Electromagnetic Wave Scattering on Nonspherical Particles: Basic Methodology and Simulations

298 Pages·2009·6.813 MB·English

by  Tom Rother (auth.)

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Tom Rother
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Basic Methodology and Simulations
With 69 Figures
ABC
Dr.Tom Rother
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SpringerSeriesinOpticalSciences ISSN0342-4111 e-ISSN1556-1534
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The photo shows a 22-degree Halo phenomenon with sun dogs. It was taken by
the author in March 1998 at the DLR site in Neustrelitz. This phenomenon is
caused by light scattering on hexagonal ice columns (the 22-degree Halo) and ice
plates (the sun dogs) in Cirrus clouds
Preface
Scatteringofelectromagneticwavesonthree-dimensional,dielectricstructures
is a basic interaction process in physics, which is also of great practical im-
portance. Most of our visual impressions are caused not by direct but by
scattered light, as everybody can experience of looking directly at the sun.
Several modern measurement technologies in technical and medical diagnos-
tics are also based on this interaction process. Atmospheric remote sensing
with lidar and radar as well as nephelometer instruments for measuring sus-
pended particulates in a liquid or gas colloid are only a few examples where
scattered electromagnetic waves provide us with information concerning the
structure and consistence of the objects under consideration. Using the infor-
mation of the elastically scattered electromagnetic wave is a common ground
of most of those measuring methods. The phrase “elastically scattered” ex-
presses the restriction that we consider such interaction processes only where
the scattered wave possesses the same wavelength as the primary incident
wave. This book addresses this special scattering problem.
The methodology part of this book is concerned with the solution of the
partial differential equations underlying this scattering process. These are
especially the scalar Helmholtz equation and the vector-wave equation. From
the mathematical point of view, we are faced with the solution of bound-
ary value problems. This becomes especially simple if the boundary values
are given along a constant coordinate line in one of the coordinate systems
that allow a separation of the partial differential equation. Such problems are
sometimes called “separable boundary value problems” in the literature. The
applied method is the so-called “Separation of Variables method.” It reduces
theprimarypartialdifferentialequationtoasetofordinarydifferentialequa-
tionswhoseeigensolutionsserveafterwardsasexpansionfunctionstoapprox-
imate the sought solution appropriately. Scattering of light on dielectric and
ideal metallic spheres was first solved by this method in 1908 by Gustav Mie.
TheMietheory,asitiscallednowadays,formsthebasisinmanyapplications
even today.
VIII Preface
On the other hand, not the least due to the possibilities of modern
computers, there can be observed a growing interest in modeling more and
more realistic scattering scenarios which goes beyond the conventional Mie
theory for spherical scatterers. If the geometry of the scatterer differs only
slightlyfromthatofaseparablegeometrysimplerperturbationmethodsmay
be applied successfully. But more often the deviation from such a separable
geometry is much stronger so that we are forced to apply more rigorous nu-
merical methods to solve the problem. A large variety of such methods have
beendevelopedinthepast.Differinginconceptandexecutionthesemethods
start from the common assumption that it is no longer possible to apply the
Separation of Variables method. A critical discussion of this assumption is a
major objective of this book. The Green functions are of special importance
forthisdiscussion.Basedonthesefunctions,wewillshowthattherecanbees-
tablished a formalism which provides a common methodological background
for a variety of different numerical methods. But the so-called “T-matrix”
method is within the focus of our interest. This special solution method was
developed by Waterman at the end of the 1960s and beginning of the 1970s
of the twentieth century. It has been proved to be very successful in many
applications.
This book at hand is restricted to the relatively simple case of model-
ingelectromagneticwavescatteringonsingle,homogeneous,butnonspherical
particlesinsphericalcoordinates.Neverthelessitprovidesasoundbasistode-
velopthemethodsformorecomplexsituationsaswehave,ifscatteringonan
ensemble of objects or on inhomogeneous objects (like multilayered particles,
for example) is considered.
Thebasicmethodologicalconsiderationsofthisbookarecomplementedin
the second last chapter with numerical simulations of a few typical scattering
scenarios. For these simulations we have developed a specific T-matrix code
whichcanbefoundintheenclosedCD.Itsstructureandusageispresentedin
detail. Thisprogramwillenablethereaderwhoismoreinterestedintangible
calculations, to perform quickly his own numerical simulations, and, most
important, to estimate the accuracy and usefulness of the obtained results.
But this software can also be used in university lectures to demonstrate the
principal aspects of light scattering. Some parts of this book began as notes
of a special lecture given by the author at the Meteorological Institute of the
University Leipzig. The considerations in the second to last chapter should
reveal some of the numerical problems which must be solved if one tries to
develop a certain T-matrix code.
Naturally, this book is not the result of only my efforts. For the multitude
of (partly very heavily) discussions over the years I sincerely thank my two
colleagues at the Remote Sensing Technology Institute, Dr. K. Schmidt and
Dr. J. Wauer. Special thanks are due to Dr. J. Wauer for his numerical work
on the Rayleigh hypothesis as discussed in Chap.6, and for his work on the
software package mieschka and the database. Special thanks are due also to
the German Aerospace Center (DLR) for the confidence in my work over the
Preface IX
years,andforthefinancialaswellastheadministrativesupportwhichcannot
be taken for granted nowadays.
C. Ascheron at Springer deserves a word of thanks for his continuous
interest,support,andencouragementafterthemanuscriptappearedviae-mail
for the first time suddenly in his office.
My gratitude is deepest, however, to my parents, who supported me in
manifold ways and pushed me into the direction of science, and to my wife
Doreen, who helped me without complaint through the nearly 7 years of
writing this book. She also carefully read the manuscript and wiped out a lot
of needless “m”-dashes – thus demonstrating that a musician and a physicist
can really benefit from each other.
Neustrelitz, Germany T. Rother
June 2009
Contents
1 Scattering as a Boundary Value Problem .................. 1
1.1 Introduction ............................................ 1
1.2 Formulation of the Boundary Value Problems ............... 5
1.3 Solving the Boundary Value Problems
with the Rayleigh Method ................................ 8
1.3.1 The Outer Dirichlet Problem ....................... 10
1.3.2 The Outer Transmission Problem.................... 12
2 Filling the Mathematical Tool Box......................... 17
2.1 Introduction ............................................ 17
2.2 Approximation of Functions and Fields
at the Scatterer Surface .................................. 18
2.2.1 Approximation by Finite Series Expansions........... 19
2.2.2 Best Approximation ............................... 21
2.2.3 The Transformation Character of the T-Matrix ....... 24
2.3 Eigensolutions of the Scalar Helmholtz Equation
in Spherical Coordinates ................................. 29
2.3.1 The Eigensolutions ................................ 30
2.3.2 The Combined Summation Index.................... 36
2.3.3 Properties of the Scalar Eigensolutions ............... 36
2.3.4 Expansion of a Scalar Plane Wave ................... 38
2.4 Eigensolutions of the Vector-Wave Equation
in Spherical Coordinates ................................. 41
2.4.1 The Vectorial Eigensolutions........................ 41
2.4.2 Properties of the Vectorial Eigensolutions ............ 46
2.4.3 Expansion of a Linearly Polarized Plane Wave ........ 51
2.5 Green Theorems and Green Functions Related
to the Scalar Boundary Value Problems .................... 57
2.5.1 The Green Theorems .............................. 58
2.5.2 The Free-Space Green Function ..................... 59
XII Contents
2.5.3 The Green Functions Related to the Outer Dirichlet
and Transmission Problem.......................... 65
2.6 Green Theorems and Green Functions Related
to the Vectorial Boundary Value Problems.................. 67
2.6.1 Dyadics .......................................... 67
2.6.2 The Green Theorems .............................. 70
2.6.3 The Dyadic Free-Space Green Function .............. 71
2.6.4 The Dyadic Green Functions Related to the Outer
Dirichlet and Transmission Problem ................. 74
3 First Approach to the Green Functions:
The Rayleigh Method ..................................... 77
3.1 Introduction ............................................ 77
3.2 The Scalar Delta Distribution at the Scatterer Surface ....... 79
3.3 The Scalar Green Functions Related to the Helmholtz
Equation ............................................... 79
3.3.1 The Outer Dirichlet Problem ....................... 79
3.3.2 The Outer Transmission Problem.................... 87
3.4 The Dyadic Delta Distribution at the Scatterer Surface....... 89
3.5 The Dyadic Green Functions Related to the Vector-Wave
Equation ............................................... 91
3.5.1 The Outer Dirichlet Problem ....................... 91
3.5.2 The Outer Transmission Problem.................... 97
4 Second Approach to the Green Functions:
The Self-Consistent Way................................... 99
4.1 Introduction ............................................ 99
4.2 The Scalar Green Functions Related
to the Helmholtz Equation................................100
4.2.1 The Outer Dirichlet Problem .......................100
4.2.2 The Outer Transmission Problem....................103
4.3 The Dyadic Green Functions Related
to the Vector-Wave Equation .............................105
4.3.1 The Outer Dirichlet Problem .......................105
4.3.2 The Outer Transmission Problem....................107
4.4 Symmetry and Unitarity .................................109
4.4.1 Symmetry ........................................110
4.4.2 Unitarity.........................................113
5 Other Solution Methods ...................................123
5.1 Introduction ............................................123
5.2 T-Matrix Methods.......................................124
5.2.1 The Extended Boundary Condition Method ..........124
5.2.2 Point Matching Methods ...........................130