CONTNTS
CHAPTER 1
INTRODUCTION TO ELECTROMAGNETIC SCATTERING
BY A SINGLE PARTICLE .................................... 1
1 Basic Scattering Parameters 2
1.1 Scattering Amplitudes and Cross Sections 2
1.2 Scattering Amplitude Matrix 6
2 Rayleigh Scattering 9
2.1 Rayleigh Scattering by a Small Particle 9
2.2 Rayleigh Scattering by a Sphere 10
2.3 Rayleigh Scattering by an Ellipsoid 12
2.4 Scattering Dyads 14
3 Integral Representations of Scattering and
Born Approximation 16
3.1 Integral Expression for Scattering Amplitude 16
3.2 Born Approximation 18
4 Plane Waves, Cylindrical Waves, and
Spherical Waves 9.1
4.1 Cartesian Coordinates: Plane Waves 21
4.2 Cylindrical Waves 22
4.3 Spherical Waves 24
5 Acoustic Scattering 30
6 Scattering by Spheres, Cylinders, and Disks 32
6.1 Mie Scattering 32
6.2 Scattering by a Finite Length Cylinder Using the Infinite
Cylinder Approximation 41
6.3 Scattering by a Disk Based on the Infinite Disk Approximation 46
References and Additional Readings 52
CHAPTER 2
BASIC THEORY OF ELECTROMAGNETIC SCATTERING 53
I Dyadic Green's Function 54
1.1 Green's Functions 54
1.2 Plane Wave Representation 55
1.3 Cylindrical Waves 57
1.4 Spherical Waves 59
2 Huygens' Principle and Extinction Theorem 60
3 Active Remote Sensing and Bistatic Scattering
Coefficients 66
4 Optical Theorem 68
5 Reciprocity and Symmetry 73
5.1 Reciprocity 73
5.2 Reciprocal Relations for Bistatic Scattering Coefficients and
Scattering Amplitudes 75
5.3 Symmetry Relations for Dyadic Green's Function 79
6 Eulerian Angles of Rotation 81
7 T-Matrix 83
7.1 T-Matrix and Relation to Scattering Amplitudes 83
7.2 Unitarity and Symmetry 88
8 Extended Boundary Condition 91
8.1 Extended Boundary Condition Technique 91
8.2 Spheres 97
8.2.1 Scattering and Absorption for Arbitrary Excitation 100
8.2.2 Mie Scattering of Coated Sphere 102
8.3 Spheroids 104
References and Additional Readings 106
CHAPTER 3
FUNDAMENTALS OF RANDOM SCATTERING .......... 107
Radar Equation for Conglomeration of Scatterers 108
Stokes Parameters and Phase Matrices 116
Elliptical Polarization, Stokes Parameters, Partial
Polarization 116
Stokes Matrix 123
Scattering per Unit Volume and Phase Matrix 124
Rayleigh Phase Matrix 127
Phase Matrix of Random Media 129
Fluctuating Fields 131
Coherent and Incoherent Fields 131
Probability Distribution of Scattered Fields and Polarimetric
Description 132
Specific Intensity 140
Passive Remote Sensing 145
Planck's Radiation Law and Brightness Temperature 145
Kirchhoff's Law 149
Fluctuation Dissipation Theorem 152
Emissivity of Four Stokes Parameters 155
Correlation Function of Fields 161
References and Additional Readings 165
CHAPTER 4
CHARACTERISTICS OF DISCRETE SCATTERERS AND
ROUGH SURFACES ..................................... ... 167
I Ice 168
2 S now 170
3 Vegetation 171
4 Atmosphere 172
CONTENTS
Correlation Function and Pair Distribution
Function 173
5.1 Correlation Function 174
5.2 Pair Distribution Function 176
6 Gaussian Rough Surface and Spectral Density 179
7 Soil and Rocky Surfaces 184
8 Ocean Surface 185
References and Additional Readings 195
CHAPTER 5
SCATTERING AND EMISSION BY LAYERED MEDIA . .. 199
I Incoherent Approach of Radiative Transfer 200
2 Wave Approach 203
2.1 Reflection and Transmission 203
2.2 Dyadic Green's Function for Stratified Medium 207
2.3 Brightness Temperatures for a Stratified Medium with
Temperature Distribution 212
3 Comparison Between Incoherent Approach and
Coherent Approach 217
4 Applications to Passive Remote Sensing of Soil 220
References and Additional Readings 229
CHAPTER 6
SINGLE SCATTERING AND APPLICATIONS ............. 231
Single Scattering and Particle Position Correlation 232
Applications of Single Scattering 237
Synthetic Aperture Radar 237
Interferometric SAR 248
Active Remote Sensing of Half-Space Random Media 252
References and Additional Readings 258
CHAPTER 7
RADIATIVE TRANSFER THEORY ........................ 259
I Scalar Radiative Transfer Theory
2 Vector Radiative Transfer Theory
2.1 Phase Matrix of Independent Scattering
2.2 Extinction Matrix
2.3 Emission Vector
2.4 Boundary Conditions
References and Additional Readings
CHAPTER 8
SOLUTION TECHNIQUES OF RADIATIVE
TRANSFER THEORY ...................................... 287
I Iterative Method 288
1.1 Iterative Procedure 288
1.2 Integral Equation for Scattering Problems 293
1.3 Active Remote Sensing of a Half-Space of Spherical Particles 298
1.4 Active Remote Sensing of a Layer of Nonspherical Particles 303
1.4.1 Numerical Illustrations with Finite Dielectric Cylinders 310
1.5 Second-Order Scattering from Isotropic Point Scatterers 322
2 Discrete Ordinate-Eigenanalysis Method 324
2.1 Radiative Transfer Solution for Laminar Structures 324
2.2 Numerical Procedure of Discrete Ordinate Method: Normal
Incidence 328
2.3 Active Remote Sensing: Oblique Incidence 337
2.4 Discrete Ordinate Method for Passive Remote Sensing 343
2.5 Passive Remote Sensing of a Three-Dimensional Random
Medium 349
2.6 Passive Remote Sensing of a Layer of Mie Scatterers
Overlying a Dielectric Half-Space 352
CONTENTS
Invariant Imbedding 362
One-Dimensional Problem 363
Passive Remote Sensing of a Three-Dimensional Scattering
Medium with Inhomogeneous Profiles 370
Passive Remote Sensing of a Three-Dimensional Random
Medium 373
Thermal Emission of Layers of Spherical Scatterers in the
Presence of Inhomogeneous Absorption and Temperature
Profiles 374
Diffusion Approximation 380
References and Additional Readings 386
CHAPTER 9
ONE-DIMENSIONAL RANDOM ROUGH SURFACE
SCATTERING .............................................. 389
1 Introduction 390
2 Statistics of Random Rough Surface 392
2.1 Statistics, Correlation Function and Spectral Density 392
2.2 Characteristic Functions 396
3 Small Perturbation Method 397
3.1 Dirichlet Problem for One-Dimensional Surface 397
3.2 Neumann Problem for One-Dimensional Surface 403
4 Kirchhoff Approach 407
4.1 Dirichlet Problem for One-Dimensional Surface 408
4.2 Neumann Problem for One-Dimensional Surface 415
References and Additional Readings 417
INDEX ...................................................... 419
PREFACE
Electromagnetic wave scattering is an active, interdisciplinary area of
research with myriad practical applications in fields ranging from atomic
physics to medical imaging to geoscience and remote sensing. In particular,
the subject of wave scattering by random discrete scatterers and rough sur-
faces presents great theoretical challenges due to the large degrees of freedom
in these systems and the need to include multiple scattering effects accu-
rately. In the past three decades, considerable theoretical progress has been
made in elucidating and understanding the scattering processes involved in
such problems. Diagrammatic techniques and effective medium theories re-
main essential for analytical studies; however, rapid advances in computer
technology have opened new doors for researchers with the full power of
Monte Carlo simulations in the numerical analysis of random media scatter-
ing. Numerical simulations allow us to solve the Maxwell equations exactly
without the limitations of analytical approximations, whose regimes of va-
lidity are often difficult to assess. Thus it is our aim to present in these three
volumes a balanced picture of both theoretical and numerical methods that
are commonly used for tackling electromagnetic wave scattering problems.
Whi.le our book places an emphasis on remote sensing applications, the ma-
terials covered here should be useful for students and researchers from a
variety of backgrounds as in, for example, composite materials, photonic de-
vices, optical thin films, lasers, optical tomography, and X-ray lithography.
Introductory chapters and sections are also added so that the materials can
be readily understood by graduate students. We hope that our book would
help stimulate new ideas and innovative approaches to electromagnetic wave
scattering in the years to come.
The increasingly important role of numerical simulations in solving elec-
tromagnetic wave scattering problems has motivated us to host a companion
web site that contains computer codes on topics relevant to the book. These
computer codes are written in the MATLAB' programming language and
are available for download from our web site at www. emwave. com. They are
provided to serve two main purposes. The first is to supply our readers a
hands-on laboratory for performing numerical experiments, through which
the concepts in the book can be more dynamically relayed. The second is
to give new researchers a set of basic tools with which they could quickly
build on projects of their own. The fluid nature of the web site would also
allow us to regularly update the contents and keep pace with new research
developments. The present volume covers the basic principles and applications of elec-
tromagnetic wave scattering and lays the groundwork for the study of more
advanced topics in Volumes II and III. We start in Chapter 1 with exact
and approximate solutions of wave scattering by a single particle of simple
shape. Such problems can be solved exactly by expanding the fields in terms
of scalar or vector waves in separable coordinates, depending on the geometry
of the scatterer. When the size of the scatterer is small, Rayleigh scatter-
ing represents a simple and valid approximation. When scattering is weak,
the Born approximation can be applied to the volume integral equation for
the internal field. Approximate solutions are also useful when the scatterer
lacks perfect symmetry as in the case of a finite cylinder. In Chapter 2, we
discuss basic scattering theory. We introduce the Green's function for the
wave equation and its various coordinate representations. From the vector
Green's theorem, we derive the Hugyens' Principle and the extinction the-
orem, which are especially useful for formulating surface integral equalions
in scattering problems. The reciprocity principle leads to useful symmetry
relations in the scattering amplitudes and the Green's function, while energy
conservation leads to the optical theorem. The T-matrix formulation with
the extended boundary condition technique is a popular method that can
be used to calculate scattering from an arbitrarily shaped object. We give
explicit results for dielectric spheres and spheroids.
In Chapter 3, we begin the study of electromagnetic scattering by a ran-
dom collection of scatterers. The concepts of fluctuating fields and ensemble
averaging are of central importance in random media scattering. These and
related ideas are explored in this chapter. The specific intensity is often used
to describe energy transport through a random medium. The fully polari-
metric description of the specific intensity is provided by the Stokes vector.
As an application to passive remote sensing, we derive the emissivity of
the four Stokes parameters using the fluctuation dissipation theorem. Basic
radiative transfer (RT) theory elements including the extinction coefficient
and scattering phase matrix are also introduced. In contrast to conventional
RT theory, the phase functions are defined in terms of bistatic scattering
cross sections. This allows for the development of the dense medium radia-
tive transfer theory (DMRT) to be discussed in Volumes II and III. Many
natural media, e.g., snow, vegetation, and ocean surfaces, can be effectively
modeled in terms of simple random media. Chapter 4 is devoted to the sta-
tistical characterizations of such random discrete media and rough surfaces.
Useful characterizations include the pair distribution function for volume
scatterers and the power spectrum for rough surfaces.
In Chapter 5, we consider scattering and emission by plane-parallel lay-
ered media, which provide simple but very useful models for geophysical
remote sensing. We solve this problem in two different ways: the coherent
or wave approach, which is exact, versus the incoherent or radiative transfer
approach. This gives us some insights into the approximations involved in
RT theory. In Chapter 6, we discuss the single scattering approximation,
where each particle is assumed to scatter independently. However, we take
into account of the phase coherence in the addition of scattered fields. We
demonstrate the existence of an interesting correlation effect in random me-
dia scattering known as the memory effect. As will be shown in Volumes II
and III, this effect persists even when multiple scattering is included. Appli-
cations of single scattering to synthetic aperture radar (SAR) and random
media scattering are also discussed.
In Chapters 7 and 8, we take a closer look at the radiative transfer
equation and its solutions. The iterative method is useful when scattering is
weak and provides physical correspondence with different orders of multiple
scattering. When scattering is strong, the discrete ordinate eigenanalysis
approach can be used to obtain numerically exact solutions. For scattering
media with inhomogeneous profiles, the method of invariant imbedding can
be applied. Diffusion approximation is useful when, upon multiple scatter-
ing, the intensities have been diffused almost uniformly in all directions. We
illustrate these solution techniques with extensive examples from active and
passive microwave remote sensing.
In Chapter 9, we discuss wave scattering by random rough surfaces. De-
spite much theoretical and numerical efforts, the two "classical" analytical
approximations of small perturbation method and Kirchhoff approach are
still the simplest and most widely used analytical methods for solving rough
surface problems. Here they are illustrated using one-dimensional rough sur-
faces with Dirichlet and Neumann boundary conditions. Two-dimensional
rough surface scattering problems are discussed extensively in Volumes II
and III.
Acknowledgments
We would like to acknowledge the collaboration with our colleagues and grad-
uate students. In particular, we wish to thank Professor Chi Chan of City
University of Hong Kong, Professor Joel T. Johnson of Ohio State University,
Dr. Robert T. Shin of MIT Lincoln Laboratory, and Dr. Dale Winebrenner
of University of Washington. The graduate students who completed their
Ph.D. theses from the University of Washington on random media scatter-
ing include Boheng Wen (1989), Kung-Hau Ding (1989), Shu-Hsiang Lou
(1991), Charles E. Mandt (1992), Richard D. West (1994), Zhengxiao Chen
(1994), Lisa M. Zurk (1995), Kyung Pak (1996), Guifu Zhang (1998), and
Qin Li (2000). Much of their dissertation works are included in this book.
Financial supports from the Air Force Office of Scientific Research, Army
Research Office, National Aeronautics and Space Administration, National
Science Foundation, Office of Naval Research, and Schlumberger-Doll Re-
search Center for research materials included in this book are gratefully
acknowledged. Special thanks are due to Chite Chen' for her contributions
on the MATLAB programs, Henning Braunisch for careful proofreading on
parts of the manuscript, and Bae-Ian Wu for production assistance. We
would also like to thank Chi .On Ao for his help in editing and typsetting
the manuscript.
Leung Tsang
Seattle, Washington
Jin Au Kong
Cambridge, Massachusetts
Kung-Hau Ding
Hanscorn AFB, Massachusetts
May 2000
No comments:
Post a Comment