors and , beam splitter , compensator , and detector .
Figure 5.2 Normalized intensity incident on detector vs. normalized mirror displacement . The spectrum shape has been assumed to be Gaussian, centered at , for this plot. The envelope of the fringe pattern is drawn dotted.
Figure 5.3 Michelson interferometer based on single-mode fiber.
Figure 5.4 Normalized power spectral densities for three spectral line shapes.
Figure 5.5 Visibility vs. for three spectral line shapes.
Figure 5.6 Typical mid-infrared interferogram plotted with two different scales. The vertical axis represents detected intensity, and the horizontal axis represents optical path difference. The maximum path difference is 0.125 cm. Courtesy of Peter R. Griffiths and the American Association for the Advancement of Science. Copyright 1983 by the American Association for the Advancement of Science. Reprinted from Ref. [89].
Figure 5.7 The Fourier transform of Figure 5.6, representing the spectrum of the source. The vertical axis represents power spectral density, and the horizontal axis represents optical wavenumber, , in inverse centimeters. Copyright 1983 by the American Association for the Advancement of Science. Reprinted from Ref. [89].
Figure 5.8 The fiber-based Michelson interferometer for use in OCT. The axial scanning mirror changes the path length delay in the reference arm to select axial depth and the transverse scanning mirror selects the transverse coordinates to be imaged.
Figure 5.9 Image of hamster skin and underlying tissue obtained by time-domain OCT. Image courtesy of Prof. Jennifer Kehlet Barton of the University of Arizona. Reprinted from [9] with permission of the Optical Society of America.
Figure 5.10 Diagrammatic representation of a Fourier domain OCT system.
Figure 5.11 Coherence multiplexed sensor system (after [28], Figure 1).
Figure 5.12 Young's interference experiment.
Figure 5.13 Physical explanation for loss of fringe visibility at large pinhole spacings: (a) small pinhole spacing and (b) large pinhole spacing.
Figure 5.14 Interference geometry for Young's experiment.
Figure 5.15 Geometric properties of the fringes.
Figure 5.16 Fringe patterns obtained for various values of the complex coherence factor ( assumed equal to ).
Figure 5.17 Effects of finite pinhole size. (a) Geometry of the experiment. (b) Partially overlapping diffraction patterns. The diffraction patterns of the individual pinholes are shown with a dashed line, while the interference between the light from the two pinholes is shown with a solid line. has been assumed to be unity.
Figure 5.18 Optical system for interference experiment.
Figure 5.19 Interference pattern produced by the modified system. The diffraction patterns of both pinholes overlap and are indicated by the dashed line, while the interference pattern is shown with a solid line. has been assumed to be unity.
Figure 5.20 Measurement of the mutual coherence function of light transmitted by a moving diffuser.
Figure 5.21 Geometry for propagation of mutual coherence, where and represent, respectively, the angle between the line joining to and the surface normal at and the corresponding angle for the line joining to .
5.5 Special Forms of the Mutual Coherence Function
Figure 5.22 Geometry for diffraction calculation.
Figure 5.23 Log plot of the normalized intensity in the diffraction pattern of a circular aperture for various states of illumination coherence. A Gaussian-shaped complex coherence factor with radius and a circular aperture with radius are assumed. The intensity is normalized to maintain total constant energy. The parameter represents . Small values of imply highly coherent illumination, and large values of imply low coherence illumination.
Figure 5.24 Geometry for derivation of the Van Cittert–Zernike theorem.
Figure 5.25 The complex coherence factor vs. normalized spacing, under the assumption that .
Figure 5.26 Normalized intensity distributions in the fringe patterns produced by two circular pinholes separated in a direction parallel to the -axis. Coordinates in the fringe plane are assumed to be . The coordinate has been normalized so that for a value of , the diffraction pattern of an individual pinhole falls to its first zero. The pinhole separation is held constant at value and the diameter of the incoherent source is changed. In the four cases shown, the diameter of the source is (i) (a point source), (ii) , (iii) , and (iv) . The total power emitted by the source has been held constant.
Figure 5.27 Fourier transform relations for the generalized Van Cittert–Zernike theorem. The plane is on the left and the plane is on the right.
Figure 5-4p
Figure 5-5p
Figure 5.6p
Figure 5-7p
Figure 5-8p
Figure 5-9p
Chapter 6: Some Problems Involving Higher-Order Coherence
Figure 6.1 Plots of vs. , exact solutions for Gaussian, Lorentzian and rectangular spectral profiles.
Figure 6.2 Approximation of (a) a smoothly varying instantaneous intensity by (b) a “boxcar” approximation.
Figure 6.3 Approximate probability density function of the integrated intensity of a polarized thermal wave for various values of .
Figure 6.4 Plot of for the case of unpolarized thermal light and various values of .
Figure 6.5 Plot of the first five eigenvalues obtained from a discretized version of Eq. (6.1-54).
Figure 6.6 Approximate (dotted line) and “exact” (solid line) probability density functions for integrated intensity for a rectangular power spectral density and (a) , , ; (b) , , ; (c) , , ; (d) , , .
Figure 6.7 Normalized variances of the real and imaginary parts of as a function of .
Figure 6.8 Probability clouds for when and . Both plots are for .
Figure 6.9 Intensity interferometer. PM, photomultiplier.
Chapter 7: Effects of Partial Coherence in Imaging Systems
Figure 7.1 A thin transmitting object. The -axis points into the paper. and represent refractive indices. is the thickness of the object at coordinates and is the normal distance between the two parallel bounding planes.
Figure 7.2 Geometry for calculation of Hopkins’ formula.
Figure 7.3 A generic optical system.
Figure 7.4 Illumination optics.
Figure 7.5 Up to multiplicative constants, the transmission cross-coefficient for arguments and is the area of overlap of the three circles shown. is the radius of the condenser lens and is the radius of the imaging lens. The circles are drawn in frequency space; hence, the dimensions 1/length for their radii.
7.5 Some Examples
Figure 7.6 A telecentric imaging system. All lenses are assumed to have the same focal length .
Figure 7.7 Normalized image intensity distributions for an object consisting of two pinholes separated by the Rayleigh resolution limit for various values of the complex degree of coherence of the light illuminating the two pinholes.
Figure 7.8 Normalized intensity in the image of a step object for various coherence conditions. represents the ratio of the NA of the condenser lens to the NA of the first imaging lens, as viewed from the object.
Figure 7.9 Normalized intensity distribution along the -axis in the image of a -radian phase edge for , and . Again, .
Figure 7.10 Apparent transfer functions for the intensity terms (a) and (b) as functions of the parameter . The parameter is the coherent cutoff frequency, , where is the half width of the imaging pupil.
Figure 7.11 Region of integration for calculating the spectrum of image intensity at frequency . The two circles have radius , the radius of the exit pupil. The center of the upper circle is displaced from the center of the lower circle by vector distance and the separation of the two pinholes (shown as small black circles at the ends of the heavy line) is also this same vector spacing.
Figure 7.12 Entrance and exit pupils.
Figure 7.13 Fizeau stellar interferometer.
Figure 7.14 Michelson stellar interferometer.
Figure 7.15 Illustration of the importance of phase information. (a) The original object intensity distribution (a rect function). (b) The complex coherence factor corresponding to that object intensity (a sinc function). (c) The modulus of the sinc function. (d) The image obtained by inverse transforming the modulus information rather than the full complex coherence factor.
Figure 7.16 Determining the separation of two small sources from the autocorrelation function of their combined intensity distribution. (a) The object intensity distribution. (b) The autocorrelation function of the object intensity distribution.
Figure 7.17 Special case in which full image recovery is possible from the autocorrelation function of the object intensity distribution. (a) Object intensity distribution. (b) Autocorrelation function of the object intensity distribution.
Figure 7.18 Illustration of the results of applying iterative phase-retrieval algorithms. (a) Original object (a simulated spacecraft), (b) modulus of the Fourier spectrum of the object, and (c) image recovered by using iterative algorithms. (Courtesy J.R. Fienup and the Optical Society of America. See Ref. [33]).
Figure 7.19 Speckle in coherent imaging. (a) Object illuminated with incoherent light—no speckle discernible. (b) Object illuminated with coherent light—speckle very noticeable. (c) Magnified image of one character in the coherent case. (Photo courtesy of P. Chavel and T. Avignon, Institut d'Optique.)
Figure 7.20 Speckle formation in the image of a rough object.
Figure 7.21 Cross section of the power spectral density of a speckle pattern resulting from an image of a uniformly bright rough surface using an imaging system having an unobstructed square exit pupil of width .
Figure 7-4p
Chapter 8: Imaging Through Randomly Inhomogeneous Media
Figure 8.1 Optical system assumed in the random screen analysis.
Figure 8.2 Typical behavior of the structure function of phase for a wide-sense stationary case.
Figure 8.3 Typical OTFs for a system with a random-phase screen: (a) diffraction-limited OTF; (b) average OTF of the screen; (c) average OTF of the system. Note .
Figure 8.4 Typical average system point-spread function for various phase variances. . Note the gradual reduction of the narrow core of the PSF as the phase variance increases.
Figure 8.5 Imaging geometry.
Figure 8.6 Normalized power spectrum of the refractive index fluctuations: the Kolmogorov spectrum with the Tatarski and von Kármán modifications. Inner scale of 2 mm and outer scale of 10 m are assumed for illustration purposes.
Figure 8.7 Refractive index structure function corresponding to the Kolmogorov spectrum.
Figure 8.8 Log-normal probability density function for intensity for various values of .
8.5 The Long-Exposure OTF
Figure 8.9 (a) Long- and (b) short-exposure photographs of the star
Lambda Cratis
. (Courtesy of Gerd Weigelt and Gerhard Baier, University of Erlangen).
Figure 8.10 Imaging a distant point source through the atmosphere. The smooth curve on the right is a long-time-average intensity distribution in the image, while the atmospheric inhomogeneities and optical ray paths shown are at a fixed instantaneous time.
Figure 8.11 Geometry for phase structure function calculation.
Figure 8.12 Value of as a function of . The horizontal line represents the constant , which is the asymptotic value of the integral as .
Figure 8.13 Long-exposure atmospheric transfer functions for various values of (shown by solid curves) and diffraction-limited transfer functions for pupil sizes of 2 cm, 40 cm, and 5 m (shown by dashed curves). and assumed.
Figure 8.14 Long-time-average atmospheric MTFs. The dotted curves result from using the constant 2.91 in Eq. (8.5-39) and the solid curves from using the full expression from Eq. (8.5-38). Three different values of are assumed. The distance is 100m and the wavelength is 500nm. The values along the horizontal axis are expressed in cycles per milliradian.
Figure 8.15 Average profile of the height dependence of the structure constant . is the height above the ground in meters. Calculated from the “Hufnagel-Valley” model ([211]).
Figure 8.16 Normalized bandwidth in angular-frequency space vs. the ratio of the pupil diameter of the optics to the coherence diameter of the atmosphere. The dashed lines represent asymptotes for the resolution when and than unity.
Figure 8.17 Geometry for spherical wave propagation within the atmosphere.
Figure 8.18 Propagation geometry for filter function analysis. is a particular point in the collecting aperture.
Figure 8.19 Filter functions for refractive index in a single plane of the turbulent medium and a single transverse wave number as a function of distance in the medium.
Figure 8.20 Log-amplitude and phase filter functions for an extended turbulent medium, dependence on wavenumber, for a fixed .
Figure 8.21 Log-amplitude and phase filter functions as a function of for a fixed .
Figure 8.22 Areas (shaded) on the exit pupil that influence the spatial frequency .
Figure 8.23 Diffraction-limited OTF and three OTFs that are products of the diffraction-limited OTF and the average atmospheric OTF for the cases , , and . The ratio of to has been taken to be 10.
Figure 8.24 Normalized bandwidths achieved for a diffraction-limited system of diameter when atmospheric turbulence is present with a coherence diameter . corresponds to the long-exposure case, while and correspond to short-exposure cases. The normalizing constant is equal to .
8.7 Stellar Speckle Interferometry
Figure 8.25 Average energy spectrum produced from 120 short-exposure images of the double star
9 Pupis
, after compensation for the speckle interferometry transfer function. Photograph supplied courtesy of Gerd Weigelt and Gerhard Baier, University of Erlangen.
Figure 8.26 Mean-squared MTF for a short exposure with a system having a circular pupil and for a ratio . is the cutoff frequency of the diffraction-limited OTF.
Figure 8.27 Simulation results showing the squared magnitude of the diffraction-limited OTF , the squared magnitude of the product of the diffraction-limited OTF with the long-exposure atmospheric OTF, , and the speckle interferometer transfer function, . The symbol represents the diffraction-limited cutoff frequency.
Figure 8.28 The average squared magnitude of the system MTF for an imaging system with a circular pupil in the presence of atmospheric turbulence. The dotted curve is the square of the diffraction-limited MTF, while the solid curves correspond to three different values of : 0.95, 0.14, and 0.03.
Figure 8.29 Cross-spectrum transfer functions as a function of normalized spatial frequency. Three different frequency offsets are used. Part (a) shows as a function of when 0 pixels and 4 pixels and the slice through is in the direction orthogonal to the separation . Part (b) shows the corresponding results when the slice is in the same direction as . Parts (c) and (d) are similar to parts (a) and (b) except that the separations are 0 and 16 pixels.
Figure 8.30 Block diagram of an adaptive-optics system. Solid lines represent light paths, dashed lines represent electronic connections.
Figure 8.31 Two types of wavefront sensors. (a) The shearing interferometer. (b) The Shack–Hartmann sensor.
Figure 8.32 Laser illuminator, satellite, and 2D detector array, separated by atmospheric turbulence.
Figure 8-4p
Chapter 9: Fundamental Limits in Photoelectric Detection of Light
Figure 9.1 Probability masses associated with the Poisson distribution for .
Figure 9.2 Probability masses associated with the Bose–Einstein distribution for .
Figure 9.3 Probability masses associated for partially polarized light with and , , and .
Figure 9.4 Contours of constant degeneracy parameters in the and plane. The visible and microwave portions of the spectrum are shown in gray.
Figure 9.5 Probability density functions of the continuous count measurement . The average number of photocounts has been chosen to be 100. In part (a), the read noise standard deviation is 0.1, while in part (b) it is 10. The increase in width of the probability density function of with increasing read noise is evident. In computing these curves, 1000 terms in the sum of Eq. 9.3-24 were used.
Figure 9.6 Detection and estimation system assumed for the amplitude interferometer. and are estimates of the fringe visibility and phase.
Figure 9.7 Phasor diagram for noisy fringe estimation.
9.5 Noise Limitations of the Intensity Interferometer at Low Light Levels
Figure 9.8 Counting version of the intensity interferometer.
Figure 9.9 Example of a rate function and the corresponding compound Poisson impulse process.
Figure 9.10 (a) Normalized energy spectral density of the image intensity and (b) corresponding energy spectral density of the detected image for .
Figure 9.11 Energy spectral density estimate for a sinusoidal image intensity. The solid line represents the mean, and the shaded area represents the standard deviation of the estimate at each frequency.
Figure 9.12 Typical single-image rms signal-to-noise ratio for speckle interferometry, as a function of , the mean number of photoevents per speckle. Assumptions: , .
Appendix C: The Atmospheric Filter Functions
Figure C.1 Region of integration in the plane.
List of Tables
Chapter 5: Temporal and Spatial Coherence of Optical Waves
Table 5.1 Names and Definitions of Various Measures of Coherence
Appendix A: The Fourier Transform
Table A.1 One-Dimensional Fourier Transform Pairs
Table A.2 Two-Dimensional Fourier Transform Pairs
WILEY SERIES IN PURE AND APPLIED OPTICS.
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EDITOR: Glenn Boreman, University of North Carolina at Charlotte
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STATISTICAL OPTICS
Second Edition
Joseph W. Goodman
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Published by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
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Library of Congress Cataloging-in-Publication Data:
Goodman, Joseph W., author.
Statistical optics / Joseph W. Goodman. – Second edition.
pages cm – (Wiley series in pure and applied optics)
“Published simultaneously in Canada” – Title page verso.
Includes bibliographical references and index.
ISBN 978-1-119-00945-0 (cloth)
1. Optics–Statistical methods. 2. Mathematical statistics. I. Title. II. Series: Wiley series in pure and applied optics.
QC355.2.G66 2015
535.01′5195–dc23
2015010871
The Cover Image: The cover shows a simulation of a fringe pattern detected by a 256 × 256 photoevent counting array. The visibility (or contrast) of the fringe is 0.4, and the number of photoevents, averaged over the array, is 0.4 events per detector element. Poisson statistics are assumed for the number of events per pixel.
To Hon Mai, who provided the light.
Preface– Second Edition
The first edition of this book was published in 1985, some 30 years ago. In the meantime, there have been enormous changes in technology, not only in electronics but also in optics. These changes in optical technology have enabled new applications, many of which were simply impossible to perform in a satisfactory way in 1985 with the available technology. At the same time, there are underlying fundamental concepts that remain useful in understanding both old and new developments, and many of the concepts introduced in this book are of that kind. How does the second edition differ from the first edition? While the chapter titles and order remain the same, the contents have been changed in a number of significant ways. First, all line-drawing figures have been redrawn, in some cases, using Mathematica to obtain more accurate representations of functions. More importantly, the following major changes have been made:
Chapter 4
The section on the statistical properties of single-mode laser light has been expanded.
Chapter 5
A discussion of Fellgett's advantage has been added to the section on Fourier spectroscopy.
A section has been added covering optical coherence tomography.
A section has been added on coherence multiplexing of fiber sensors.
Sections have been added on Schell model fields and quasihomogeneous fields.
A section has been added on coherent modes of the mutual intensity function.
A section on ensemble-average coherence has been added.
Chapter 6
The discussion of the “exact” solutions for the probability density function of the integrated intensity of thermal light has been rewritten.
The discussion of the statistical properties of mutual intensity with finite measurement time has been shortened, with some results deferred to the problems.
Chapter 7
The section titled “Preliminaries” in the chapter on partial coherence in imaging has been shortened.
The section on Hopkin's formula has been improved.
Several parts of the section on frequency domain calculation of the image intensity spectrum have been improved.
Several examples of images obtained with partially coherent illumination have been added.
The section on speckle has been expanded considerably.
Chapter 8
Sections on the cross-spectrum technique and the bispectrum technique for recovering object information in the presence of atmospheric turbulence have been added.
The section on adaptive optics has been expanded.
A section on imaging coherently illuminated objects in the presence of atmospheric turbulence has been added.
Chapter 9
A section on read noise has been added.
Appendices
Two especially detailed mathematical arguments have been moved to appendices.
Problems
A number of new problems are included in this edition.
References
Many new references have been added.
A solutions manual, not available for the first edition, has been created and is available from the publisher.
I am indebted to San Ma for proof-reading the early chapters of this edition. I would also like to thank Prof.~James Fienup of the University of Rochester for the suggestions and comments he has provided to me during the long process of producing this edition. In addition, I would like to thank three of his graduate students, Zack De Santis, Matt Bergkoetter, and Dustin Moore, for carefully reading large portions of the manuscripts, finding a multitude of typographical errors and making very useful suggestions for change. I should also thank Prof. Michael Roggemann for reading Chapter 8 and providing comments. I owe considerable thanks to Prof.~Moshe Tur, who has provided suggestions for improvement over the years since the first edition was published. Finally, I am very deeply indebted to my wife, Hon Mai, for tolerating my many hours spent on the computer writing this revision. I hope the reader will find the new edition clear and helpful in his or her work.
Joseph W. GoodmanLos Altos, California
Preface– First Edition
Since the early 1960s, it has gradually become accepted that a modern academic training in optics should include a significant exposure to the concepts of Fourier analysis and linear systems theory. This book is based on the premise that a similar stage has also been reached with respect to the tools of probability and statistics and that some exposure to the area of “statistical optics” should be included as a standard part of any advanced optics curriculum. In writing this book, I have attempted to fill the need for a suitable textbook in this area.
The subjects covered in this book are very physical but often can be obscured by mathematics. As an author, I had, therefore, been faced with the dilemma of how best to utilize the powerful mathematical tools of statistics without losing sight of the underlying physics. Some compromises in mathematical rigor must be made, and to the largest extent possible, repetitive emphasis of the physical meaning of mathematical quantities is needed. Since fringe formation is the most fundamental underlying phenomenon involved in most of these subjects, I have tried to stay as close as possible to fringes in dealing with the meaning of the mathematics. I would hope that the treatment used here would be appealing to both optical and electrical engineers and also useful for physicists. The treatment is suitable for both self-study and formal presentations in the classroom. Many homework problems are included.
The material presented in this book covers a great deal of ground. An outline is presented in Chapter 1 and will not be repeated here. The course on which the text was based was taught over the 10 weeks of a single academic quarter, but there is sufficient material for a full 15-week semester, or even for two academic quarters. The problem is then to decide what material to omit in the single-quarter version.
If the material to be covered is in one quarter, it is essential that the students have some previous exposure to probability theory and stochastic processes as well as a good grasp of Fourier methods. Under these conditions my suggestion to the instructor is to allow the students to study Chapters 1–3 on their own and to begin the lectures directly with optics in Chapter 4. The instructor can then choose other sections to be omitted, depending on his/her special interests.
The book began in the form of rough notes for a course at Stanford University in 1968 and has thus had a long time in the making. In many respects it has been too long in the making (as my patient publisher will surely agree). The challenge has thus been to treat the subject matter in a manner that does not become obsolete as time progresses.
The transition from a rough set of notes to a more polished manuscript first began in the academic year 1973–1974, when I was fortunate enough to spend a sabbatical year at the Institut d'Optique in Orsay , France. The hospitality of my immediate host, Professor Serge Lowenthal, as well as the institute's Director, Professor André Marechal, was impeccable. Not only did they provide me with all the surroundings needed for productivity, but they were kind enough to relieve me of the duties normally accompanying a formal appointment. I am most grateful for their support and advice, without which this book would never have had a solid start.
One benefit from the slowness with which the book progressed was the opportunity over many years to spot the weak arguments and the outright errors in the manuscript. To the students of my statistical optics courses at Stanford, therefore, I owe an enormous debt. The evolving notes were also used at a number of other universities, and I am grateful to both William Rhodes (Georgia Institute of Technology) and Timothy Strand (University of Southern California) for providing me with feedback that improved the presentation.
The relationship between the author and publisher is often a distant one and sometimes not even pleasant. Nothing could be further from the truth in this case. Beatrice Shube, the editor at John Wiley & Sons who encouraged me to begin this book 15 years ago, has not only been exceedingly patient and understanding, but has also supplied much encouragement and has become a good personal friend. It has been the greatest pleasure to work with her.
I owe special debts to K.-C. Chin, of Beijing University, for his enormous investment of time in reading the manuscript and suggesting improvements, and to Judith Clark, who typed the manuscript, including all the difficult mathematics, in an extremely professional way.
Finally, I am unable to express adequate thanks to my wife, Hon Mai, and my daughter, Michele, not only for their encouragement, but also for the many hours they accepted being without me while I labored at writing.
Joseph W. GoodmanStanford, CA, October 1984
Chapter 1Introduction
Optics, as a field of science, is in its third millennium of life; yet in spite of its age, it remains remarkably vigorous and youthful. During the middle of the twentieth century, various events and discoveries gave new life, energy, and richness to the field. Especially important in this regard were (i) the introduction of the concepts and tools of Fourier analysis and communications theory into optics, primarily in the late 1940s and throughout the 1950s; (ii) the invention of the laser in late 1950s and its commercialization starting in the early 1960s; (iii) the origin of the field of nonlinear optics in the 1960s; (iv) the invention of the low-loss optical fiber in the early 1970s and the revolution in optical communications that followed; and (v) the rise of the young fields of nanophotonics and biophotonics. It is the thesis of this book that in parallel with these many advances, another important change has also taken place gradually but with an accelerating pace, namely, the infusion of statistical concepts and methods of analysis into the field of optics. It is to the role of such concepts in optics that this book is devoted.
The field we shall call “statistical optics” has a considerable history of its own. Many fundamental statistical problems were solved in the late nineteenth century and applied to acoustics and optics by Lord Rayleigh. The need for statistical methods in optics increased dramatically with the discovery of the quantized nature of light and, particularly, with the statistical interpretation of quantum mechanics introduced by Max Born. The introduction by E. Wolf in 1954 of an elegant and broad framework for considering the coherence properties of waves laid a foundation upon which many of the important statistical problems in optics could be treated in a unified way. Also worth mentioning is the semiclassical theory of light detection, pioneered by L. Mandel, which tied together, in a comparatively simple way, knowledge of the statistical fluctuations of classical wave quantities (fields, intensities) and fluctuations associated with the interaction of light and matter. This history is far from complete but is dealt with in more detail in the individual chapters that follow.
1.1 Deterministic Versus Statistical Phenomena and Models
In the normal course of events, a student of physics or engineering first encounters optics in an entirely deterministic framework. Physical quantities are represented by mathematical functions that are either completely specified in advance or are assumed precisely measurable. These physical quantities are subjected to well-defined transformations that modify their form in perfectly predictable ways. For example, if a monochromatic light wave with a known complex field distribution is incident on a transparent aperture in an otherwise opaque screen, the resulting complex field distribution some distance behind the screen can be calculated using the well-established diffraction formulas of wave optics. In this approach, inaccuracies in the results arise only due to inaccuracies of the deterministic models used to describe the diffraction process.
The students emerging from such an introductory course may feel confident that they have grasped the basic physical concepts and laws and are ready to find a precise answer to almost any problem that comes their way. To be sure, they have probably been warned that there are certain problems, arising particularly in the detection of weak light waves, for which a statistical approach is required. But a statistical approach to problem solving appears at first glance to be a “second class” approach, for statistics is generally used when we lack sufficient information to carry out the aesthetically more pleasing “exact” solution. The problem may be inherently too complex to be solved analytically or numerically, or the boundary conditions may be poorly defined. Surely, the preferred way to solve a problem must be the deterministic way, with statistics entering only as a sign of our weakness or limitations. Partially as a consequence of this viewpoint, the subject of statistical optics is usually left for the more advanced students, particularly those with a mathematical flair.
Although the origins of the above viewpoint are quite clear and understandable, the conclusions reached regarding the relative merits of deterministic and statistical analysis are very greatly in error, for several important reasons. First, it is difficult, if not impossible, to conceive of a real engineering problem in optics that does not contain some element of uncertainty requiring statistical analysis. Even the lens designer, who traces rays through application of precise physical laws accepted for centuries, must ultimately worry about quality control! Thus statistics is certainly not a subject to be left primarily to those more interested in mathematics than in physics and engineering.
Furthermore, the view that the use of statistics is an admission of one's limitations and thus should be avoided is based on too narrow a view of the nature of statistical phenomena. Experimental evidence indicates, and indeed the great majority of physicists believe, that the interaction of light and matter is fundamentally a statistical phenomenon, which in principle cannot be predicted with perfect precision in advance. Thus statistical phenomena play a role of the greatest importance in the world around us, independent of our particular mental capabilities or limitations.
Finally, in defense of statistical analysis, we must say that, whereas both deterministic and statistical approaches to problem solving require the construction of mathematical models of physical phenomena, the models constructed for statistical analysis are inherently more general and flexible. Indeed, they invariably contain the deterministic model as a special case! For a statistical model to be accurate and useful, it should fully incorporate the current state of our knowledge regarding the physical parameters of concern. Our solutions to statistical problems will be no more accurate than the models we use to describe both the physical laws involved and the state of knowledge or ignorance.
The statistical approach is indeed somewhat more complex than the deterministic approach, for it requires knowledge of the elements of probability theory. In the long run, however, statistical models are far more powerful and useful than deterministic models in solving physical problems of genuine practical interest. Hopefully, the reader will agree with this viewpoint by the time this book has been digested.
1.2 Statistical Phenomena in Optics
Statistical phenomena are so plentiful in optics that there is no difficulty in compiling a long list of examples. Because of the wide variety of these problems, it is difficult to find a general scheme for classifying them. Here we attempt to identify several broad aspects of optics that require statistical treatment. These aspects are conveniently discussed in the context of an optical imaging problem.
Most optical imaging problems are of the following type. Nature assumes some particular state (e.g., a certain collection of atoms and/or molecules in a distant region of space, a certain distribution of reflectance over terrain of unknown characteristics, or a certain distribution of transmittance in a sample of interest). By operating on optical waves that arise as a consequence of this state of Nature, we wish to deduce exactly what that state is.
Statistics is involved in this task in a wide variety of ways, as can be discovered by reference to Fig. 1.1.
Figure 1.1 An optical imaging system.
First, and most fundamentally, the state of Nature is known to us a priori only in a statistical sense. If it were known exactly, there would be no need for any measurement in the first place. Thus the state of Nature is random, and in order to properly assess the performance of the system, we must have a statistical model, ideally representing the set of possible states, together with associated probabilities of the occurrence of those states. Usually, a less complete description of the statistical properties of the object will suffice.
Our measurement system operates not on the state of Nature per se, but rather on, an optical representation of that state (e.g., radiated light, transmitted light, or reflected light). The representation of the state of Nature by an optical wave has statistical attributes itself, primarily as a result of the statistical or random properties of all light waves. Because of fundamentally statistical attributes of the mechanisms that generate light, all optical sources produce radiation that is to some degree random in its properties. At one extreme, we have the chaotic and unordered emission of light by a thermal source, such as an incandescent lamp; at the other extreme, we have the comparatively ordered emission of light by a continuous-wave (CW) gas laser. In the latter case, the light comes close to containing only a single frequency and traveling in a single direction. Nonetheless, any real laser emits light with statistical properties, including random fluctuations of both amplitude and phase of the radiation. Statistical fluctuations of light are of great importance in many optical experiments and indeed play a central role in the character of the image produced by the system depicted in Fig. 1.1.
After interacting with the state of Nature, the radiation travels through an intervening medium until it reaches the focusing optics. The parameters of that medium may or may not be well known. If the medium is a perfect vacuum, it introduces no additional statistical aspects to the problem. On the other hand, if the medium is the Earth's atmosphere and the optical path length is more than a few meters, random fluctuations of the atmospheric index of refraction can have dramatic effects on the wave and can seriously degrade the image obtained by the system. Statistical methods are required to quantify this degradation.
The light eventually reaches the focusing optics. How well are the exact parameters of this system known? Any lack of knowledge of the parameters of the system must be taken into account in our statistical model for the measurement system. For example, there may be unknown errors in the wavefront deformation introduced by passage through the focusing optics. Such errors can be modeled statistically over an ensemble of possible lenses and should be taken into account in assessment of the performance of the system.
The radiation finally reaches an optical detector, where there is an interaction of light and matter. Random fluctuations of the detected energy are readily observed, particularly at low light levels, and can be attributed to a variety of sources,including the discrete nature of the interaction between light and matter as well as the presence of internal additive electronic noise (e.g., thermal noise associated with resistors in the detector circuitry). The result of the measurement is thus related to the image falling on the detector only in a statistical way.
We conclude that at all stages of the optical problem, including illumination, transmission, image formation, and detection, statistical treatment may be needed to assess the performance of the system. Our goal in this book is to lay the necessary foundation and to illustrate the application of statistics to the many diverse areas of optics where it is needed.
1.3 An Outline of the Book
Eight chapters follow this chapter. Since many scientists and engineers working in the field of optics may feel the need to strengthen their basic knowledge of statistical tools, Chapter 2 presents a review of probability theory and Chapter 3 contains a review of the theory of random processes. The reader already familiar with these subjects may wish to proceed directly to Chapter 4, using the earlier material only for reference when needed.
Discussion of optical problems begins in Chapter 4, which deals with the “first-order” statistics (i.e., the statistics at a single point in space and time) of several kinds of light waves, including light generated by thermal sources and light generated by lasers. Also included is a formalism that allows characterization of the polarization properties of an optical wave.
Chapter 5 introduces the concepts of time coherence and space coherence (which are “second-order” properties of light waves) and deals at length with the propagation of coherence under various conditions. Chapter 6 extends this theory to coherence of order higher than two and illustrates the need for fourth-order coherence functions in a variety of different optical problems, including a classical analysis of the intensity interferometer.
Chapter 7 is devoted to the theory of image formation in partially coherent light. Several analytical approaches to the problem are introduced, including the approach widely used in microlithography. The concept of interferometric imaging, as widely practiced in radio astronomy, is also introduced in this chapter and is used to lend insight into the incoherent image formation process.
Chapter 8 is concerned with the effects of transparent random media, such as the Earth's atmosphere, on the quality of images formed by optical systems. The origin of the random refractive-index fluctuations in the atmosphere is reviewed, and statistical models for such fluctuations are introduced. The effects of these fluctuations on optical waves are dealt with, and image degradations introduced by the atmosphere are treated from a statistical viewpoint. Stellar speckle interferometry, a method for partially overcoming the effects of atmospheric turbulence, is discussed in some detail, as are several related methods for achieving more complete image recovery.
Finally, Chapter 9 treats the semiclassical theory of light detection and illustrates the theory with analyses of the sensitivity limitations of amplitude interferometry, intensity interferometry, and speckle interferometry.
Appendices A–E present supplemental background material and analysis.
Chapter 2Random Variables
Since this book deals primarily with statistical problems in optics, it is essential that we start with a clear understanding of the mathematical methods used to analyze random or statistical phenomena. We shall assume at the start that the reader has been exposed previously to at least some of the basic elements of probability theory. The purpose of this chapter is to provide a review of the most important material, establish notation, and present a few specific results that will be useful in later applications of the theory to optics. The emphasis is not on mathematical rigor but rather on physical plausibility. For more rigorous treatment of the theory of probability, the reader may consult various texts on statistics (see, e.g., Refs. [162] and [53]). In addition, there are many excellent engineering-oriented books that discuss the theory of random variables and random processes (see, e.g., [148], [159] and [195]).
2.1 Definitions of Probability and Random Variables
By a random experiment, we mean an experiment with an outcome that cannot be predicted in advance. Let the collection of possible outcomes be represented by the set of events . For example, if the experiment consists of tossing two coins side by side, the possible “elementary events” are , and TT, where H indicates “heads” and T denotes “tails.” However, the set contains more than four elements, since events such as “at least one head occurs in two tosses” (HH or HT or TH) are included. If and are any two events, then the set must also include andor, not, and not. In this way, the complete set is derived from the underlying elementary events.
If we repeat the experiment N times and observe the specific event A to occur n
