Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006

founded by H.K.V. Lotsch

Editor-in-Chief: W. T. Rhodes, Atlanta

Editorial Board: A. Adibi, Atlanta

T. Asakura, Sapporo

T. W. Hansch,¨ Garching

T. Kamiya, Tokyo

F. Krausz, Garching

B. Monemar, Linkoping¨

H. Venghaus, Berlin

H. Weber, Berlin

H. Weinfurter, Munchen¨

Adrian Doicu

Institut fur¨ Methodik der Fernerkundung

Deutsches Institut fur¨ Luftund Raumfahrt e.V.

D-82234 Oberpfaffenhofen, Germany

E-mail: adrian.doicu@dlr.de

Thomas Wriedt

University of Bremen, FB4, VT

Badgasteiner Str. 3, 28359 Bremen, Germany

E-mail: thw@iwt.uni-bremen.de

Yuri A. Eremin

Applied Mathematics and Computer Science Faculty Moscow State University

119899 Moscow, Russia E-mail: eremin@cs.msu.su

ISSN 0342-4111

ISBN-10 3-540-33696-6 Springer Berlin Heidelberg New York

ISBN-13 978-3-540-33696-9 Springer Berlin Heidelberg New York

Library of Congress Control Number: 2006929447

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To our families: Aniela and Alexandru

Ursula and Jannis Natalia, Elena and Oleg

Preface

Since the classic paper by Mie [159] or even the papers by Clebsch [37] and Lorenz [146] there is a permanent preoccupation in light scattering theory. Mie was interested in the varied colors exhibited by colloidal suspensions of noble metal spheres, but nowadays, the theory of light scattering by particles covers a much broader and diverse field. Particles encountered in practical applications are no longer considered spherical; they are nonspherical, nonrotational symmetric, inhomogeneous, coated, chiral or anisotropic.

Light scattering simulation is needed in optical particle characterization, to understand new physical phenomena or to design new particle diagnostics systems. Other examples of applications are climatology and remote sensing of Earth and planetary atmospheres, which rely on the analysis of the parameters of radiation scattered by aerosols, clouds, and precipitation. Similar electromagnetic modeling methods are needed to investigate microwave scattering by raindrops and ice crystals, while electromagnetic scattering is also encountered in astrophysics, ocean and biological optics, optical communications engineering, and photonics technology. Specifically, in near-field- or nano-optics and the design of optical sensor, biosensors or particle surface scanners, light scattering by particles on or near infinite surfaces is of interest.

Many techniques have been developed for analyzing scattering problems. Each of the available methods generally has a range of applicability that is determined by the size of the particle relative to the wavelength of the incident radiation. Classical methods of solution like the finite-di erence method, finite element method or integral equation method, owing to their universality, lead to computational algorithms that are expensive in computer resources. This significantly restricts their use in studying electromagnetic scattering by large particles. In the last years, the null-field method has become an e cient and powerful tool for rigorously computing electromagnetic scattering by single and compounded particles significantly larger than a wavelength. In many applications, it compares favorably to other techniques in terms of e ciency, accuracy, and size parameter range and is the only method that has been used in computations for thousand of particles in random orientation.

VIII Preface

The null-field method (otherwise known as the extended boundary condition method, Schelkuno equivalent current method, Eswald–Oseen extinction theorem and T-matrix method) has been developed by Waterman [253, 254] as a technique for computing electromagnetic scattering by perfectly conducting and dielectric particles. In time, the null-field method has been applied to a wide range of scattering problems. A compilation of T-matrix publications and a classification of various references into a set of narrower subject categories has recently been given by Mishchenko et al. [172]. Peterson and Str¨om [187, 189], Varadan [233], and Str¨om and Zheng [219] extended the null-field method to the case of an arbitrary number of particles and to multilayered and composite particles. Lakhtakia et al. [135] applied the null-field method to chiral particles, while Varadan et al. [236] treated multiple scattering in random media. A number of modifications to the null-field method have been suggested, especially to improve the numerical stability in computations for particles with extreme geometries. These techniques include formal modifications of the single spherical coordinate-based null-field method [25, 109], di erent choices of basis functions and the application of the spheroidal coordinate formalism [12,89] and the use of discrete sources [49]. Mishchenko [163] developed analytical procedures for averaging scattering characteristics over particle orientations and increased the e ciency of the method. At the same time, several computer programs for computing electromagnetic scattering by axisymmetric particles in fixed and random orientations have been designed. In this context, we mention the Fortran programs included with the book by Barber and Hill [8] and the Internet available computer programs developed by Mishchenko et al. [169]. For specific applications, other computer codes have been developed by various research groups, but these programs are currently not yet publicly available.

This monograph is based on our own research activity over the last decade and is intended to provide an exhaustive analysis of the null-field method and to present appropriate computer programs for solving various scattering problems. The following outline should provide a fair idea of the main intent and content of the book.

In the first chapter, we recapitulate the fundamentals of classical electromagnetics and optics which are required to present the theory of the null-field method. This part contains explicit derivations of all important results and is mainly based on the textbooks of Kong [122] and Mishchenko et al. [169].

The next chapter provides a comprehensive analysis of the null-field method for various electromagnetic scattering problems. This includes scattering by

–Homogeneous, dielectric (isotropic, uniaxial anisotropic, chiral), and perfectly conducting particles with axisymmetric and nonaxisymmetric surfaces

–Inhomogeneous, layered and composite particles,

–Clusters of arbitrarily shaped particles, and

–Particles on or near a plane surface.

Preface IX

The null-field method is used to compute the T matrix of each individual particle and the T-matrix formalism is employed to analyze systems of particles. For homogeneous, composite and layered, axisymmetric particles, the null-field method with discrete sources is applied to improve the numerical stability of the conventional method. Evanescent wave scattering and scattering by a half-space with randomly distributed particles are also discussed. To extend the domain of applicability of the method, plane waves and Gaussian laser beams are considered as external excitations.

The last chapter covers the numerical analysis of the null-field method by presenting some exemplary computational results. For all scattering problems discussed in the preceding chapters we developed a Fortran software package which is provided on a CD-ROM with the book. After a description of the Fortran programs we present a number of exemplary computational results with the intension to demonstrate the broad range of applicability of the method. These should enable the readers to adapt and extend the programs to other specific applications and to gain some practical experience with the methods outlined in the book. Because it is hardly possible to comprehensively address all aspects and computational issues, we choose those topics that we think are currently the most interesting applications in the growing field of light scattering theory. As we are continuously working in this field, further extensions of the programs and more computational results will hopefully become available at our web page www.t-matrix.de. The computer programs have been extensively tested, but we cannot guarantee that the programs are free of errors. In this regard, we like to encourage the readers to communicate us any errors in the program or documentation. This software is published under German Copyright Law (Urheberrechtsgesetz, UrhG) and in this regard the readers are granted the right to apply the software but not to copy, to sell or distribute it nor to make it available to the public in any form. We provide the software without warranty of any kind. No liability is taken for any loss or damages, direct or indirect, that may result through the use of the programs.

This volume is intended for engineering and physics students as well as researchers in scattering theory, and therefore we decided to leave out rigorous mathematical details. The properties of scalar and vector spherical wave functions, addition theorems under translation and rotation of the coordinate systems and some completeness results are presented in appendices. These can be regarded as a collection of necessary formulas.

We would like to thank Elena Eremina, Jens Hellmers, Sorin Pulbere, Norbert Riefler, and Roman Schuh for many useful discussions and for performing extensive simulation and validation tests with the programs provided with the book. We also thankfully acknowledge support from DFG (Deutsche Forschungsgemeinschaft) and RFBR (Russian Foundation of Basic Research) which funded our research in light scattering theory.

XPreface

Researching for and writing of this book were both very personal but shared experience. My new research activity in the field of inversion methods for atmospheric remote sensing has left me little free time for writing. Fortunately, I have had assistance of my wife Aniela. She read what I had written, spent countless hours editing the manuscript and helped me in the testing and comparing of computer codes. Without the encouragement and the stimulus given by Aniela, this book might never have been completed. For her love and support, which buoyed me through the darkest time of self-doubt and fear, I sincerely thank.

Contents

1 Basic Theory of Electromagnetic Scattering . . . . . . . . . . . . . . . 1 1.1 Maxwell’s Equations and Constitutive Relations . . . . . . . . . . . . . 1 1.2 Incident Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.1 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.2 Vector Spherical Wave Expansion . . . . . . . . . . . . . . . . . . . 15 1.3 Internal Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.3.1 Anisotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3.2 Chiral Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.4 Scattered Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.4.1 Stratton–Chu Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.4.2 Far-Field Pattern and Amplitude Matrix . . . . . . . . . . . . . 40 1.4.3 Phase and Extinction Matrices . . . . . . . . . . . . . . . . . . . . . . 44 1.4.4 Extinction, Scattering and Absorption Cross-Sections . . 48 1.4.5 Optical Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 1.4.6 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

1.5 Transition Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 1.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 1.5.2 Unitarity and Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 1.5.3 Randomly Oriented Particles . . . . . . . . . . . . . . . . . . . . . . . 66

2 Null-Field Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2.1 Homogeneous and Isotropic Particles . . . . . . . . . . . . . . . . . . . . . . . 84

2.1.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

2.1.2 Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

2.1.3 Symmetries of the Transition Matrix . . . . . . . . . . . . . . . . . 93

2.1.4 Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

2.1.5 Surface Integral Equation Method . . . . . . . . . . . . . . . . . . . 97

2.1.6 Spherical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

2.2 Homogeneous and Chiral Particles . . . . . . . . . . . . . . . . . . . . . . . . . 102

2.3 Homogeneous and Anisotropic Particles . . . . . . . . . . . . . . . . . . . . 104