Light Scattering by Systems of Particles: Null-FieldMethodwithDiscrete Sources: Theory and Programs

Springer Series in

optical sciences 124

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Theodor W. HanschMax-Planck-Institut fur QuantenoptikHans-Kopfermann-Straße 185748 Garching, GermanyE-mail: [email protected]

Takeshi KamiyaMinistry of Education, Culture, SportsScience and TechnologyNational Institution for Academic Degrees3-29-1 Otsuka, Bunkyo-kuTokyo 112-0012, JapanE-mail: [email protected]

Ferenc KrauszLudwig-Maximilians-Universitat MunchenLehrstuhl fur Experimentelle PhysikAm Coulombwall 185748 Garching, GermanyandMax-Planck-Institut fur QuantenoptikHans-Kopfermann-Straße 185748 Garching, GermanyE-mail: [email protected]

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Horst WeberTechnische Universitat BerlinOptisches InstitutStraße des 17. Juni 13510623 Berlin, GermanyE-mail: [email protected]

Harald WeinfurterLudwig-Maximilians-Universitat MunchenSektion PhysikSchellingstraße 4/III80799 Munchen, GermanyE-mail: [email protected]

A. Doicu T. Wriedt Y.A. Eremin

Light Scatteringby Systemsof ParticlesNull-Field Method with Discrete Sources:Theory and Programs

123

With 123 Figures, 4 in Color and 9 Tables

Adrian DoicuInstitut fur Methodik der FernerkundungDeutsches Institut fur Luft- und Raumfahrt e.V.

E-mail: [email protected]

Thomas WriedtUniversity of Bremen, FB4, VT


Yuri A. EreminApplied Mathematics and Computer Science Faculty


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

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Ursula and JannisNatalia, Elena and Oleg

Preface

Since the classic paper by Mie [159] or even the papers by Clebsch [37] andLorenz [146] there is a permanent preoccupation in light scattering theory. Miewas interested in the varied colors exhibited by colloidal suspensions of noblemetal spheres, but nowadays, the theory of light scattering by particles coversa much broader and diverse field. Particles encountered in practical applica-tions are no longer considered spherical; they are nonspherical, nonrotationalsymmetric, inhomogeneous, coated, chiral or anisotropic.

Light scattering simulation is needed in optical particle characterization,to understand new physical phenomena or to design new particle diagnosticssystems. Other examples of applications are climatology and remote sens-ing of Earth and planetary atmospheres, which rely on the analysis of theparameters of radiation scattered by aerosols, clouds, and precipitation. Sim-ilar electromagnetic modeling methods are needed to investigate microwavescattering by raindrops and ice crystals, while electromagnetic scattering isalso encountered in astrophysics, ocean and biological optics, optical com-munications engineering, and photonics technology. Specifically, in near-field-or nano-optics and the design of optical sensor, biosensors or particle surfacescanners, 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 isdetermined by the size of the particle relative to the wavelength of the incidentradiation. Classical methods of solution like the finite-difference method, finiteelement method or integral equation method, owing to their universality, leadto computational algorithms that are expensive in computer resources. Thissignificantly restricts their use in studying electromagnetic scattering by largeparticles. In the last years, the null-field method has become an efficient andpowerful tool for rigorously computing electromagnetic scattering by singleand compounded particles significantly larger than a wavelength. In manyapplications, it compares favorably to other techniques in terms of efficiency,accuracy, and size parameter range and is the only method that has been usedin computations for thousand of particles in random orientation.

VIII Preface

The null-field method (otherwise known as the extended boundary condi-tion method, Schelkunoff equivalent current method, Eswald–Oseen extinctiontheorem and T-matrix method) has been developed by Waterman [253, 254]as a technique for computing electromagnetic scattering by perfectly conduct-ing and dielectric particles. In time, the null-field method has been appliedto a wide range of scattering problems. A compilation of T-matrix publica-tions and a classification of various references into a set of narrower subjectcategories has recently been given by Mishchenko et al. [172]. Peterson andStrom [187, 189], Varadan [233], and Strom and Zheng [219] extended thenull-field method to the case of an arbitrary number of particles and to mul-tilayered and composite particles. Lakhtakia et al. [135] applied the null-fieldmethod to chiral particles, while Varadan et al. [236] treated multiple scatter-ing in random media. A number of modifications to the null-field method havebeen suggested, especially to improve the numerical stability in computationsfor particles with extreme geometries. These techniques include formal mod-ifications of the single spherical coordinate-based null-field method [25, 109],different choices of basis functions and the application of the spheroidal coor-dinate formalism [12,89] and the use of discrete sources [49]. Mishchenko [163]developed analytical procedures for averaging scattering characteristics overparticle orientations and increased the efficiency of the method. At the sametime, several computer programs for computing electromagnetic scattering byaxisymmetric particles in fixed and random orientations have been designed.In this context, we mention the Fortran programs included with the book byBarber and Hill [8] and the Internet available computer programs developedby Mishchenko et al. [169]. For specific applications, other computer codeshave been developed by various research groups, but these programs are cur-rently not yet publicly available.

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

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

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

– Homogeneous, dielectric (isotropic, uniaxial anisotropic, chiral), and per-fectly conducting particles with axisymmetric and nonaxisymmetric sur-faces– 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 individualparticle and the T-matrix formalism is employed to analyze systems of par-ticles. For homogeneous, composite and layered, axisymmetric particles, thenull-field method with discrete sources is applied to improve the numericalstability of the conventional method. Evanescent wave scattering and scatter-ing by a half-space with randomly distributed particles are also discussed. Toextend the domain of applicability of the method, plane waves and Gaussianlaser beams are considered as external excitations.

The last chapter covers the numerical analysis of the null-field method bypresenting some exemplary computational results. For all scattering problemsdiscussed in the preceding chapters we developed a Fortran software packagewhich is provided on a CD-ROM with the book. After a description of theFortran programs we present a number of exemplary computational resultswith the intension to demonstrate the broad range of applicability of themethod. These should enable the readers to adapt and extend the programsto other specific applications and to gain some practical experience with themethods outlined in the book. Because it is hardly possible to comprehensivelyaddress all aspects and computational issues, we choose those topics that wethink are currently the most interesting applications in the growing field oflight scattering theory. As we are continuously working in this field, furtherextensions of the programs and more computational results will hopefullybecome available at our web page www.t-matrix.de. The computer programshave been extensively tested, but we cannot guarantee that the programs arefree of errors. In this regard, we like to encourage the readers to communicateus any errors in the program or documentation. This software is publishedunder German Copyright Law (Urheberrechtsgesetz, UrhG) and in this regardthe readers are granted the right to apply the software but not to copy, to sellor distribute it nor to make it available to the public in any form. We providethe software without warranty of any kind. No liability is taken for any loss ordamages, direct or indirect, that may result through the use of the programs.

This volume is intended for engineering and physics students as well asresearchers in scattering theory, and therefore we decided to leave out rigor-ous mathematical details. The properties of scalar and vector spherical wavefunctions, addition theorems under translation and rotation of the coordinatesystems and some completeness results are presented in appendices. Thesecan 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 per-forming extensive simulation and validation tests with the programs providedwith the book. We also thankfully acknowledge support from DFG (DeutscheForschungsgemeinschaft) and RFBR (Russian Foundation of Basic Research)which funded our research in light scattering theory.

Bremen, Adrian DoicuApril 2006 Thomas Wriedt

Yuri Eremin

X Preface

Researching for and writing of this book were both very personal butshared experience. My new research activity in the field of inversion methodsfor atmospheric remote sensing has left me little free time for writing. Fortu-nately, 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 andcomparing of computer codes. Without the encouragement and the stimulusgiven by Aniela, this book might never have been completed. For her love andsupport, which buoyed me through the darkest time of self-doubt and fear,I sincerely thank.

Munchen, Adrian DoicuApril 2006

Contents

1 Basic Theory of Electromagnetic Scattering . . . . . . . . . . . . . . . 11.1 Maxwell’s Equations and Constitutive Relations . . . . . . . . . . . . . 11.2 Incident Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.1 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.2 Vector Spherical Wave Expansion . . . . . . . . . . . . . . . . . . . 15

1.3 Internal Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.3.1 Anisotropic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.3.2 Chiral Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.4 Scattered Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.4.1 Stratton–Chu Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.4.2 Far-Field Pattern and Amplitude Matrix . . . . . . . . . . . . . 401.4.3 Phase and Extinction Matrices . . . . . . . . . . . . . . . . . . . . . . 441.4.4 Extinction, Scattering and Absorption Cross-Sections . . 481.4.5 Optical Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531.4.6 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

1.5 Transition Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581.5.2 Unitarity and Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 611.5.3 Randomly Oriented Particles . . . . . . . . . . . . . . . . . . . . . . . 66

2 Null-Field Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832.1 Homogeneous and Isotropic Particles . . . . . . . . . . . . . . . . . . . . . . . 84

2.1.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852.1.2 Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892.1.3 Symmetries of the Transition Matrix . . . . . . . . . . . . . . . . . 932.1.4 Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952.1.5 Surface Integral Equation Method . . . . . . . . . . . . . . . . . . . 972.1.6 Spherical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

2.2 Homogeneous and Chiral Particles . . . . . . . . . . . . . . . . . . . . . . . . . 1022.3 Homogeneous and Anisotropic Particles . . . . . . . . . . . . . . . . . . . . 104

XII Contents

2.4 Inhomogeneous Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052.4.1 Formulation with Addition Theorem . . . . . . . . . . . . . . . . . 1062.4.2 Formulation without Addition Theorem . . . . . . . . . . . . . . 112

2.5 Layered Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1152.5.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1152.5.2 Practical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1182.5.3 Formulation with Discrete Sources . . . . . . . . . . . . . . . . . . . 1202.5.4 Concentrically Layered Spheres . . . . . . . . . . . . . . . . . . . . . 122

2.6 Multiple Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1242.6.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1242.6.2 Formulation for a System with N Particles . . . . . . . . . . . 1312.6.3 Superposition T -matrix Method . . . . . . . . . . . . . . . . . . . . . 1322.6.4 Formulation with Phase Shift Terms . . . . . . . . . . . . . . . . . 1362.6.5 Recursive Aggregate T -matrix Algorithm . . . . . . . . . . . . . 137

2.7 Composite Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1392.7.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1392.7.2 Formulation for a Particle with N Constituents . . . . . . . 1432.7.3 Formulation with Discrete Sources . . . . . . . . . . . . . . . . . . . 145

2.8 Complex Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1462.9 Effective Medium Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

2.9.1 T -matrix Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1502.9.2 Generalized Lorentz–Lorenz Law . . . . . . . . . . . . . . . . . . . . 1592.9.3 Generalized Ewald–Oseen Extinction Theorem . . . . . . . . 1612.9.4 Pair Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . 162

2.10 Particle on or near an Infinite Surface . . . . . . . . . . . . . . . . . . . . . . 1642.10.1 Particle on or near a Plane Surface . . . . . . . . . . . . . . . . . . 1642.10.2 Particle on or near an Arbitrary Surface . . . . . . . . . . . . . . 173

3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1833.1 T -matrix Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

3.1.1 Complete Uniform Distribution Function . . . . . . . . . . . . . 1853.1.2 Incomplete Uniform Distribution Function . . . . . . . . . . . . 186

3.2 Electromagnetics Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1883.2.1 T -matrix Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1883.2.2 MMP Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1893.2.3 DDSCAT Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1923.2.4 CST Microwave Studio Program . . . . . . . . . . . . . . . . . . . . 198

3.3 Homogeneous, Axisymmetric and Nonaxisymmetric Particles . . 2013.3.1 Axisymmetric Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2013.3.2 Nonaxisymmetric Particles . . . . . . . . . . . . . . . . . . . . . . . . . 2123.3.3 Triangular Surface Patch Model . . . . . . . . . . . . . . . . . . . . . 216

3.4 Inhomogeneous Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2213.5 Layered Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2253.6 Multiple Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2303.7 Composite Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

Contents XIII

3.8 Complex Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2423.9 Particle on or Near a Plane Surface . . . . . . . . . . . . . . . . . . . . . . . . 2453.10 Effective Medium Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

A Spherical Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253A.1 Spherical Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254A.2 Legendre Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

B Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261B.1 Scalar Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261B.2 Vector Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265B.3 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270B.4 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

C Computational Aspects in Effective Medium Theory . . . . . . . 289C.1 Computation of the Integral I1

mm′n′′ . . . . . . . . . . . . . . . . . . . . . . . 289C.2 Computation of the Integral I2

mm′n′′ . . . . . . . . . . . . . . . . . . . . . . . 292C.3 Computation of the Terms S1

1nn′ and S21nn′ . . . . . . . . . . . . . . . . . 293

D Completeness of Vector Spherical Wave Functions . . . . . . . . . 295

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

1

Basic Theory of Electromagnetic Scattering

This chapter is devoted to present the fundamentals of the electromagneticscattering theory which are relevant in the analysis of the null-field method.We begin with a brief discussion on the physical background of Maxwell’sequations and establish vector spherical wave expansions for the incident field.We then derive new systems of vector functions for internal field approxi-mations by analyzing wave propagation in isotropic, anisotropic and chiralmedia, and present the T -matrix formulation for electromagnetic scattering.We decided to leave out some technical details in the presentation. Therefore,the integral and orthogonality relations, the addition theorems and the basicproperties of the scalar and vector spherical wave functions are reviewed inAppendices A and B.

1.1 Maxwell’s Equations and Constitutive Relations

In this section, we formulate the Maxwell equations that govern the behav-ior of the electromagnetic fields. We present the fundamental laws of elec-tromagnetism, derive the boundary conditions and describe the propertiesof isotropic, anisotropic and chiral media by constitutive relations. Our pre-sentation follows the treatment of Kong [122] and Mishchenko et al. [169].Other excellent textbooks on classical electrodynamics and optics have beengiven by Stratton [215], Tsang et al. [228], Jackson [110], van de Hulst [105],Kerker [115], Bohren and Huffman [17], and Born and Wolf [19].

The behavior of the macroscopic field at interior points in material mediais governed by Maxwell’s equations:

∇× E = −∂B

∂t(Faraday’s induction law) , (1.1)

∇× H = J +∂D

∂t(Maxwell–Ampere law) , (1.2)

∇ · D = ρ (Gauss’ electric field law) , (1.3)

2 1 Basic Theory of Electromagnetic Scattering

∇ · B = 0 (Gauss’ magnetic field law) , (1.4)

where t is time, E the electric field, H the magnetic field, B the magneticinduction, D the electric displacement and ρ and J the electric charge densityand current density, respectively. The first three equations in Maxwell’s theoryare independent, because the Gauss magnetic field law can be obtained fromFaraday’s law by taking the divergence and by setting the integration constantwith respect to time equal to zero. Analogously, taking the divergence ofMaxwell–Ampere law and using the Gauss electric field law we obtain thecontinuity equation:

∇ · J +∂ρ

∂t= 0 , (1.5)

which expresses the conservation of electric charge. The Gauss magnetic fieldlaw and the continuity equation should be treated as auxiliary or dependentequations in the entire system of equations (1.1)–(1.5). The charge and currentdensities are associated with the so-called “free” charges, and for a source-free medium, J = 0 and ρ = 0. In this case, the Gauss electric field law canbe obtained from Maxwell–Ampere law and only the first two equations inMaxwell’s theory are independent.

In our analysis we will assume that all fields and sources are time harmonic.With ω being the angular frequency and j =

√−1, we write

E(r, t) = ReE(r)e−jωt

and similarly for other field quantities. The vector field E(r) in the frequencydomain is a complex quantity, while E(r, t) in the time domain is real. As a re-sult of the Fourier component Ansatz, the Maxwell equations in the frequencydomain become

∇× E = jωB ,

∇× H = J − jωD ,

∇ · D = ρ ,

∇ · B = 0 .

Taking into account the continuity equation in the frequency domain ∇ · J−jωρ = 0, we may express the Maxwell–Ampere law and the Gauss electricfield law as

∇× H = −jωDt ,

∇ · Dt = 0 ,

where

Dt = D +jω

J

is the total electric displacement.

1.1 Maxwell’s Equations and Constitutive Relations 3

Across the interface separating two different media the fields may be dis-continuous and a boundary condition is associated with each of Maxwell’sequations. To derive the boundary conditions, we consider a regular domainD enclosed by a surface S with outward normal unit vector n, and use thecurl theorem

∫

D

∇× a dV =∫

S

n × a dS ,

to obtain∫

S

n × E dS = jω∫

D

B dV ,

∫

S

n × H dS =∫

D

J dV − jω∫

D

D dV ,

and the Gauss theorem∫

D

∇ · a dV =∫

S

n · a dS ,

to derive∫

S

n · D dS =∫

D

ρdV ,

∫

S

n · B dS = 0 .

Note that the curl theorem follows from Gauss theorem applied to the vectorfield c × a, where c is a constant vector, and the identity ∇ · (c × a) =−c ·(∇× a). We then consider a surface boundary joining two different media1 and 2, denote by n1 the surface normal pointing toward medium 2, andassume that the surface of discontinuity is contained in D. We choose thedomain of analysis in the form of a thin slab with thickness h and area ∆S,and let the volume approach zero by letting h go to zero and then letting∆S go to zero (Fig. 1.1). Terms involving vector or dot product by n willbe dropped except when n is in the direction of n1 or −n1. Assuming thatD and B are finite in the region of integration, and that the boundary maysupport a surface current J s such that J s = limh→0 hJ , and a surface charge

∆

n1

Medium 1

Medium 2

S

h

Fig. 1.1. The surface of discontinuity and a thin slab of thickness h and area ∆S


density ρs such that ρs = limh→0 hρ, we see that the tangential component ofE is continuous:

n1 × (E2 − E1) = 0 ,

the tangential component of H is discontinuous:

n1 × (H2 − H1) = J s ,

the normal component of B is continuous:

n1 · (B2 − B1) = 0 ,

and the normal component of D is discontinuous:

n1 · (D2 − D1) = ρs .

Energy conservation follows from Maxwell’s equations. The vector identity

∇ · (a × b) = b · (∇× a) − a · (∇× b)

yields the Poynting theorem in the time domain:

∇ · (E × H) + H · ∂B

∂t+ E · ∂D

∂t= −E · J ,

and the Poynting vector defined as

S = E × H

is interpreted as the power flow density. Integrating over a finite domain Dwith boundary S, and using the Gauss theorem, yields

−∫

D

E · J dV =∫

S

S · n dS +∫

D

(H · ∂B

∂t+ E · ∂D

∂t

)dV ,

where as before, n is the outward normal unit vector to the surface S. Theabove equation states that the power supplied by the sources within a volumeis equal to the sum of the increase in electromagnetic energy and the Poynt-ing’s power flowing out through the volume boundary. Poynting’s theoremcan also be derived in the frequency domain:

∇ · (E × H∗) = jω (B · H∗ − E · D∗) − E · J∗ ,

where the asterisk denotes a complex-conjugate value. The complex Poyntingvector is defined as S = E × H∗ and the term −

∫D

E · J∗ dV is interpretedas the complex power supplied by the source.

In practice, the angular frequency ω is such high that a measuring instru-ment is not capable of following the rapid oscillations of the power flow but


rather responds to a time average power flow. Considering the time-harmonicvector fields a and b,

a(r, t) =12[a(r)e−jωt + a∗(r)ejωt

],

b(r, t) =12[b(r)e−jωt + b∗(r)ejωt

],

we express the dot product of the vectors as

c(r, t) = a(r, t) · b(r, t)

=12Rea(r) · b∗(r) + a(r) · b(r)e−2jωt

.

Defining the time average of c as

〈c(r)〉 = limT→∞

1T

∫ T

0

c(r, t) dt ,

where T is a time interval, we derive

〈c(r)〉 =12Re a(r) · b∗(r) ,

while for the cross product of the vectors

c(r, t) = a(r, t) × b(r, t) ,

we similarly obtain

〈c(r)〉 =12Re a(r) × b∗(r) .

Thus, the time average of the dot or cross product of two time-harmoniccomplex quantities is equal to half of the real part of the respective productof one quantity and the complex conjugate of the other. In this regard, thetime-averaged Poynting vector is given by

〈S〉 =12Re E × H∗ .

The three independent vector equations (1.1)–(1.3) are equivalent to sevenscalar differential equations, while the number of unknown scalar functionsis 16. Obviously, the three independent equations are not sufficient to forma complete systems of equations to solve for the unknown functions, and forthis reason, the equations given by (1.1)–(1.4) are known as the indefiniteform of the Maxwell equations. Note that for a free-source medium, we havesix scalar differential equations with 12 unknown scalar functions. To makethe Maxwell equations definite we need more information and this additional


information is given by the constitutive relations. The constitutive relationsprovide a description of media and give functional dependence among vectorfields. For isotropic media, the constitutive relations read as

D = εE ,

B = µH ,

J = σE (Ohm’s law) , (1.6)

where ε is the electric permittivity, µ is the magnetic permeability and σ isthe electric conductivity. The above equations provide nine scalar relationsthat make the number of unknowns and the number of equations compatible,while for a source-free medium, the first two constitutive relations guaranteethis compatibility. When the constitutive relations between the vector fieldsare specified, Maxwell equations become definite. In free space ε0 = 8.85 ×10−12 F m−1 and µ0 = 4π × 10−7 Hm−1, while in a material medium, thepermittivity and permeability are determined by the electrical and magneticproperties of the medium. A dielectric material can be characterized by afree-space part and a part depending on the polarization vector P such that

D = ε0E + P .

The polarization P symbolizes the average electric dipole moment per unitvolume and is given by

P = ε0χeE ,

where χe is the electric susceptibility. A magnetic material can also be char-acterized by a free-space part and a part depending on the magnetizationvector M ,

B = µ0H+µ0M ,

where M symbolizes the average magnetic dipole moment per unit volume,

M = χmH ,

and χm is the magnetic susceptibility. A medium is diamagnetic if µ < µ0 andparamagnetic if µ > µ0, while for a nonmagnetic medium we have µ = µ0.The permittivity and permeability of isotropic media can be written as

ε = ε0εr = ε0 (1 + χe) ,

µ = µ0µr = µ0 (1 + χm) ,

where εr and µr stand for the corresponding relative quantities. The consti-tutive relation for the total electric displacement is

Dt = εtE ,


where the complex permittivity εt is given by

εt = ε0εrt = ε0

(1 + χe +

jσωε0

)

with εrt being the complex relative permittivity. Both the conductivityand the susceptibility contribute to the imaginary part of the permittivity,Imεt = ε0Imχe + Reσ/ω, and a complex value for εt means that themedium is absorbing. Usually, Imχe is associated with the “bound” chargecurrent density and Reσ/ω with the “free” charge current density, and ab-sorption is determined by the sum of these two quantities. Note that for afree-source medium, σ = 0 and εrt = εr = 1 + χe. The simplest solution toMaxwell’s equations in source-free media is the vector plane wave solution.The behavior of a vector plane wave in an isotropic medium is characterizedby the dispersion relation

k = ω√

εµ ,

which relates the wave number k to the properties of the medium and to theangular frequency ω of the wave. The dimensionless quantity

m = c√

εµ

is the refractive index of the medium, where c = 1/√

ε0µ0 is the speed of lightin vacuum, and if k0 = ω

√ε0µ0 is the wave number in free space, we see that

m =k

k0.

The constitutive relations for anisotropic media are

D = εE ,

B = µH , (1.7)

where ε and µ are the permittivity and permeability tensors, respectively. Inour analysis we will consider electrically anisotropic media for which the per-mittivity is a tensor and the permeability is a scalar. Except for amorphousmaterials and crystals with cubic symmetry, the permittivity is always a ten-sor, and in general, the permittivity tensor of a crystal is symmetric. Sincethere exists a coordinate transformation that transforms a symmetric matrixinto a diagonal matrix, we can take this coordinate system as reference frameand we have

ε =

⎡

⎢⎣

εx 0 00 εy 00 0 εz

⎤

⎥⎦ . (1.8)

This reference frame is called the principal coordinate system and the threecoordinate axes are known as the principal axes of the crystal. If εx = εy = εz,


the medium is biaxial, and if ε = εx = εy and ε = εz, the medium is uniaxial.Orthorhombic, monoclinic and triclinic crystals are biaxial, while tetragonal,hexagonal and rhombohedral crystals are uniaxial. For uniaxial crystals, theprincipal axis that exhibits the anisotropy is called the optic axis. The crystalis positive uniaxial if εz > ε and negative uniaxial if εz < ε.

In our analysis, we will investigate the electromagnetic response of isotropic,chiral media exposed to arbitrary external excitations. The lack of geometricsymmetry between a particle and its mirror image is referred to as chiralityor optical activity. A chiral medium is characterized by either a left- or aright-handedness in its microstructure, and as a result, left- and right-handcircularly polarized fields propagate through it with differing phase veloc-ities. For a source-free, isotropic, chiral medium, the constitutive relationsread as

D = εE + βε∇× E ,

B = µH + βµ∇× H ,

where the real number β is known as the chirality parameter. The Maxwellequations can be written compactly in matrix form as

∇×[

EH

]= K

[EH

], ∇ ·

[EH

]= 0 , (1.9)

where

K =1

1 − β2k2

[βk2 jωµ

−jωε βk2

]

and k = ω√

εµ.Without loss of generality and so as to simplify our notations we make the

following transformations:

E → 1√ε0

E,H → 1√

µ0H ,

D → √ε0D,B → √

µ0B .

As a result, the Maxwell equations for a free-source medium become more“symmetric”:

∇× E = jk0B ,

∇× H = −jk0D ,

∇ · D = 0 ,

∇ · B = 0 , (1.10)

the constitutive relations are given by (1.6) and (1.7) with ε and µ beingthe relative permittivity and permeability, respectively, the wave number isk = k0

√εµ, and the K matrix in (1.9) takes the form

K =1

1 − β2k2

[βk2 jk0µ

−jk0ε βk2

]. (1.11)

1.2 Incident Field 9

1.2 Incident Field

In this section, we characterize the polarization state of vector plane wavesand derive vector spherical wave expansions for the incident field. The firsttopic is relevant in the analysis of the scattered field, while the second oneplays an important role in the derivation of the transition matrix.

1.2.1 Polarization

In addition to intensity and frequency, a monochromatic (time harmonic)electromagnetic wave is characterized by its state of polarization. This conceptis useful when we discuss the polarization of the scattered field since thepolarization state of a beam is changed on interaction with a particle.

We consider a right-handed Cartesian coordinate system OXY Z with afixed spatial orientation. This reference frame will be referred to as the globalcoordinate system or the laboratory coordinate system. The direction of prop-agation of the vector plane wave is specified by the unit vector ek, or equiva-lently, by the zenith and azimuth angles β and α, respectively (Fig. 1.2). Thepolarization state of the incident wave will be described in terms of the verticalpolarization unit vector eα = ez×ek/|ez×ek| and the horizontal polarizationvector eβ = eα × ek. Note that other names for vertical polarization are TMpolarization, parallel polarization and p polarization, while other names forhorizontal polarization are TE polarization, perpendicular polarization and spolarization.

eβ

e

ek

x

yO

z

ke

β

α

epol

α

α

pol

Fig. 1.2. Wave vector in the global coordinate system


In the frequency domain, a vector plane wave propagating in a mediumwith constant wave number ks = k0

√εsµs is given by

Ee(r) = Ee0ejke·r , Ee0 · ek = 0 , (1.12)

where k0 is the wave number in free space, ke is the wave vector, ke = ksek,Ee0 is the complex amplitude vector,

Ee0 = Ee0,βeβ + Ee0,αeα ,

and Ee0,β and Ee0,α are the complex amplitudes in the β- and α-direction,respectively. An equivalent representation for Ee0 is

Ee0 = |Ee0| epol , (1.13)

where epol is the complex polarization unit vector, |epol| = 1, and

epol =1

|Ee0|(Ee0,βeβ + Ee0,αeα) .

Inserting (1.13) into (1.12), gives the representation

Ee(r) = |Ee0| epolejke·r , epol · ek = 0 ,

and obviously, |Ee(r)| = |Ee0|.There are three ways of describing the polarization state of vector plane

waves.

1. Setting

Ee0,β = aβejδβ ,

Ee0,α = aαejδα , (1.14)

where aβ and aα are the real non-negative amplitudes, and δβ and δα are thereal phases, we characterize the polarization state of a vector plane wave byaβ , aα and the phase difference ∆δ = δβ − δα.

2. Taking into account the representation of a vector plane wave in the timedomain

Ee(r, t) = ReEe(r)e−jωt

= Re

Ee0ej(ke·r−ωt)

,

where Ee(r, t) is the real electric vector, we deduce that (cf. (1.14))

Ee,β(r, t) = aβ cos (δβ + ke · r − ωt) ,

Ee,α(r, t) = aα cos (δα + ke · r − ωt) ,

where

Ee(r, t) = Ee,β(r, t)eβ + Ee,α(r, t)eα .


At any fixed point in space the endpoint of the real electric vector describesan ellipse which is also known as the vibration ellipse [17]. The vibrationellipse can be traced out in two opposite senses: clockwise and anticlock-wise. If the real electric vector rotates clockwise, as viewed by an observerlooking in the direction of propagation, the polarization of the ellipse isright-handed and the polarization is left-handed if the electric vector rotatesanticlockwise. The two opposite senses of rotation lead to a classification ofvibration ellipses according to their handedness. In addition to its handed-ness, a vibration ellipse is characterized by E0 =

√a2 + b2, where a and

b are the semi-major and semi-minor axes of the ellipse, the orientationangle ψ and the ellipticity angle χ (Fig. 1.3). The orientation angle ψ isthe angle between the α-axis and the major axis, and ψ ∈ [0, π). The ellipticityangle χ is usually expressed as tan χ = ±b/a, where the plus sign correspondsto right-handed elliptical polarization, and χ ∈ [−π/4, π/4].

We now proceed to relate the complex amplitudes Ee0,β and Ee0,α tothe ellipsometric parameters E0, ψ and χ. Representing the semi-axes of thevibration ellipse as

b = ±E0 sin χ ,

a = E0 cos χ , (1.15)

where the plus sign corresponds to right-handed polarization, and taking intoaccount the parametric representation of the ellipse in the principal coordinatesystem Oα′β′

E′e,β(r, t) = ±b sin (ke · r − ωt) ,

E′e,α(r, t) = a cos (ke · r − ωt) ,

O

α

α

ββ

ab ψ

χ

Fig. 1.3. Vibration ellipse


we obtain

E′e,β(r, t) = E0 sinχ sin (ke · r − ωt) = E0 sin χ cos

(ke · r − ωt − π

2

),

E′e,α(r, t) = E0 cos χ cos (ke · r − ωt) .

In the frequency domain, the complex amplitude vector E′e0 defined as

E′e(r, t) = Re

E′

e0ej(ke·r−ωt)

,

where

E′e(r, t) = E′

e,β(r, t)e′β + E′

e,α(r, t)e′α ,

has the components

E′e0,β = −jE0 sinχ ,

E′e0,α = E0 cos χ .

Using the transformation rule for rotation of a two-dimensional coordinatesystem we obtain the desired relations

Ee0,β = E0 (cos χ sinψ − j sin χ cos ψ) ,

Ee0,α = E0 (cos χ cos ψ + j sinχ sin ψ) ,

and

epol = (cos χ sin ψ − j sin χ cos ψ) eβ

+ (cos χ cos ψ + j sinχ sin ψ)eα . (1.16)

If b = 0, the ellipse degenerates into a straight line and the wave is linearlypolarized. In this specific case χ = 0 and

Ee0,β = E0 sin ψ = E0 cos(π

2− ψ)

= E0 cos αpol ,

Ee0,α = E0 cos ψ = E0 sin(π

2− ψ)

= E0 sinαpol ,

where αpol is the polarization angle and

αpol = π/2 − ψ , αpol ∈ (−π/2, π/2] . (1.17)

In view of (1.16) and (1.17) it is apparent that the polarization unit vector isreal and is given by

epol = cos αpoleβ + sin αpoleα . (1.18)


If a = b, the ellipse is a circle and the wave is circularly polarized. We havetan χ = ±1, which implies χ = ±π/4, and choosing ψ = π/2, we obtain

Ee0,β =√

22

E0 ,

Ee0,α = ±j√

22

E0 .

The polarization unit vectors of right- and left-circularly polarized waves thenbecome

eR =√

22

(eβ + jeα) ,

eL =√

22

(eβ − jeα) ,

and we see that these basis vectors are orthonormal in the sense thateR · e∗

R = 1, eL · e∗L = 1 and eR · e∗

L = 0.

3. The polarization characteristics of the incident field can also be describedby the coherency and Stokes vectors. Although the ellipsometric parameterscompletely specify the polarization state of a monochromatic wave, they aredifficult to measure directly (with the exception of the intensity E2

0). In con-trast, the Stokes parameters are measurable quantities and are of greaterusefulness in scattering problems. The coherency vector is defined as

Je =12

√εs

µs

⎡

⎢⎢⎢⎣

Ee0,βE∗e0,β

Ee0,βE∗e0,α

Ee0,αE∗e0,β

Ee0,αE∗e0,α

⎤

⎥⎥⎥⎦

, (1.19)

while the Stokes vector is given by

Ie =

⎡

⎢⎢⎢⎣

Ie

Qe

Ue

Ve

⎤

⎥⎥⎥⎦

= DJe =12

√εs

µs

⎡

⎢⎢⎢⎢⎣

|Ee0,β |2 + |Ee0,α|2

|Ee0,β |2 − |Ee0,α|2

−Ee0,αE∗e0,β − Ee0,βE∗

e0,α

j(Ee0,αE∗

e0,β − Ee0,βE∗e0,α

)

⎤

⎥⎥⎥⎥⎦

, (1.20)

where D is a transformation matrix and

D =

⎡

⎢⎢⎣

1 0 0 11 0 0 −10 −1 −1 00 −j j 0

⎤

⎥⎥⎦ . (1.21)


The first Stokes parameter Ie,

Ie =12

√εs

µs|Ee0|2

is the intensity of the wave, while the Stokes parameters Qe, Ue and Ve describethe polarization state of the wave. The Stokes parameters are defined withrespect to a reference plane containing the direction of wave propagation, andQe and Ue depend on the choice of the reference frame. If the unit vectors eβ

and eα are rotated through an angle ϕ (Fig. 1.4), the transformation from theStokes vector Ie to the Stokes vector I ′

e (relative to the rotated unit vectorse′

β and e′α) is given by

I ′e = L (ϕ) Ie , (1.22)

where the Stokes rotation matrix L is

L (ϕ) =

⎡

⎢⎢⎣

1 0 0 00 cos 2ϕ − sin 2ϕ 00 sin 2ϕ cos 2ϕ 00 0 0 1

⎤

⎥⎥⎦ . (1.23)

The Stokes parameters can be expressed in terms of aβ , aα and ∆δ as (omit-ting the factor 1

2

√εs/µs)

Ie = a2β + a2

α ,

Qe = a2β − a2

α ,

Ue = −2aβaα cos ∆δ ,

Ve = 2aβaα sin ∆δ ,

eβ

eα

ϕ

eβ

eα

O

Fig. 1.4. Rotation of the polarization unit vectors through the angle ϕ


and in terms of E0, ψ and χ, as

Ie = E20 ,

Qe = −E20 cos 2χ cos 2ψ ,

Ue = −E20 cos 2χ sin 2ψ ,

Ve = −E20 sin 2χ .

The above relations show that the Stokes parameters carry information aboutthe amplitudes and the phase difference, and are operationally defined in termsof measurable quantities (intensities). For a linearly polarized plane wave,χ = 0 and Ve = 0, while for a circularly polarized plane wave, χ = ±π/4and Qe = Ue = 0. Thus, the Stokes vector of a linearly polarized wave ofunit amplitude is given by Ie = [1, cos 2αpol,− sin 2αpol, 0]T, while the Stokesvector of a circularly polarized wave of unit amplitude is Ie = [1, 0, 0,∓1]T.

The Stokes parameters of a monochromatic plane wave are not indepen-dent since

I2e = Q2

e + U2e + V 2

e , (1.24)

and we may conclude that only three parameters are required to characterizethe state of polarization. For quasi-monochromatic light, the amplitude of theelectric field fluctuate in time and the Stokes parameters are expressed interms of the time-averaged quantities 〈Ee0,pE

∗e0,q〉, where p and q stand for β

and α. In this case, the equality in (1.24) is replaced by the inequality

I2e ≥ Q2

e + U2e + V 2

e ,

and the quantity

P =

√Q2

e + U2e + V 2

e

Ie

is known as the degree of polarization of the quasi-monochromatic beam. Fornatural (unpolarized) light, P = 0, while for fully polarized light P = 1. TheStokes vector defined by (1.20) is one possible representation of polarization.Other representations are discussed by Hovenier and van der Mee [101], while adetailed discussion of the polarimetric definitions can be found in [17,169,171].

1.2.2 Vector Spherical Wave Expansion

The derivation of the transition matrix in the framework of the null-fieldmethod requires the expansion of the incident field in terms of (localized)vector spherical wave functions. This expansion must be provided in the parti-cle coordinate system, where in general, the particle coordinate system Oxyzis obtained by rotating the global coordinate system OXY Z through theEuler angles αp, βp and γp (Fig. 1.5). In our analysis, vector plane waves andGaussian beams are considered as external excitations.


y

2

Y

Z, z

O

1

βp

αppγ 1

2

y1, y2

x

X x

x

z, z

Fig. 1.5. Euler angles αp, βp and γp specifying the orientation of the particlecoordinate system Oxyz with respect to the global coordinate system OXY Z. Thetransformation OXY Z → Oxyz is achieved by means of three successive rotations:(1) rotation about the Z-axis through αp, OXY Z → Ox1y1z1, (2) rotation aboutthe y1-axis through βp, Ox1y1z1 → Ox2y2z2 and (3) rotation about the z2-axisthrough γp, Ox2y2z2 → Oxyz

Vector Plane Wave

We consider a vector plane wave of unit amplitude propagating in the direction(βg, αg) with respect to the global coordinate system. Passing from sphericalcoordinates to Cartesian coordinates and using the transformation rules undercoordinate rotations we may compute the spherical angles β and α of thewave vector in the particle coordinate system. Thus, in the particle coordinatesystem we have the representation

Ee(r) = epolejke·r , epol · ek = 0 ,

where as before, ke = ksek.The vector spherical waves expansion of the incident field reads as

Ee(r) =∞∑

n=1

n∑

m=−n

amnM1mn (ksr) + bmnN1

mn (ksr) , (1.25)

where the expansion coefficients are given by [9,228]

amn = 4jnepol · m∗mn(β, α)

= − 4jn√

2n(n + 1)epol ·

[jmπ|m|

n (β)eβ + τ |m|n (β)eα

]e−jmα ,


bmn = −4jn+1epol · n∗mn(β, α)

= − 4jn+1

√2n(n + 1)

epol ·[τ |m|n (β)eβ − jmπ|m|

n (β)eα

]e−jmα . (1.26)

To give a justification of the above expansion we consider the integral repre-sentation

epol(β, α)ejke(β,α)·r =∫ 2π

0

∫ π

0

epol(β, α)ejk(β′,α′)·r

×δ(α′ − α)δ(cos β′ − cos β) sin β′ dβ′ dα′ , (1.27)

and expand the tangential field

f (β, α, β′, α′) = epol(β, α)δ(α′ − α)δ(cos β′ − cos β) (1.28)

in vector spherical harmonics

f (β, α, β′, α′) =1

4πjn

∞∑

n=1

n∑

m=−n

amnmmn (β′, α′)+jbmnnmn (β′, α′) . (1.29)

Using the orthogonality relations of vector spherical harmonics we see thatthe expansion coefficients amn and bmn are given by

amn = 4jn∫ 2π

0

∫ π

0

f (β, α, β′, α′) · m∗mn (β′, α′) sin β′ dβ′ dα′

= 4jnepol(β, α) · m∗mn(β, α) ,

bmn = −4jn+1

∫ 2π

0

∫ π

0

f (β, α, β′, α′) · n∗mn (β′, α′) sin β′ dβ′ dα′

= −4jn+1epol(β, α) · n∗mn(β, α) .

Substituting (1.28) and (1.29) into (1.27) and taking into account the integralrepresentations for the regular vector spherical wave functions (cf. (B.26) and(B.27)) yields (1.25).

The polarization unit vector of a linearly polarized vector plane wave isgiven by (1.18). If the vector plane wave propagates along the z-axis we haveβ = α = 0 and for β = 0, the spherical vector harmonics are zero unless m =±1. Using the special values of the angular functions π1

n and τ1n when β = 0,

π1n (0) = τ1

n (0) =1

2√

2

√n(n + 1)(2n + 1) ,

we obtain

a±1n = −jn√

2n + 1 (±j cos αpol + sin αpol) ,

b±1n = −jn+1√

2n + 1 (cos αpol ∓ j sin αpol) .


Thus, for a vector plane wave polarized along the x-axis we have

a1n = −a−1n = jn−1√

2n + 1 ,

b1n = b−1n = jn−1√

2n + 1 ,

while for a vector plane wave polarized along the y-axis we have

a1n = a−1n = jn−2√

2n + 1 ,

b1n = −b−1n = jn−2√

2n + 1 .

Gaussian Beam

Many optical particle sizing instruments and particle characterization meth-ods are based on scattering by particles illuminated with laser beams.A laser beam has a Gaussian intensity distribution and the often usedappellation Gaussian beam appears justified. A mathematical description ofa Gaussian beam relies on Davis approximations [45]. An nth Davis beamcorresponds to the first n terms in the series expansion of the exact solutionto the Maxwell equations in power of the beam parameter s,

s =w0

l,

where w0 is the waist radius and l is the diffraction length, l = ksw20. According

to Barton and Alexander [9], the first-order approximation is accurate tos < 0.07, while the fifth-order is accurate to s < 0.02, if the maximum percenterror of the solution is less than 1.2%. Each nth Davis beam appears underthree versions which are: the mathematical conservative version, the L-versionand the symmetrized version [145]. None of these beams are exact solutionsto the Maxwell equations, so that each nth Davis beam can be considered asa “pseudo-electromagnetic” field.

In the T -matrix method a Gaussian beam is expanded in terms of vectorspherical wave functions by replacing the pseudo-electromagnetic field of annth Davis beam by an equivalent electromagnetic field, so that both fieldshave the same values on a spherical surface [81, 83, 85]. As a consequenceof the equivalence method, the expansion coefficients (or the beam shapecoefficients) are computed by integrating the incident field over the spher-ical surface. Because these fields are rapidly varying, the evaluation of thecoefficients by numerical integration requires dense grids in both the θ- andϕ-direction and the computer run time is excessively long.

For weakly focused Gaussian beams, the generalized localized approxima-tion to the beam shape coefficients represents a pleasing alternative (see, forinstance, [84, 87]). The form of the analytical approximation was found inpart by analogy to the propagation of geometrical light rays and in part bynumerical experiments. This is not a rigorous method but its use simplifiesand significantly speeds up the numerical computations. A justification of


the localized approximations for both on- and off-axis beams has been givenby Lock and Gouesbet [145] and Gouesbet and Lock [82]. We note that thefocused beam generated by the localized approximation is a good approxima-tion to a Gaussian beam for s ≤ 0.1.

We consider the geometry depicted in Fig. 1.6 and assume that the middleof the beam waist is located at the point Ob. The particle coordinate systemOxyz and the beam coordinate system Obxbybzb have identical spatial orien-tation, and the position vector of the particle center O in the system Obxbybzb

is denoted by r0. The Gaussian beam is of unit amplitude, propagates alongthe zb-axis and is linearly polarized along the xb-axis. In the particle coor-dinate system, the expansion of the Gaussian beam in vector spherical wavefunctions is given by

Ee(r) =∞∑

n=1

n∑

m=−n


mn (ksr) ,

and the generalized localized approximation to the Davis first-order beam is

amn = KmnΨ0ejksz0

[ej(m−1)ϕ0Jm−1 (u) − ej(m+1)ϕ0Jm+1 (u)

],

bmn = KmnΨ0ejksz0

[ej(m−1)ϕ0Jm−1 (u) + ej(m+1)ϕ0Jm+1 (u)

],

x b

by bO

O

zb

r0

ϕ0

z0

ρ0

y

x

z

2w0

Fig. 1.6. The particle coordinate system Oxyz and the beam coordinate systemOxbybzb have the same spatial orientation


where (ρ0, ϕ0, z0) are the cylindrical coordinates of r0,

Ψ0 = jQ exp(−jQ

ρ20 + ρ2

n

w20

), Q =

1j − 2z0/l

, ρn =1ks

(n +

12

),

and

u = 2Qρ0ρn

w20

.

The normalization constant Kmn is given by

Kmn = 2jn√

n(n + 1)2n + 1

for m = 0, and by

Kmn = (−1)|m| jn+|m|(n + 1

2

)|m|−1

√2n + 1

n(n + 1)· (n + |m|)!(n − |m|)!

for m = 0. If both coordinate systems coincide (ρ0 = 0), all expansion coeffi-cients are zero unless m = ±1 and

a1n = −a−1n = jn−1√

2n + 1 exp(− ρ2

n

w20

),

b1n = b−1n = jn−1√

2n + 1 exp(− ρ2

n

w20

).

The Gaussian beam becomes a plane wave if w0 tends to infinity and for thisspecific case, the expressions of the expansion coefficients reduce to those ofa vector plane wave.

We next consider the general situation depicted in Fig. 1.7 and assumethat the auxiliary coordinate system ObXbYbZb and the global coordinatesystem OXY Z have the same spatial orientation. The Gaussian beam propa-gates in a direction characterized by the zenith and azimuth angles βg and αg,respectively, while the polarization unit vector encloses the angle αpol withthe xb-axis of the beam coordinate system Obxbybzb. As before, the parti-cle coordinate system is obtained by rotating the global coordinate systemthrough the Euler angles αp, βp and γp. The expansion of the Gaussian beamin the particle coordinate system is obtained by using the addition theoremfor vector spherical wave functions under coordinate rotations (cf. (B.52) and(B.53)), and the result is

Ee(r) =∞∑

n=1

n∑

m=−n


mn (ksr) ,

1.3 Internal Field 21

xb

yb

O b

zb

y

x

z

O

X

Y

Z

Yb

Xb

Zb

βp

βg

r0

Fig. 1.7. General orientations of the particle and beam coordinate systems

where

amn =n∑

m′=−n

n∑

m′′=−n

Dnm′m′′ (−αpol,−βg,−αg) Dn

m′′m (αp, βp, γp) am′n ,

bmn =n∑

m′=−n

n∑

m′′=−n

Dnm′m′′ (−αpol,−βg,−αg) Dn

m′′m (αp, βp, γp) bm′n ,

and the Wigner D-functions Dnm′′m are given by (B.34).

Remark. Another representation of a Gaussian beam is the integral repre-sentation over plane waves [48, 116]. This can be obtained by using Fourieranalysis and by replacing the pseudo-vector potential of a nth Davis beamby an equivalent vector potential (satisfying the wave equation), so that bothvector fields have the same values in a plane z = const.

1.3 Internal Field

To solve the scattering problem in the framework of the null-field method itis necessary to approximate the internal field by a suitable system of vectorfunctions. For isotropic particles, regular vector spherical wave functions ofthe interior wave equation are used for internal field approximations. In thissection we derive new systems of vector functions for anisotropic and chiralparticles by representing the electromagnetic fields (propagating in anisotropic


and chiral media) as integrals over plane waves. For each plane wave, we solvethe Maxwell equations and derive the dispersion relation following the treat-ment of Kong [122]. The dispersion relation which relates the amplitude ofthe wave vector k to the properties of the medium enable us to reduce thethree-dimensional integrals to two-dimensional integrals over the unit sphere.The integral representations are then transformed into series representationsby expanding appropriate tangential vector functions in vector spherical har-monics. The new basis functions are the vector quasi-spherical wave functions(for anisotropic media) and the vector spherical wave functions of left- andright-handed type (for chiral media).

1.3.1 Anisotropic Media

Maxwell equations describing electromagnetic wave propagation in a source-free, electrically anisotropic medium are given by (1.10), while the constitutiverelations are given by (1.7) with the scalar permeability µ in place of thepermeability tensor µ. In the principal coordinate system, the first constitutiverelation can be written as

E = λD ,

where the impermittivity tensor λ is given by

λ =

⎡

⎣λx 0 00 λy 00 0 λz

⎤

⎦ ,

and λx = 1/εx, λy = 1/εy and λz = 1/εz.The electromagnetic fields can be expressed as integrals over plane waves

by considering the inverse Fourier transform (excepting the factor 1/(2π)3):

A(r) =∫

A (k) ejk·r dV (k) ,

where A stands for E, D, H and B, and A stands for the Fourier transformsE , D, H and B. Using the identities

∇× A(r) = j∫

k ×A (k) ejk·r dV (k) ,

∇ · A(r) = j∫

k · A (k) ejk·r dV (k) ,

we see that the Maxwell equations for the Fourier transforms take the forms

k × E = k0B , k ×H = −k0D ,

k · D = 0 , k · B = 0 ,


and the plane wave solutions read as

Eβ =k0

kBα , Eα = −k0

kBβ ,

Hβ = −k0

kDα , Hα =

k0

kDβ ,

Dk = 0 , Bk = 0 , (1.30)

where (k, β, α) and (ek,eβ ,eα) are the spherical coordinates and the sphericalunit vectors of the wave vector k, respectively, and in general, (Ak,Aβ ,Aα)are the spherical coordinates of the vector A. The constitutive relations forthe transformed fields E = λD and H = (1/µ)B can be written in sphericalcoordinates by using the transformation

⎡

⎣Ax

Ay

Az

⎤

⎦ =

⎡

⎣cos α sin β cos α cos β − sin αsin α sinβ sin α cos β cos α

cos β − sin β 0

⎤

⎦

⎡

⎣Ak

Aβ

Aα

⎤

⎦ ,

and the result is

Ek = λkβDβ + λkαDα ,

Eβ = λββDβ + λβαDα ,

Eα = λαβDβ + λααDα , (1.31)

andHk = 0 , Hβ =

1µBβ , Hα =

1µBα , (1.32)

where

λkβ =(λx cos2 α + λy sin2 α − λz

)sin β cos β ,

λkα = (λy − λx) sin α cos α sinβ ,

λββ =(λx cos2 α + λy sin2 α

)cos2 β + λz sin2 β ,

λβα = (λy − λx) sin α cos α cos β ,

and

λαβ = λβα ,

λαα = λx sin2 α + λy cos2 α .

Equations (1.30) and (1.32) are then used to express Eβ , Eα and Hβ , Hα interms of Dβ , Dα, and we obtain

Eβ = µk20

k2Dβ , Eα = µ

k20

k2Dα ,

Hβ = −k0

kDα , Hα =

k0

kDβ . (1.33)


The last two equations in (1.31) and the first two equations in (1.33) yield ahomogeneous system of equations for Dβ and Dα

[λββ − µ

k20

k2 λβα

λβα λαα − µk20

k2

] [Dβ

Dα

]= 0 . (1.34)

Requiring nontrivial solutions we set the determinant equal to zero and obtaintwo values for the wave number k2,

k21,2 = k2

0

µ

λ1,2,

where

λ1 =12

[(λββ + λαα) −

√(λββ − λαα)2 + 4λ2

βα

],

and

λ2 =12

[(λββ + λαα) +

√(λββ − λαα)2 + 4λ2

βα

].

The above relations are the dispersion relations for the extraordinary waves,which are the permissible characteristic waves in anisotropic media. For anextraordinary wave, the magnitude of the wave vector depends on the direc-tion of propagation, while for an ordinary wave, k is independent of β andα. Straightforward calculations show that for real values of λx, λy and λz,λββλαα > λ2

βα and as a result λ1 > 0 and λ2 > 0. The two characteristicwaves, corresponding to the two values of k2, have the D vectors orthogonalto each other, i.e., D(1) · D(2) = 0. In view of (1.34) it is apparent that thecomponents of the vectors D(1) and D(2) can be expressed in terms of twoindependent scalar functions Dα and Dβ . For k1 = k0

√µ/λ1 we set

D(1)β = fDα , D(1)

α = Dα ,

while for k2 = k0

√µ/λ2 we choose

D(2)β = −Dβ , D(2)

α = fDβ ,

where

f = −λβα

∆λ

and

∆λ =12

[(λββ − λαα) +

√(λββ − λαα)2 + 4λ2

βα

].

Next, we define the tangential fields

vα = feβ + eα ,

vβ = −eβ + feα ,


and note that the vectors vα and vβ are orthogonal to each other, vα ·vβ = 0,and vα = −ek × vβ and vβ = ek × vα. Taking into account that k1 (ek) =k1 (ek) ek and k2 (ek) = k2 (ek) ek, we find the following integral representa-tion for the electric displacement:

D(r) =∫

Ω

[Dα (ek) vα (ek) ejk1(ek)·r + Dβ (ek) vβ (ek) ejk2(ek)·r

]dΩ (ek) ,

with Ω being the unit sphere. The result, the integral representation for theelectric field is

E(r) =1

εxy

×∫

Ω

[Dα (ek) we

α (ek) ejk1(ek)·r −Dβ (ek) weβ (ek) ejk2(ek)·r

]dΩ (ek) ,

where

εxy =12

(εx + εy) ,

and

weα = εxy [(λkβf + λkα) ek + λ1vα] ,

weβ = εxy [(λkβ − λkαf) ek − λ2vβ ] ,

while for the magnetic field, we have

H(r) = − 1√

εxyµ

∫

Ω

[Dα (ek) wh

α (ek) ejk1(ek)·r

+ Dβ (ek) whβ (ek) ejk2(ek)·r

]dΩ (ek) ,

where

whα = −

√εxyλ1vβ ,

whβ =√

εxyλ2vα .

For uniaxial anisotropic media we set λ = λx = λy, derive the relations

λkβ = (λ − λz) sinβ cos β , λkα = 0 ,

λββ = λ cos2 β + λz sin2 β , λβα = 0 ,

λαβ = 0 , λαα = λ ,

and use the identities λ1 = λαα and λ2 = λββ , to obtain

k21 = k2

0εµ , (1.35)

k22 = k2

0

εµ

cos2 β + εεz

sin2 β, (1.36)


where λ = 1/ε and λz = 1/εz. Equation (1.35) is the dispersion relation forordinary waves, while (1.36) is the dispersion relation for extraordinary waves.For f = 0 it follows that D(1)

β = 0 and D(2)α = 0, and further that D(1)

α = Dα

and D(2)β = −Dβ . The electric displacement is then given by

D(r) =∫ 2π

0

∫ π

0

[Dα(β, α)eαejk1(β,α)·r −Dβ(β, α)eβejk2(β,α)·r

]sinβ dβ dα ,

(1.37)where k1(β, α) = k1ek(β, α), k2(β, α) = k2(β)ek(β, α), and for notation sim-plification, the dependence of the spherical unit vectors eα and eβ on thespherical angles β and α is omitted. For εxy = ε, the integral representationsfor the electric and magnetic fields become

E(r) =1ε

∫ 2π

0

∫ π

0

Dα(β, α)eαejk1(β,α)·r (1.38)

− ε [λkβ(β)ek + λββ(β)eβ ]Dβ(β, α)ejk2(β,α)·r

sinβ dβ dα ,

and

H(r) = − 1√

εµ

∫ 2π

0

∫ π

0

[Dα(β, α)eβejk1(β,α)·r

+√

ελββ(β)Dβ(β, α)eαejk2(β,α)·r]

sinβ dβ dα , (1.39)

respectively. For isotropic media, the only nonzero λ functions are λββ andλαα, and we have λββ = λαα = λ. The two waves degenerate into one (ordi-nary) wave, i.e., k1 = k2 = k, and the dispersion relation is

k2 = k20εµ .

Next we proceed to derive series representations for the electric and mag-netic fields propagating in uniaxial anisotropic media. On the unit sphere, thetangential vector function Dα(β, α)eα −Dβ(β, α)eβ can be expanded in termsof the vector spherical harmonics mmn and nmn as follows:

Dα(β, α)eα −Dβ(β, α)eβ = −ε∞∑

n=1

n∑

m=−n

14πjn+1

[−jcmnmmn(β, α)

+ dmnnmn(β, α)] . (1.40)

Because the system of vector spherical harmonics is orthogonal and completein L2(Ω), the series representation (1.40) is valid for any tangential vectorfield. Taking into account the expressions of the vector spherical harmonics(cf. (B.8) and (B.9)) we deduce that the expansions of Dβ and Dα are given by


−Dβ(β, α) = −ε∞∑

n=1

n∑

m=−n

14πjn+1

1√

2n(n + 1)

[mπ|m|

n (β)cmn

+ τ |m|n (β)dmn

]ejmα

and

Dα(β, α) = −ε∞∑

n=1

n∑

m=−n

14πjn+1

1√

2n(n + 1)

[jτ |m|

n (β)cmn

+ jmπ|m|n (β)dmn

]ejmα ,

respectively. Inserting the above expansions into (1.38) and (1.39), yields theseries representations

E(r) =∞∑

n=1

n∑

m=−n

cmnXemn(r) + dmnY e

mn(r) , (1.41)

H(r) = −j√

ε

µ

∞∑

n=1

n∑

m=−n

cmnXhmn(r) + dmnY h

mn(r) , (1.42)

where the new vector functions are defined as

Xemn(r) = − 1

4πjn+1

1√

2n(n + 1)

∫ 2π

0

∫ π

0

jτ |m|

n (β)ejk1(β,α)·reα

+ ε [λkβ(β)ek + λββ(β)eβ ] mπ|m|n (β)ejk2(β,α)·r

×ejmα sinβ dβ dα , (1.43)

Y emn(r) = − 1

4πjn+1

1√

2n(n + 1)

∫ 2π

0

∫ π

0

jmπ|m|

n (β)ejk1(β,α)·reα (1.44)

+ ε [λkβ(β)ek + λββ(β)eβ ] τ |m|n (β)ejk2(β,α)·r

ejmα sin β dβ dα ,

Xhmn(r) = − 1

4πjn+1

1√

2n(n + 1)

∫ 2π

0

∫ π

0

[τ |m|n (β)ejk1(β,α)·reβ (1.45)

+ j√

ελββ(β)mπ|m|n (β)ejk2(β,α)·reα

]ejmα sin β dβ dα ,

and

Y hmn(r) = − 1

4πjn+1

1√

2n(n + 1)

∫ 2π

0

∫ π

0

[mπ|m|

n (β)ejk1(β,α)·reβ

+ j√

ελββ(β)τ |m|n (β)ejk2(β,α)·reα

]ejmα sin β dβ dα . (1.46)


In (1.38)–(1.39), the electromagnetic fields are expressed in terms of theunknown scalar functions Dα and Dβ , while in (1.41) and (1.42), the electro-magnetic fields are expressed in terms of the unknown expansion coefficientscmn and dmn. These unknowns will be determined from the boundary condi-tions for each specific scattering problem. The vector functions Xe,h

mn and Y e,hmn

can be regarded as a generalization of the regular vector spherical wave func-tions M1

mn and N1mn. For isotropic media, we have ελββ = 1, λkβ = 0 and

k1 = k2 = k, and we see that both systems of vector functions are equivalent:

Xemn(r) = Y h

mn(r) = M1mn(kr) ,

Y emn(r) = Xh

mn(r) = N1mn(kr) . (1.47)

As a result, we obtain the familiar expansions of the electromagnetic fields interms of vector spherical wave functions of the interior wave equation:

E(r) =∞∑

n=1

n∑

m=−n

cmnM1mn (kr) + dmnN1

mn(kr) ,

H(r) = −j√

ε

µ

∞∑

n=1

n∑

m=−n

cmnN1mn (kr) + dmnM1

mn(kr) .

Although the derivation of Xe,hmn and Y e,h

mn differs from that of Kiselev et al.[119], the resulting systems of vector functions are identical except for a mul-tiplicative constant. Accordingly to Kiselev et al. [119], this system of vectorfunctions will be referred to as the system of vector quasi-spherical wave func-tions. In (1.43)–(1.46) the integration over α can be analytically performedby using the relations

ek = sin β cos αex + sinβ sin αey + cos βez ,

eβ = cos β cos αex + cos β sin αey − sin βez ,

eα = − sin αex + cos αey ,

and the standard integrals

Im (x, ϕ) =∫ 2π

0

ejx cos(α−ϕ)ejmα dα = 2πjmejmϕJm (x) ,

Icm (x, ϕ) =

∫ 2π

0

cos αejx cos(α−ϕ)ejmα dα = π[jm+1ej(m+1)ϕJm+1 (x)

+ jm−1ej(m−1)ϕJm−1 (x)]

,

Ism (x, ϕ) =

∫ 2π

0

sin αejx cos(α−ϕ)ejmα dα = −jπ[jm+1ej(m+1)ϕJm+1 (x)

− jm−1ej(m−1)ϕJm−1 (x)]

,


where (ex,ey,ez) are the Cartesian unit vectors and Jm is the cylindricalBessel functions of order m. The expressions of the Cartesian components ofthe vector function Xe

mn read as

Xemn,x(r) = − 1

4πjn+1

1√

2n(n + 1)

∫ π

0

−jτ |m|

n (β)Ism(x1, ϕ)ejy1(r,θ,β)

+ε [λββ(β) cos β + λkβ(β) sin β] mπ|m|n (β)

× Icm(x2, ϕ)ejy2(r,θ,β)

sinβ dβ (1.48)

Xemn,y(r) = − 1

4πjn+1

1√

2n(n + 1)

∫ π

0

jτ |m|

n (β)Icm(x1, ϕ)ejy1(r,θ,β)

+ε [λββ(β) cos β + λkβ(β) sin β] mπ|m|n (β)

× Ism(x2, ϕ)ejy2(r,θ,β)

sinβ dβ , (1.49)

Xemn,z(r) = − 1

4πjn+1

1√

2n(n + 1)

∫ π

0

ε [λkβ(β) cos β − λββ(β) sin β]

×mπ|m|n (β)Im(x2, ϕ)ejy2(r,θ,β) sin β dβ , (1.50)

where x1(r, θ, β) = k1r sin β sin θ, x2(r, θ, β) = k2(β)r sin β sin θ, y1(r, θ, β) =k1r cos β cos θ and y2(r, θ, β) = k2(β)r cos β cos θ, while the expressions of theCartesian components of the vector functions Y e

mn are given by (1.48)–(1.50)with mπ

|m|n and τ

|m|n interchanged. Similarly, the Cartesian components of the

vector function Xhmn are

Xhmn,x(r) = − 1

4πjn+1

1√

2n(n + 1)

∫ π

0

[τ |m|n (β) cos βIc

m(x1, ϕ)ejy1(r,θ,β)

− j√

ελββ(β)mπ|m|n (β)Is


]sinβ dβ , (1.51)

Xhmn,y(r) = − 1

4πjn+1

1√

2n(n + 1)

∫ π

0

[τ |m|n (β) cos βIs


+ j√

ελββ(β)mπ|m|n (β)Ic


]sin β dβ , (1.52)

Xhmn,z(r) =

14πjn+1

1√

2n(n + 1)

∫ π

0

τ |m|n (β)Im(x1, ϕ)ejy1(r,θ,β) sin2 β dβ ,

(1.53)


and as before, the components of the vector functions Y hmn are given by

(1.51)–(1.53) with mπ|m|n and τ

|m|n interchanged.

In the above analysis, Xe,hmn and Y e,h

mn are expressed in the principal coor-dinate system, but in general, it is necessary to transform these vector func-tions from the principal coordinate system to the particle coordinate systemthrough a rotation. The vector quasi-spherical wave functions can also bedefined for biaxial media (εx = εy = εz) by considering the expansion ofthe tangential vector function Dα(β, α)vα + Dβ(β, α)vβ in terms of vectorspherical harmonics.

1.3.2 Chiral Media

For a source-free, isotropic, chiral medium, the Maxwell equations are givenby (1.10), with the K matrix defined by (1.11). Following Bohren [16], theelectromagnetic field is transformed to

[EH

]= A

[LR

],

where A is a transformation matrix and

A =

⎡

⎣1 −j

√µε

−j√

εµ 1

⎤

⎦ .

The transformed fields L and R are the left- and right-handed circularly po-larized waves, or simply the waves of left- and right-handed types. Explicitly,the electromagnetic field transformation is

E = L − j√

µ

εR ,

H = −j√

ε

µL + R

and note that this linear transformation diagonalizes the matrix K,

Λ = A−1KA =

[k

1−βk 00 − k

1+βk

]

.

Defining the wave numbers

kL =k

1 − βk,

kR =k

1 + βk,


we see that the waves of left- and right-handed types satisfy the equations

∇× L = kLL , ∇ · L = 0 (1.54)

and∇× R = −kRR , ∇ · R = 0, (1.55)

respectively.For chiral media, we use the same technique as for anisotropic media and

express the left- and right-handed circularly polarized waves as integrals overplane waves. For the Fourier transform corresponding to waves of left-handedtype,

L(r) =∫

L (k) ejk·r dV (k) ,

the differential equations (1.54) yield

Lβ = −jkL

kLα ,

Lα = jkL

kLβ ,

and Lk = 0. The above set of equations form a system of homogeneous equa-tions and setting the determinant equal to zero, gives the dispersion relationfor the waves of left-handed type

k2 = k2L .

Choosing Lβ as an independent scalar function, we express L as

L(r) =∫ 2π

0

∫ π

0

(eβ + jeα)Lβ(β, α)ejkL(β,α)·r sinβ dβ dα , (1.56)

where kL(β, α) = kLek(β, α). The tangential field (eβ + jeα)Lβ is orthogonalto the vector spherical harmonics of right-handed type with respect to thescalar product in L2

tan(Ω) (cf. (B.16) and (B.17)), and as result, (eβ +jeα)Lβ

possesses an expansion in terms of vector spherical harmonics of left-handedtype (cf. (B.14))

(eβ + jeα)Lβ(β, α) =∞∑

n=1

n∑

m=−n

12√

2πjncmnlmn(β, α) (1.57)

=∞∑

n=1

n∑

m=−n

14πjn

cmn [mmn(β, α) + jnmn(β, α)] .


Inserting (1.57) into (1.56), yields

L(r) = −∞∑

n=1

n∑

m=−n

14πjn+1

cmn

∫ 2π

0

∫ π

0

[−jmmn(β, α)

+ nmn(β, α)] ejkL(β,α)·r sinβ dβ dα ,

whence, using the integral representation for the vector spherical wave func-tions (cf. (B.26) and (B.27)), gives

L(r) =∞∑

n=1

n∑

m=−n

cmnLmn (kLr)

=∞∑

n=1

n∑

m=−n

cmn

[M1

mn (kLr) + N1mn (kLr)

],

where the vector spherical wave functions of left-handed type Lmn are de-fined as

Lmn = M1mn + N1

mn . (1.58)

For the waves of right-handed type we proceed analogously. We obtain theintegral representation

R(r) =∫ 2π

0

∫ π

0

(eβ − jeα)Rβ(β, α)ejkR(β,α)·r sinβ dβ dα

with Rβ being an independent scalar function, and the expansion

R(r) =∞∑

n=1

n∑

m=−n

dmnRmn (kRr)

=∞∑

n=1

n∑

m=−n

dmn

[M1

mn (kRr) − N1mn (kRr)

]

with the vector spherical wave functions of right-handed type Rmn beingdefined as

Rmn = M1mn − N1

mn . (1.59)

In conclusion, the electric and magnetic fields propagating in isotropic, chiralmedia possess the expansions [16,17,135]

E(r) =∞∑

n=1

n∑

m=−n

cmnLmn (kLr) − j√

µ

εdmnRmn (kRr) ,

H(r) =∞∑

n=1

n∑

m=−n

−j√

ε

µcmnLmn (kLr) + dmnRmn (kRr) .

1.4 Scattered Field 33

An exhaustive treatment of electromagnetic wave propagation in isotropic,chiral media has been given by Lakhtakia et al. [136]. This analysis deals withthe conservation of energy and momentum, properties of the infinite-mediumGreen’s function and the mathematical expression of Huygens’s principle.

1.4 Scattered Field

In this section we consider the basic properties of the scattered field as theyare determined by energy conservation and by the propagation properties ofthe fields in source-free regions. The results are presented for electromag-netic scattering by dielectric particles, which is modeled by the transmissionboundary-value problem. To formulate the transmission boundary-value prob-lem we consider a bounded domain Di (of class C2) with boundary S andexterior Ds, and denote by n the unit normal vector to S directed into Ds

(Fig. 1.8). The relative permittivity and relative permeability of the domainDt are εt and µt, where t = s, i, and the wave number in the domain Dt iskt = k0

√εtµt, where k0 is the wave number in free space. The unbounded

domain Ds is assumed to be lossless, i.e., εs > 0 and µs > 0, and the externalexcitation is considered to be a vector plane wave

Ee(r) = Ee0ejke·r , He(r) =√

εs

µsek × Ee0ejke·r ,

where Ee0 is the complex amplitude vector and ek is the unit vector in thedirection of the wave vector ke. The transmission boundary-value problemhas the following formulation.

Given Ee,He as an entire solution to the Maxwell equations representingthe external excitation, find the vector fields Es,Hs ∈ C1(Ds) ∩ C(Ds) andEi,H i ∈ C1(Di) ∩ C(Di) satisfying the Maxwell equations

∇× Et = jk0µtHt, ∇× Ht = −jk0εtEt, (1.60)

i s

n

εs, µs,

ε i, µi,

ke

k i

k s

Ee

Es

O D D

S

Fig. 1.8. The domain Di with boundary S and exterior Ds


in Dt, t = s, i, and the two transmission conditions

n × Ei − n × Es = n × Ee ,

n × H i − n × Hs = n × He , (1.61)

on S. In addition, the scattered field Es,Hs must satisfy the Silver–Mullerradiation condition

r

r×√

µsHs +√

εsEs = o(

1r

), as r → ∞ , (1.62)

uniformly for all directions r/r.It should be emphasized that for the assumed smoothness conditions, the

transmission boundary-value problem possesses an unique solution [177].Our presentation is focused on the analysis of the scattered field in the

far-field region. We begin with a basic representation theorem for electro-magnetic scattering and then introduce the primary quantities which definethe single-scattering law: the far-field patterns and the amplitude matrix. Be-cause the measurement of the amplitude matrix is a complicated experimentalproblem, we characterize the scattering process by other measurable quanti-ties as for instance the optical cross-sections and the phase and extinctionmatrices.

In our analysis, we will frequently use the Green second vector theorem∫

D

[a · (∇×∇× b) − b · (∇×∇× a)] dV

=∫

S

n · [b × (∇× a) − a × (∇× b)] dS ,

where D is a bounded domain with boundary S and n is the outward unitnormal vector to S.

1.4.1 Stratton–Chu Formulas

Representation theorems for electromagnetic fields have been given by Strat-ton and Chu [216]. If Es, Hs is a radiating solution to Maxwell’s equationsin Ds, then we have the Stratton–Chu formulas(

Es(r)0

)= ∇×

∫

S

es (r′) g (ks, r, r′) dS(r′) (1.63)

+j

k0εs∇×∇×

∫

S

hs (r′) g (ks, r, r′) dS(r′) ,

(r ∈ Ds

r ∈ Di

)


and(

Hs(r)0

)= ∇×

∫

S

hs (r′) g (ks, r, r′) dS(r′)

− jk0µs

∇×∇×∫

S

es (r′) g (ks, r, r′) dS(r′) ,

(r ∈ Ds

r ∈ Di

),

where g is the Green function and the surface fields es and hs are the tangentialcomponents of the electric and magnetic fields on the particle surface, i.e.,es = n × Es and hs = n × Hs, respectively. In the above equations we use acompact way of writing two formulas (for r ∈ Ds and for r ∈ Di) as a singleequation.

A similar result holds for vector functions satisfying the Maxwell equationsin bounded domains. With Ei, H i being a solution to Maxwell’s equations inDi we have(−Ei(r)

0

)= ∇×

∫

S

ei (r′) g (ki, r, r′) dS(r′)

+j

k0εi∇×∇×

∫

S

hi (r′) g (ki, r, r′) dS(r′) ,

(r ∈ Di

r ∈ Ds

)

and(−H i(r)

0

)= ∇×

∫

S

hi (r′) g (ki, r, r′) dS(r′)

− jk0µi

∇×∇×∫

S

ei (r′) g (ki, r, r′) dS(r′) ,

(r ∈ Di

r ∈ Ds

),

where ei = n × Ei and hi = n × H i.A rigorous proof of these representation theorems on the assumptions Es,

Hs ∈ C1(Ds) ∩C(Ds) and Ei, H i ∈ C1(Di) ∩C(Di) can be found in Coltonand Kress [39]. An alternative proof can be given if we accept the validityof Green’s second vector theorem for generalized functions such as the three-dimensional Dirac delta function δ(r − r′). To prove the representation the-orem for vector fields satisfying the Maxwell equations in bounded domainswe use the Green second vector theorem for a divergence free vector field a(∇ · a = 0). Using the vector identities ∇ × ∇ × b = −∆b + ∇∇ · b anda · (∇∇ · b) = ∇ · (a∇ · b) for ∇ · a = 0, and the Gauss divergence theoremwe see that∫

D

[a · ∆b + b · (∇×∇× a)] dV =∫

S

n · a (∇ · b) + n · [a × (∇× b)

+ (∇× a) × b]dS . (1.64)

In the above equations, the simplified notations ∇∇ · a and a∇ · b shouldbe understood as ∇(∇ · a) and a(∇ · b), respectively. Next, we choose an


arbitrary constant unit vector u and apply Green’s second vector theorem(1.64) to a(r′) = Ei(r′) and b(r′) = g(ki, r

′, r)u, for r ∈ Di. Recalling that

∆′g (ki, r′, r) + k2

i g (ki, r′, r) = −δ (r′ − r) , (1.65)

and ∇′ ×∇′ × Ei = k2i Ei, we see that the left-hand side of (1.64) is

∫

Di

Ei (r′) · ∆′g (ki, r′, r) u + g (ki, r

′, r) u · [∇′ ×∇′ × Ei (r′)]

×dV (r′)

= −∫

Di

u · Ei (r′) δ (r′ − r) dV (r′) = −u · Ei(r) .

Taking into account that for r′ = r,

∇′ ×∇′ × g (ki, r′, r) u = k2

i g (ki, r′, r) u + ∇′∇′ · g (ki, r

′, r) u ,

we rewrite the right-hand side of (1.64) as∫

S

n · Ei [∇′ · g (ki, ·r) u] + n · [Ei × (∇′ × g (ki, ·r) u)

+ (∇′ × Ei) × g (ki, ·r) u]dS

=∫

S

n · Ei [∇′ · g (ki, ·r) u] + n · [Ei × (∇′ × g (ki, ·r) u)]

+1k2i

n · [(∇′ × Ei) × (∇′ ×∇′ × g (ki, ·r) u)

− (∇′ × Ei) ×∇′∇′ · g (ki, ·r) u]dS .

From Stokes theorem we have∫

S

n · ∇′ × [H i∇′ · g (ki, ·r) u]dS = 0 ,

whence, using the vector identity ∇×(αb) = ∇α×b+α∇×b and the Maxwellequations, we obtain

∫

S

n · Ei [∇′ · g (ki, ·, r) u] dS

=1k2i

∫

S

n · [(∇′ × Ei) ×∇′∇′ · g (ki, ·, r) u] dS .

Finally, using the vector identity a ·(b×c) = (a×b) ·c, the symmetry relation∇′g(ki, r

′, r) = −∇g(ki, r′, r), the Maxwell equations and the identities


[∇′ × g (ki, r′, r) u] · [n (r′) × Ei (r′)]

= ∇ × g (ki, r′, r) [n (r′) × Ei (r′)] · u

and

[∇′ ×∇′ × g (ki, r′, r) u] · [n (r′) × H i (r′)]

= ∇ ×∇× g (ki, r′, r) [n (r′) × H i (r′)] · u ,

we arrive at

−u · Ei(r) = u ·∇×

∫

S

n (r′) × Ei (r′) g (ki, r, r′) dS(r′)

+j

k0εi∇×∇×

∫

S

n (r′) × H i (r′) g (ki, r, r′) dS(r′)

.

Since u is arbitrary, we have established the Stratton–Chu formula for r ∈ Di.If r ∈ Ds, we have

∫

Di

u · Ei (r′) δ (r′ − r) dV (r′) = 0 ,

and the proof follows in a similar manner. For radiating solutions to theMaxwell equations, we see that the proof is established if we can show that

∫

SR

n · [Es × (∇′ × g (ks, ·, r) u)

+1k2s

(∇′ × Es) × (∇′ ×∇′ × g (ks, ·, r) u)]

dS → 0 ,

as R → ∞, where SR is a spherical surface situated in the far-field region.To prove this assertion we use the Silver–Muller radiation condition and thegeneral assumption Imks ≥ 0.

Alternative representations for Stratton–Chu formulas involve the freespace dyadic Green function G instead of the fundamental solution g [228].A dyad D serves as a linear mapping from one vector to another vector,and in general, D can be introduced as the dyadic product of two vectors:D = a ⊗ b. The dot product of a dyad with a vector is another vector:D · c = (a⊗ b) · c = a(b · c) and c ·D = c · (a⊗ b) = (c ·a)b, while the crossproduct of a dyad with a vector is another dyad: D×c = (a⊗b)×c = a⊗(b×c)and c×D = c× (a⊗ b) = (c×a)⊗ b. The free space dyadic Green functionis defined as

G (k, r, r′) =(

I +1k2

∇⊗∇)

g (k, r, r′) ,


where I is the identity dyad (D · I = I ·D = D). Multiplying the differentialequation (1.65) by I and using the identities

∇×∇×(Ig)

= ∇⊗∇g − I g ,

∇×∇× (∇⊗∇g) = 0 ,

gives the differential equation for the free space dyadic Green function

∇×∇× G (k, r, r′) = k2G (k, r, r′) + δ (r − r′) I . (1.66)

The Stratton–Chu formula for vector fields satisfying the Maxwell equationsin bounded domains read as

(−Ei(r)

0

)=∫

S

ei (r′) ·[∇′ × G (ki, r, r′)

]dS(r′)

+jk0µi

∫

S

hi (r′) · G (ki, r, r′) dS(r′) ,

(r ∈ Di

r ∈ Ds

)

and this integral representation follows from the second vector-dyadic Greentheorem [220]:

∫

D

[a ·(∇×∇× D

)− (∇×∇× a) · D

]dV

= −∫

S

n ·[a ×(∇× D

)+ (∇× a) × D

]dS ,

applied to a(r′) = Ei(r′) and D(r′) = G(ki, r′, r), the differential equation

for the free space dyadic Green function, and the identity a·(b×D) = (a×b)·D. For radiating solutions to the Maxwell equations, we use the asymptoticbehavior of the free space dyadic Green function in the far-field region

r

r×[∇× G (ks, r, r′)

]+ jksG (ks, r, r′) = o

(1r

), as r → ∞ ,

to show that the integral over the spherical surface vanishes at infinity.

Remark. The Stratton–Chu formulas are surface-integral representations forthe electromagnetic fields and are valid for homogeneous particles. For inho-mogeneous particles, a volume-integral representation for the electric field canbe derived. For this purpose, we consider the nonmagnetic domains Ds andDi (µs = µi = 1), rewrite the Maxwell equations as

∇× Et = jk0Ht , ∇× Ht = −jk0εtEt in Dt , t = s, i ,

and assume that the domain Di is isotropic, linear and inhomogeneous, i.e.,εi = εi(r). The Maxwell curl equation for the magnetic field H i can bewritten as


∇× H i = −jk0εsEi − jk0εs

(εi

εs− 1)

Ei

= −jk0εsEi −jk0

k2s

(m2

r − 1)Ei ,

where mr = mr(r) is the relative refractive index and ks = k0√

εs. Definingthe total electric and magnetic fields everywhere in space by

E =

Es + Ee

Ei

in Ds ,in Di ,

and

H =

Hs + He

H i

in Ds ,in Di ,

respectively, and the forcing function J by

J = k2s

(m2

rt − 1)E ,

where

mrt =

1mr

in Ds ,in Di ,

we see that the total electric and magnetic fields satisfy the Maxwell curlequations,

∇× E = jk0H , ∇× H = −jk0εsE − jk0

J in Ds ∪ Di .

By taking the curl of the first equation we obtain an inhomogeneous differen-tial equation for total electric field

∇×∇× E − k2s E = J in Ds ∪ Di . (1.67)

Making use of the differential equation for the free space dyadic Green function(1.66) and the identity

∇×[G (ks, r, r′) · J (r′)

]=[∇× G (ks, r, r′)

]· J (r′) ,

we derive

∇×∇×[G (ks, r, r′) · J (r′)

]

−k2s G (ks, r, r′) · J (r′) = I · J (r′) δ (r − r′) .

Integrating this equation over all r′ and using the identity δ(r−r′) = δ(r′−r),gives [131]

(∇×∇× I − k2

s I)·∫

R3G (ks, r, r′) · J (r′) dV (r′) = J(r) . (1.68)


Because (1.67) and (1.68) have the same right-hand side we deduce that

E(r) =∫

R3G (ks, r, r′) · J (r′) dV (r′) , r ∈ Ds ∪ Di

and, since J = 0 in Ds, we obtain

E(r) =∫

Di

G (ks, r, r′) · J (r′) dV (r′) , r ∈ Ds ∪ Di .

This vector field is the particular solution to the differential equation (1.67)that depends on the forcing function. For r ∈ Ds, the particular solutionsatisfies the Silver–Muller radiation condition and gives the scattered field.The solution to the homogeneous equation or the complementary solutionsatisfies the equation

∇×∇× Ee − k2s Ee = 0 in Ds ∪ Di .

and describes the field that would exist in the absence of the scattering object,i.e., the incident field. Thus, the complete solution to (1.67) can be written as

E(r) = Ee(r) +∫

Di

G (ks, r, r′) · J (r′) dV (r′)

= Ee(r) + k2s

∫

Di

G (ks, r, r′) ·[m2

r (r′) − 1]E (r′) dV (r′) ,

r ∈ Ds ∪ Di .

We note that for a nontrivial magnetic permeability of the particle, a volume-surface integral equation has been derived by Volakis [245], and a “pure”volume-integral equation has been given by Volakis et al. [246].

1.4.2 Far-Field Pattern and Amplitude Matrix

Application of Stratton–Chu representation theorem to the vector fields Es

and Ee in the domain Ds together with the boundary conditions es +ee = ei

and hs + he = hi, yield

Es(r) = ∇×∫

S

ei (r′) g (ks, r, r′) dS(r′)

+j

k0εs∇×∇×

∫

S

hi (r′) g (ks, r, r′) dS(r′) , r ∈ Ds ,

where Es,Hs and Ei,H i solve the transmission boundary-value problem.The above equation is known as the Huygens principle and it expresses thefield in the domain Ds in terms of the surface fields on the surface S (see, for


example, [229]). Application of Stratton–Chu representation theorem in thedomain Di gives the (general) null-field equation or the extinction theorem:

Ee(r) + ∇×∫

S

ei (r′) g (ks, r, r′) dS(r′)

+j

k0εs∇×∇×

∫

S

hi (r′) g (ks, r, r′) dS(r′) = 0 , r ∈ Di ,

which shows that the radiation of the surface fields into Di extinguishes theincident wave [229]. In the null-field method, the extinction theorem is usedto derive a set of integral equations for the surface fields, while the Huygensprinciple is employed to compute the scattered field.

Every radiating solution Es,Hs to the Maxwell equations has the asymp-totic form

Es(r) =ejksr

r

Es∞(er) + O

(1r

), r → ∞ ,

Hs(r) =ejksr

r

Hs∞(er) + O

(1r

), r → ∞ ,

uniformly for all directions er = r/r. The vector fields Es∞ and Hs∞ definedon the unit sphere are the electric and magnetic far-field patterns, respectively,and satisfy the relations:

Hs∞ =√

εs

µser × Es∞ ,

er · Es∞ = er · Hs∞ = 0 .

Because Es∞ also depends on the incident direction ek, Es∞ is known as thescattering amplitude from the direction ek into the direction er [229]. Usingthe Huygens principle and the asymptotic expressions

∇×[

a (r′)ejks|r−r′||r − r′|

]

= jksejksr

r

e−jkser·r′

er × a (r′) + O(a

r

),

∇×∇×[

a (r′)ejks|r−r′||r − r′|

]

= k2s

ejksr

r

×

e−jkser·r′er × [a (r′) × er] + O

(a

r

),

as r → ∞, we obtain the following integral representations for the far-fieldpatterns [40]

Es∞(er) =jks

4π

∫

S

er × es(r′)

+√

µs

εser × [hs (r′) × er]

e−jkser·r′

dS (r′) , (1.69)


Hs∞(er) =jks

4π

∫

S

er × hs(r′)

−√

εs

µser × [es (r′) × er]

e−jkser·r′

dS (r′) . (1.70)

The quantity σd = |Es∞|2 is called the differential scattering cross-section anddescribes the angular distribution of the scattered light. The differential scat-tering cross-section depends on the polarization state of the incident field andon the incident and scattering directions. The quantities σdp = |Es∞,θ|2 andσds = |Es∞,ϕ|2 are referred to as the differential scattering cross-sections forparallel and perpendicular polarizations, respectively. The differential scat-tering cross-section has the dimension of area, and a dimensionless quantityis the normalized differential scattering cross-section σdn = σd/πa2

c , where ac

is a characteristic dimension of the particle.To introduce the concepts of tensor scattering amplitude and amplitude

matrix it is necessary to choose an orthonormal unit system for polariza-tion description. In Sect. 1.2 we chose a global coordinate system and usedthe vertical and horizontal polarization unit vectors eα and eβ , to describethe polarization state of the incident wave (Fig. 1.9a). For the scattered wavewe can proceed analogously by considering the vertical and horizontal po-larization unit vectors eϕ and eθ. Essentially, (ek,eβ ,eα) are the sphericalunit vectors of ke, while (er,eθ,eϕ) are the spherical unit vectors of ks in

eβ

eα

ek

X

YO

Z

keβ

α

eθ

eϕ

er

X

YO

Z

ks

Θ

Ψ

er

eθ

eϕθ

ks

ϕeβ

eα

e⊥

e⎜⎜

Θ

ek

Fig. 1.9. Reference frames: (a) global coordinate system and (b) beam coordinatesystem


the global coordinate system. A second choice is the system based on thescattering plane. In this case we consider the beam coordinate system withthe Z-axis directed along the incidence direction, and define the Stokes vectorswith respect to the scattering plane, that is, the plane through the direction ofincidence and scattering (Fig. 1.9b). For the scattered wave, the polarizationdescription is in terms of the vertical and horizontal polarization unit vectorseϕ and eθ, while for the incident wave, the polarization description is in termsof the unit vectors e⊥ = eϕ and e‖ = e⊥ × ek. The advantage of this sys-tem is that the scattering amplitude can take simple forms for particles withsymmetry, and the disadvantage is that e⊥ and e‖ depend on the scatter-ing direction. Furthermore, any change in the direction of light incidence alsochanges the orientation of the particle with respect to the reference frame.In our analysis we will use a fixed global coordinate system to specify boththe direction of propagation and the states of polarization of the incident andscattered waves and the particle orientation (see also, [169,228]).

The tensor scattering amplitude or the scattering dyad is given by [169]

Es∞(er) = A (er,ek) · Ee0 , (1.71)

and since er · Es∞ = 0, it follows that:

er · A (er,ek) = 0 . (1.72)

Because the incident wave is a transverse wave, ek ·Ee0 = 0, the dot productA(er,ek) ·ek is not defined by (1.71), and to complete the definition, we take

A (er,ek) · ek = 0 . (1.73)

Although the scattering dyad describes the scattering of a vector plane wave,it can be used to describe the scattering of any incident field, because anyregular solution to the Maxwell equations can be expressed as an integralover vector plane waves. As a consequence of (1.72) and (1.73), only fourcomponents of the scattering dyad are independent and it is convenient tointroduce the 2× 2 amplitude matrix S to describe the transformation of thetransverse components of the incident wave into the transverse componentsof the scattered wave in the far-field region. The amplitude matrix is givenby [17,169,228]

[Es∞,θ(er)Es∞,ϕ(er)

]= S (er,ek)

[Ee0,β

Ee0,α

], (1.74)

where Ee0,β and Ee0,α do not depend on the incident direction. Essentially,the amplitude matrix is a generalization of the scattering amplitudes includingpolarization effects. The amplitude matrix provides a complete description ofthe far-field patterns and it depends on the incident and scattering directionsas well on the size, optical properties and orientation of the particle. Theelements of the amplitude matrix


S =[

Sθβ Sθα

Sϕβ Sϕα

]

are expressed in terms of the scattering dyad as follows:

Sθβ = eθ · A · eβ ,

Sθα = eθ · A · eα ,

Sϕβ = eϕ · A · eβ ,

Sϕα = eϕ · A · eα . (1.75)

1.4.3 Phase and Extinction Matrices

As in optics the electric and magnetics fields cannot directly be measured be-cause of their high frequency oscillations, other measurable quantities describ-ing the change of the polarization state upon scattering have to be defined.The transformation of the polarization characteristic of the incident light intothat of the scattered light is given by the phase matrix. The coherency phasematrix Zc relates the coherency vectors of the incident and scattered fields

J s (rer) =1r2

Zc (er,ek) Je ,

where the coherency vector of the incident field Je is given by (1.19) and thecoherency vector of the scattered field J s is defined as

J s (rer) =1

2r2

√εs

µs

⎡

⎢⎢⎢⎣

Es∞,θ (er) E∗s∞,θ (er)

Es∞,θ (er) E∗s∞,ϕ (er)

Es∞,ϕ (er) E∗s∞,θ (er)

Es∞,ϕ (er) E∗s∞,ϕ (er)

⎤

⎥⎥⎥⎦

.

Explicitly, the coherency phase matrix is given by

Zc =

⎡

⎢⎢⎢⎢⎣

|Sθβ |2 SθβS∗θα SθαS∗

θβ |Sθα|2

SθβS∗ϕβ SθβS∗

ϕα SθαS∗ϕβ SθαS∗

ϕα

SϕβS∗θβ SϕβS∗

θα SϕαS∗θβ SϕαS∗

θα

|Sϕβ |2 SϕβS∗ϕα SϕαS∗

ϕβ |Sϕα|2

⎤

⎥⎥⎥⎥⎦

.

The phase matrix Z describes the transformation of the Stokes vector of theincident field into that of the scattered field

Is (rer) =1r2

Z (er,ek) Ie (1.76)

and we have

Z (er,ek) = DZc (er,ek) D−1 ,


where the transformation matrix D is given by (1.21), the Stokes vector ofthe incident field Ie is given by (1.20) and the Stokes vector of the scatteredfield Is is defined as

Is (rer) =1r2

⎡

⎢⎢⎢⎣

Is (er)Qs (er)Us (er)Vs (er)

⎤

⎥⎥⎥⎦

= DJ s (rer)

=1

2r2

√εs

µs

⎡

⎢⎢⎢⎢⎢⎣

|Es∞,θ (er)|2 + |Es∞,ϕ (er)|2

|Es∞,θ (er)|2 − |Es∞,ϕ (er)|2

−Es∞,ϕ (er) E∗s∞,θ (er) − Es∞,θ (er) E∗

s∞,ϕ (er)

j[Es∞,ϕ (er) E∗

s∞,θ (er) − Es∞,θ (er) E∗s∞,ϕ (er)

]

⎤

⎥⎥⎥⎥⎥⎦

.

Explicit formulas for the elements of the phase matrix are:

Z11 =12

(|Sθβ |2 + |Sθα|2 + |Sϕβ |2 + |Sϕα|2

),

Z12 =12

(|Sθβ |2 − |Sθα|2 + |Sϕβ |2 − |Sϕα|2

),

Z13 = −ReSθβS∗

θα + SϕαS∗ϕβ

,

Z14 = −ImSθβS∗

θα − SϕαS∗ϕβ

,

Z21 =12

(|Sθβ |2 + |Sθα|2 − |Sϕβ |2 − |Sϕα|2

),

Z22 =12

(|Sθβ |2 − |Sθα|2 − |Sϕβ |2 + |Sϕα|2

),


θα − SϕαS∗ϕβ

,

Z24 = −ImSθβS∗

θα + SϕαS∗ϕβ

,


ϕβ + SϕαS∗θα

,


ϕβ − SϕαS∗θα

,

Z33 = ReSθβS∗

ϕα + SθαS∗ϕβ

,

Z34 = ImSθβS∗

ϕα + SϕβS∗θα

,

Z41 = −ImSϕβS∗

θβ + SϕαS∗θα

,

Z42 = −ImSϕβS∗

θβ − SϕαS∗θα

,

Z43 = ImSϕαS∗

θβ − SθαS∗ϕβ

,

Z44 = ReSϕαS∗

θβ − SθαS∗ϕβ

. (1.77)


The above phase matrix is also known as the pure phase matrix, becauseits elements follow directly from the corresponding amplitude matrix thattransforms the two electric field components [100]. The phase matrix of aparticle in a fixed orientation may contain sixteen nonvanishing elements.Because only phase differences occur in the expressions of Zij , i, j = 1, 2, 3, 4,the phase matrix elements are essentially determined by no more than sevenreal numbers: the four moduli |Spq| and the three differences in phase betweenthe Spq, where p = θ, ϕ and q = β, α. Consequently, only seven phase matrixelements are independent and there are nine linear relations among the sixteenelements. These linear dependent relations show that a pure phase matrix hasa certain internal structure. Several linear and quadratic inequalities for thephase matrix elements have been reported by exploiting the internal structureof the pure phase matrix, and the most important inequalities are Z11 ≥ 0and |Zij | ≤ Z11 for i, j = 1, 2, 3, 4 [102–104]. In principle, all scalar and matrixproperties of pure phase matrices can be used for theoretical purposes or totest whether an experimentally or numerically determined matrix can be apure phase matrix.

Equation (1.76) shows that electromagnetic scattering produces light withpolarization characteristics different from those of the incident light. If theincident beam is unpolarized, Ie = [Ie, 0, 0, 0]T , the Stokes vector of thescattered field has at least one nonvanishing component other than inten-sity, Is = [Z11Ie, Z21Ie, Z31Ie, Z41Ie]T . When the incident beam is linearlypolarized, Ie = [Ie, Qe, Ue, 0]T , the scattered light may become ellipticallypolarized since Vs may be nonzero. However, if the incident beam is fully po-larized (Pe = 1), then the scattered light is also fully polarized (Ps = 1) [104].

As mentioned before, a scattering particle can change the state of polar-ization of the incident beam after it passes the particle. This phenomenon iscalled dichroism and is a consequence of the different values of attenuationrates for different polarization components of the incident light. A completedescription of the extinction process requires the introduction of the so-calledextinction matrix. In order to derive the expression of the extinction matrixwe consider the case of the forward-scattering direction, er = ek, and definethe coherency vector of the total field E = Es + Ee by

J (rek) =12

√εs

µs

⎡

⎢⎢⎢⎣

Eβ (rek) E∗β (rek)

Eβ (rek) E∗α (rek)

Eα (rek) E∗β (rek)

Eα (rek) E∗α (rek)

⎤

⎥⎥⎥⎦

.

Using the decomposition

Ep(r)E∗q (r) = Ee0,pE

∗e0,q + Ee0,pejke·rE∗

s,q(r)

+Es,p(r)E∗e0,qe

−jke·r + Es,p(r)E∗s,q(r) ,

where p and q stand for β and α, we approximate the integral of the genericterm EpE

∗q in the far-field region and over a small solid angle ∆Ω around the


O

D i

Ds

S

keEe

Es

∆∆ Ω Sek rM

Fig. 1.10. Elementary surface ∆S in the far-field region

direction ek by∫

∆Ω

Ep(r)E∗q (r)r2 dΩ (er) ≈ Ep (rek) E∗

q (rek) ∆S ,

where ∆S = r2∆Ω (Fig. 1.10). On the other hand, using the far-field repre-sentation for the scattered field

Es,p(r) =ejksr

r

Es∞,p(er) + O

(1r

)

and the asymptotic expression of the plane wave exp(jke · r) (cf. (B.7)) weapproximate the integrals of each term composing EpE

∗q as follows:

∫

∆Ω

Ee0,pE∗e0,qr

2 dΩ (er) ≈ Ee0,pE∗e0,q∆S ,

∫

∆Ω

Ee0,pejke·rE∗s,q(r)r2 dΩ (er) ≈ −2πj

ksEe0,pE

∗s∞,q (ek) ,

∫

∆Ω

Es,p(r)E∗e0,qe

−jke·rr2 dΩ (er) ≈2πjks

Es∞,p (ek) E∗e0,q ,

∫

∆Ω

Es,p(r)E∗s,q(r)r2 dΩ (er) ≈ Es∞,p (ek) E∗

s∞,q (ek)∆S

r2.

Neglecting the term proportional to r−2, we see that

Ep (rek) E∗q (rek) ∆S ≈ Ee0,pE

∗e0,q∆S

−2πjks

[Ee0,pE

∗s∞,q (ek) − Es∞,p (ek) E∗

e0,q

]

and the above relation gives

J (rek) ∆S ≈ Je∆S − Kc (ek) Je ,


where the coherency extinction matrix Kc is defined as

Kc =2πjks

⎡

⎢⎢⎣

S∗θβ − Sθβ S∗

θα −Sθα 0S∗

ϕβ S∗ϕα − Sθβ 0 −Sθα

−Sϕβ 0 S∗θβ − Sϕα S∗

θα

0 −Sϕβ S∗ϕβ S∗

ϕα − Sϕα

⎤

⎥⎥⎦ .

For the Stokes parameters we have

I (rek) ∆S ≈ Ie∆S − K (ek) Ie (1.78)

with the extinction matrix K being defined as

K (ek) = DKc (ek) D−1 .

The explicit formulas for the elements of the extinction matrix are

Kii =2π

ksIm Sθβ + Sϕα , i = 1, 2, 3, 4 ,

K12 = K21 =2π

ksIm Sθβ − Sϕα ,

K13 = K31 = −2π

ksIm Sθα + Sϕβ ,

K14 = K41 =2π

ksRe Sϕβ − Sθα ,

K23 = −K32 =2π

ksIm Sϕβ − Sθα ,

K24 = −K42 = −2π

ksRe Sθα + Sϕβ ,

K34 = −K43 =2π

ksRe Sϕα − Sθβ . (1.79)

The elements of the extinction matrix have the dimension of area and onlyseven components are independent. Equation (1.78) is an interpretation ofthe so-called optical theorem which will be discussed in the next section. Thisrelation shows that the particle changes not only the total electromagneticpower received by a detector in the forward scattering direction, but also itsstate of polarization.

1.4.4 Extinction, Scattering and Absorption Cross-Sections

Scattering and absorption of light changes the characteristics of the incidentbeam after it passes the particle. Let us assume that the particle is placed in abeam of electromagnetic radiation and a detector located in the far-field regionmeasures the radiation in the forward scattering direction (er = ek). Let W


be the electromagnetic power received by the detector downstream from theparticle, and W0 the electromagnetic power received by the detector if the par-ticle is removed. Evidently, W0 > W and we say that the presence of theparticle has resulted in extinction of the incident beam. For a nonabsorbingmedium, the electromagnetic power removed from the incident beam W0−Wis accounted for by absorption in the particle and scattering by the particle.

We now consider extinction from a computational point of view. In order tosimplify the notations we will use the conventional expressions of the Poyntingvectors and the electromagnetic powers in terms of the transformed fieldsintroduced in Sect. 1.1 (we will omit the multiplicative factor 1/

√ε0µ0). The

time-averaged Poynting vector 〈S〉 can be written as [17]

〈S〉 =12Re E × H∗ = 〈Sinc〉 + 〈Sscat〉 + 〈Sext〉 ,

where E = Es + Ee and H = Hs + He are the total electric and magneticfields,

〈Sinc〉 =12Re Ee × H∗

e

is the Poynting vector associated with the external excitation,

〈Sscat〉 =12Re Es × H∗

s

is the Poynting vector corresponding to the scattered field and

〈Sext〉 =12Re Ee × H∗

s + Es × H∗e

is the Poynting vector caused by the interaction between the scattered andincident fields.

Taking into account the boundary conditions n×Ei = n×E and n×H i =n×H on S, we express the time-averaged power absorbed by the particle as

Wabs = −12

∫

S

n · Re Ei × H∗i dS

= −12

∫

S

n · Re E × H∗dS .

With Sc being an auxiliary surface enclosing S (Fig. 1.11), we apply the Greensecond vector theorem to the vector fields E and E∗ in the domain D boundedby S and Sc. We obtain

∫

S

n · (E × H∗ + E∗ × H) dS =∫

Sc

n · (E × H∗ + E∗ × H) dS ,

and further∫

S

n · Re E × H∗dS =∫

Sc

n · Re E × H∗dS .


O

D i

D

S

ke

Ee

Es

Sc

E = Ee+Es

Fig. 1.11. Auxiliary surface Sc

The time-averaged power absorbed by the particle then becomes

Wabs = −12

∫

Sc

n · Re E × H∗dS = −∫

Sc

n · 〈S〉dS

= Winc − Wscat + Wext ,

where

Winc = −∫

Sc

n · 〈Sinc〉dS = −12

∫

Sc

n · Re Ee × H∗edS , (1.80)

Wscat =∫

Sc

n · 〈Sscat〉dS =12

∫

Sc

n · Re Es × H∗sdS , (1.81)

Wext = −∫

Sc

n · 〈Sext〉dS = −12

∫

Sc

n · Re Ee × H∗s + Es × H∗

edS .

(1.82)

The divergence theorem applied to the excitation field in the domain Dc

bounded by Sc gives∫

Dc

∇ · (Ee × H∗e) dV =

∫

Sc

n · (Ee × H∗e) dS ,

whence, using

∇ · (Ee × H∗e) = jk0

(µs |He|2 − εs |Ee|2

),

andRe ∇ · (Ee × H∗

e) = 0 ,

yieldWinc = 0 .


Thus, Wext is the sum of the electromagnetic scattering power and the elec-tromagnetic absorption power

Wext = Wscat + Wabs .

For a plane wave incidence, the extinction and scattering cross-sections aregiven by

Cext =Wext

12

√εsµs

|Ee0|2, (1.83)

Cscat =Wscat

12

√εsµs

|Ee0|2, (1.84)

the absorption cross-section is

Cabs = Cext − Cscat ≥ 0 ,

while the single-scattering albedo is

ω =Cscat

Cext≤ 1 .

Essentially, Cscat and Cabs represent the electromagnetic powers removed fromthe incident wave as a result of scattering and absorption of the incident ra-diation, while Cext gives the total electromagnetic power removed from theincident wave by the combined effect of scattering and absorption. The op-tical cross-sections have the dimension of area and depend on the directionand polarization state of the incident wave as well on the size, optical proper-ties and orientation of the particle. The efficiencies (or efficiency factors) forextinction, scattering and absorption are defined as

Qext =Cext

G, Qscat =

Cscat

G, Qabs =

Cabs

G,

where G is the particle cross-sectional area projected onto a plane perpendicu-lar to the incident beam. In view of the definition of the normalized differentialscattering cross-section, we set G = πa2

c , where ac is the area-equivalent-circleradius. From the point of view of geometrical optics we expect that the ex-tinction efficiency of all particles would be identically equal to unity. In fact,there are many particles which can scatter and absorb more light than isgeometrically incident upon them [17].

The scattering cross-section is the integral of the differential scatter-ing cross-section over the unit sphere. To prove this assertion, we expressCscat as

Cscat =1

|Ee0|2√

µs

εs

∫

Sc

er · Re Es × H∗sdS ,


where Sc is a spherical surface situated at infinity and use the far-field repre-sentation

er · (Es × H∗s ) =

1r2

√εs

µs

[|Es∞|2 + O

(1r

)], r → ∞

to obtainCscat =

1|Ee0|2

∫

Ω

|Es∞|2 dΩ . (1.85)

The scattering cross-section can be expressed in terms of the elements of thephase matrix and the Stokes parameters of the incident wave. Taking intoaccount the expressions of Ie and Is, and using (1.76) we obtain

Cscat =1Ie

∫

Ω

Is (er) dΩ (er)

=1Ie

∫

Ω

[Z11 (er,ek) Ie + Z12 (er,ek) Qe

+ Z13 (er,ek) Ue + Z14 (er,ek) Ve] dΩ (er) . (1.86)

The phase function is related to the differential scattering cross-section by therelation

p (er,ek) =4π

Cscat |Ee0|2|Es∞ (er)|2

and in view of (1.85) we see that p is dimensionless and normalized, i.e.,

14π

∫

Ω

p dΩ = 1 .

The mean direction of propagation of the scattered field is defined as

g =1

Cscat |Ee0|2∫

Ω

|Es∞ (er)|2 er dΩ (er) (1.87)

and obviously

g =1

CscatIe

∫

Ω

Is (er) er dΩ (er)

=1

CscatIe

∫

Ω

[Z11 (er,ek) Ie + Z12 (er,ek) Qe

+ Z13 (er,ek) Ue + Z14 (er,ek) Ve] er dΩ (er) .

The asymmetry parameter 〈cos Θ〉 is the dot product between the vector gand the incident direction ek [17, 169],


〈cos Θ〉 = g · ek =1

Cscat |Ee0|2∫

Ω

|Es∞ (er)|2 er · ek dΩ (er)

=14π

∫

Ω

p (er,ek) cos Θ dΩ (er) ,

where cos Θ = er · ek, and it is apparent that the asymmetry parameter isthe average cosine of the scattering angle Θ. If the particle scatters morelight toward the forward direction (Θ = 0), 〈cos Θ〉 is positive and 〈cos Θ〉 isnegative if the scattering is directed more toward the backscattering direction(Θ = 180). If the scattering is symmetric about a scattering angle of 90,〈cos Θ〉 vanishes.

1.4.5 Optical Theorem

The expression of extinction has been derived by integrating the Poynting vec-tor over an auxiliary surface around the particle. This derivation emphasizedthe conservation of energy aspect of extinction: extinction is the combinedeffect of absorption and scattering. A second derivation emphasizes the inter-ference aspect of extinction: extinction is a result of the interference betweenthe incident and forward scattered light [17]. Applying Green’s second vectortheorem to the vector fields Es and E∗

e in the domain D bounded by S andSc, we obtain∫

S

n · (Es × H∗e + E∗

e × Hs) dS =∫

Sc

n · (Es × H∗e + E∗

e × Hs) dS

and further∫

S

n · Re Es × H∗e + E∗

e × HsdS =∫

Sc


e × HsdS .

This result together with (1.82) and the identity ReEe×H∗s = ReE∗

e×Hsgive

Wext = −12

∫

S


e × HsdS ,

whence, using the explicit expressions for Ee and He, we derive

Wext =12Re∫

S

[E∗e0 · hs (r′)

−√

εs

µs(ek × E∗

e0) · es (r′)]

e−jke·r′dS (r′)

.

In the integral representation for the electric far-field pattern (cf. (1.69)) weset er = ek, take the dot product between Es∞(ek) and E∗

e0, and obtain


E∗e0 · Es∞ (ek) =

jks

4π

√µs

εs

∫

S

E∗e0 · hs (r′)

−√

εs

µs(ek × E∗

e0) · es (r′)

e−jke·r′dS (r′) .

The last two relations imply that

Wext =12Re√

εs

µs

4π

jksE∗

e0 · Es∞ (ek)

and further that

Cext =4π

ks |Ee0|2Im E∗

e0 · Es∞ (ek) . (1.88)

The above relation is a representation of the optical theorem, and since theextinction cross-section is in terms of the scattering amplitude in the forwarddirection, the optical theorem is also known as the extinction theorem orthe forward scattering theorem. This fundamental relation can be used tocompute the extinction cross-section when the imaginary part of the scatteringamplitude in the forward direction is known accurately. In view of (1.88) and(1.74), and taking into account the explicit expressions of the elements of theextinction matrix we see that

Cext =1Ie

[K11 (ek) Ie + K12 (ek) Qe + K13 (ek) Ue + K14 (ek) Ve] . (1.89)

1.4.6 Reciprocity

The tensor scattering amplitude satisfies a useful symmetry property whichis referred to as reciprocity. As a consequence, reciprocity relations for theamplitude, phase and extinction matrices can be derived. Reciprocity is amanifestation of the symmetry of the scattering process with respect to aninversion of time and holds for particles in arbitrary orientations [169]. Inorder to derive this property we use the following result: if E1, H1 and E2,H2 are the total fields generated by the incident fields Ee1, He1 and Ee2,He2, respectively, we have∫

S

n · (E2 × H1 − E1 × H2) dS =∫

Sc

n · (E2 × H1 − E1 × H2) dS ,

where as before, Sc is an auxiliary surface enclosing S. Since E1 and E2 aresource free in the domain bounded by S and Sc, the above equation followsimmediately from Green second vector theorem. Further, applying Green’ssecond vector theorem to the internal fields Ei1 and Ei2 in the domain Di,and taking into account the boundary conditions n × Ei1,2 = n × E1,2 andn × H i1,2 = n × H1,2 on S, yields


∫

S

n · (E2 × H1 − E1 × H2) dS = 0 ,

whence ∫

Sc

n · (E2 × H1 − E1 × H2) dS = 0 , (1.90)

follows. We take the surface Sc as a large sphere of outward unit normalvector er, consider the limit when the radius R becomes infinite, and writethe integrand in (1.90) as

E2 × H1 − E1 × H2

= Ee2 × He1 − Ee1 × He2 + Es2 × Hs1 − Es1 × Hs2

+Es2 × He1 − Ee1 × Hs2 + Ee2 × Hs1 − Es1 × He2 .

The Green second vector theorem applied to the incident fields Ee1 and Ee2 inany bounded domain shows that the vector plane wave terms do not contributeto the integral. Furthermore, using the far-field representation

Es2 × Hs1 − Es1 × Hs2

=e2jksr

r2

Es∞2 × Hs∞1 − Es∞1 × Hs∞2 + O

(1r

),

and taking into account the transversality of the far-field patterns

Es∞2 × Hs∞1 − Es∞1 × Hs∞2 = 0 ,

we see that the integral over the scattered wave terms also vanishes. Thus,(1.90) implies that∫

Sc

er ·(Es2 × He1 − Ee1 × Hs2) dS =∫

Sc

er ·(Es1 × He2 − Ee2 × Hs1) dS .

(1.91)For plane wave incidence,

Eeu(r) = Ee0u exp (jkeu · r) , keu = kseku , u = 1, 2 ,

the integrands in (1.91) contain the term exp(jksRek1,2 ·er). Since R is large,the stationary point method can be used to compute the integrals accordinglyto the basic result

kR

2πj

∫ 2π

0

∫ π

0

g (θ, ϕ) ejkRf(θ,ϕ)dθ dϕ =1

√fθθfϕϕ − f2

θϕ

g (θst, ϕst) ejkRf(θst,ϕst) ,

(1.92)


where (θst, ϕst) is the stationary point of f , fθθ = ∂2f/∂θ2, fϕϕ = ∂2f/∂ϕ2

and fθϕ = ∂2f/∂θ∂ϕ. The integrals in (1.91) are then given by∫

Sc

er · (Es2 × He1 − Ee1 × Hs2) dS

= −4πjR

ks

√εs

µsEs∞2 (−ek1) · Ee01e−jksR ,

∫

Sc

er · (Es1 × He2 − Ee2 × Hs1) dS

= −4πjR

ks

√εs

µsEs∞1 (−ek2) · Ee02e−jksR ,

and we deduce the reciprocity relation for the far-field pattern (Fig. 1.12):

Es∞2 (−ek1) · Ee01 = Es∞1 (−ek2) · Ee02 .

The above relation gives

Ee01 · A (−ek1,ek2) · Ee02 = Ee02 · A (−ek2,ek1) · Ee01

and since a · D · b = b · DT · a, and Ee01 and Ee02 are arbitrary transversevectors, the following constraint on the tensor scattering amplitude:

A (−ek2,−ek1) = AT

(ek1,ek2)

follows. This is the reciprocity relation for the tensor scattering amplitudewhich relates scattering from the direction −ek1 into −ek2 to scattering fromek2 to ek1. Taking into account the representation of the amplitude matrixelements in terms of the tensor scattering amplitude and the fact that for

O

D i

Ske1

Ee1

Es1

Ee2

ke2

Es2(-ek1

(-ek2)

)

Fig. 1.12. Illustration of the reciprocity relation

1.5 Transition Matrix 57

e′k = −ek we have e′

β = eβ and e′α = −eα, we obtain the reciprocity

relation for the amplitude matrix:

S (−ek2,−ek1) =

[Sθβ (ek1,ek2) −Sϕβ (ek1,ek2)−Sθα (ek1,ek2) Sϕα (ek1,ek2)

]

.

If we choose ek1 = −ek2 = −ek, we obtain

Sϕβ (−ek,ek) = −Sθα (−ek,ek) ,

which is a representation of the backscattering theorem [169].From the reciprocity relation for the amplitude matrix we easily derive the

reciprocity relation for the phase and extinction matrices:

Z (−ek,−er) = QZT (er,ek) Q

and

K (−ek) = QKT (ek) Q ,

respectively, where Q = diag[1, 1,−1, 1].The reciprocity relations can be used in practice for testing the results of

theoretical computations and laboratory measurements. It should be remarkedthat reciprocity relations give also rise to symmetry relations for the dyadicGreen functions [229].

1.5 Transition Matrix

The transition matrix relates the expansion coefficients of the incident andscattered fields. The existence of the transition matrix is “postulated” by theT -Matrix Ansatz and is a consequence of the series expansions of the incidentand scattered fields and the linearity of the Maxwell equations. Historically,the transition matrix has been introduced within the null-field method for-malism (see [253, 256]), and for this reason, the null-field method has oftenbeen referred to as the T -matrix method. However, the null-field methodis only one among many methods that can be used to compute the transi-tion matrix. The transition matrix can also be derived in the framework ofthe method of moments [88], the separation of variables method [208], thediscrete dipole approximation [151] and the point matching method [181].Rother et al. [205] found a general relation between the surface Green func-tion and the transition matrix for the exterior Maxwell problem, which inprinciple, allows to compute the transition matrix with the finite-differencetechnique.

In this section we review the general properties of the transition matrixsuch as unitarity and symmetry and discuss analytical procedures for aver-aging scattering characteristics over particle orientations. These procedures


relying on the rotation transformation rule for vector spherical wave func-tions are of general use because an explicit expression of the transition matrixis not required. In order to simplify our analysis we consider a vector planewave of unit amplitude

Ee(r) = epolejke·r , He(r) =√

εs

µsek × epolejke·r ,

where epol · ek = 0 and |epol| = 1.

1.5.1 Definition

Everywhere outside the (smallest) sphere circumscribing the particle it is ap-propriate to expand the scattered field in terms of radiating vector sphericalwave functions

Es(r) =∞∑

n=1

n∑

m=−n

fmnM3mn (ksr) + gmnN3

mn (ksr) (1.93)

and the incident field in terms of regular vector spherical wave functions

Ee(r) =∞∑

n=1

n∑

m=−n


mn (ksr) . (1.94)

Within the vector spherical wave formalism, the scattering problem is solvedby determining fmn and gmn as functions of amn and bmn. Due to the lin-earity relations of the Maxwell equations and the constitutive relations, therelation between the scattered and incident field coefficients must be linear.This relation is given by the so-called transition matrix T as follows [256]

[fmn

gmn

]= T

[amn

bmn

]=[

T 11 T 12

T 21 T 22

] [amn

bmn

]. (1.95)

Essentially, the transition matrix depends on the physical and geometricalcharacteristics of the particle and is independent on the propagation directionand polarization states of the incident and scattered field.

If the transition matrix is known, the scattering characteristics (introducedin Sect. 1.4) can be readily computed. Taking into account the asymptoticbehavior of the vector spherical wave functions we see that the far-field patterncan be expressed in terms of the elements of the transition matrix by therelation

Es∞ (er) =1ks

∑

n,m

(−j)n+1 [fmnmmn(er) + jgmnnmn(er)]

=1ks

∑

n,m

∑

n1,m1

(−j)n+1

×[(

T 11mn,m1n1

am1n1 + T 12mn,m1n1

bm1n1

)mmn(er)

+ j(T 21

mn,m1n1am1n1 + T 22

mn,m1n1bm1n1

)nmn(er)

]. (1.96)


To derive the expressions of the tensor scattering amplitude and amplitudematrix, we consider the scattering and incident directions er and ek, andexpress the vector spherical harmonics as

xmn(er) = xmn,θ (er) eθ + xmn,ϕ (er) eϕ ,

xmn(ek) = xmn,β (ek) eβ + xmn,α (ek) eα ,

where xmn stands for mmn and nmn. Recalling the expressions of the incidentfield coefficients for a plane wave excitation (cf. (1.26))

amn = 4jnepol · m∗mn (ek) ,

bmn = −4jn+1epol · n∗mn (ek) ,

and using the definition of the tensor scattering amplitude (cf. (1.71)), weobtain

A (er,ek) =4ks

∑

n,m

∑

n1,m1

(−j)n+1 jn1[

T 11mn,m1n1

mmn (er)

+ jT 21mn,m1n1

nmn (er)]⊗ m∗

m1n1(ek) +

[−jT 12

mn,m1n1mmn (er)

+ T 22mn,m1n1

nmn (er)]⊗ n∗

m1n1(ek)

.

In view of (1.75), the elements of the amplitude matrix are given by

Spq (er,ek) =4ks

∑

n,m

∑

n1,m1

(−j)n+1 jn1[

T 11mn,m1n1

m∗m1n1,q (ek)

− jT 12mn,m1n1

n∗m1n1,q (ek)

]mmn,p (er)

+j[T 21

mn,m1n1m∗

m1n1,q (ek) ,

− jT 22mn,m1n1

n∗m1n1,q (ek)

]nmn,p (er)

(1.97)

for p = θ, ϕ and q = β, α. For a vector plane wave linearly polarized in theβ-direction, amn = 4jnm∗

mn,β and bmn = −4jn+1n∗mn,β , and Sθβ = Es∞,θ and

Sϕβ = Es∞,ϕ. Analogously, for a vector plane wave linearly polarized in theα-direction, amn = 4jnm∗

mn,α and bmn = −4jn+1n∗mn,α, and Sθα = Es∞,θ

and Sϕα = Es∞,ϕ. In practical computer calculations, this technique, relyingon the computation of the far-field patterns for parallel and perpendicularpolarizations, can be used to determine the elements of the amplitude matrix.

For our further analysis, it is more convenient to express the above equa-tions in matrix form. Defining the vectors

s =[

fmn

gmn

], e =

[amn

bmn

]


and the “augmented” vector of spherical harmonics

v (er) =[

(−j)nmmn (er)

j (−j)nnmn (er)

]

we see that

Es∞ (er) = − jks

vT (er) s = − jks

vT (er) Te = − jks

eTT Tv (er) , (1.98)

and, since e = 4epol · v∗(ek), we obtain

Spq (er,ek) = − 4jks

vTp (er) Tv∗

q (ek) = − 4jks

v†q (ek) T Tvp (er) . (1.99)

The superscript † means complex conjugate transpose, and

vp (·) =[

(−j)nmmn,p (·)

j (−j)nnmn,p (·)

],

where p = θ, ϕ for the er-dependency and p = β, α for the ek-dependency.The extinction and scattering cross-sections can be expressed in terms

of the expansion coefficients amn, bmn, fmn and gmn. Denoting by Sc thecircumscribing sphere of outward unit normal vector er and radius R, andusing the definition of the extinction cross-section (cf. (1.82) and (1.83) with|Ee0| = 1), yields

Cext = −√

µs

εs

∫

Sc

er · Re Ee × H∗s + Es × H∗

edS

= −Re

j∑

n,m

∑

n1,m1

∫

Sc

(fmna∗

m1n1+ gmnb∗m1n1

)

×[(

er × M3mn

)· N1∗

m1n1+(er × N3

mn

)· M1∗

m1n1

]

+(fmnb∗m1n1

+ gmna∗m1n1

)

×[(

er × M3mn

)· M1∗

m1n1+(er × N3

mn

)· N1∗

m1n1

]dS

.

Taking into account the orthogonality relations of the vector spherical wavefunctions on a spherical surface (cf. (B.18) and (B.19)) we obtain

Cext = −Re

jπR

ks

∞∑

n=1

n∑

m=−n

(fmna∗mn + gmnb∗mn)

×

h(1)n (ksR) [ksRjn(ksR)]′ − jn(ksR)

[ksRh(1)

n (ksR)]′

,

whence, using the Wronskian relation

h(1)n (ksR) [ksRjn(ksR)]′ − jn(ksR)

[ksRh(1)

n (ksR)]′

= − jksR

,


we end up with

Cext = − π

k2s

∞∑

n=1

n∑

m=−n

Re fmna∗mn + gmnb∗mn . (1.100)

For the scattering cross-section, the expansion of the far-field pattern in termsof vector spherical harmonics (cf. (1.96)) and the orthogonality relations ofthe vector spherical harmonics (cf. (B.12) and (B.13)), yields

Cscat =π

k2s

∞∑

n=1

n∑

m=−n

|fmn|2 + |gmn|2 . (1.101)

Thus, the extinction cross-section is given by the expansion coefficients of theincident and scattered field, while the scattering cross-section is determinedby the expansion coefficients of the scattered field.

1.5.2 Unitarity and Symmetry

It is of interest to investigate general constraints of the transition matrix suchas unitarity and symmetry. These properties can be established by applyingthe principle of conservation of energy to nonabsorbing particles (εi > 0 andµi > 0). We begin our analysis by defining the S matrix in terms of the Tmatrix by the relation

S = I + 2T ,

where I is the identity matrix. In the literature, the S matrix is also knownas the scattering matrix but in our analysis we avoid this term because thescattering matrix will have another significance.

First we consider the unitarity property. Application of the divergencetheorem to the total fields E = Es +Ee and H = Hs +He in the domain Dbounded by the surface S and a spherical surface Sc situated in the far-fieldregion, yields∫

D

∇·(E × H∗) dV = −∫

S

n ·(E × H∗) dS+∫

Sc

er ·(E × H∗) dS . (1.102)

We consider the real part of the above equation and since the bounded domainD is assumed to be lossless (εs > 0 and µs > 0) it follows that:

Re ∇ · (E × H∗) = Re

jk0µs |H|2 − jk0εs |E|2

= 0 in D . (1.103)

On the other hand, taking into account the boundary conditions n × Ei =n × E and n × H i = n × H on S, we have∫

S

n · (E × H∗) dS =∫

S

n · (Ei × H∗i ) dS =

∫

Di

∇ · (Ei × H∗i ) dV


and since for nonabsorbing particles

Re ∇ · (Ei × H∗i ) = Re

jk0µi |H i|2 − jk0εi |Ei|2

= 0 in Di ,

we obtain ∫

S

n · Re E × H∗dS = 0 . (1.104)

Combining (1.102), (1.103) and (1.104) we deduce that∫

Sc

er · Re E × H∗dS = 0 . (1.105)

We next seek to find a series representation for the total electric field. For thispurpose, we use the decomposition

(M1

mn

N1mn

)

=12

[(M3

mn

N3mn

)

+

(M2

mn

N2mn

)]

,

where the vector spherical wave functions M2mn and N2

mn have the sameexpressions as the vector spherical wave functions M3

mn and N3mn, but with

the spherical Hankel functions of the second kind h(2)n in place of the spherical

Hankel functions of the first kind h(1)n . It should be remarked that for real

arguments x, h(2)n (x) = [h(1)

n (x)]∗. In the far-field region

M2mn(kr) =

e−jkr

kr

jn+1mmn(θ, ϕ) + O

(1r

),

N2mn(kr) =

e−jkr

kr

jnnmn(θ, ϕ) + O

(1r

),

as r → ∞, and we see that M2mn and N2

mn behave as incoming transversevector spherical waves.

The expansion of the incident field then becomes

Ee =∑

n,m

amnM1mn + bmnN1

mn

=12

∑

n,m

amnM3mn + bmnN3

mn +12

∑

n,m

amnM2mn + bmnN2

mn ,

whence

E =∑

n,m

(fmn +

12amn

)M3

mn +(

gmn +12bmn

)N3

mn

+12

∑

n,m

amnM2mn + bmnN2

mn ,


follows. Setting

cmn = 2fmn + amn ,

dmn = 2gmn + bmn ,

and using the T -matrix equation, yields[

cmn

dmn

]= S[

amn

bmn

]= (I + 2T )

[amn

bmn

].

The coefficients amn and bmn are determined by the incoming field. Since inthe far-field region M2

mn and N2mn become incoming vector spherical waves,

we see that the S matrix determines how an incoming vector spherical waveis scattered into the same one.

In the far-field region, the total electric field

E =12

∑

n,m

cmnM3mn + dmnN3

mn +12

∑

n,m

amnM2mn + bmnN2

mn

can be expressed as a superposition of outgoing and incoming transversespherical waves

E(r) =ejksr

r

E(1)

∞ (er) + O(

1r

)+

e−jksr

r

E(2)

∞ (er) + O(

1r

),

r → ∞ (1.106)

with

E(1)∞ (er) =

12ks

∑

n,m

(−j)n+1 [cmnmmn(er) + jdmnnmn(er)] ,

E(2)∞ (er) =

12ks

∑

n,m

jn+1 [amnmmn(er) − jbmnnmn(er)] .

For the total magnetic field we proceed analogously and obtain

H(r) =ejksr

r

H(1)

∞ (er) + O(

1r

)+

e−jksr

r

H(2)

∞ (er) + O(

1r

)

r → ∞

with

H(1)∞ =

√εs

µser × E(1)

∞ ,

H(2)∞ = −

√εs

µser × E(2)

∞ .


Thus

Re er · (E × H∗) =1r2

√εs

µs

∣∣∣E(1)

∞

∣∣∣2

−∣∣∣E(2)

∞

∣∣∣2

+ O(

1r

), r → ∞

and (1.105) yields∫

Ω

(∣∣∣E(1)

∞

∣∣∣2

−∣∣∣E(2)

∞

∣∣∣2)

dΩ = 0 . (1.107)

The orthogonality relations of the vector spherical harmonics on the unitsphere give

∫

Ω

∣∣∣E(1)

∞

∣∣∣2

dΩ =π

4k2s

[a∗mn, b∗mn]S†S

[amn

bmn

],

∫

Ω

∣∣∣E(2)

∞

∣∣∣2

dΩ =π

4k2s

[a∗mn, b∗mn]

[amn

bmn

](1.108)

and since the incident field is arbitrarily, (1.107) and (1.108) implies that[217,228,256]

S†S = I . (1.109)

The above relation is the unitary condition for nonabsorbing particles. Interms of the transition matrix, this condition is

T †T = −12(T + T †) ,

or explicitly

2∑

k=1

∞∑

n′=1

n′∑

m′=−n′

T ki∗m′n′,mnT kj

m′n′,m1n1= −1

2(T ji∗

m1n1,mn + T ijmn,m1n1

). (1.110)

For absorbing particles, the integral in (1.107) is negative. Consequently, theequality in (1.110) transforms into an inequality which is equivalent to thecontractivity of the S matrix [169]. Taking the trace of (1.110), Mishchenkoet al. [169] derived an equality (inequality) between the T -matrix elements ofan axisymmetric particle provided that the z-axis of the particle coordinatesystem is directed along the axis of symmetry.

To obtain the symmetry relation we proceed as in the derivation of thereciprocity relation for the tensor scattering amplitude, i.e., we consider theelectromagnetic fields Eu, Hu generated by the incident fields Eeu, Heu, withu = 1, 2. The starting point is the integral (cf. (1.90))

∫

Sc

er · (E2 × H1 − E1 × H2) dS = 0


over a spherical surface Sc situated in the far-field region. Then, using theasymptotic form (cf. (1.106))

Eu(r) =ejksr

r

E(1)

u∞(er) + O(

1r

)+

e−jksr

r

E(2)

u∞(er) + O(

1r

),

r → ∞

for u = 1, 2, we obtain∫

Ω

(er × E

(1)2∞

)·(er × E

(2)1∞

)dΩ =

∫

Ω

(er × E

(1)1∞

)·(er × E

(2)2∞

)dΩ .

(1.111)Taking into account the vector spherical harmonic expansions of the far-fieldpatterns E

(1)u∞ and E

(2)u∞, u = 1, 2, and the relations er × mmn = nmn and

er × nmn = −mmn, we see that

∫

Ω

(er × E

(1)2∞

)·(er × E

(2)1∞

)dΩ =

π

4k2s

[a1,m1n1 , b1,m1n1 ]

[c2,−m1n1

d2,−m1n1

]

,

∫

Ω

(er × E

(1)1∞

)·(er × E

(2)2∞

)dΩ =

π

4k2s

[c1,mn, d1,mn]

[a2,−mn

b2,−mn

]

,

where au,mn, bu,mn are the expansion coefficients of the far-field pattern E(2)u∞,

while cu,mn, du,mn are the expansion coefficients of the far-field pattern E(1)u∞.

Consequently, (1.111) can be written in matrix form as

[a1,m1n1 , b1,m1n1 ]

[S11−m1n1,−mn S12

−m1n1,−mn

S21−m1n1,−mn S22

−m1n1,−mn

][a2,−mn

b2,−mn

]

= [a1,m1n1 , b1,m1n1 ]

[S11

mn,m1n1S21

mn,m1n1

S12mn,m1n1

S22mn,m1n1

][a2,−mn

b2,mn

]

and since the above equation holds true for any incident field, we find that

Sijmn,m1n1

= Sji−m1n1,−mn

and further thatT ij

mn,m1n1= T ji

−m1n1,−mn (1.112)

for i, j = 1, 2. This relation reflects the symmetry property of the transitionmatrix and is of basic importance in practical computer calculations. Wenote that the symmetry relation (1.112) can be obtained directly from thereciprocity relation for the tensor scattering amplitude

A (θ1, ϕ1; θ2, ϕ2) = AT

(π − θ2, π + ϕ2;π − θ1, π + ϕ1)


and the identities

mmn (π − θ, π + ϕ) = (−1)nmmn (θ, ϕ) ,

nmn (π − θ, π + ϕ) = (−1)n+1nmn (θ, ϕ) ,

and

m−mn (θ, ϕ) = m∗mn (θ, ϕ) ,

n−mn (θ, ϕ) = n∗mn (θ, ϕ) .

Additional properties of the transition matrix for particles with specific sym-metries will be discussed in the next chapter. The “exact” infinite transitionmatrix satisfies the unitarity and symmetry conditions (1.110) and (1.112),respectively. However, in practical computer calculations, the truncated tran-sition matrix may not satisfies these conditions and we can test the unitarityand symmetry conditions to get a rough idea regarding the convergence to beexpected in the solution computation.

Remark. In the above analysis, the incident field is a vector plane wavewhose source is situated at infinity. Other incident fields than vector planewaves can be considered, but we shall assume that the source of the inci-dent field lies outside the circumscribing sphere Sc. In this case, the incidentfield is regular everywhere inside the circumscribing sphere, and both expan-sions (1.93) and (1.94) are valid on Sc. The T -matrix equation holds trueat finite distances from the particle (not only in the far-field region), andtherefore, the transition matrix is also known as the “nonasymptotic vector-spherical-wave transition matrix”. The properties of the transition matrix(unitarity and symmetry) can also be established by considering the energyflow through a finite sphere Sc [238]. If we now let the source of the incidentfield recede to infinity, we can let the surface Sc follow, i.e., we can considerthe case of an arbitrary large sphere and this brings us to the precedentanalysis.

1.5.3 Randomly Oriented Particles

In the following analysis we consider scattering by an ensemble of ran-domly oriented, identical particles. Random particle orientation means thatthe orientation distribution of the particles is uniform. As a consequenceof random particle orientation, the scattering medium is macroscopicallyisotropic, i.e., the scattering characteristics are independent of the incidentand scattering directions ek and er, and depend only on the angle bet-ween the unit vectors ek and er. For this type of scattering problem, itis convenient to direct the Z-axis of the global coordinate system alongthe incident direction and to choose the XZ-plane as the scattering plane(Fig. 1.13).


eβ

eα

ek

X

YO

Z

epolαpol

er

θ

Fig. 1.13. The Z-axis of the global coordinate system is along the incident directionand the XZ-plane is the scattering plane

General Considerations

The phase matrix of a volume element containing randomly oriented particlescan be written as

Z (er,ek) = Z (θ, ϕ = 0, β = 0, α = 0) ,

where, in general, θ and ϕ are the polar angles of the scattering directioner, and β and α are the polar angles of the incident direction ek. The phasematrix Z(θ, 0, 0, 0) is known as the scattering matrix F and relates the Stokesparameters of the incident and scattered fields defined with respect to thescattering plane. Taking into account that for an incident direction (β, α),the backscattering direction is (π − β, α + π), the complete definition of thescattering matrix is [169]

F (θ) =

Z(θ, 0, 0, 0) ,

Z (π, π, 0, 0) ,

θ ∈ [0, π),θ = π.

The scattering matrix of a volume element containing randomly oriented par-ticles has the following structure:

F (θ) =

⎡

⎢⎢⎢⎣

F11(θ) F12(θ) F13(θ) F14(θ)F12(θ) F22(θ) F23(θ) F24(θ)−F13(θ) −F23(θ) F33(θ) F34(θ)F14(θ) F24(θ) −F34(θ) F44(θ)

⎤

⎥⎥⎥⎦

. (1.113)

If each particle has a plane of symmetry or, equivalently, the particles andtheir mirror-symmetric particles are present in equal numbers, the scattering


medium is called macroscopically isotropic and mirror-symmetric. Note thatrotationally symmetric particles are obviously mirror-symmetric with respectto the plane through the axis of symmetry. Because of symmetry, the scat-tering matrix of a macroscopically isotropic and mirror-symmetric scatteringmedium has the following block-diagonal structure [103,169]:

F (θ) =

⎡

⎢⎢⎢⎣

F11(θ) F12(θ) 0 0F12(θ) F22(θ) 0 0

0 0 F33(θ) F34(θ)0 0 −F34(θ) F44(θ)

⎤

⎥⎥⎥⎦

. (1.114)

The phase matrix can be related to the scattering matrix by using the rotationtransformation rule (1.22), and this procedure involves two rotations as shownin Fig. 1.14. Taking into account that the scattering matrix relates the Stokesvectors of the incident and scattered fields specified relative to the scatteringplane, I ′

s = (1/r2)F (Θ)I ′e, and using the transformation rule of the Stokes

vectors under coordinate rotations I ′e = L(σ1)Ie and Is = L(−σ2)I ′

s, weobtain

Z (θ, ϕ, β, α) = L (−σ2) F (Θ) L (σ1) ,

where

cos Θ = ek · er = cos β cos θ + sin β sin θ cos (ϕ − α) ,

eβ

ek

O

Z

β

X

er

eθ

θ Θ

e’θ

e’β

σ1

σ2

Y

Fig. 1.14. Incident and scattering directions ek and er. The scattering matrixrelates the Stokes vectors of the incident and scattered fields specified relative tothe scattering plane


cos σ1 = e′α · eα = − sin β cos θ − cos β sin θ cos (ϕ − α)

sinΘ,

cos σ2 = e′ϕ · eϕ =

cos β sin θ − sin β cos θ cos (ϕ − α)sin Θ

.

For an ensemble of randomly positioned particles, the waves scattered bydifferent particles are random in phase, and the Stokes parameters of theseincoherent waves add up. Therefore, the scattering matrix for the ensemble isthe sum of the scattering matrices of the individual particles:

F = N 〈F 〉 ,

where N is number of particles and 〈F 〉 denotes the ensemble-average scat-tering matrix per particle. Similar relations hold for the extinction matrixand optical cross-sections. Because the particles are identical, the ensemble-average of a scattering quantity X is the orientation-averaged quantity

〈X〉 =1

8π2

∫ 2π

0

∫ 2π

0

∫ π

0

X (αp, βp, γp) sinβp dβp dαp dγp ,

where αp, βp and γp are the particle orientation angles.In the following analysis, the T matrix formulation is used to derive

efficient analytical techniques for computing 〈X〉. These methods work muchfaster than the standard approaches based on the numerical averaging ofresults computed for many discrete orientations of the particle. We begin withthe derivation of the rotation transformation rule for the transition matrixand then compute the orientation-averaged transition matrix, optical cross-sections and extinction matrix. An analytical procedure for computing theorientation-averaged scattering matrix will conclude our analysis.

Rotation Transformation of the Transition Matrix

To derive the rotation transformation rule for the transition matrix we assumethat the orientation of the particle coordinate system Oxyz with respect tothe global coordinate system OXY Z is specified by the Euler angles αp, βp

and γp.In the particle coordinate system, the expansions of the incident and scat-

tered field are given by

Ee (r, θ, ϕ) =∞∑

n=1

n∑

m=−n

amnM1mn (ksr, θ, ϕ) + bmnN1

mn (ksr, θ, ϕ) ,

Es (r, θ, ϕ) =∞∑

n=1

n∑

m=−n

fmnM3mn (ksr, θ, ϕ) + gmnN3

mn (ksr, θ, ϕ) ,


while in the global coordinate system, these expansions take the form

Ee (r, Φ, Ψ) =∞∑

n=1

n∑

m=−n

amnM1mn (ksr, Φ, Ψ) + bmnN1

mn (ksr, Φ, Ψ) ,

Es (r, Φ, Ψ) =∞∑

n=1

n∑

m=−n

fmnM3mn (r, Φ, Ψ) + gmnN3

mn (ksr, Φ, Ψ) .

Assuming the T -matrix equations s = Te and s = T e, our task is to expressthe transition matrix in the global coordinate system T in terms of the tran-sition matrix in the particle coordinate system T . Defining the “augmented”vectors of spherical wave functions in each coordinate system

w1,3 (ksr, θ, ϕ) =

[M1,3

mn (ksr, θ, ϕ)N1,3

mn (ksr, θ, ϕ)

]

and

w1,3 (ksr, Φ, Ψ) =

[M1,3

mn (ksr, Φ, Ψ)N1,3

mn (ksr, Φ, Ψ)

]

,

and using the rotation addition theorem for vector spherical wave functions,we obtain

Ee = eTw1 = eTw1 = eTR (αp, βp, γp) w1 ,

Es = sTw3 = sTw3 = sTR (−γp,−βp,−αp) w3 .

Consequently

e = RT (αp, βp, γp) e ,

s = RT (−γp,−βp,−αp) s ,

and therefore,

T (αp, βp, γp) = RT (−γp,−βp,−αp) TRT (αp, βp, γp) . (1.115)

The explicit expression of the matrix elements is [169,228,233]

T ijmn,m1n1

(αp, βp, γp) =n∑

m′=−n

n1∑

m′1=−n1

Dnm′m (−γp,−βp,−αp) T ij

m′n,m′1n1

×Dn1m1m′

1(αp, βp, γp) (1.116)

for i, j = 1, 2.Because the elements of the amplitude matrix can be expressed in terms of

the elements of the transition matrix, the above relation can be used to expressthe elements of the amplitude matrix as functions of the particle orientationangles αp, βp and γp. The properties of the Wigner D-functions can then beused to compute the integrals over the particle orientation angles.


Orientation-Averaged Transition Matrix

The elements of the orientation-averaged transition matrix with respect tothe global coordinate system are given by

⟨T ij

mn,m1n1

⟩=

18π2

∫ 2π

0

∫ 2π

0

∫ π

0

T ijmn,m1n1

(αp, βp, γp) sinβp dβp dαp dγp

=1

8π2

n∑

m′=−n

n1∑

m′1=−n1

T ijm′n,m′

1n1

∫ 2π

0

∫ 2π

0

∫ π

0

Dnm′m (−γp,−βp,−αp)

×Dn1m1m′

1(αp, βp, γp) sinβp dβp dαp dγp . (1.117)

Using the definition of the Wigner D-functions (cf. (B.34)), the symmetryrelation of the Wigner d-functions dn

m′m(−βp) = dnmm′(βp), and integrating

over αp and γp, yields

⟨T ij

mn,m1n1

⟩=

12

n∑

m′=−n

n1∑

m′1=−n1

∆m′m∆m1m′1δm′m′

1δmm1

×T ijm′n,m′

1n1

∫ π

0

dnmm′ (βp) dn1

m1m′1(βp) sinβp dβp ,

where ∆mm′ is given by (B.36). Taking into account the identities: ∆m′m =∆mm′ and (∆mm′)2 = 1, and using the orthogonality property of the d-functions (cf. (B.43)), we obtain [168,169]

⟨T ij

mn,m1n1

⟩= δmm1δnn1t

ijn (1.118)

with

tijn =1

2n + 1

n∑

m′=−n

T ijm′n,m′n . (1.119)

The above relation provides a simple analytical expression for the orientation-averaged transition matrix in terms of the transition matrix in the particlecoordinate system. The orientation-averaged 〈T ij〉 matrices are diagonal andtheir elements do not depend on the azimuthal indices m and m1.

Orientation-Averaged Extinction and Scattering Cross-Sections

In view of the optical theorem, the orientation-averaged extinction cross-section is (cf. (1.88) with |Ee0| = 1)

〈Cext〉 =4π

ksIm⟨

e∗pol · Es∞ (ez)

⟩.


Considering the expansion of the far-field pattern in the global coordinate sys-tem (cf. (1.96)), taking the average and using the expression of the orientation-averaged transition matrix (cf. (1.118) and (1.119)), gives

⟨e∗

pol · Es∞ (ez)⟩

=1ks

∑

n,m

(−j)n+1[(

t11n amn + t12n bmn

)e∗

pol · mmn (ez)

+ j(t21n amn + t22n bmn

)e∗

pol · nmn (ez)]

,

where the summation over the index m involves the values −1 and 1. Inthe next chapter we will show that for axisymmetric particles, T ij

−mn,−mn =−T ij

mn,mn and T ij0n,0n = 0 for i = j, while for particles with a plane of sym-

metry, T ijmn,mn = 0 for i = j. Thus, for macroscopically isotropic and mirror-

symmetric media, (1.119) gives t12n = t21n = 0. Further, using the expressionsof the incident field coefficients (cf. (1.26))

amn = 4jnepol · m∗mn (ez) ,

bmn = −4jn+1epol · n∗mn (ez) ,

(1.120)

and the special values of the vector spherical harmonics in the forward direc-tion

mmn (ez) =√

2n+14 (jmex − ey) ,

nmn (ez) =√

2n+14 (ex + jmey) ,

(1.121)

we obtain [163]

〈Cext〉 = −2π

k2s

Re

∞∑

n=1

(2n + 1)(t11n + t22n

)

,

= −2π

k2s

Re

∞∑

n=1

n∑

m=−n

T 11mn,mn + T 22

mn,mn

. (1.122)

The above relation shows that the orientation-averaged extinction cross-section for macroscopically isotropic and mirror-symmetric media is deter-mined by the diagonal elements of the transition matrix in the particle coor-dinate system. The same result can be established if we consider an ensembleof randomly oriented particles (with t12n = 0 and t21n = 0) illuminated by alinearly polarized plane wave (with real polarization vector epol).

For an arbitrary excitation, the scattering cross-section can be expressedin the global coordinate system as

Cscat =π

k2s

∞∑

n=1

n∑

m=−n

∣∣∣fmn

∣∣∣2

+ |gmn|2 =π

k2s

s†s ,


whence, using the T -matrix equation s = T e, we obtain

〈Cscat〉 =π

k2s

e†⟨T

†(αp, βp, γp) T (αp, βp, γp)

⟩e .

Since

T (αp, βp, γp) = RT (−γp,−βp,−αp) TRT (αp, βp, γp) ,

T†(αp, βp, γp) = R∗ (αp, βp, γp) T †R∗ (−γp,−βp,−αp) ,

and in view of (B.54) and (B.55),

R∗ (−γp,−βp,−αp) =(RT (−γp,−βp,−αp)

)−1,

R∗ (αp, βp, γp) = RT (−γp,−βp,−αp) ,

we see that

T†(αp, βp, γp) T (αp, βp, γp) = RT (−γp,−βp,−αp) T †TRT (αp, βp, γp) .

The above equation is similar to (1.115), and taking the average, we obtain⟨(

T †T)ijmn,m1n1

⟩= δmm1δnn1 t

ijn ,

where

tijn =1

2n + 1

n∑

m′=−n

(T †T

)ijm′n,m′n

or explicitly,

t11n =1

2n + 1

n∑

m′=−n

∞∑

n1=1

n1∑

m1=−n1

∣∣T 11

m1n1,m′n

∣∣2 +

∣∣T 21

m1n1,m′n

∣∣2 ,

t12n =1

2n + 1

n∑

m′=−n

∞∑

n1=1

n1∑

m1=−n1

T 11∗m1n1,m′nT 12

m1n1,m′n

+T 21∗m1n1,m′nT 22

m1n1,m′n ,

t21n = t12∗n ,

t22n =1

2n + 1

n∑

m′=−n

∞∑

n1=1

n1∑

m1=−n1

∣∣T 12

m1n1,m′n

∣∣2 +

∣∣T 22

m1n1,m′n

∣∣2 .


The orientation-averaged scattering cross-section then becomes

〈Cscat〉 =π

k2s

∑

n,m

t11n |amn|2 + t12n a∗mnbmn

+t21n amnb∗mn + t22n

∣∣∣bmn

∣∣∣2

, (1.123)

where as before, the summation over the index m involves the values −1 and1. For macroscopically isotropic and mirror-symmetric media, t12n = t21n = 0,and using (1.120) and (1.121), we obtain [120,162]

〈Cscat〉 =2π

k2s

∞∑

n=1

(2n + 1)(t11n + t22n

)

=2π

k2s

∞∑

n=1

n∑

m=−n

∞∑

n1=1

n1∑

m1=−n1

∣∣T 11

m1n1,mn

∣∣2 +

∣∣T 12

m1n1,mn

∣∣2

+∣∣T 21

m1n1,mn

∣∣2 +

∣∣T 22

m1n1,mn

∣∣2 . (1.124)

Thus, the orientation-averaged scattering cross-section for macroscopicallyisotropic and mirror-symmetric media is proportional to the sum of thesquares of the absolute values of the transition matrix in the particle coordi-nate system. The same result holds true for an ensemble of randomly orientedparticles illuminated by a linearly polarized plane wave.

Despite the derivation of simple analytical formulas, the above analysisshows that the orientation-averaged extinction and scattering cross-sectionsfor macroscopically isotropic and mirror-symmetric media do not depend onthe polarization state of the incident wave. The orientation-averaged extinc-tion and scattering cross-sections are invariant with respect to rotations andtranslations of the coordinate system and using these properties, Mishchenkoet al. [169] have derived several invariants of the transition matrix.

Orientation-Averaged Extinction Matrix

To compute the orientation-averaged extinction matrix it is necessary to eval-uate the orientation-averaged quantities 〈Spq(ez,ez)〉. Taking into account theexpressions of the elements of the amplitude matrix (cf. (1.97)), the equationof the orientation-averaged transition matrix (cf. (1.118) and (1.119)) andthe expressions of the vector spherical harmonics in the forward direction (cf.(1.121)), we obtain

〈Sθβ (ez,ez)〉 = 〈Sϕα (ez,ez)〉 = − j2ks

∞∑

n=1

(2n + 1)(t11n + t22n

),

〈Sθα (ez,ez)〉 = −〈Sϕβ (ez,ez)〉 = − 12ks

∞∑

n=1

(2n + 1)(t12n + t21n

).


Inserting these expansions into the equations specifying the elements of theextinction matrix (cf. (1.79)), we see that the nonzero matrix elements are

〈Kii〉 = −2π

k2s

Re

∞∑

n=1

(2n + 1)(t11n + t22n

)

, i = 1, 2, 3, 4 (1.125)

and

〈K14〉 = 〈K41〉 = 2πk2sRe ∞∑

n=1(2n + 1)

(t12n + t21n

)

,

〈K23〉 = −〈K32〉 = 2πk2sIm ∞∑

n=1(2n + 1)

(t12n + t21n

)

.(1.126)

In terms of the elements of the extinction matrix, the orientation-averagedextinction cross-section is (cf. (1.89))

〈Cext〉 =1Ie

[〈K11〉 Ie + 〈K14〉Ve] ,

while for macroscopically isotropic and mirror-symmetric media, the identitiest12n = t21n = 0, imply

〈K14〉 = 〈K41〉 = 〈K23〉 = 〈K32〉 = 0 .

In this specific case, the orientation-averaged extinction matrix becomes diag-onal with diagonal elements being equal to the orientation-averaged extinctioncross-section per particle, 〈K〉 = 〈Cext〉I.

Orientation-Averaged Scattering Matrix

By definition, the orientation-averaged scattering matrix is the orientation-averaged phase matrix with β = 0 and α = ϕ = 0. In the present analysiswe consider the calculation of the general orientation-averaged phase matrix〈Z(er,ek;αp, βp, γp)〉 without taking into account the specific choice of theincident and scattering directions. We give guidelines for computing the quan-tities of interest, but we do not derive a final formula for the average phasematrix.

According to the definition of the phase matrix we see that the orientation-averaged quantities 〈Spq(er,ek)S∗

p1q1(er,ek)〉, with p, p1 = θ, ϕ and q, q1 =

β, α, need to be computed. In view of (1.99), we have⟨Spq (er,ek) S∗

p1q1(er,ek)

⟩

=16k2s

⟨(v†

q1(ek) T

T(αp, βp, γp) vp1 (er)

)∗vT

p (er) T (αp, βp, γp) v∗q (ek)

⟩

=16k2s

vTq1

(ek)⟨T

†(αp, βp, γp) v∗

p1(er) vT

p (er) T (αp, βp, γp)⟩

v∗q (ek) ,


where, as before, T stands for the transition matrix in the global coordinatesystem. Defining the matrices

V pp1 (er) = v∗p1

(er) vTp (er)

and

App1 (er) =⟨T

†(αp, βp, γp) V pp1(er)T (αp, βp, γp)

⟩,

we see that

⟨Spq (er,ek) S∗

p1q1(er,ek)

⟩=

16k2

s

vTq1

(ek) App1 (er) v∗q (ek) . (1.127)

Using the block-matrix decomposition

X =

[X11 X12

X21 X22

]

,

where X stands for V pp1 and App1 , we express the submatrices of App1 as

A11pp1

=⟨T

11†V 11

pp1T

11+ T

11†V 12

pp1T

21+ T

21†V 21

pp1T

11+ T

21†V 22

pp1T

21⟩

,

A12pp1

=⟨T

11†V 11

pp1T

12+ T

11†V 12

pp1T

22+ T

21†V 21

pp1T

12+ T

21†V 22

pp1T

22⟩

,

A21pp1

=⟨T

12†V 11

pp1T

11+ T

12†V 12

pp1T

21+ T

22†V 21

pp1T

11+ T

22†V 22

pp1T

21⟩

,

A22pp1

=⟨T

12†V 11

pp1T

12+ T

12†V 12

pp1T

22+ T

22†V 21

pp1T

12+ T

22†V 22

pp1T

22⟩

.

(1.128)

It is apparent that each matrix product in the above equations is of the form

W pp1 (er) =⟨T

kl†(αp, βp, γp) V uv

pp1(er) T

ij(αp, βp, γp)

⟩, (1.129)

where the permissive values of the index pairs (i, j), (k, l) and (u, v) followfrom (1.128). The elements of the W pp1 matrix are given by

(Wpp1)m1n1,m1n1(er) =

∑

n,m

∑

n,m

⟨T ij

mn,m1n1(αp, βp, γp) T kl∗

mn,m1n1(αp, βp, γp)

⟩

×(V uv

pp1

)mn,mn

(er) , (1.130)

and the rest of our analysis concerns with the computation of the term

T =⟨T ij

mn,m1n1(αp, βp, γp) T kl∗

mn,m1n1(αp, βp, γp)

⟩. (1.131)


It should be mentioned that for notation simplification we omit to indicatethe dependency of T on the matrix indices.

Using the rotation transformation rule for the transition matrix (cf.(1.116)), we obtain

T =1

8π2

n∑

m′=−n

n1∑

m′1=−n1

n∑

m′=−n

n1∑

m′1=−n1

[∫ 2π

0

∫ 2π

0

∫ π

0

Dnm′m (−γp,−βp,−αp)

×Dn1m1m′

1(αp, βp, γp) Dn∗

m′m(−γp,−βp,−αp)

× Dn1∗m1m′

1

(αp, βp, γp) sinβp dβp dαp dγp

]T ij

m′n,m′1n1

T kl∗m′n,m′

1n1.

Taking into account the definition of the Wigner D-functions (cf. (B.34)) andintegrating over αp and γp, yields

T =12

n∑

m′=−n

n1∑

m′1=−n1

n∑

m′=−n

n1∑

m′1=−n1

δm1−m,m1−m

δm′

1−m′,m′1−m′∆

×[∫ π

0

dnm′m (−βp) dn

m′m(−βp) dn1

m1m′1(βp) dn1

m1m′1

(βp) sinβp dβp

]

×T ijm′n,m′

1n1T kl∗

m′n,m′1n1

,

where dnmm′ are the Wigner d-functions defined in Appendix B,

∆ = ∆m′m∆m′m

∆m1m′1∆

m1m′1,

and ∆mm′ is given by (B.36). To compute the integral

I =12δm1−m,m1−m

δm′

1−m′,m′1−m′

×∫ π

0

dnm′m (−βp) dn

m′m(−βp) dn1

m1m′1(βp) dn1

m1m′1

(βp) sinβp dβp ,

we use the symmetry relations (cf. (B.39) and (B.41))

dnm′m(−βp) = dn

mm′(βp) = (−1)m+m′dn−m−m′(βp) ,

dn

m′m(−βp) = dn

mm′(βp) = (−1)m+m′dn

−m−m′(βp) ,

the expansions of the d-functions products dn1m1m′

1dn−m−m′ and dn1

m1m′1

dn

−m−m′

given by (B.47), and the orthogonality property of the d-functions (cf. (B.43)).


We obtain

I = (−1)m+m′+m+m′(−1)n+n1+n+n1δ

m1−m,m1−mδm′

1−m′,m′1−m′

×umax∑

u=umin

12u + 1

Cm1−mum1n1,−mnC

m′−m′1u

−m′1n1,m′nCm1−mu

m1n1,−mnC

m′−m′1u

−m′1n1,m′n

,

where Cm+m1umn,m1n1

are the Clebsch–Gordan coefficients defined in Appendix B,and

umin = max (|n − n1| , |n − n1| , |m1 − m| , |m1 − m| , |m′ − m′1| , |m′ − m′

1|) ,

umax = min (n + n1, n + n1) .

Further, using the symmetry properties of the Clebsch–Gordan coefficients(cf. (B.48) and (B.51)) we arrive at

I = (−1)n+n1+n+n1δm1−m,m1−m

δm′

1−m′,m′1−m′

×umax∑

u=umin

2u + 1(2n1 + 1) (2n1 + 1)

Cm1n1m1−mu,mnC

−m′1n1

m′−m′1u,−m′n

×Cm1n1

m1−mu,mnC

−m′1n1

m′−m′1u,−m′n

and

T =n∑

m′=−n

n1∑

m′1=−n1

n∑

m′=−n

n1∑

m′1=−n1

∆IT ijm′n,m′

1n1T kl∗

m′n,m′1n1

. (1.132)

The orientation-averaged quantities 〈Spq(er,ek)S∗p1q1

(er,ek)〉 can be com-puted from the set of equations (1.127)–(1.132).

For an incident wave propagating along the Z-axis, the augmented vectorof spherical harmonics vq(ez) can be computed by using (1.121). Choosing theXZ-plane as the scattering plane, i.e., setting ϕ = 0, we see that the matricesV pp1(er) involve only the normalized angular functions π

|m|n (θ) and τ

|m|n (θ).

The resulting orientation-averaged scattering matrix can be computed at aset of polar angles θ and polynomial interpolation can be used to evaluate theorientation-averaged scattering matrix at any polar angle θ.

For macroscopically isotropic media, the orientation-averaged scatteringmatrix has sixteen nonzero elements (cf. (1.113)) but only ten of themare independent. For macroscopically isotropic and mirror-symmetric me-dia, the orientation-averaged scattering matrix has a block-diagonal structure(cf. (1.114)), so that only eight elements are nonzero and only six of themare independent. In this case we determine the six quantities 〈|Sθβ(θ)|2〉,


〈|Sθα(θ)|2〉, 〈|Sϕβ(θ)|2〉, 〈|Sϕα(θ)|2〉, 〈Sθβ(θ)S∗ϕα(θ)〉 and 〈Sθα(θ)S∗

ϕβ(θ)〉, andcompute the eight nonzero elements by using the relations

〈F11(θ)〉 =12

(⟨|Sθβ(θ)|2

⟩+⟨|Sθα(θ)|2

⟩+⟨|Sϕβ(θ)|2

⟩+⟨|Sϕα(θ)|2

⟩),

〈F12(θ)〉 =12

(⟨|Sθβ(θ)|2

⟩−⟨|Sθα(θ)|2

⟩+⟨|Sϕβ(θ)|2

⟩−⟨|Sϕα(θ)|2

⟩),

〈F21(θ)〉 = 〈F12(θ)〉 ,

〈F22(θ)〉 =12

(⟨|Sθβ(θ)|2

⟩−⟨|Sθα(θ)|2

⟩−⟨|Sϕβ(θ)|2

⟩+⟨|Sϕα(θ)|2

⟩),

〈F33(θ)〉 = Re⟨

Sθβ(θ)S∗ϕα(θ)

⟩+⟨Sθα(θ)S∗

ϕβ(θ)⟩

,

〈F34(θ)〉 = Im⟨


⟩+⟨Sθα(θ)S∗

ϕβ(θ)⟩∗

,

〈F43(θ)〉 = −〈F34(θ)〉 ,

〈F44(θ)〉 = Re⟨


⟩∗ −⟨Sθα(θ)S∗

ϕβ(θ)⟩

.

Other scattering characteristics as for instance the orientation-averaged scat-tering cross-section and the orientation-averaged mean direction of propaga-tion of the scattered field can be expressed in terms of the elements of theorientation-averaged scattering matrix. To derive these expressions we con-sider the scattering plane characterized by the azimuth angle ϕ as shown inFig. 1.15. In the scattering plane, the Stokes vector of the scattered wave isgiven by 〈Is(rer)〉 = (1/r2)〈F (θ)〉I ′

e, whence, using the transformation ruleof the Stokes vector under coordinate rotation I ′

e = L(ϕ)Ie, we obtain

〈Is (rer)〉 =1r2

〈F (θ)〉L (ϕ) Ie .

Further, taking into account the expression of the Stokes rotation matrix L(cf. (1.23)) we derive

〈Is (er)〉 = 〈F11 (θ)〉 Ie + [〈F12(θ)〉 cos 2ϕ + 〈F13(θ)〉 sin 2ϕ] Qe

− [〈F12(θ)〉 sin 2ϕ − 〈F13(θ)〉 cos 2ϕ] Ue + 〈F14(θ)〉Ve .

Integrating over ϕ, we find that the orientation-averaged scattering cross-section and the orientation-averaged mean direction of propagation of thescattered field are given by

〈Cscat〉 =1Ie

∫

Ω

〈Is (er)〉dΩ (er)

=2π

Ie

∫ π

0

[〈F11(θ)〉 Ie + 〈F14(θ)〉Ve] sin θ dθ


O

Z

X

er

eθ

Y

eϕ

eβ

eα

e’β

e’α

θ

ϕ

ek

Fig. 1.15. Incident and scattering directions ek and er. The incident direction isalong the Z-axis and the scattering matrix relates the Stokes vectors of the incidentand scattered fields specified relative to the scattering plane characterized by theazimuth angle ϕ

and

〈g〉 =1

〈Cscat〉 Ie

∫

Ω

〈Is (er)〉 er dΩ (er)

=2π

〈Cscat〉 Ie

∫ π

0

[〈F11(θ)〉 Ie + 〈F14(θ)〉Ve] sin θ cos θ dθ

ez ,

respectively. Because the incident wave propagates along the Z-axis, thenonzero component of 〈g〉 is the orientation-averaged asymmetry parameter〈cos Θ〉. In practical computer simulations, we use the decomposition

〈Cscat〉 =1Ie

(〈Cscat〉I Ie + 〈Cscat〉V Ve) ,

and compute the quantities 〈Cscat〉I and 〈Cscat〉V by using (1.124) and therelation

〈Cscat〉V = 2π

∫ π

0

〈F14(θ)〉 sin θ dθ , (1.133)

respectively. These quantities do not depend on the polarization state of theincident wave and can be used to compute the orientation-averaged scatteringcross-section for any incident polarization. For the asymmetry parameter weproceed analogously; we use the decomposition


〈cos Θ〉 =〈Cscat〉I〈Cscat〉

1Ie

(〈cos Θ〉I Ie + 〈cos Θ〉V Ve) ,

and compute 〈cos Θ〉I and 〈cos Θ〉V by using the relations

〈cos Θ〉I =2π

〈Cscat〉I

∫ π

0

〈F11(θ)〉 sin θ cos θ dθ (1.134)

and

〈cos Θ〉V =2π

〈Cscat〉I

∫ π

0

〈F14(θ)〉 sin θ cos θ dθ , (1.135)

respectively. For macroscopically isotropic and mirror-symmetric media,

〈F14(θ)〉 = 0

and consequently, 〈Cscat〉 = 〈Cscat〉I and 〈cos Θ〉 = 〈cos Θ〉I.Another important scattering characteristic is the angular distribution of

the scattered field. For an ensemble of randomly oriented particles illuminatedby a vector plane wave of unit amplitude and polarization vector epol =epol,βeβ + epol,αeα, the differential scattering cross-sections in the scatteringplane ϕ are given by

〈σdp(θ)〉 =⟨|Es∞,θ(θ)|2

⟩=⟨|Sθβ(θ)|2

⟩ ∣∣E′

e0,β

∣∣2

+⟨|Sθα(θ)|2

⟩ ∣∣E′

e0,α

∣∣2

+2Re〈Sθβ(θ)S∗

θα(θ)〉E′e0,βE′∗

e0,α

(1.136)

and

〈σds(θ)〉 =⟨|Es∞,ϕ(θ)|2

⟩=⟨|Sϕβ(θ)|2

⟩ ∣∣E′

e0,β

∣∣2

+⟨|Sϕα(θ)|2

⟩ ∣∣E′

e0,α

∣∣2

+2Re⟨

Sϕβ(θ)S∗ϕα(θ)

⟩E′

e0,βE′∗e0,α

, (1.137)

where

E′e0,β = epol,β cos ϕ + epol,α sin ϕ ,

E′e0,α = −epol,β sin ϕ + epol,α cos ϕ .

It should be noted that for macroscopically isotropic and mirror-symmetricmedia,

〈Sθβ(θ)S∗θα(θ)〉 = 0 ,

⟨Sϕβ(θ)S∗

ϕα(θ)⟩

= 0 ,

and the expressions of 〈σdp(θ)〉 and 〈σds(θ)〉 simplify considerably.


In practice, the inequalities

〈F11〉 ≥ |〈Fij〉| , i, j = 1, 2, 3, 4 ,

(〈F11〉 + 〈F22〉)2 − 4 〈F12〉2 ≥ (〈F33〉 + 〈F44〉)2 + 4 〈F34〉2 ,

〈F11〉 − 〈F22〉 ≥ |〈F33〉 − 〈F44〉| ,

〈F11〉 − 〈F12〉 ≥ |〈F22〉 − 〈F12〉| ,

〈F11〉 + 〈F12〉 ≥ |〈F22〉 + 〈F12〉| , (1.138)

can be used to test the numerically obtained orientation-averaged scatteringmatrix [104,169].

It should be emphasized that we do not expand the elements of theorientation-averaged scattering matrix in generalized spherical functions (orWigner d-functions) and do not exploit the advantage of performing as muchwork analytically as possible. Therefore, the above averaging procedure iscomputationally not so fast as the scattering matrix expansion method givenby Mishchenko [162]. As noted by Mishchenko et al. [169], the analyticity ofthe T -matrix formulation can be connected with the formalism of expand-ing scattering matrices in generalized spherical functions to derive an efficientprocedure that does not involve any angular variable.

Khlebtsov [120] and Fucile et al. [71] developed a similar formalism thatexploit the rotation property of the transition matrix but avoids the expansionof the scattering matrix in generalized spherical functions. Paramonov [182]and Borghese et al. [23] extended the analytical orientation-averaging proce-dure to arbitrary orientation distribution functions, while the standard aver-aging approach employing numerical integrations over the orientation angleshas been used by Wiscombe and Mugnai [263, 264] and Barber and Hill [8].A method to compute light scattering by arbitrarily oriented rotationally sym-metric particles has been given by Skaropoulos and Russchenberg [212].

2

Null-Field Method

The standard scheme for computing the transition matrix relies on the null-field method. The null-field method has been introduced by Waterman [253] asa technique for computing electromagnetic scattering by perfectly conductingparticles. Later, Bates [10] studied the same problem, followed by Waterman’sapplications of his method to electromagnetic scattering from dielectric par-ticles [254, 256] and to acoustic scattering [255]. The null-field method hassubsequently been extended to multiple scattering problems [187,188] and toelectromagnetic scattering by multilayered and composite particles [189,190].Peterson and Strom [189] derived a recurrence relation for the transition ma-trix of a multilayered particle and analyzed particles which contains severalenclosures, each with arbitrary but constant electromagnetic properties. Thenull-field treatment of a composite particle with two constituents uses a par-ticular projection of the boundary conditions on the interface and treats thetwo constituents as separate particles by exploiting the translation additiontheorem for vector spherical wave functions. However, for a large class of com-posite particles it is not natural to treat the two parts as separate particles.The geometric constraints which appear in the two-particles formalism (orig-inating in the use of translations of radiating vector spherical wave functions)are eliminated, partially or completely by Strom and Zheng [219]. The case of acomposite particle consisting of three or more constituents has been analyzedby Zheng [276], while the extension of the method to composite particles withconcavo-convex constituents has been discussed by Zheng and Strom [279].Applications of the null-field method to electromagnetic, acoustic and elasto-dynamic scattering by single and aggregated particles, to multiple scatteringin random media, and to scattering by periodical and infinite surfaces can befound in [168,169,228,234,235].

Essentially, the null-field method involves the following steps:

1. Derivation of an infinite system of integral equations for the surface fieldsby using the general null-field equation.

84 2 Null-Field Method

2. Derivation of integral representations for the scattered field coefficients interms of surface fields by using the Huygens principle.

3. Approximation of the surface fields by a complete and linearly indepen-dent system of (tangential) vector functions.

4. Truncation of the infinite system of null-field equations and of the scattered-field expansion.

5. Computation of the truncated transition matrix by matrix inversion.

These steps explicitly avoid invoking the Rayleigh hypothesis [11, 34] and re-lies on the approximation of the surface fields by a complete system of vectorfunctions (the localized vector spherical wave functions). The completenessand linear independence of various systems of vector functions on closed sur-faces, including the vector spherical wave functions, have been established byDoicu et al. [49]. For scattering problems involving inhomogeneous particlesit is necessary to establish the completeness and linear independence of thissystem of vector functions on two enclosing surfaces and this result is givenin Appendix D.

Several studies have addressed the convergence of the null-field method.Ramm [200,201] and Kristensson et al. [126] have derived sufficiency criteriafor the convergence of the null-field method, but this criteria are not satisfiedfor nonspherical surfaces. A major progress has been achieved by Dallas [44],who showed that for ellipsoidal surfaces, the far-field pattern converges in theleast-squares sense on the unit sphere.

In this chapter, we present applications of the null-field method toelectromagnetic scattering by homogeneous, isotropic, anisotropic and chi-ral particles, inhomogeneous, layered and composite particles and clusters ofparticles. Our presentation closely follows the analysis developed by Stromand his coworkers. In order to preserve the generality we assume that theparticles and the infinite, exterior medium are magnetic, but in practice wealways deal with nonmagnetic media (µs = µi = 1). In the last two sections,we treat multiple scattering in random media and scattering by a particle onor near a plane surface.

2.1 Homogeneous and Isotropic Particles

Most practical implementations of the null-field method pertain to homoge-nous and isotropic particles. T -matrix codes for axisymmetric particles havebeen developed by Wiscombe and Mugnai [263], Barber and Hill [8] andMishchenko and Travis [167]. Light scattering calculations for nonaxisym-metric particles have been reported by Barber [7], Schneider and Peden [207],Laitinen and Lumme [127], Wriedt and Comberg [269], Baran et al. [6], Have-mann and Baran [98], Kahnert et al. [111,112], and Wriedt [268]. In this sec-tion, we provide a detailed description of the null-field method in applicationto homogenous and isotropic particles.

2.1 Homogeneous and Isotropic Particles 85

OD i

Ds

S

nεs, µs

ε i, µ i

Fig. 2.1. Geometry of a homogeneous particle

2.1.1 General Formulation

The transmission boundary-value problem for homogeneous and isotropic par-ticles has been formulated in Sect. 1.4 but we mention it in order for ouranalysis to be complete. We consider an homogeneous, isotropic particle oc-cupying a domain Di with boundary S and exterior Ds (Fig. 2.1). The unitnormal vector to S directed into Ds is denoted by n. The exterior domainDs is assumed to be homogeneous, isotropic, and nonabsorbing, and if εt andµt are the relative permittivity and permeability of the domain Dt, wheret = s, i, we have εs > 0 and µs > 0. The wave number in the domain Dt iskt = k0

√εtµt, where k0 is the wave number in the free space. The transmis-

sion boundary-value problem for a homogeneous and isotropic particle has thefollowing formulation.

Given Ee,He as an entire solution to the Maxwell equations representingthe external excitation, find the vector fields Es,Hs and Ei,H i satisfying theMaxwell equations

∇× Et = jk0µtHt , ∇× Ht = −jk0εtEt , (2.1)

in Dt, t = s, i, the two transmission conditions

n × Ei − n × Es = n × Ee ,

n × H i − n × Hs = n × He , (2.2)

on S, and the Silver–Muller radiation condition

r

r×√

µsHs +√

εsEs = O(

1r

), as r → ∞ , (2.3)

uniformly for all directions r/r.The standard scheme for computing the transition matrix in the framework

of the null-field method relies on the solution of the general null-field equation


Ee(r) + ∇×∫

S

ei (r′) g (ks, r, r′) dS(r′)

+j

k0εs∇×∇×

∫

S

hi (r′) g (ks, r, r′) dS(r′) = 0 , r ∈ Di (2.4)

for the surface fields ei = n × Ei and hi = n × H i, and the calculation ofthe scattered field from Huygens principle

Es(r) = ∇×∫

S

ei (r′) g (ks, r, r′) dS(r′)

+j

k0εs∇×∇×

∫

S

hi (r′) g (ks, r, r′) dS(r′) , r ∈ Ds . (2.5)

Considering the general null-field equation (2.4), we restrict r to lie on aspherical surface enclosed in Di, expand the incident field and the dyad gIin terms of regular vector spherical wave functions (cf. (1.25), (B.21) and(B.22)), and use the orthogonality of the vector spherical wave functions onspherical surfaces to obtain

jk2s

π

∫

S

[

ei (r′) ·(

N3ν (ksr

′)M3

ν (ksr′)

)

(2.6)

+ j√

µs

εshi (r′) ·

(M3

ν (ksr′)

N3ν (ksr

′)

)]

dS (r′) = −(

aν

bν

), ν = 1, 2, . . .

For notation simplification we introduced the multi-indices ν = (m,n) andν = (−m,n), and used the convention ν = 1, 2, . . . , when n = 1, 2, . . . , andm = −n, . . . , n. The above set of integral equations will be referred to as thenull-field equations.

An approximate solution to the null-field equations can be obtained byapproximating the surface fields ei and hi by the complete set of regularvector spherical wave functions for the interior domain (or the interior waveequation)

(eN

i (r′)hN

i (r′)

)

=N∑

µ=1

cNµ

⎛

⎝n (r′) × M1

µ (kir′)

−j√

εiµi

n (r′) × N1µ (kir

′)

⎞

⎠

+dNµ

⎛

⎝n (r′) × N1

µ (kir′)

−j√

εiµi

n (r′) × M1µ (kir

′)

⎞

⎠ , (2.7)

where N is a truncation multi-index and µ = 1, 2, . . . , N , when n = 1, 2, . . .Nrank, and m = −n, . . . , n. In all applications of the null-field method weapproximate the surface fields by finite expansions, and the coefficients cN

µ anddN

µ on the right-hand side of the above equation then depend on the number of


terms in the expansion. The completeness of the system n×M1µ,n×N1

µµ

implies the existence of the set of expansion coefficients cNµ , dN

µ Nµ=1 and of

the integer N = N(ε), such that ‖ei − eNi ‖2,S < ε, for any ε > 0. In (2.7), eN

i

and hNi are expressed in terms of the same set of expansion coefficients and

this choice shows that ei and hi are not independent unknowns. In fact, thesurface fields ei and hi solve the general null-field equations in Di and Ds asgiven by (2.4) and

∇×∫

S

ei (r′) g (ki, r, r′) dS(r′)

+j

k0εi∇×∇×

∫

S

hi (r′) g (ki, r, r′) dS(r′) = 0 , r ∈ Ds , (2.8)

respectively, or equivalently, the infinite systems of null-field equations as givenby (2.7) and

∫

S

[

ei (r′) ·(

N1ν (kir

′)M1

ν (kir′)

)

+ j√

µi

εihi (r′) ·

(M1

ν (kir′)

N1ν (kir

′)

)]

dS (r′) = 0 , ν = 1, 2, ... , (2.9)

respectively. Note that (2.8) is the Stratton–Chu representation theorem forthe internal field in Ds, while (2.9) follows from the spherical wave expan-sion of the dyad gI outside a sphere enclosing Di. When ei and hi aretreated as independent unknowns and are approximated by sub-boundarybases, the expansion coefficients are obtained by solving (2.6) and (2.9).In (2.7), the expansion of hN

i has a physical meaning: the surface fields ei

and hi are the tangential components of the electric and magnetic fieldsEi and H i, respectively. Taking into account the orthogonality relation(cf. (B.25))

∫

S

[(n × M1

µ

)·(

M1ν

N1ν

)

+(n × N1

µ

)·(

N1ν

M1ν

)]

dS = 0 ,

we see that eNi and hN

i as given by (2.7) solve the null-field equations in Ds

and therefore, the expansion coefficients cNµ and dN

µ have to be determinedfrom the null-field equations in Di.

Inserting (2.7) into the first 2N null-field equations, yields

Q31 (ks, ki) i = −e, , (2.10)

where i = [cNµ , dN

µ ]T and e = [aν , bν ]T are vectors containing the expan-sion coefficients of the surface and incident fields, respectively. In general,


for nonmagnetic media with k1 = k0√

ε1 and k2 = k0√

ε2, the Qpq(k1, k2)matrix,

Qpq (k1, k2) =

[(Qpq)11νµ (Qpq)12νµ

(Qpq)21νµ (Qpq)22νµ

]

,

is defined as

(Qpq)11νµ =jk2

1

π

∫

S

[n (r′) × M q

µ (k2r′)]· Np

ν (k1r′)

+√

ε2

ε1

[n (r′) × N q

µ (k2r′)]· Mp

ν (k1r′)

dS (r′) , (2.11)

(Qpq)12νµ =jk2

1

π

∫

S

[n (r′) × N q

µ (k2r′)]· Np

ν (k1r′)

+√

ε2

ε1

[n (r′) × M q

µ (k2r′)]· Mp

ν (k1r′)

dS (r′) , (2.12)

(Qpq)21νµ =jk2

1

π

∫

S

[n (r′) × M q

µ (k2r′)]· Mp

ν (k1r′)

+√

ε2

ε1

[n (r′) × N q

µ (k2r′)]· Np

ν (k1r′)

dS (r′) , (2.13)

and

(Qpq)22νµ =jk2

1

π

∫

S

[n (r′) × N q

µ (k2r′)]· Mp

ν (k1r′)

+√

ε2

ε1

[n (r′) × M q

µ (k2r′)]· Np

ν (k1r′)

dS (r′) . (2.14)

Considering the scattered field representation (2.5), we replace the surfacefields by their approximations and use the vector spherical wave expansion ofthe dyad gI on a sphere enclosing Di to obtain

ENs (r) =

N∑

ν=1

fNν M3

ν (ksr) + gNν N3

ν (ksr) , (2.15)

where the expansion coefficients of the scattered field are given by(

fNν

gNν

)

=jk2

s

π

∫

S

[

eNi (r′) ·

(N1

ν (ksr′)

M1ν (ksr

′)

)

+ j√

µs

εshN

i (r′) ·(

M1ν (ksr

′)N1

ν (ksr′)

)]

dS (r′) (2.16)


for ν = 1, 2, . . . , N . Inserting (2.7) into (2.16) yields

s = Q11 (ks, ki) i , (2.17)

where s = [fNν , gN

ν ]T is the vector containing the expansion coefficients of thescattered field. Combining (2.10) and (2.17) we deduce that the transitionmatrix relating the scattered field coefficients to the incident field coefficients,s = Te, is given by

T = −Q11 (ks, ki)[Q31 (ks, ki)

]−1. (2.18)

In view of (2.11)–(2.14), we see that the transition matrix computed in theparticle coordinate system depends only on the physical and geometrical char-acteristics of the particle, such as particle shape and relative refractive indexand is independent of the propagation direction and polarization states of theincident and scattered fields. In contrast to the infinite transition matrix dis-cussed in Sect. 1.5, the transition matrix given by (2.18) is of finite dimension,and the often used appellation approximate or truncated transition matrix isjustified.

2.1.2 Instability

Instability of the null-field method occurs for strongly deformed particlesand large size parameters. A number of modifications to the conventionalapproach has been suggested, especially to improve the numerical stabil-ity in computations for particles with extreme geometries. These techniquesinclude formal modifications of the single spherical coordinate-based null-fieldmethod by using discrete sources [49, 109], different choices of basis func-tions and the application of the spheroidal coordinate formalism [12,89], andthe orthogonalization method [133, 256]. For large particles, the maximumconvergent size parameter can be increased by using a special form of theLU-factorization method [261] and by performing the matrix inversion usingextended-precision [166]. For strongly absorbing metallic particles, Gaussianelimination with backsubstitution gives improved convergence results [174].

Formulation with Discrete Sources

The conventional derivation of the T matrix relies on the approximation ofthe surface fields by the system of localized vector spherical wave functions.Although these wave functions appear to provide a good approximation tothe solution when the surface is not extremely aspherical, they are disadvan-tageous when this is not the case. The numerical instability of the T -matrixcalculation arises because the elements of the Q31 matrix differ by manyorders of magnitude and the inversion process is ill-conditioned. As a result,slow convergence or divergence occur. If instead of localized vector spherical


functions we use distributed sources, it is possible to extend the applicabil-ity range of the single spherical coordinate-based null-field method. Discretesources were used for the first time in the iterative version of the null-fieldmethod [108, 109, 132]. This iterative approach utilizes multipole sphericalexpansions to represent the internal fields in different overlapping regions,while the various expansions are matched in the overlapping regions to enforcethe continuity of the fields throughout the entire interior volume.

In the following analysis we summarize the basic concepts of the null-field method with distributed sources. The distributed vector spherical wavefunctions are defined as

M1,3mn(kr) = M1,3

m,|m|+l [k(r − znez)] ,

N 1,3mn(kr) = N1,3

m,|m|+l [k(r − znez)] ,

where zn∞n=1 is a dense set of points situated on the z-axis and in the interiorof S, ez is the unit vector in the direction of the z-axis, n = 1, 2, . . . ,m ∈ Z,and l = 1 if m = 0 and l = 0 if m = 0. Taking into account the Stratton–Chu representation theorem for the incident field in Di we rewrite the generalnull-field equation as

∇×∫

S

[ei (r′) − ee (r′)] g (ks, r, r′) dS(r′) (2.19)

+j

k0εs∇×∇×

∫

S

[hi (r′) − he (r′)] g (ks, r, r′) dS(r′) = 0 , r ∈ Di ,

and derive the following set of null-field equations:

jk2s

π

∫

S

[

[ei (r′) − ee (r′)] ·(

N3ν (ksr

′)M3

ν (ksr′)

)

+ j√

µs

εs[hi (r′) − he (r′)] ·

(M3

ν (ksr′)

N3ν (ksr

′)

)]

dS (r′) = 0 , ν = 1, 2, . . .

(2.20)

The formulation of the null-field method with distributed vector sphericalwave functions relies on two basic results. The first result states that if ei andhi solve the set of null-field equations (compare to (2.20))

jk2s

π

∫

S

[

[ei (r′) − ee (r′)] ·(

N 3ν (ksr

′)M3

ν (ksr′)

)

+ j√

µs

εs[hi (r′) − he (r′)] ·

(M3

ν (ksr′)

N 3ν (ksr

′)

)]

dS (r′) = 0 ,

ν = 1, 2, . . . , (2.21)


where ν = (m,n), ν = (−m,n) and ν = 1, 2, ..., when n = 1, 2, ..., and m ∈ Z,then ei and hi solve the general null-field equation (2.20). The second resultstates the completeness and linear independence of the system of distributedvector spherical wave functions n ×M1

µ,n × N 1µ on closed surfaces. Con-

sequently, approximating the surface fields by the finite expansions

(eN

i (r′)hN

i (r′)

)

=N∑

µ=1

cNµ

⎛

⎝n (r′) ×M1

µ (kir′)

−j√

εiµi

n (r′) ×N 1µ (kir

′)

⎞

⎠

+dNµ

⎛

⎝n (r′) ×N 1

µ (kir′)

−j√

εiµi

n (r′) ×M1µ (kir

′)

⎞

⎠ , (2.22)

inserting (2.22) into (2.21) and using the vector spherical wave expansion ofthe incident field, yields

Q31

(ks, ki) i = −Q31

(ks, ks) e .

The Q31

(ks, ki) matrix has the same structure as the Q31(ks, ki) matrix, but itcontains as rows and columns the vectors M3

ν(ksr′), N 3

ν (ksr′) and M1

µ(kir′),

N 1µ(kir

′), respectively, while Q31

(ks, ks) contains as rows and columns the vec-tors M3

ν(ksr′), N 3

ν (ksr′) and M1

µ(ksr′), N1

µ(ksr′), respectively. To compute

the scattered field we proceed as in the case of localized sources. Applicationof the Huygens principle yields the expansion of the scattered field in termsof localized vector spherical wave functions as in (2.15) and (2.16). Inserting(2.22) into (2.16) gives

s = Q11

(ks, ki) i ,

and we obtain

T = −Q11

(ks, ki)[Q

31(ks, ki)

]−1

Q31

(ks, ks) ,

where Q11

(ks, ki) contains as rows and columns the vectors M1ν(ksr

′),N1

ν(ksr′) and M1

µ(kir′), N 1

µ(kir′), respectively. Taking into account the defin-

ition of the distributed vector spherical wave functions (cf. (B.30)), we see that

the Q31

(ks, ki) matrix includes Hankel functions of low order which results ina better conditioned system of equations as compared to that obtained in thesingle spherical coordinate-based null-field method.

The use of distributed vector spherical wave functions is most effective foraxisymmetric particles because, in this case, the T matrix is diagonal withrespect to the azimuthal indices. For elongated particles, the sources are dis-tributed on the axis of rotation, while for flattened particles, the sources aredistributed in the complex plane (which is the dual of the symmetry plane).


The expressions of the distributed vector spherical wave functions with theorigins located in the complex plane are given by (B.31) and (B.32).

Various systems of vector functions can be used instead of localized anddistributed vector spherical wave functions. Formulations of the null-fieldmethod with multiple vector spherical wave functions, electric and magneticdipoles and vector Mie potentials have been given by Doicu et al. [49]. Numer-ical experiments performed by Strom and Zheng [218] demonstrated that thesystem of tangential fields constructed from the vector spherical harmonics issuitable for analyzing particles with pronounced concavities. When the sur-face fields have discontinuities in their continuity or any of their derivatives,straightforward application of the global basis functions provides poor con-vergence of the solution. Wall [247] showed that sub-boundary bases or localbasis approximations of the surface fields results in larger condition numbersof the matrix equations.

Orthogonalization Method

To ameliorate the numerical instability of the null-field method for nonabsorb-ing particles, Waterman [256,257] and Lakhtakia et al. [133,134] proposed toexploit the unitarity property of the transition matrix. To summarize this tech-nique we consider nonabsorbing particles and use the identity Q11 = ReQ31to rewrite the T -matrix equation

QT = −Re Q , (2.23)

as

QS = −Q∗ ,

where Q = Q31 and S = I + 2T . Applying the Gramm–Schmidt orthogonal-ization procedure on the row vectors of Q, we construct the unitarity matrixQ (Q

†Q = QQ

†= I) such that

Q = MQ ,

and derive

S = −Q† [

M (M∗)−1]Q

∗= −Q

†MQ

∗,

where M is an upper-triangular matrix with real diagonal elements and M =M(M∗)−1. Taking into account the unitarity condition for the S matrix (cf.(1.109)),

I = S†S = QTM

†Q · Q

†MQ

∗= Q

TM

†MQ

∗,


and using the relation QTQ

∗= I, we deduce that

M†M = I . (2.24)

Because M is upper-triangular with real diagonal elements, it follows thatM is unit upper-triangular (all diagonal elements are equal 1). Then, (2.24)implies that M = I, and therefore

S = −Q†Q

∗.

In terms of the transition matrix, the above equation read as

T = −Q†Re

Q

and this new sequence of truncated transition matrices might be expected toconverge more rapidly than (2.23) because the unitarity condition is satisfiedfor each truncation index.

2.1.3 Symmetries of the Transition Matrix

Symmetry properties of the transition matrix can be derived for specific par-ticle shapes. These symmetry relations can be used to test numerical codesas well as to simplify many equations of the T -matrix method and developefficient numerical procedures. In fact, the computer time for the numericalevaluation of the surface integrals (which is the most time consuming part ofthe T -matrix calculation) can be substantially reduced. The surface integralsare usually computed in spherical coordinates and for a surface defined by

r = r (θ, ϕ) ,

we have

n (r) dS (r) = σ (r) r2 sin θ dθ dϕ ,

where

σ(r) = er −1r

∂r

∂θeθ −

1r sin θ

∂r

∂ϕeϕ

is a vector parallel to the unit normal vector n. In the following analysis, wewill investigate the symmetry properties of the transition matrix for axisym-metric particles and particles with azimuthal and mirror symmetries.

1. For an axisymmetric particle, the equation of the surface does not de-pend on ϕ, and therefore ∂r/∂ϕ = 0. The vector σ has only er- and eθ-components and is independent of ϕ. The integral over the azimuthal anglecan be performed analytically and we see that the result is zero unless m = m1.


Therefore, the T matrix is diagonal with respect to the azimuthal indices mand m1, i.e.,

T ijmn,m1n1

= T ijmn,mn1

δmm1 .

Direct calculation shows that

T ijmn,mn1

= (−1)i+jT ij−mn,−mn1

(2.25)

and the symmetry relation (1.112) takes the form

T ijmn,mn1

= T ji−mn1,−mn = (−1)i+jT ji

mn1,mn .

2. We consider a particle with a principal N -fold axis of rotation symmetryand assume that the axis of symmetry coincides with the z-axis of the particlecoordinate system. In this case, r, ∂r/∂θ and ∂r/∂ϕ are periodic in ϕ withperiod 2π/N . Taking into account the definition of the Qpq matrices, we seethat the surface integrals are of the form

∫ 2π

0

f (ϕ) ej(m1−m)ϕdϕ ,

where f is a periodic function of ϕ,

f (ϕ) = f

(ϕ + k

2π

N

).

As a consequence of the rotation symmetry around the z-axis, the T matrixis invariant with respect to discrete rotations of angles αk = 2πk/N , k =1, 2, . . . , N − 1. A necessary and sufficient condition for the equation

∫ 2π

0

f (ϕ) ej(m1−m)ϕdϕ = ej(m1−m)k 2πN

∫ 2π

0

f (ϕ) ej(m1−m)ϕdϕ

to hold is |m1 − m| = lN , l = 0, 1, . . ., and we conclude that

T ijmn,m1n1

= 0

Further,∫ 2π

0

f (ϕ) ej(m1−m)ϕdϕ =N∑

k=1

ej(m1−m)(k−1) 2πN

∫ 2π/N

0

f (ϕ) ej(m1−m)ϕdϕ

= N

∫ 2π/N

0

f (ϕ) ej(m1−m)ϕdϕ ,

for |m1 − m| = lN , l = 0, 1, . . ., and we see that

T ijmn,m1n1

=

N (Trotsym)ij

mn,m1n1,

0 ,

|m1 − m| = lN, l = 0, 1, . . .

rest,

where (Trotsym)ijmn,m1n1

are the elements of the transition matrix computedby integrating ϕ over the interval [0, 2π/N ].


3. We consider a particle with mirror symmetry and assume that the xy-planeis the horizontal plane of reflection. The surface parameterization of a particlewith mirror symmetry has the property

r (θ, ϕ) = r (π − θ, ϕ)

and therefore

∂r

∂θ(θ, ϕ) = −∂r

∂θ(π − θ, ϕ) ,

∂r

∂ϕ(θ, ϕ) =

∂r

∂ϕ(π − θ, ϕ)

for 0 ≤ θ ≤ π/2 and 0 ≤ ϕ < 2π. Taking into account the symmetry relationsof the normalized angular functions π

|m|n and τ

|m|n (cf. (A.24) and (A.25)),

πmn (π − θ) = (−1)n−m

πmn (θ) ,

τmn (π − θ) = (−1)n−m+1

τmn (θ) ,

we find that

T ijmn,m1n1

=[1 + (−1)n+n1+|m|+|m1|+i+j

](Tmirrsym)ij

mn,m1n1,

where (Tmirrsym)ijmn,m1n1

are the elements of the transition matrix computedby integrating θ over the interval [0, π/2]. The above relation also shows that

T ijmn,m1n1

= 0 ,

if n + n1 + |m| + |m1| + i + j is an odd number.Exploitation of these symmetry relations leads to a reduction in CPU time

by three orders of magnitude from that of a standard implementation with nogeometry-specific adaptations. Additional properties of the transition matrixfor particles with specific symmetries are discussed by Kahnert et al. [111]and by Havemann and Baran [98].

2.1.4 Practical Considerations

The integrals over the particle surface are usually computed by using appro-priate quadrature formulas. For particles with piecewise smooth surfaces, thenumerical stability and accuracy of the T -matrix calculations can be improvedby using separate Gaussian quadratures on each smooth section [8, 170].

In practice, it is more convenient to interchange the order of summationin the surface field representations, i.e., the sum

Nrank∑

n=1

n∑

m=−n

(·)


is replaced by

Mrank∑

m=−Mrank

Nrank∑

n=max(1,|m|)(·) ,

where Nrank and Mrank are the maximum expansion order and the number ofazimuthal modes, respectively. Consequently, the dimension of the transitionmatrix T is

dim (T ) = 2Nmax × 2Nmax ,

where

Nmax = Nrank + Mrank (2Nrank − Mrank + 1) ,

is the truncation multi-index appearing in (2.7). The interchange of summa-tion orders is effective for axisymmetric particles because, in this case, thescattering problem can be reduced to a sequence of subproblems for eachazimuthal mode.

An important part of the numerical analysis is the convergence proce-dure that checks whether the size of the transition matrix and the number ofquadrature points for surface integral calculations are sufficiently large thatthe scattering characteristics are computed with the desired accuracy. Theconvergence tests presented in the literature are based on the analysis of thedifferential scattering cross-section [8] or the extinction and scattering cross-sections [164]. The procedure used by Barber and Hill [8] solves the scatteringproblem for consecutive values of Nrank and Mrank, and checks the convergenceof the differential scattering cross-section at a number of scattering angles. Ifthe calculated results converge within a prescribed tolerance at 80% of thescattering angles, then convergence is achieved. The procedure developed byMishchenko [164] is applicable to axisymmetric particles and finds reliableestimates of Nrank by checking the convergence of the quantities

Ce = −2π

k2s

Nrank∑

n=1

(2n + 1) ReT 11

0n,0n + T 220n,0n

and

Cs =2π

k2s

Nrank∑

n=1

(2n + 1)(∣∣T 11

0n,0n

∣∣2 +

∣∣T 22

0n,0n

∣∣2)

.

The null-field method is a general technique and is applicable for arbitrarilyshaped particles. However, for nonaxisymmetric particles, a semi-convergent


behavior is usually attended: the relative variations of the differential scat-tering cross-sections decrease with increasing the maximum expansion order,attain a relative constant level and afterwards increase. In this case, the mainproblem of the convergence analysis is the localization of the region of sta-bility. The extinction and scattering cross-sections can also give informationon the convergence process. Although the convergence of Cext and Cscat doesnot guarantee that the differential scattering cross-section converges, the di-vergence of Cext and Cscat implies the divergence of the T -matrix calculation.

2.1.5 Surface Integral Equation Method

The null-field method leads to a nonsingular integral equation of the firstkind. However, in the framework of the surface integral equation method, thetransmission boundary-value problem can be reduced to a pair of singularintegral equations of the second kind [97]. These equations are formulatedin terms of two surface fields which are treated as independent unknowns. Inorder to elucidate the difference between the null-field method and the surfaceintegral equation method we follow the analysis of Martin and Ola [155] andreview the basic boundary integral equations for the transmission boundary-value problem. We consider the vector potential Aa with density a

Aa(r) =∫

S

a(r′)g(k, r, r′) dS(r′) , r ∈ R3 − S

and evaluate the tangential components of the curl and double curl of thevector potential on S. For continuous tangential density, we have [39]

limh→0+

n(r) × [∇× Aa(r ± hn(r))] = ±12a(r) + (Ma)(r) , r ∈ S ,

where M is the magnetic dipole operator,

(Ma)(r) = n(r) ×[∇×

∫

S

a(r′)g(k, r, r′) dS(r′)]

, r ∈ S .

For a sufficiently smooth tangential density, we also have

limh→0+

n(r) × [∇×∇× Aa(r ± hn(r))] = (Pa)(r) , r ∈ S ,

where the principal value singular integral operator P, called the electric di-pole operator is given by

(Pa)(r) = n(r) ×[∇×∇×

∫

S

a(r′)g(k, r, r′)dS(r′)]

, r ∈ S .

Considering the null-field equations for the electric and magnetic fields inDi and Ds, and passing to the boundary along a normal direction we obtain


(12I −Ms

)ei −

jk0εs

Pshi = ee , (2.26)

(12I −Ms

)hi +

jk0µs

Psei = he , (2.27)

and(

12I + Mi

)ei +

jk0εi

Pihi = 0 , (2.28)

(12I + Mi

)hi −

jk0µi

Piei = 0 , (2.29)

respectively. These are four boundary integral equations for the unknowns ei

and hi, and we consider two linear combinations of equations, i.e.,

α1 (2.26) + α2 (2.27) + α3 (2.29) andβ1 (2.26) + β2(2.27) + β3(2.28) ,

where αi and βi, i = 1, 2, 3, are constants to be chosen. Harrington [97] de-scribes several possible choices, as shown in Table 2.1, and for all these choiceswe always have existence and unique solvability [155].

The above approach for deriving a pair of boundary integral equations isknown as the direct method. In contrast to the null-field method, the directmethod considers the null-field equations in both domains Di and Ds, andtreats the surface fields as independent unknowns. The indirect method forderiving a pair of boundary integral equations relies on the representation ofthe electromagnetic fields in terms of four surface fields. Passing to the bound-ary, using the boundary conditions on the particle surface and imposing twoconstraints on the surface fields, yields the desired pair of boundary integralequations [97]. It should be emphasized that single integral equations for thetransmission boundary-value problem have also been derived by Marx [156],Mautz [157] and Martin and Ola [155]. Another major difference to the null-field method is the discretization of the boundary integral equations, whichis achieved by using the method of moments [96]. The boundary surface isdiscretized into a set of surface elements and on each element, the surfacefields are approximated by finite expansions of basis functions. Next, a set of

Table 2.1. Choice of constants for the boundary integral equations

Formulation α1 α2 α3 β1 β2 β3

E-field 1 0 0 0 0 1

H-field 0 0 1 0 1 0

Combined field 0 1 −1 1 0 −1

Mautz–Harrington 0 1 −β 1 0 −α

Muller 0 µs µi εs 0 εi


testing functions are defined and the scalar product of each testing function isformed with both sides of the equation being solved. This results in a systemof equations which is referred to as the element matrix equations. The ele-ment matrices are assembled into the global matrix of the entire “structure”and the resulting system of equations is solved for the unknown expansioncoefficients.

2.1.6 Spherical Particles

An interesting feature of the null-field method is that all matrix equationsbecome considerably simpler and reduce to the corresponding equations of theLorenz–Mie theory when the particle is spherically. For a spherical particle ofradius R, the orthogonality relations of the vector spherical harmonics showthat the Qpq matrices are diagonal

(Qpq)12mn,m1n1= (Qpq)21mn,m1n1

= 0

for all values of m, n, m1 and n1, and

(Q31)11mn,m1n1

= jx

jn (mrx)[xh(1)

n (x)]′

− h(1)n (x) [mrxjn (mrx)]′

δmm1δnn1 , (2.30)

(Q31)22mn,m1n1

=jxmr

−h(1)

n (x) [mrxjn (mrx)]′

+ m2r jn (mrx)

[xh(1)

n (x)]′

δmm1δnn1 , (2.31)

(Q33)11mn,m1n1

= jx

h(1)n (mrx)

[xh(1)

n (x)]′

− h(1)n (x)

[mrxh(1)

n (mrx)]′

δmm1δnn1 , (2.32)

(Q33)22mn,m1n1

=jxmr

−h(1)

n (x)[mrxh(1)

n (mrx)]′

+ m2rh

(1)n (mrx)

[xh(1)

n (x)]′

δmm1δnn1 , (2.33)

(Q11)11mn,m1n1

= jxjn (mrx) [xjn (x)]′

− jn (x) [mrxjn (mrx)]′

δmm1δnn1 , (2.34)


(Q11)22mn,m1n1

=jxmr

−jn (x) [mrxjn (mrx)]′

+ m2r jn (mrx) [xjn (x)]′

δmm1δnn1 , (2.35)

and(Q13)11mn,m1n1

= jx

h(1)n (mrx) [xjn (x)]′

− jn (x)[mrxh(1)

n (mrx)]′

δmm1δnn1 , (2.36)

(Q13)22mn,m1n1

=jxmr

−jn (x)

[mrxh(1)

n (mrx)]′

+ m2rh

(1)n (mrx) [xjn (x)]′

δmm1δnn1 , (2.37)

where x = ksR is the size parameter and

mr =√

εi

εs

is the relative refractive index of the particle with respect to the ambientmedium. The above relations are not suitable for computing the Qpq matrices.Denoting by An and Bn the logarithmic derivatives [2, 17]

An(x) =ddx

ln [xjn (x)] =[xjn (x)]′

xjn(x),

Bn(x) =ddx

ln[xh(1)

n (x)]

=

[xh

(1)n (x)

]′

xh(1)n (x)

,

and using the recurrence relation (cf. (A.8))

[xzn (x)]′ = xzn−1(x) − nzn(x) ,

where zn stands for jn or h(1)n , we rewrite (2.30)–(2.37) as

(Q31)11mn,m1n1

= −jx2jn (mrx)[

mrAn (mrx) +n

x

]h(1)

n (x)

− h(1)n−1 (x)

δmm1δnn1 , (2.38)

(Q31)22mn,m1n1

= −jmrx2jn (mrx)

[An (mrx)

mr+

n

x

]h(1)

n (x)

− h(1)n−1 (x)

δmm1δnn1 , (2.39)


(Q33)11mn,m1n1

= −jx2h(1)n (mrx)

[mrBn (mrx) +

n

x

]h(1)

n (x)

− h(1)n−1 (x)

δmm1δnn1 , (2.40)

(Q33)22mn,m1n1

= −jmrx2h(1)

n (mrx)[

Bn (mrx)mr

+n

x

]h(1)

n (x)

− h(1)n−1 (x)

δmm1δnn1 , (2.41)

(Q11)11mn,m1n1

= −jx2jn (mrx)[

mrAn (mrx) +n

x

]jn (x)

− jn−1 (x) δmm1δnn1 , (2.42)

(Q11)22mn,m1n1

= −jmrx2jn (mrx)

[An (mrx)

mr+

n

x

]jn (x)

− jn−1 (x) δmm1δnn1 , (2.43)

and(Q13)11mn,m1n1

= −jx2h(1)n (mrx)

[mrBn (mrx) +

n

x

]jn (x)

− jn−1 (x) δmm1δnn1 , (2.44)

(Q13)22mn,m1n1

= −jmrx2h(1)

n (mrx)[

Bn (mrx)mr

+n

x

]jn (x)

− jn−1 (x) δmm1δnn1 . (2.45)

The functions An and Bn satisfy the recurrence relation

Ψn(x) =n + 1

x− 1

Ψn+1(x) + n+1x

,

where Ψn stands for An and Bn, and a stable scheme for computing Ψn relieson a downward recursion. Beginning with an estimate for Ψn, where n is largerthat the number of terms required for convergence, successively lower-orderlogarithmic derivatives can be generated by downward recursion. It should benoted that the downward stability of Ψn is a consequence of the downwardstability of the spherical Bessel functions jn (see Appendix A).

The transition matrix of a spherical particle is diagonal with entries

T 11mn,m1n1

= T 1nδmm1δnn1 ,

T 22mn,m1n1

= T 2nδmm1δnn1 ,


where

T 1n = −

[mrAn (mrx) + n

x

]jn (x) − jn−1 (x)

[mrAn (mrx) + n

x

]h

(1)n (x) − h

(1)n−1 (x)

(2.46)

and

T 2n = −

[An(mrx)

mr+ n

x

]jn (x) − jn−1 (x)

[An(mrx)

mr+ n

x

]h

(1)n (x) − h

(1)n−1 (x)

. (2.47)

Equations (2.46) and (2.47) relating the transition matrix to the size pa-rameter and relative refractive index are identical to the expressions of theLorenz–Mie coefficients given by Bohren and Huffman [17].

2.2 Homogeneous and Chiral Particles

The problem of scattering by isotropic, chiral spheres has been treated byBohren [16], and Bohren and Huffman [17] using rigorous electromagneticfield-theoretical calculations, while the analysis of nonspherical, isotropic, chi-ral particles has been rendered by Lakhtakia et al. [135]. To account for chi-rality, the surface fields have been approximated by left- and right-circularlypolarized fields and the same technique is employed in our analysis. The trans-mission boundary-value problem for a homogeneous and isotropic, chiral par-ticle has the following formulation.


∇× Es = jk0µsHs , ∇× Hs = −jk0εsEs (2.48)

in Ds, and

∇×[

Ei

H i

]= K

[Ei

H i

](2.49)

in Di, where

K =1

1 − β2k2i

[βk2

i jk0µi

−jk0εi βk2i

]. (2.50)

In addition, the vector fields must satisfy the transmission conditions (2.2)and the Silver–Muller radiation condition (2.3).

Applications of the extinction theorem and Huygens principle yield thenull-field equations (2.6) and the integral representations for the scatteredfield coefficients (2.16). Taking into account that the electromagnetic fieldspropagating in an isotropic, chiral medium can be expressed as a superpositionof vector spherical wave functions of left- and right-handed type (cf. Sect. 1.3),we represent the approximate surface fields as

2.2 Homogeneous and Chiral Particles 103

(eN

i (r′)hN

i (r′)

)

=N∑

µ=1

cNµ

⎛

⎝n (r′) × Lµ (kLir

′)

−j√

εiµi

n (r′) × Lµ (kLir′)

⎞

⎠

+dNµ

⎛

⎝n (r′) × Rµ (kRir

′)

j√

εiµi

n (r′) × Rµ (kRir′)

⎞

⎠ ,

where Lµ and Rµ are given by (1.58) and (1.59), respectively, and

kLi =ki

1 − βki, kRi =

ki

1 + βki

are the wave numbers of the left- and right-handed type waves. The transitionmatrix of a homogeneous, chiral particle then becomes

T = −Q11chiral (ks, ki)

[Q31

chiral (ks, ki)]−1

,

where, for µi = µs, the elements of the Q31chiral matrix are given by

(Q31

chiral

)11νµ

=jk2

s

π

∫

S

[n (r′) × Lµ (kLir

′)] · N3ν (ksr

′)

+√

εi

εs[n (r′) × Lµ (kLir

′)] · M3ν (ksr

′)

dS (r′) , (2.51)

(Q31

chiral

)12νµ

=jk2

s

π

∫

S

[n (r′) × Rµ (kRir

′)] · N3ν (ksr

′)

−√

εi

εs[n (r′) × Rµ (kRir

′)] · M3ν (ksr

′)

dS (r′) , (2.52)

(Q31

chiral

)21νµ

=jk2

s

π

∫

S

[n (r′) × Lµ (kLir

′)] · M3ν (ksr

′)

+√

εi

εs[n (r′) × Lµ (kLir

′)] · N3ν (ksr

′)

dS (r′) , (2.53)

and

(Q31

chiral

)22νµ

=jk2

s

π

∫

S

[n (r′) × Rµ (kRir

′)] · M3ν (ksr

′)

−√

εi

εs[n (r′) × Rµ (kRir

′)] · N3ν (ksr

′)

dS (r′) . (2.54)

The expressions of the elements of the Q11chiral matrix are similar but with M1

ν

and N1ν in place of M3

ν and N3ν , respectively. It must be noted that in case the


particle is spherical, the transition matrix becomes diagonal, and the resultingsolution tallies exactly with that given by Bohren [16] for chiral spheres. Inaddition, if β = 0, i.e., the particle becomes a nonchiral sphere, the Lorenz–Mie series solution is obtained. Finally, simply by setting kLi = kRi = ki andβ = 0, the solution for a nonspherical, nonchiral particles is recovered. In ouranalysis, the ambient medium is nonchiral and a generalization of the presentapproach to the scattering by a chiral particle in a chiral host medium hasbeen addressed by Lakhtakia [128].

2.3 Homogeneous and Anisotropic Particles

The scattering by anisotropic particles is mostly restricted to simple shapessuch as cylinders [232] or spheres [265]. Liu et al. [143] solved the electromag-netic fields in a rotationally uniaxial medium by using the method of separa-tion of variables, while Piller and Martin [193] analyzed three-dimensionalanisotropic particles by using the generalized multipole technique. In ouranalysis we follow the treatment of Kiselev et al. [119] which solved the scatter-ing problem of radially and uniformly anisotropic spheres by using the vectorquasi-spherical wave functions for internal field representation. The trans-mission boundary-value problem for a homogeneous and uniaxial anisotropicparticle has the following formulation.


∇× Es = jk0µsHs , ∇× Hs = −jk0εsEs (2.55)

in Ds, and

∇× Ei = jk0Bi , ∇× H i = −jk0Di ,

∇ · Bi = 0 , ∇ · Di = 0 (2.56)

in Di, where

Di = εiEi , Bi = µiH i , (2.57)

and

εi =

⎡

⎣εi 0 00 εi 00 0 εiz

⎤

⎦ . (2.58)

In addition, the vector fields must satisfy the transmission conditions (2.2)and the Silver–Muller radiation condition (2.3).

Considering the null-field equations (2.6) and the integral representationsfor the scattered field coefficients (2.16), and taking into account the results

2.4 Inhomogeneous Particles 105

established in Sect. 1.3 regarding the series representations of the electromag-netic fields propagating in anisotropic media, we see that the scattering prob-lem can be solved if we approximate the surface fields by finite expansions ofvector quasi-spherical wave functions Xe,h

mn and Y e,hmn (cf. (1.48)–(1.53)),

(eN

i (r′)hN

i (r′)

)

=N∑

µ=1

cNµ

⎛

⎝n (r′) × Xe

µ (r′)

−j√

εiµi

n (r′) × Xhµ (r′)

⎞

⎠

+dNµ

⎛

⎝n (r′) × Y e

µ (r′)

−j√

εiµi

n (r′) × Y hµ (r′)

⎞

⎠ .

The transition matrix of an uniaxial anisotropic particle then becomes

T = −Q11anis (ks, ki,mrz)

[Q31

anis (ks, ki,mrz)]−1

,

where mrz =√

εiz/εs, and for µi = µs, the elements of the Q31anis matrix are

given by

(Q31

anis

)11νµ

=jk2

s

π

∫

S

[(n × Xe

µ

)· N3

ν +√

εi

εs

(n × Xh

µ

)· M3

ν

]dS , (2.59)

(Q31

anis

)12νµ

=jk2

s

π

∫

S

[(n × Y e

µ

)· N3

ν +√

εi

εs

(n × Y h

µ

)· M3

ν

]dS , (2.60)

(Q31

anis

)21νµ

=jk2

s

π

∫

S

[(n × Xe

µ

)· M3

ν +√

εi

εs

(n × Xh

µ

)· N3

ν

]dS , (2.61)

and

(Q31

anis

)22νµ

=jk2

s

π

∫

S

[(n × Y e

µ

)· M3

ν +√

εi

εs

(n × Y h

µ

)· N3

ν

]dS . (2.62)

The expressions of the elements of the Q11anis matrix are similar but with M1

ν

and N1ν in place of M3

ν and N3ν , respectively. Using the properties of the

vector quasi-spherical wave functions (cf. (1.47)) it is simple to show that forεiz = εi, the present approach leads to the T -matrix solution of an isotropicparticle.

2.4 Inhomogeneous Particles

In this section, we consider electromagnetic scattering by an arbitrarilyshaped, inhomogeneous particle with an irregular inclusion. Our treatmentfollows the analysis of Peterson and Strom [189] for multilayered particlesand is similar to the approach used by Videen et al. [243] for a sphere with anirregular inclusion. Note that an alternative derivation using the Schelkunoff’sequivalence principle has been given by Wang and Barber [248].


2.4.1 Formulation with Addition Theorem

In the present analysis we will derive the expression of the transition matrixby using the translation properties of the vector spherical wave functions. Thecompleteness property of the vector spherical wave functions on two enclosingsurfaces, which is essential in our analysis, is established in Appendix D.

The scattering problem is depicted in Fig. 2.2. The surface S1 is definedwith respect to a Cartesian coordinate system O1x1y1z1, while the surfaceS2 is defined with respect to a Cartesian coordinate system O2x2y2z2. Byassumption, the coordinate system O2x2y2z2 is obtained by translating thecoordinate system O1x1y1z1 through r12 and by rotating the translated co-ordinate system through the Euler angles α, β and γ. The boundary-valueproblem for the inhomogeneous particle depicted in Fig. 2.2 has the followingformulation.

Given the external excitation Ee,He as an entire solution to the Maxwellequations, find the scattered field Es, Hs and the internal fields Ei,1, H i,1andEi,2, H i,2 satisfying the Maxwell equations

∇× Es = jk0µsHs , ∇× Hs = −jk0εsEs in Ds , (2.63)

∇× Ei,1 = jk0µi,1H i,1 , ∇× H i,1 = −jk0εi,1Ei,1 in Di,1 , (2.64)

and


the boundary conditions

n1 × Ei,1 − n1 × Es = n1 × Ee ,

n1 × H i,1 − n1 × Hs = n1 × He , (2.66)

O1

D i,1

Ds

S1

n1

O2

M1

M 2

P r199

r299r19

r29

r1

r2

S2D i,2

r12

n2

Fig. 2.2. Geometry of an inhomogeneous particle


on S1 and

n2 × Ei,1 = n2 × Ei,2 ,

n2 × H i,1 = n2 × H i,2 , (2.67)

onS2, and the Silver–Muller radiation condition for the scattered field (2.3).The Stratton–Chu representation theorem for the scattered field Es in Di,

where Di = Di,1 ∪ Di,2 ∪ S2 = Di,1 ∪ Di,2 = R3 − Ds, together with theboundary conditions (2.66), yield the general null-field equation

Ee(r1) + ∇1 ×∫

S1

ei,1 (r′1) g (ks, r1, r

′1) dS (r′

1) (2.68)

+j

k0εs∇1 ×∇1 ×

∫

S1

hi,1 (r′1) g (ks, r1, r

′1) dS (r′

1) = 0 , r1 ∈ Di ,

while the Stratton–Chu representation theorem for the internal field Ei,1 inDs and Di,2 together with the boundary conditions (2.67), give the generalnull-field equation

−∇1 ×∫

S1

ei,1 (r′1) g (ki,1, r1, r

′1) dS (r′

1)

− jk0εi,1

∇1 ×∇1 ×∫

S1

hi,1 (r′1) g (ki,1, r1, r

′1) dS (r′

1)

+∇1 ×∫

S2

ei,2 (r′′1) g (ki,1, r1, r

′′1) dS (r′′

1) (2.69)

+j

k0εi,2∇1 ×∇1 ×

∫

S2

hi,2 (r′′1) g (ki,1, r1, r

′′1) dS (r′′

1) = 0 , r1 ∈ Ds ∪ Di,2 .

In (2.68) and (2.69), the surface fields are the tangential components of theelectromagnetic fields in Di,1 and Di,2

ei,1 = n1 × Ei,1 , hi,1 = n1 × H i,1 on S1

and

ei,2 = n2 × Ei,2, hi,2 = n2 × H i,2 on S2 ,

respectively.Considering the general null-field equation (2.68), we use the vector spher-

ical wave expansions of the incident field and of the dyad gI on a sphereenclosed in Di, to obtain

jk2s

π

∫

S1

[

ei,1 (r′1) ·(

N3ν (ksr

′1)

M3ν (ksr

′1)

)

(2.70)

+ j√

µs

εshi,1 (r′

1) ·(

M3ν (ksr

′1)

N3ν (ksr

′1)

)]

dS (r′1) = −

(aν

bν

)

, ν = 1, 2, . . .


For the general null-field equation (2.69) in Ds, we proceed analogously butrestrict r1 to lie on a sphere enclosing Di. We then have

−jk2

i,1

π

∫

S1

[

ei,1 (r′1) ·(

N1ν(ki,1r

′1)

M1ν(ki,1r

′1)

)

+ j√

µi,1

εi,1hi,1 (r′

1) ·(

M1ν(ki,1r

′1)

N1ν(ki,1r

′1)

)]

dS (r′1)

+jk2

i,1

π

∫

S2

[

ei,2 (r′′2) ·(

N1ν (ki,1r

′′1)

M1ν(ki,1r

′′1)

)

(2.71)

+ j√

µi,1

εi,1hi,2 (r′′

2) ·(

M1ν(ki,1r

′′1)

N1ν(ki,1r

′′1)

)]

dS (r′′2) = 0 , ν = 1, 2, . . . ,

where the identities ei,2(r′′2) = ei,2(r′′

1) and hi,2(r′′2) = hi,2(r′′

1) have beenused. Finally, for the general null-field equation (2.69) in Di,2, we pass fromthe origin O1 to the origin O2 by taking into account that the gradient isinvariant to a translation of the coordinate system, use the relations

g (ki,1, r1, r′1) = g (ki,1, r2, r

′2) ,

g (ki,1, r1, r′′1) = g (ki,1, r2, r

′′2) ,

and restrict r2 to lie on a sphere enclosed in Di,2. We obtain

−jk2

i,1

π

∫

S1

[

ei,1 (r′1) ·(

N3ν(ki,1r

′2)

M3ν(ki,1r

′2)

)

+ j√

µi,1

εi,1hi,1 (r′

1) ·(

M3ν(ki,1r

′2)

N3ν(ki,1r

′2)

)]

dS (r′1)

+jk2

i,1

π

∫

S2

[

ei,2 (r′′2) ·(

N3ν(ki,1r

′′2)

M3ν(ki,1r

′′2)

)

(2.72)

+ j√

µi,1

εi,1hi,2 (r′′

2) ·(

M3ν(ki,1r

′′2)

N3ν(ki,1r

′′2)

)]

dS (r′′2) = 0 , ν = 1, 2, . . . ,

where, as before, the identities ei,1(r′1) = ei,1(r′

2) and hi,1(r′1) = hi,1(r′

2)have been employed. The set of integral equations (2.70)–(2.72) represent thenull-field equations for the scattering problem under examination.

The surface fields ei,1 and hi,1 are the tangential components of the elec-tric and magnetic fields in the domain Di,1 bounded by the closed surfaces S1

and S2. Taking into account the completeness property of the system of reg-ular and radiating vector spherical wave functions on two enclosing surfaces


(Appendix D), we approximate the surface fields ei,1 and hi,1 by the finiteexpansions

(eN

i,1(r′1)

hNi,1(r

′1)

)

=N∑

µ=1

cN1,µ

⎛

⎝n1(r′

1) × M1µ(ki,1r

′1)

−j√

εi,1µi,1

n1(r′1) × N1

µ(ki,1r′1)

⎞

⎠

+dN1,µ

⎛

⎝n1(r′

1) × N1µ(ki,1r

′1)

−j√

εi,1µi,1

n1(r′1) × M1

µ(ki,1r′1)

⎞

⎠

+cN1,µ

⎛

⎝n1(r′

1) × M3µ(ki,1r

′1)

−j√

εi,1µi,1

n1(r′1) × N3

µ(ki,1r′1)

⎞

⎠

+dN1,µ

⎛

⎝n1(r′

1) × N3µ(ki,1r

′1)

−j√

εi,1µi,1

n1(r′1) × M3

µ(ki,1r′1)

⎞

⎠ . (2.73)

The surface fields ei,2 and hi,2 are the tangential components of the electricand magnetic fields in the domain Di,2 and the surface fields approximationscan be expressed as linear combinations of regular vector spherical wave func-tions:

(eN

i,2(r′′2)

hNi,2(r

′′2)

)

=N∑

µ=1

cN2,µ

⎛

⎝n2(r′′

2) × M1µ(ki,2r

′′2)

−j√

εi,2µi,2

n2(r′′2) × N1

µ(ki,2r′′2)

⎞

⎠

+dN2,µ

⎛

⎝n2(r′′

2) × N1µ(ki,2r

′′2)

−j√

εi,2µi,2

n2(r′′2) × M1

µ(ki,2r′′2)

⎞

⎠ . (2.74)

To express the resulting system of equations in matrix form we introduce theQpq

t (k1, k2) matrix as the Qpq(k1, k2) matrix of the surface St. The elementsof the Qpq

t matrix are given by (2.11)–(2.14) but with St in place of S. Forexample, the elements (Qpq

t )11νµ read as

(Qpqt )11νµ =

jk21

π

∫

St

[n (r′

t) × M qµ (k2r

′t)]· Np

ν (k1r′t)

+√

ε2

ε1

[n (r′

t) × N qµ (k2r

′t)]· Mp

ν (k1r′t)

dS (r′t) .

Inserting (2.73) and (2.74) into (2.70), (2.71) and (2.72), using the identities(cf. (B.23), (B.24) and (B.25)),

Qpp1 (k, k) = 0 , for p = 1 or p = 3 ,

Q131 (k, k) = I ,

Q311 (k, k) = −I ,


and taking into account the transformation rules[

M1ν (ki,1r

′′1)

N1ν (ki,1r

′′1)

]

=[(Str

12

)νµ

][

M1µ (ki,1r

′′2)

N1µ (ki,1r

′′2)

]

and[

M3ν (ki,1r

′2)

N3ν (ki,1r

′2)

]

=[(Srt

21

)νµ

][

M3µ (ki,1r

′1)

N3µ (ki,1r

′1)

]

,

we obtain the system of matrix equations

Q311 (ks, ki,1)i1 + Q33

1 (ks, ki,1)i1 = −e ,

−i1 + Str12Q

112 (ki,1, ki,2)i2 = 0 ,

Srt21i1 + Q31

2 (ki,1, ki,2)i2 = 0 , (2.75)

where i1 = [cN1,µ, dN

1,µ]T, i1 = [cN1,µ, dN

1,µ]T, i2 = [cN2,µ, dN

2,µ]T, and as before,e = [aν , bν ]T is the vector containing the expansion coefficients of the incidentfield. The Str

12 matrix relates the vector spherical wave functions defined withrespect to the coordinate system O1x1y1z1 to those defined with respect tothe coordinate system O2x2y2z2, and can be expressed as the product of atranslation and a rotation matrix:

Str12 = T 11 (ki,1r12)R (α, β, γ) , (2.76)

where T and R are defined in Appendix B. The Srt21 matrix describes the

inverse transformation and is given by

Srt21 = R (−γ,−β,−α) T 33 (−ki,1r12) for r′1 > r12 . (2.77)

Since R is a block-diagonal matrix and T is a block-symmetric matrix itfollows that Str

12 and Srt21 are also block-symmetric matrices.

Solving the system of matrix equations gives

i1 = −[Q31

1 (ks, ki,1) + Q331 (ks, ki,1)Str

12T 2Srt21

]−1e ,

i1 = Str12T 2Srt

21i1 ,

i2 = −[Q31

2 (ki,1, ki,2)]−1 Srt

21i1 , (2.78)

where

T 2 = −Q112 (ki,1, ki,2)

[Q31

2 (ki,1, ki,2)]−1

is the transition matrix of the inhomogeneity imbedded in a medium withrelative media constants εi,1 and µi,1.


The Stratton–Chu representation theorem for the scattered field Es in Ds,yields the expansion of the approximate scattered field EN

s in the exterior ofa sphere enclosing the particle

ENs (r1) =

N∑

ν=1

fNν M3

ν (ksr1) + gNν N3

ν (ksr1) ,

where the expansion coefficients are given by(

fNν

gNν

)

=jk2

s

π

∫

S1

[

eNi,1 (r′) ·

(N1

ν (ksr′1)

M1ν (ksr

′1)

)

+ j√

µs

εshN

i,1 (r′) ·(

M1ν (ksr

′1)

N1ν (ksr

′1)

)]

dS (r′1) .

Taking into account the expressions of the approximate surface fields givenby (2.73), we derive the matrix equation

s = Q111 (ks, ki,1) i1 + Q13

1 (ks, ki,1) i1 , (2.79)


ν ]T is the vector containing the expansion coefficients ofthe scattered field. Combining (2.78) and (2.79) we see that the transitionmatrix relating the scattered field coefficients to the incident field coefficients,s = Te, is given by

T = −[Q11

1 (ks, ki,1) + Q131 (ks, ki,1) T 2

] [Q31

1 (ks, ki,1) + Q331 (ks, ki,1)T 2

]−1

,

(2.80)where

T 2 = Str12T 2Srt

21 . (2.81)

For a homogeneous particle T 2 = 0, and we obtain the result established inSect. 2.1

T = −Q111 (ks, ki)

[Q31

1 (ks, ki)]−1

.

The expression of the transition matrix can also be written as

T =

T 1 − Q131 (ks, ki,1) T 2

[Q31

1 (ks, ki,1)]−1

×

I + Q331 (ks, ki,1)T 2

[Q31

1 (ks, ki,1)]−1−1

, (2.82)

where

T 1 = −Q111 (ks, ki)

[Q31

1 (ks, ki)]−1

is the transition matrix of the host particle. If the coordinate systems O1x1y1z1

and O2x2y2z2 coincide, (2.82) is identical to the result given by Peterson and


Strom [189]. The various terms obtained by a formal expansion of the inversein (2.82) can be interpreted as various multiple-scattering contributions to thetotal transition matrix. Indeed, using the representation

T =T 1 − Q13

1 T 2

(Q31

1

)−1 I − Q33

1 T 2

(Q31

1

)−1+ ...

= T 1 − Q131 T 2

(Q31

1

)−1 − T 1Q331 T 2

(Q31

1

)−1

+Q131 T 2

(Q31

1

)−1Q33

1 T 2

(Q31

1

)−1+ . . .

we see that the term T 1 represents a reflection at S1, Q131 T 2(Q31

1 )−1 repre-sents a passage of a wave through S1 and a reflection at S2, T 1Q

331 T 2(Q31

1 )−1

represents a refraction of a wave through S1 and two consecutive reflectionsat S2 and S1, etc.

The expressions of the total transition matrix given by (2.80) or (2.82)are important in practical applications. As it has been shown by Petersonand Strom [189], this result can be extended to the case of S1 containing anarbitrary number of separate enclosures by simply replacing T 2 with the sys-tem transition matrix of the particles. In the later sections we will derive thetransition matrix for a system of particles and the present formalism will en-able us to analyze scattering by an arbitrarily shaped, inhomogeneous particlewith an arbitrary number of irregular inclusions. In this context it should bementioned that the separation of variables solution for a single sphere (theLorenz–Mie theory) can be extended to spheres with one or more eccentri-cally positioned spherical inclusions by using the translation addition theoremfor vector spherical wave functions. Theories of scattering by eccentricallystratified spheres have been derived by Fikioris and Uzunoglu [64], Borgheseet al. [22], Fuller [73], Mackowski and Jones [152] and Ngo et al. [180], whiletreatments for a sphere with multiple spherical inclusions have been renderedby Borghese et al. [20], Fuller [75] and Ioannidou and Chrissoulidis [107].A detailed review of the separation of variable method for inhomogeneousspheres has been given by Fuller and Mackowski [77]. The separation of vari-able method has also been employed in spheroidal coordinate systems byCooray and Ciric [41] and Li et al. [141] to analyze the scattering by inhomo-geneous spheroids.

2.4.2 Formulation without Addition Theorem

In our previous analysis we assumed the geometric constraint r′1 > r12 (cf.(2.77)), which originates in the use of the translation addition theorem forradiating vector spherical wave functions. In this section we present a formal-ism that avoids the use of any local origin translation. To simplify our analy-sis, we assume that the coordinate systems O1x1y1z1 and O2x2y2z2 have thesame spatial orientation and set α = β = γ = 0. We begin by defining theQpq

t (k1, i, k2, j) matrix


Qpqt (k1, i, k2, j) =

⎡

⎣(Qpq

t )11νµ (Qpqt )12νµ

(Qpqt )21νµ (Qpq

t )22νµ

⎤

⎦ , (2.83)

as

(Qpqt )11νµ =

jk21

π

∫

St

[n(r′

j

)× M q

µ

(k2r

′j

)]· Np

ν (k1r′i)

+√

ε2

ε1

[n(r′

j

)× N q

µ

(k2r

′j

)]· Mp

ν (k1r′i)

dS (r′) , (2.84)

(Qpqt )12νµ =

jk21

π

∫

St

[n(r′

j

)× N q

µ

(k2r

′j

)]· Np

ν (k1r′i)

+√

ε2

ε1

[n(r′

j

)× M q

µ

(k2r

′j

)]· Mp

ν (k1r′i)

dS (r′) , (2.85)

(Qpqt )21νµ =

jk21

π

∫

St

[n(r′

j

)× M q

µ

(k2r

′j

)]· Mp

ν (k1r′i)

+√

ε2

ε1

[n(r′

j

)× N q

µ

(k2r

′j

)]· Np

ν (k1r′i)

dS (r′) , (2.86)

and

(Qpqt )22νµ =

jk21

π

∫

St

[n(r′

j

)× N q

µ

(k2r

′j

)]· Mp

ν (k1r′i)

+√

ε2

ε1

[n(r′

j

)× M q

µ

(k2r

′j

)]· Np

ν (k1r′i)

dS (r′) . (2.87)

The vectors r′, r′i and r′

j are the position vectors of a point M on the surfaceSt, and are defined with respect to the origins O, Oi and Oj , respectively(Fig. 2.3). In terms of the new Qpq

t matrices, the system of matrix equations(2.75) can be written as

Q331 (ks, 1, ki,1, 1)i1 + Q31

1 (ks, 1, ki,1, 1)i1 = −e ,

−i1 + Q112 (ki,1, 1, ki,2, 2)i2 = 0 ,

−Q311 (ki,1, 2, ki,1, 1)i1 + Q31

2 (ki,1, 2, ki,2, 2)i2 = 0 , (2.88)

while the matrix equation (2.79) reads as

s = Q131 (ks, 1, ki,1, 1) i1 + Q11

1 (ks, 1, ki,1, 1) i1 . (2.89)

Using the substitution method we eliminate the unknown vector i2 from thelast two matrix equations in (2.88) and obtain


O i

O j

M

St

O

rirj

r

Fig. 2.3. Position vectors of a point M on the surface St with respect to the originsO, Oi and Oj

i1 = T 2i1 (2.90)

with

T 2 = Q112 (ki,1, 1, ki,2, 2)

[Q31

2 (ki,1, 2, ki,2, 2)]−1

Q311 (ki,1, 2, ki,1, 1) . (2.91)

Equation (2.89) and the first matrix equation in (2.88), written in compactmatrix notation as

[se

]= Q1 (ks, ki,1)

[i1i1

], (2.92)

where

Q1 (ks, ki,1) =

[Q13

1 (ks, 1, ki,1, 1) Q111 (ks, 1, ki,1, 1)

−Q331 (ks, 1, ki,1, 1) −Q31

1 (ks, 1, ki,1, 1)

]

, (2.93)

then yield

T = −[Q11

1 (ks, 1, ki,1, 1) + Q131 (ks, 1, ki,1, 1) T 2

]

×[Q31

1 (ks, 1, ki,1, 1) + Q331 (ks, 1, ki,1, 1)T 2

]−1

. (2.94)

The expression of the transition matrix is identical to that given by (2.80),but the T 2 matrix is now given by (2.91) instead of (2.81). If the origins O1

and O2 coincide

Q311 (ki,1, 1, ki,1, 1) = −I

and we see that the T 2 matrix is the transition matrix of the inhomogeneity:

T 2 = T 2 = −Q112 (ki,1, 1, ki,2, 1)

[Q31

2 (ki,1, 1, ki,2, 1)]−1

.

This formalism will be used in Sect. 2.5 to derive a recurrence relation for thetransition matrix of a multilayered particle.

2.5 Layered Particles 115

2.5 Layered Particles

A layered particle is an inhomogeneous particle consisting of several consec-utively enclosing surfaces Sl, l = 1, 2, . . . ,N . Each surface Sl is defined withrespect to a coordinate system Olxlylzl and we assume that the coordinatesystems Olxlylzl have the same spatial orientation. The layered particle isimmersed in a medium with optical constants εs and µs, while the relativemedia constants and the wave number in the domain between Sl and Sl+1

are εi,l, µi,l and ki,l, respectively. The geometry of a (multi)layered particle isshown in Fig. 2.4. The case N = 2 has been treated in the previous sectionand the objective of the present analysis is to extend the results establishedfor two-layered particles to multilayered particles.


For a particle with N layers, the system of matrix equations consists in thenull-field equations in the interior of S1,

Q331 (ks, 1, ki,1, 1)i1 + Q31

1 (ks, 1, ki,1, 1)i1 = −e , (2.95)

the null-field equations in the exterior of Sl−1 and the interior of Sl

−il−1 + Q13l (ki,l−1, l − 1, ki,l, l)il

+Q11l (ki,l−1, l − 1, ki,l, l)il = 0 , (2.96)

−Q31l−1(ki,l−1, l, ki,l−1, l − 1)il−1 + Q33

l (ki,l−1, l, ki,l, l)il

+Q31l (ki,l−1, l, ki,l, l)il = 0 (2.97)

l = 2, 3, . . . ,N − 1

the null-field equations in the exterior of SN−1 and the interior of SN

O1

D i,l-1

Ds

S1

O l

D i,l-2

SlSN

ON

O l

O l-1

Sl

Sl-1

D i,l

Fig. 2.4. Geometry of a multilayered particle


−iN−1 + Q11N (ki,N−1,N − 1, ki,N ,N )iN = 0 , (2.98)

−Q31N−1(ki,N−1,N , ki,N−1,N − 1)iN−1 + Q31

N (ki,N−1,N , ki,N ,N )iN = 0(2.99)

and the matrix equation corresponding to the scattered field representation

s = Q131 (ks, 1, ki,1, 1) i1 + Q11

1 (ks, 1, ki,1, 1) i1 . (2.100)

For two consecutive layers, the surface fields il−1 and il−1 are related to thesurface fields il and il by the matrix equation

[il−1

il−1

]= Ql (ki,l−1, ki,l)

[il

il

], (2.101)

where

Ql (ki,l−1, ki,l) =

[I 0

0[Q31

l−1(ki,l−1, l, ki,l−1, l − 1)]−1

]

(2.102)

×[

Q13l (ki,l−1, l − 1, ki,l, l) Q11

l (ki,l−1, l − 1, ki,l, l)Q33

l (ki,l−1, l, ki,l, l) Q31l (ki,l−1, l, ki,l, l)

]

and l = 2, 3, . . . ,N − 1. For the surface fields iN−1and iN−1, that is, forl = N , we have

iN−1 = TN iN−1 (2.103)

with

TN = Q11N (ki,N−1,N − 1, ki,N ,N )

[Q31

N (ki,N−1,N , ki,N ,N )]−1

×Q31N−1(ki,N−1,N , ki,N−1,N − 1) . (2.104)

Then, using the matrix equation[

se

]= Q1 (ks, ki,1)

[i1i1

]

with Q1 being given by (2.93), we see that[

se

]= Q

[iN−1

iN−1

],

where

Q = Q1 (ks, ki,1)Q2 (ki,1, ki,2) . . .QN−1 (ki,N−2, ki,N−1) .

Finally, using (2.103) and denoting by (Q)ij , i, j = 1, 2, the block-matrixcomponents of Q, we obtain the expression of the transition matrix in termsof Q and TN :


T =[(Q)12 + (Q)11 TN

] [(Q)22 + (Q)21 TN

]−1

.

The structure of the above equations is such that a recurrence relation forcomputing the transition matrix can be established. For this purpose, wedefine the matrix T l+1,l+2,...,N as

il = T l+1,l+2,...,N il .

and use (2.101) to obtain

il−1 =[(Ql)

12 + (Ql)11

T l+1,l+2,...,N]il ,

il−1 =[(Ql)

22 + (Ql)21

T l+1,l+2,...,N]il ,

where (Ql)ij , i, j = 1, 2, are the block-matrix components of Ql(ki,l−1, ki,l).Hence, the matrix T l,l+1,...,N , satisfying il−1 = T l,l+1,...,N il−1, can be com-puted by using the downward recurrence relation

T l,l+1,...,N =[(Ql)

12 + (Ql)11

T l+1,l+2,...,N]

×[(Ql)

22 + (Ql)21

T l+1,l+2,...,N]−1

(2.105)

for l = N − 1,N − 2, . . . , 1. For l = N − 1, TN is given by (2.104), while forl = 1, Ql is the matrix Q1(ks, ki,1) and T l,l+1,...,N is the transition matrix ofthe layered particle

T = T 1,2,...,N = −[Q11

1 (ks, 1, ki,1, 1) + Q131 (ks, 1, ki,1, 1) T 2,3,...,N

]

×[Q31

1 (ks, 1, ki,1, 1) + Q331 (ks, 1, ki,1, 1)T 2,3,...,N

]−1

.

If the origins coincide, the above relations simplify considerably, since

Ql (ki,l−1, ki,l) =

[Q13

l (ki,l−1, 1, ki,l, 1) Q11l (ki,l−1, 1, ki,l, 1)

−Q33l (ki,l−1, 1, ki,l, 1) −Q31

l (ki,l−1, 1, ki,l, 1)

]

(2.106)

and

TN = TN = −Q11N (ki,N−1, 1, ki,N , 1)

[Q31

N (ki,N−1, 1, ki,N , 1)]−1

.

We obtain

T l,l+1,...,N = −[Q11

l (ki,l−1, 1, ki,l, 1) + Q13l (ki,l−1, 1, ki,l, 1)T l+1,l+2,...,N

]

×[Q31

l (ki,l−1, 1, ki,l, 1) + Q33l (ki,l−1, 1, ki,l, 1)T l+1,l+2,...,N

]−1

(2.107)


and further

T l,l+1,...,N =T l − Q13

l (ki,l−1, 1, ki,l, 1)T l+1,l+2,...,N

×[Q31

l (ki,l−1, 1, ki,l, 1)]−1

I + Q33l (ki,l−1, 1, ki,l, 1)

× T l+1,l+2,...,N[Q31

l (ki,l−1, 1, ki,l, 1)]−1−1

(2.108)

of which (2.82) is the simplest special case. Note that in (2.107) and (2.108),T l,l+1,...,N is the total transition matrix of the layered particle with outersurface Sl.

2.5.2 Practical Formulation

In practical computer calculations it is simpler to solve the system of matrixequations (2.95)–(2.99) for all unknown vectors il and il, l = 1, 2, . . . ,N − 1,and iN . For this purpose, we consider the global matrix

A =

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

A1 0 0 ... 0 0

A12 A21 0 ... 0 0

0 A23 A32 ... 0 0

0 0 0 ... AN−1,N−2 0

0 0 0 ... AN−1,N AN

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

,

where

A1 =[Q33

1 (ks, 1, ki,1, 1)Q311 (ks, 1, ki,1, 1)

], (2.109)

Al,l−1 =

[Q13

l (ki,l−1, l − 1, ki,l, l) Q11l (ki,l−1, l − 1, ki,l, l)

Q33l (ki,l−1, l, ki,l, l) Q31

l (ki,l−1, l, ki,l, l)

]

, (2.110)

Al−1,l =

[−I 0

0 −Q31l−1(ki,l−1, l, ki,l−1, l − 1)

]

, (2.111)

and

AN =

[Q11

N (ki,N−1,N − 1, ki,N ,N )Q31

N (ki,N−1,N , ki,N ,N )

]

. (2.112)

Then, denoting by A the inverse of A,

A = A−1 =

⎡

⎢⎢⎣

A11 A12 ... A1,2N−1

A21 A22 ... A2,2N−1

A2N−1,1 A2N−1,2 ... A2N−1,2N−1

⎤

⎥⎥⎦ ,


we express i1 and i1 as

i1 = −A11e , i1 = −A21e ,

and use (2.100) to obtain

T = −[Q13

1 (ks, 1, ki,1, 1)A11 + Q111 (ks, 1, ki,1, 1)A21

].

For axisymmetric layers and axial positions of the origins Ol (along the z-axisof rotation), the scattering problem decouples over the azimuthal modes andthe transition matrix can be computed separately for each m. Specifically,for each layer l, we compute the Ql matrices and assemble these matricesinto the global matrix A. The matrix A is inverted, and the blocks 11 and21 of the inverse matrix are used for T -matrix calculation. Because A is asparse matrix, appropriate LU–factorization routines (for sparse systems ofequations) can be employed.

An important feature of this solution method is that the expansion ordersof the surface field approximations can be different. To derive the dimensionof the global matrix A, we consider an axisymmetric particle. If Nrank(l) isthe maximum expansion order of the layer l and, for a given azimuthal modem, 2Nmax(l) × 2Nmax(l) is the dimension of the corresponding Q matrices,where

Nmax(l) =

Nrank(l) ,

Nrank(l) − |m| + 1 ,

m = 0m = 0 ,

then, the dimension of the global matrix A is given by

dim (A) = 2Nmax × 2Nmax ,

with

Nmax = Nmax (N ) + 2N−1∑

l=1

Nmax(l) .

The dimension and occupation of the matrix A is shown in Table 2.2 for threelayers.

Since

dim(A11)

= dim(A21)

= dim(Q13

1

)= dim

(Q11

1

)= 2Nmax(1) × 2Nmax(1) ,

it follows that

dim (T ) = 2Nmax(1) × 2Nmax(1) .

Thus, the dimension of the transition matrix is given by the maximum ex-pansion order corresponding to the first layer, while the maximum expansion


Table 2.2. Occupation of the global matrix

2Nmax(1) 2Nmax(1) 2Nmax(2) 2Nmax(2) 2Nmax(3)

2Nmax(1) Q331 Q31

1 0 0 0

2Nmax(1) −I(−Q131 ) 0 Q13

2 Q112 0

2Nmax(2) 0 −Q311 Q33

2 Q312 0

2Nmax(2) 0 0 −I(−Q132 ) 0 Q11

3

2Nmax(3) 0 0 0 −Q312 Q31

3

For distributed sources, the identity matrix I is replaced by the Q13 matrix

orders corresponding to the subsequent layers are in descending order. In con-trast to these prescriptions, the solution method using a recurrence relationfor T -matrix calculation requires all matrices to be of the same order, i.e.,

dim (T ) = dim(T l,l+1,...,N

)=

12dim (Ql) = 2Nmax(1) × 2Nmax(1) .

This requirement implies that the same number of basis functions must beused to approximate the surface fields on each layer. For concentrically lay-ered spheres, this requirement is not problematic because the basis functionsare orthogonal on spherical surfaces. For nonspherical layered particles, weapproximate the surface fields by a complete system of vector functions and itis natural to use fewer basis functions for smaller layer surfaces. However, forconvergence tests it is simpler to consider a single truncation index [181,248].

2.5.3 Formulation with Discrete Sources

For a two-layered particle as shown in Fig. 2.2, the null-field equations for-mulated in terms of distributed vector spherical wave functions (compare to(2.70), (2.71) and (2.72))

jk2s

π

∫

S1

[ei,1 (r′1) − ee (r′

1)] ·(

N 3ν (ksr

′1)

M3ν (ksr

′1)

)

(2.113)

+ j√

µs

εs[hi,1 (r′

1) − he (r′1)] ·(M3

ν (ksr′1)

N 3ν (ksr

′1)

)

dS (r′1) = 0 , ν = 1, 2, . . . ,

−jk2

i,1

π

∫

S1

[

ei,1 (r′1) ·(

N 1ν (ki,1r

′1)

M1ν(ki,1r

′1)

)

+ j√

µi,1

εi,1hi,1 (r′

1) ·(M1

ν(ki,1r′1)

N 1ν (ki,1r

′1)

)]

dS (r′1)


+jk2

i,1

π

∫

S2

[

ei,2 (r′′2) ·(

N 1ν (ki,1r

′′1)

M1ν(ki,1r

′′1)

)

(2.114)

+ j√

µi,1

εi,1hi,2 (r′′

2) ·(M1

ν(ki,1r′′1)

N 1ν (ki,1r

′′1)

)]

dS (r′′2) = 0 , ν = 1, 2, . . . ,

and

−jk2

i,1

π

∫

S1

[

ei,1 (r′1) ·(

N 3ν (ki,1r

′2)

M3ν(ki,1r

′2)

)

+ j√

µi,1

εi,1hi,1 (r′

1) ·(M3

ν(ki,1r′2)

N 3ν (ki,1r

′2)

)]

dS (r′1)

+jk2

i,1

π

∫

S2

[

ei,2 (r′′2) ·(

N 3ν (ki,1r

′′2)

M3ν(ki,1r

′′2)

)

(2.115)

+ j√

µi,1

εi,1hi,2 (r′′

2) ·(M3

ν(ki,1r′′2)

N 3ν (ki,1r

′′2)

)]

dS (r′′2) = 0 , ν = 1, 2, . . .

are equivalent to the general null-field equations (2.68) and (2.69). The dis-tributed vector spherical wave functions in (2.113)–(2.115) are defined as

M1,3mn(kr1) = M1,3

m,|m|+l [k(r1 − z1,nez)] ,

N 1,3mn(kr1) = N1,3

m,|m|+l [k(r1 − z1,nez)] ,

and

M1,3mn(kr2) = M1,3

m,|m|+l [k(r2 − z2,nez)] ,

N 1,3mn(kr2) = N1,3

m,|m|+l [k(r2 − z2,nez)] ,

where z1,n∞n=1 is a dense set of points situated on the z-axis and in theinterior of S1, while z2,n∞n=1 is a dense set of points situated on the z-axis andin the interior of S2. Due to their completeness property, the distributed vectorspherical wave functions can be used to approximate the surface fields as in(2.73) and (2.74) but with M1,3

µ (ki,1r′1), N 1,3

µ (ki,1r′1) in place of M1,3

µ (ki,1r′1),

N1,3µ (ki,1r

′1) and M1

µ(ki,2r′′2), N 1

µ(ki,2r′′2) in place of M1

µ(ki,2r′′2), N1

µ(ki,2r′′2),

respectively.For a multilayered particle, it is apparent that the solution methods with

distributed sources use essentially the same matrix equations as the solutionmethods with localized sources. The matrices A1, Al,l−1 and AN are givenby (2.109), (2.110) and (2.112), respectively, with Q

pq

l in place of Qpql , while

the matrix Al−1,l is


Al−1,l =

⎡

⎣−Q13

l−1(ki,l−1, l − 1, ki,l−1, l − 1) 0

0 −Q31

l−1(ki,l−1, l, ki,l−1, l − 1)

⎤

⎦ .

The expressions of the elements of the Qpq

l matrix are given by (2.84)–(2.87)with the localized vector spherical wave functions replaced by the distributedvector spherical wave functions. The transition matrix is

T =[Q

13

1 (ks, 1, ki,1, 1)A11 + Q11

1 (ks, 1, ki,1, 1)A21]

×Q31

1 (ks, 1, ks, 1) ,

where the Q11

1 (ks, 1, ki,1, 1) and Q13

1 (ks, 1, ki,1, 1) matrices contain as rowsthe vectors M1

ν(ksr′1), N1

ν(ksr′1) and as columns the vectors M1

µ(ki,1r′1),

N 1µ(ki,1r

′1) and M3

µ(ki,1r′1), N 3

µ(ki,1r′1), respectively, while the Q

31(ks, 1, ks, 1)

matrix contains as rows and columns the vectors M3ν(ksr

′1), N 3

ν (ksr′1) and

M1µ(ksr

′1), N1

µ(ksr′1), respectively.

The use of distributed vector spherical wave functions improves the nu-merical stability of the null-field method for highly elongated and flattenedlayered particles. Although the above formalism is valid for nonaxisymmetricparticles, the method is most effective for axisymmetric particles, in which casethe z-axis of the particle coordinate system is the axis of rotation. Applica-tions of the null-field method with distributed sources to axisymmetric layeredspheroids with large aspect ratios have been given by Doicu and Wriedt [50].

2.5.4 Concentrically Layered Spheres

For a concentrically layered sphere, the precedent relations simplify to thoseobtained in the framework of the Lorenz–Mie theory. In this specific case, weuse the recurrence relation (2.107) with Ql given by (2.106). All matrices arediagonal and denoting by (Tl,l+1,...,N )1n and (Tl,l+1,...,N )2n the elements of thematrix T l,l+1,...,N , we rewrite the recurrence relation as

(Tl,l+1,...,N )1n

= − jn (xl) [mr,lxljn (mr,lxl)]′rn − jn (mr,lxl) [xljn (xl)]

′pn

h(1)n (xl) [mr,lxljn (mr,lxl)]

′rn − jn (mr,lxl)

[xlh

(1)n (xl)

]′pn

,

(2.116)

(Tl,l+1,...,N )2n

= −jn (xl) [mr,lxljn (mr,lxl)]

′qn − m2

r,ljn (mr,lxl) [xljn (xl)]′sn

h(1)n (xl) [mr,lxljn (mr,lxl)]

′qn − m2

r,ljn (mr,lxl)[xlh

(1)n (xl)

]′sn

,

(2.117)


where

pn = 1 + (Tl+1,l+2,...,N )1nh

(1)n (mr,lxl)jn (mr,lxl)

, (2.118)

sn = 1 + (Tl+1,l+2,...,N )2nh

(1)n (mr,lxl)jn (mr,lxl)

, (2.119)

rn = 1 + (Tl+1,l+2,...,N )1n

[mr,lxlh

(1)n (mr,lxl)

]′

[mr,lxljn (mr,lxl)]′ , (2.120)

qn = 1 + (Tl+1,l+2,...,N )2n

[mr,lxlh

(1)n (mr,lxl)

]′

[mr,lxljn (mr,lxl)]′ , (2.121)

xl = ki,lRl = k0mlRl is the size parameter of the layer l and

mr,l =ml

ml−1

is the relative refractive index of the layer l with respect to the layer l − 1.To obtain a stable scheme for computing the T matrix, we express the aboverecurrence relation in terms of the logarithmic derivatives An and Bn:

(Tl,l+1,...,N )1n = −

[mr,lAn (mr,lxl) rn + n

xlpn

]jn (xl) − jn−1 (xl) pn

[mr,lAn (mr,lxl) rn + n

xlpn

]h

(1)n (xl) − h

(1)n−1 (xl) pn

,

(2.122)

(Tl,l+1,...,N )2n = −

[An(mr,lxl)

mr,lqn + n

xlsn

]jn (xl) − jn−1 (xl) sn

[An(mr,lxl)

mr,lqn + n

xlsn

]h

(1)n (xl) − h

(1)n−1 (xl) sn

, (2.123)

where pn and sn are given by (2.118) and (2.119), respectively, and rn and qn

are now given by

rn = 1 + (Tl+1,l+2,...,N )1nBn (mr,lxl)An (mr,lxl)

h(1)n (mr,lxl)jn (mr,lxl)

and

qn = 1 + (Tl+1,l+2,...,N )2nBn (mr,lxl)An (mr,lxl)

h(1)n (mr,lxl)jn (mr,lxl)

,

respectively.The computation of Lorenz–Mie coefficients for concentrically layered

spheres has been considered by Kerker [115], Toon and Ackerman [222], andFuller [76], while recursive algorithms for multilayered spheres have been de-veloped by Bhandari [13] and Mackowski et al. [154].


2.6 Multiple Particles

In this section we extend the null-field method to the case of an arbitrarynumber of particles by using the translation properties of the vector sphericalwave functions. Taking into account the geometric restriction that the parti-cles do not overlap in space, we derive the expression of the transition matrixfor restricted values of translations. Our treatment closely follows the originalderivation given by Peterson and Strom [187,188].


For reasons of clarity of the presentation we first consider the generic case oftwo homogeneous particles immersed in a homogeneous medium with a rela-tive permittivity εs and a relative permeability µs. The scattering geometryis depicted in Fig. 2.5. The surfaces S1 and S2 are defined with respect to theparticle coordinate systems O1x1y1z1 and O2x2y2z2, respectively, while thecoordinate system of the ensemble or the global coordinate system is denotedby Oxyz. The coordinate system O1x1y1z1 is obtained by translating the co-ordinate system Oxyz through r01 and by rotating the translated coordinatesystem through the Euler angles α1, β1 and γ1. Similarly, the coordinate sys-tem O2x2y2z2 is obtained by translating the coordinate system Oxyz throughr02 and by rotating the translated coordinate system through the Euler an-gles α2, β2 and γ2. The main assumption of our analysis is that the smallestcircumscribing spheres of the particles centered at O1 and O2, respectively,do not overlap. The boundary-value problem for the two scattering particlesdepicted in Fig. 2.5 has the following formulation.

Given the external excitation Ee,He as an entire solution to the Maxwellequations, find the scattered field Es, Hs and the internal fields Ei,1, H i,1

O1

Di,1

Ds

S1

n1

O2

M1M2

P

r199

r299r19

r29

r01 r02 S2Di,2

r12

n2

O

r1 r2r

r9 r99

Fig. 2.5. Geometry of two scattering particles

2.6 Multiple Particles 125

and Ei,2, H i,2 satisfying the Maxwell equations



and

∇× Ei,2 = jk0µi,2H i,2 , quad∇× H i,2 = −jk0εi,2Ei,2 in Di,2 , (2.126)


n1 × Ei,1 − n1 × Es = n1 × Ee ,

n1 × H i,1 − n1 × Hs = n1 × He (2.127)

on S1 and

n2 × Ei,2 − n2 × Es = n2 × Ee ,

n2 × H i,2 − n2 × Hs = n2 × He (2.128)

on S2, and the Silver–Muller radiation condition for the scattered field (2.3).The Stratton–Chu representation theorem for the scattered field Es in Di,1

and Di,2 together with the boundary conditions (2.127) and (2.128) yield thegeneral null-field equation

Ee(r) + ∇×∫

S1

ei,1 (r′) g (ks, r, r′) dS (r′)

+j

k0εs∇×∇×

∫

S1

hi,1 (r′) g (ks, r, r′) dS (r′)

+∇×∫

S2

ei,2 (r′′) g (ks, r, r′′) dS (r′′)

+j

k0εs∇×∇×

∫

S2

hi,2 (r′′) g (ks, r, r′′) dS (r′′) = 0 , r ∈ Di,1 ∪ Di,2 .

Before we derive the null-field equations, we seek to find a relation betweenthe expansion coefficients of the incident field in the global coordinate systemOxyz,

Ee(r) =∑

ν

aνM1ν (ksr) + bνN1

ν (ksr)

and the expansion coefficients of the incident field in the particle coordinatesystem O1x1y1z1,

Ee (r1) =∑

ν

a1,νM1ν (ksr1) + b1,νN1

ν (ksr1) .


O1

Ds

S

M

r1

r01 O

r

Fig. 2.6. Auxiliary surface S

For this purpose we choose a sufficiently large auxiliary surface S enclosingO and O1 (Fig. 2.6) and in each coordinate system we use the Stratton–Churepresentation theorem for the incident field in the interior of S. We obtain

(aν

bν

)

= − jk2s

π

∫

S

[

ee (r′)

(N3

ν (ksr′)

M3ν (ksr

′)

)

+ j√

µs

εshe (r′)

(M3

ν (ksr′)

N3ν (ksr

′)

)]

dS (r′) ,

and(

a1,ν

b1,ν

)

= − jk2s

π

∫

S

[

ee (r′1)

(N3

ν (ksr′1)

M3ν (ksr

′1)

)

+ j√

µs

εshe (r′

1)

(M3

ν (ksr′1)

N3ν (ksr

′1)

)]

dS (r′1) ,

respectively. Using the addition theorem for radiating vector spherical wavefunctions

[M3

ν (ksr′1)

N3ν (ksr

′1)

]

=[(Srt

10

)νµ

][

M3µ (ksr

′)N3

µ (ksr′)

]

,

where

Srt10 = R (−γ1,−β1,−α1) T 33 (−ksr01) , for r′ > r01

and taking into account that Srt10 is a block-symmetric matrix, yields

[a1,ν

b1,ν

]=[(Srt

10

)νµ

] [aµ

bµ

]. (2.129)


The condition r′ > r01 can always be satisfied in practice by an appropriatechoice of the auxiliary surface S, whence, using the identity T 33(−ksr01) =T 11(−ksr01), we see that

Srt10 = R (−γ1,−β1,−α1) T 11 (−ksr01) .

We proceed now to derive the set of null-field equations. Passing from theorigin O to the origin O1, using the relations

g (ks, r, r′) = g (ks, r1, r′1) ,

g (ks, r, r′′) = g (ks, r1, r′′1) ,

and restricting r1 to lie on a sphere enclosed in Di,1, gives

jk2s

π

∫

S1

[

ei,1 (r′1) ·(

N3ν(ksr

′1)

M3ν(ksr

′1)

)

+ j√

µs

εshi,1 (r′

1) ·(

M3ν(ksr

′1)

N3ν(ksr

′1)

)]

dS (r′1)

+jk2

s

π

∫

S2

[

ei,2 (r′′2) ·(

N3ν (ksr

′′1)

M3ν(ksr

′′1)

)

(2.130)

+ j√

µs

εshi,2 (r′′

2) ·(

M3ν(ksr

′′1)

N3ν(ksr

′′1)

)]

dS (r′′2) = −

(a1,ν

b1,ν

)

, ν = 1, 2, ... ,

where the identities ei,2(r′′2) = ei,2(r′′

1) and hi,2(r′′2) = hi,2(r′′

1) have beenused. For the general null-field equation in Di,2 we proceed analogously butrestrict r2 to lie on a sphere enclosed in Di,2. We obtain

jk2s

π

∫

S1

[

ei,1 (r′1) ·(

N3ν(ksr

′2)

M3ν(ksr

′2)

)

+ j√

µs

εshi,1 (r′

1) ·(

M3ν(ksr

′2)

N3ν(ksr

′2)

)]

dS (r′1)

+jk2

s

π

∫

S2

[

ei,2 (r′′2) ·(

N3ν (ksr

′′2)

M3ν(ksr

′′2)

)

(2.131)

+ j√

µs

εshi,2 (r′′

2) ·(

M3ν(ksr

′′2)

N3ν(ksr

′′2)

)]

dS (r′′2) = −

(a2,ν

b2,ν

)

, ν = 1, 2, ... ,

where, as before, we have taken into account that ei,1(r′1) = ei,1(r′

2) andhi,1(r′

1) = hi,1(r′2).


The surface fields ei,1, hi,1 and ei,2, hi,2 are the tangential componentsof the electric and magnetic fields in the domains Di,1 and Di,2, respectively,and the surface fields approximations can be expressed as linear combinationsof regular vector spherical wave functions,

(eN

i,1(r′1)

hNi,1(r

′1)

)

=N∑

µ=1

cN1,µ

⎛

⎝n1(r′

1) × M1µ(ki,1r

′1)

−j√

εi,1µi,1

n1(r′1) × N1

µ(ki,1r′1)

⎞

⎠

+dN1,µ

⎛

⎝n1(r′

1) × N1µ(ki,1r

′1)

−j√

εi,1µi,1

n1(r′1) × M1

µ(ki,1r′1)

⎞

⎠ (2.132)

and(

eNi,2(r

′′2)

hNi,2(r

′′2)

)

=N∑

µ=1

cN2,µ

⎛

⎝n2(r′′

2) × M1µ(ki,2r

′′2)

−j√

εi,2µi,2

n2(r′′2) × N1

µ(ki,2r′′2)

⎞

⎠

+dN2,µ

⎛

⎝n2(r′′

2) × N1µ(ki,2r

′′2)

−j√

εi,2µi,2

n2(r′′2) × M1

µ(ki,2r′′2)

⎞

⎠ . (2.133)

Inserting (2.132) and (2.133) into (2.130) and (2.131), using the addition the-orem for vector spherical wave functions

[M3

ν (ksr′′1)

N3ν (ksr

′′1)

]

=[(

Srtr12

)

νµ

] [M1

µ (ksr′′2)

N1µ (ksr

′′2)

]

with

Srtr12 = R (−γ1,−β1,−α1) T 31 (ksr12)R (α2, β2, γ2) , for r′′2 < r12 ,

and[

M3ν (ksr

′2)

N3ν (ksr

′2)

]

=[(

Srtr21

)

νµ

] [M1

µ (ksr′1)

N1µ (ksr

′1)

]

,

with

Srtr21 = R (−γ2,−β2,−α2) T 31 (−ksr12)R (α1, β1, γ1) for r′1 < r12 ,

and taking into account the transformation rule for the incident field coeffi-cients (2.129), yields the system of matrix equations

Q311 (ks, ki,1)i1 + Srtr

12 Q112 (ks, ki,2)i2 = −Srt

10e ,

Srtr21 Q11

1 (ks, ki,1)i1 + Q312 (ks, ki,2)i2 = −Srt

20e , (2.134)


where i1 = [cN1,µ, dN

1,µ]T, i2 = [cN2,µ, dN

2,µ]T, and as usually, e = [aν , bν ]T is thevector containing the expansion coefficients of the incident field in the globalcoordinate system. Further, defining the scattered field coefficients

s1 = Q111 (ks, ki,1)i1 ,

s2 = Q112 (ks, ki,2)i2 ,

and introducing the individual transition matrices,

T 1 = −Q111 (ks, ki,1)

[Q31

1 (ks, ki,1)]−1

,

T 2 = −Q112 (ks, ki,2)

[Q31

2 (ks, ki,2)]−1

,

we rewrite the matrix system (2.134) as

s1 − T 1Srtr12 s2 = T 1Srt

10e ,

s2 − T 2Srtr21 s1 = T 2Srt

20e ,

and find the solutions

s1 = T 1

(I − Srtr

12 T 2Srtr21 T 1

)−1 (Srt

10 + Srtr12 T 2Srt

20

),

s2 = T 2

(I − Srtr

21 T 1Srtr12 T 2

)−1 (Srt

20 + Srtr21 T 1Srt

10

). (2.135)

To compute the T matrix of the two-particles system and to derive a scattered-field expansion centered at the origin O of the global coordinate system weuse the Stratton–Chu representation theorem for the scattered field Es inDs. In the exterior of a sphere enclosing the particles, the expansion of theapproximate scattered field EN

s in terms of radiating vector spherical wavefunctions reads as

ENs (r) =

N∑

ν=1

fNν M3

ν (ksr) + gNν N3

ν (ksr) ,

where the expansion coefficients are given by(

fNν

gNν

)

=jk2

s

π

∫

S1

[

eNi,1 (r′

1)

(N1

ν (ksr′)

M1ν (ksr

′)

)

+ j√

µs

εshN

i,1 (r′1)

(M1

ν (ksr′)

N1ν (ksr

′)

)]

dS (r′1)

+jk2

s

π

∫

S2

[

eNi,2 (r′′

2)

(N1

ν (ksr′′)

M1ν (ksr

′′)

)

+ j√

µs

εshN

i,2 (r′′2)

(M1

ν (ksr′′)

N1ν (ksr

′′)

)]

dS (r′′2) . (2.136)


Finally, using the addition theorem for the regular vector spherical wave func-tions

[M1

ν (ksr′)

N1ν (ksr

′)

]

=[(Str

01

)νµ

][

M1µ (ksr

′1)

N1µ (ksr

′1)

]

with

Str01 = T 11 (ksr01)R (α1, β1, γ1) ,

and[

M1ν (ksr

′′)N1

ν (ksr′′)

]

=[(Str

02

)νµ

][

M1µ (ksr

′′2)

N1µ (ksr

′′2)

]

with

Str02 = T 11 (ksr02)R (α2, β2, γ2) ,

we obtain

s = Str01Q

111 (ks, ki,1)i1 + Str

02Q112 (ks, ki,2)i2

= Str01s1 + Str

02s2 , (2.137)


ν ]T is the vector containing the expansion coefficients ofthe scattered field in the global coordinate system. Combining (2.135) and(2.137), and using the identities Srt

20(Srt10)

−1 = Srtr21 and Srt

10(Srt20)

−1 = Srtr12 ,

yields [187]

T = Str01T 1

(I − Srtr

12 T 2Srtr21 T 1

)−1 (I + Srtr

12 T 2Srtr21

)Srt

10

+Str02T 2

(I − Srtr

21 T 1Srtr12 T 2

)−1 (I + Srtr

21 T 1Srtr12

)Srt

20 , (2.138)

where the explicit expressions of the transformation matrices Srtr21 and Srtr

12

are given by

Srtr12 = R (−γ1,−β1,−α1) T 11 (ksr12)R (α2, β2, γ2) ,

and

Srtr21 = R (−γ2,−β2,−α2) T 11 (−ksr12)R (α1, β1, γ1) ,

respectively. Equation (2.138) gives the system transition matrix T in termsof the individual transition matrices T 1 and T 2, and the transformation ma-trices S and S. S and S involve translations of the regular and radiatingvector spherical wave functions, respectively, and geometric constraints areintroduced by the S matrices. Obviously, the geometric restrictions r12 > r′′2


and r12 > r′1 introduced by the S matrices are fulfilled if the smallest cir-cumscribing spheres of the particles do not overlap. The following feature ofequation (2.138) is apparent: if T 2 = 0, then T = Str

01T 1Srt10, and if T 1 = 0,

then T = Str02T 2Srt

20, as it should. In general, T is a sum of two terms, eachof which is a modification of these limiting values. The various terms in aformal expansion of the inverses occurring in (2.138) can be considered asmultiple-scattering contributions. The terms involving only T 1 and T 2 repre-sent reflections at S1 and S2, respectively, the terms involving only T 1Srtr

12 T 2

and T 2Srtr21 T 1 represent consecutive reflections at S2 and S1, and S1 and S2,

respectively, etc.

2.6.2 Formulation for a System with N Particles

The generalization of the T -matrix relation to a system with more than twoconstituents is straightforward. The system of matrix equations consists inthe null-field equations in the interior of Sl

sl − T l

N∑

p=l

Srtrlp sp = T lSrt

l0e for l = 1, 2, . . . ,N (2.139)


s =N∑

l=1

Str0lsl . (2.140)

In practical computer calculations it is convenient to consider the global ma-trix A with block-matrix components

All = I , l = 1, 2, . . . ,N ,

Alp = −T lSrtrlp , l = p , l, p = 1, 2, . . . ,N ,

and to express the solution to the system of matrix equations (2.139) as

sl =

( N∑

p=1

AlpT pSrtp0

)

e , l = 1, 2, . . . ,N , (2.141)

where A stay for A−1, and Alp, l, p = 1, 2, . . . ,N , are the block-matrix com-ponents of A. In view of (2.140), the system T -matrix becomes

T =N∑

l=1

N∑

p=1

Str0lAlpT pSrt

p0

and this transition matrix can be used to compute the scattering characteris-tics for fixed or random orientations of the ensemble [153,165].


In order to estimate the computer memory requirement it is important toknow the dimensions of the matrices involved in the calculation. If Nrank(l)and Mrank(l) are the maximum expansion order and the number of azimuthalmodes for the lth particle, then the dimension of the transition matrix T l isdim(T l) = 2Nmax(l) × 2Nmax(l), where

Nmax(l) = Nrank (l) + Mrank (l) [2Nrank(l) − Mrank(l) + 1] .

The dimension of the global matrix A is

dim (A) = 2N∑

l=1

Nmax(l) × 2N∑

l=1

Nmax(l) ,

since

dim(Alp)

= dim(Alp)

= 2Nmax(l) × 2Nmax(p) .

If Nmax gives the dimension of the system T -matrix

dim (T ) = 2Nmax × 2Nmax ,

we have

dim(Srt

p0

)= 2Nmax(p) × 2Nmax ,

dim(Str

0l

)= 2Nmax × 2Nmax(l) .

In our computer code, the parameters Nmax and Nmax(l), l = 1, 2, . . . ,N , areindependent. Nmax(l) is given by the size parameter of the lth particle, whileNmax is given by the size parameter of a sphere centered at O and enclosingthe particles.

2.6.3 Superposition T -matrix Method

For a system of N particles with αl = βl = γl = 0, l = 1, 2, . . . ,N , thetransformations of the vector spherical vector wave functions involve only theaddition theorem under coordinate translations, i.e.,

Str0l = T 11 (ksr0l) ,

Srtp0 = T 11 (−ksr0p) ,

and

Srtrlp = T 31 (ksrlp)

for l, p = 1, 2, . . . ,N . Consequently, the matrix equation (2.139) takes theform


sl = T l

⎡

⎣T 11 (−ksr0l) e +N∑

p=l

T 31 (ksrlp) sp

⎤

⎦ , (2.142)

while (2.140) becomes

s =N∑

l=1

T 11 (ksr0l) sl . (2.143)

The precedent equations can be written in explicit form by indicating thevector and matrix indices

[(sl)ν ] = [(Tl)νν′ ]

⎧⎨

⎩

[T 11

ν′µ(−ksr0l)

][eµ]

N∑

p=l

[T 31

ν′µ(ksrlp)

] [(sp)µ

]⎫⎬

⎭,

(2.144)

[sν ] =N∑

l=1

[T 11

νµ (ksr0l)] [

(sl)µ

], (2.145)

and can also be derived by using the so-called superposition T -matrix method[169]. The superposition T -matrix method reproduces the two cooperative ef-fects characterizing aggregate scattering: interaction between particles andfar-field interference [273]. The interaction effect take into account that eachparticle is excited by the initial incident field and the fields scattered byall other particles, while the far-field interference is a result of the incidentand scattered phase differences from different particles. The superpositionT -matrix method involves the following general steps:

1. The expansions of the incident and scattered fields in each particle coor-dinate system.

2. The representation of the field exciting a particle by a single-field expan-sion (which includes the incident field and the scattered fields from allother particles).

3. The solution of the transmission boundary-value problem for each particle.4. The derivation of the scattered-field expansion in the global coordinate

system.

The interaction effect is taken into account in steps 2 and 3, where the scat-tered fields from other particles are transformed and included in the incidentfield on each particle, while the interference effect is taken into account insteps 1 and 4 that address the incident and scattered path differences, respec-tively. For a system of N spheres, the individual component T matrices arediagonal with the standard Lorenz–Mie coefficients along their main diago-nal, and in this case, the superposition T -matrix method is also known as themultisphere separation of variables technique or the multisphere superposi-tion method [21, 24, 29, 72, 150]. Solution of (2.144) have been obtained usingdirect matrix inversion, method of successive orders of scattering, conjugate


gradient methods and iterative approaches [74, 95, 198, 244, 271]. Originallyall methods were implemented for spherical particles, but Xu [274] recentlyextended his computer code for axisymmetric particles.

In the following analysis, we derive (2.144) and (2.145) by using the su-perposition T -matrix method. The field exciting the lth particle can be ex-pressed as

Eexc,l (rl) = Ee (rl) +N∑

p=l

Es,p (rl) ,

where Ee is the incident field and Es,p is the field scattered by the pth par-ticle. In the null-field method, transformation rules between the expansioncoefficients of the incident and scattered fields in different coordinate systemshave been derived by using the integral representations for the expansion co-efficients. In the superposition T -matrix method, these transformations areobtained by using the series representations for the electromagnetic fields indifferent coordinate systems. For the external excitation, we consider the vec-tor spherical wave expansion

Ee(r) =∑

µ

aµM1µ (ksr) + bµN1

µ (ksr)

and use the addition theorem[

M1µ (ksr)

N1µ (ksr)

]

=[T 11

µν (ksr0l)][

M1ν (ksrl)

N1ν (ksrl)

]

,

to obtain

Ee (rl) =∑

ν

al,νM1ν (ksrl) + bl,νN1

ν (ksrl)

with[

al,ν

bl,ν

]=[T 11

µν (ksr0l)][

aµ

bµ

].

Similarly, for the field scattered by the pth particle, we consider the seriesrepresentation

Es,p (rp) =∑

µ

fp,µM3µ (ksrp) + gp,µN3

µ (ksrp)

and use the addition theorem[

M3µ (ksrp)

N3µ (ksrp)

]

=[T 31

µν (ksrpl)][

M1ν (ksrl)

N1ν (ksrl)

]

for rl < rpl ,


to derive

Es,p (rl) =∑

ν

flp,νM1ν (ksrl) + glp,νN1

ν (ksrl)

with[

flp,ν

glp,ν

]=[T 31

µν (ksrpl)][

fp,µ

gp,µ

].

Thus, the field exciting the lth particle can be expressed in terms of regularvector spherical wave functions centered at the origin Ol

Eexc,l (rl) =∑

ν


ν (ksrl) ,

where the expansion coefficients are given by

[(el)ν ] =[T 11

µν (ksr0l)][eµ] +

N∑

p=l

[T 31

µν (ksrpl)] [

(sp)µ

]

and as usually, el = [al,ν , bl,ν ]T. Further, using the T -matrix equation sl =T lel, we obtain

[(sl)ν ] = [(Tl)νν′ ]

⎡

⎣[T 11

µν′ (ksr0l)][eµ] +

N∑

p=l

[T 31

µν′ (ksrpl)] [

(sp)µ

]⎤

⎦ , (2.146)

and since (cf. (B.74) and (B.75))[T 31

ν′µ(ksrlp)

]=[T 31

µν′ (ksrpl)]

and[T 11

ν′µ(−ksr0l)

]=[T 11

ν′µ(ksrl0)

]=[T 11

µν′ (ksr0l)]

,

we see that (2.144) and (2.146) coincide.The scattered field is a superposition of fields that are scattered from the

individual particles,

Es(r) =N∑

l=1

Es,l (rl)

=N∑

l=1

∑

µ

fl,µM3µ (ksrl) + gl,µN3

µ (ksrl) , (2.147)

whence, using the addition theorem


[M3

µ (ksrl)N3

µ (ksrl)

]

=[T 33

µν (−ksr0l)][

M3ν (ksr)

N3ν (ksr)

]

for r > r0l ,

we derive the scattered-field expansion centered at O

Es(r) =∑

ν

fνM3ν (ksr) + gνN3

ν (ksr)

with

[sν ] =N∑

l=1

[T 33

µν (−ksr0l)] [

(sl)µ

]. (2.148)

Since,[T 11

νµ (ksr0l)]

=[T 11

µν (−ksr0l)]

=[T 33

µν (−ksr0l)]

,

we see that (2.145) and (2.148) are identical.

2.6.4 Formulation with Phase Shift Terms

For an ensemble of N particles with αl = βl = γl = 0, l = 1, 2, . . . ,N , thesystem T -matrix is given by

T =N∑

l=1

N∑

p=1

T 11 (ksr0l)AlpT pT 11 (−ksr0p) . (2.149)

In (2.149), the translation matrices T 11(−ksr0p) and T 11(ksr0l) give the re-lations between the expansion coefficients of the incident and scattered fieldsin different coordinate systems. In the present analysis, these relations are de-rived by using the direct phase differences between the electromagnetic fieldsin different coordinate systems.

For a vector plane wave of unit amplitude and wave vector ke, ke = ksek,we have

Ee (rl) = epolejksek·r0lejksek·(r−r0l)

= ejksek·r0l

∑

ν

aνM1ν (ksrl) + bνN1

ν (ksrl)

=∑

ν


ν (ksrl)

and therefore[

al,ν

bl,ν

]= ejksek·r0l

[aν

bν

].

Further, using the far-field representation of the field scattered by the lthparticle,


Es,l(rl) =ejksrl

rl

Es∞,l(er) + O

(1rl

)

and the approximation

ejksrl

rl=

ejksre−jkser·r0l

r

[1 + O

(1r

)],

we see that the angular-dependent vector of scattering coefficients

s (er) =N∑

l=1

e−jkser·r0lsl

approximates the s vector in the far-field region, and that the angular-dependent transition matrix

T (er) =N∑

l=1

N∑

p=1

ejks(ek·r0p−er·r0l)AlpT p (2.150)

approximates the T matrix in the far-field region. For spherical particles, thismethod is known as the generalized multiparticle Mie-solution [272,273].

Equation (2.150) shows that we have to impose

dim (T ) = dim(AlpT p

)= dim (T lp) ,

where T lp = AlpT p. Assuming dim(T ) = 2Nmax × 2Nmax, we can setNmax(p) = Nmax for all p = 1, 2, . . . ,N , and in this case, dim(T ) = dim(T p).Alternatively, we can set Nmax = maxp Nmax(p) and use the convention(Tlp)νµ ≡ 0 whenever ν > 2Nmax(l) and µ > 2Nmax(p). This formulationavoids the computation of the translation matrices T 11 and involves only theinversion of the global matrix A. The dimensions of T 11 depend on translationdistances and are proportional to the overall dimension of the cluster, whilethe dimension of A is determined by the size parameters of the individualparticles. Therefore, the generalized multiparticle Mie-solution is limited bythe largest possible individual particle size in a cluster and not by the over-all cluster size. However, the angular-dependent transition matrix can not beused in the analytical orientation-averaging procedure described in Sect. 1.5and in view of our computer implementation, this method is not effective forT -matrix calculations.

2.6.5 Recursive Aggregate T -matrix Algorithm

If the number of particles increases, the dimension of the global matrix Abecomes excessively large. Wang and Chew [250,251] proposed a recursive T -matrix algorithm, which computes the T matrix of a system of n componentsby using the transition matrices of the newly added q components and the


T matrix of the previous system of n − q components. In this section weuse the recursive T -matrix algorithm to analyze electromagnetic scatteringby a system of identical particles randomly distributed inside an “imaginary”spherical surface.

Let us consider Ncs circumscribing spheres with radii Rcs(k), k = 1, 2, . . . ,Ncs, in increasing order. Inside the sphere of radius Rcs(1) there are N (1)particles having their centers located at r

(1)0l , l = 1, 2, . . . ,N (1), and in each

spherical shell bounded by the radii Rcs(k − 1) and Rcs(k) there are N (k)particles having their centers located at r

(k)0l , l = 1, 2, . . . ,N (k). Obviously,

N =∑Ncs

p=1 N (p) is the total number of particles and Rcs(Ncs) is the radius ofthe circumscribing sphere containing the particles. For simplicity, we assumethat the particles are identical and have the same spatial orientation. Thescattering geometry for the problem under examination is depicted in Fig. 2.7.

At the first iteration step, we compute the system T -matrix T (1) of allparticles situated inside the sphere of radius Rcs(1). At the iteration step k,we compute the system T -matrix T (k) of all particles situated inside the sphereof radius Rcs(k) by considering a [N (k) + 1]-scatterer problem. In fact, thetransition matrix T (k) is computed by using the system T -matrix T (k−1) of theprevious

∑k−1p=1 N (p) particles situated inside the sphere of radius Rcs(k − 1)

and the individual transition matrix T p of all N (k) particles situated in thespherical shell between Rcs(k− 1) and Rcs(k). In this specific case, the block-matrix components of the global matrix A are given by

All = I , l = 1, 2, . . . ,N (k) + 1 ,

Alp = −T pT 31(ksr

(k)lp

), l = p , l, p = 1, 2, . . . ,N (k) ,

Al,N (k)+1 = −T pT 31(ksr

(k)l0

)= −T pT 31

(−ksr

(k)0l

), l = 1, 2, . . . ,N (k) ,

AN (k)+1,l = −T (k−1)T 31(ksr

(k)0l

), l = 1, 2, . . . ,N (k) .

O

Rcs

Rcs

Rcs

Rcs

Orol

(k)

(1)(k-1)

(k)(Ncs)

Fig. 2.7. Illustration of the recursive T -matrix algorithm for many spheres

2.7 Composite Particles 139

If A = A−1, and Alp, l, p = 1, 2, . . . ,N (k) + 1, are the block-matrix compo-nents of A, the recurrence relation for T -matrix calculation read as

T (k) = AN (k)+1,N (k)+1T (k−1) +N (k)∑

l=1

AN (k)+1,lT pT 11(−ksr

(k)0l

)

+N (k)∑

l=1

T 11(ksr

(k)0l

)⎡

⎣Al,N (k)+1T (k−1) +N (k)∑

p=1

AlpT pT 11(−ksr

(k)0p

)⎤

⎦ .

The procedure is repeated until all N particles are exhausted. At each itera-tion step, only a [N (k)+1]-scatterer problem needs to be solved, where N (k)usually is much smaller than N . Thus, we can keep the dimension of the prob-lem manageable even when N is very large. The geometric constraint of theT -matrix method requires that the N (k) particles completely reside insidethe spherical shell between Rcs(k − 1) and Rcs(k). However, for a large num-ber of particles and small size parameters, numerical simulations certify theaccuracy of the recursive scheme even if this geometric constraint is violated.

As mentioned before, the T 11(−ksr0l) matrices are used to translate theincident field from the global coordinate system to the lth particle coordinatesystem, while the T 11(ksr0l) matrices are used to translate the scattered fieldsfrom the basis of the lth particle to the global coordinate system. A recursiveT -matrix algorithm using phase shift terms instead of translation matriceshas been proposed by Auger and Stout [4].

2.7 Composite Particles

A composite particle consists of several nonenclosing parts, each character-ized by arbitrary but constant values of electric permittivity and magneticpermeability. The null-field analysis of composite particles using the additiontheorem for regular and radiating vector spherical wave functions is equivalentto the multiple scattering formalism [189]. The translation addition theoremfor radiating vector spherical wave functions introduces geometric constraintswhich are not fulfilled for composite particles. Several alternative expressionsfor the transition matrix have been derived by Strom and Zheng [219] usingthe Q matrices for open surfaces (the interfaces between the different parts ofa composite particle). In the present analysis we avoid the use of any local ori-gin translation and consider a formalism based on closed-surface Q matrices.


To simplify our presentation we first consider a composite particle with twohomogeneous parts as shown in Fig. 2.8. We assume that the surfaces S1c =S1 ∪ S12 and S2c = S2 ∪ S12 are star-shaped with respect to O1 and O2,


r19

r29r 9

r199

r299

r99

r1

r2r

O1

O

O2

M1

M2

P

Di,1

Di,2

S1

S2

S12Ds

O

O1

O2

S1

S2

S12

n1

n2

n12

n21

Fig. 2.8. Geometry of a composite particle

respectively. The main reason for introducing the star-shapedness restrictionis that we want to keep the individual constituents reasonably simple. A novelfeature is the appearance of edges, which are allowed by the basic regularityassumptions of the T -matrix formalism [256]. A composite particle can betreated by the multiple scattering formalism under appropriate geometricalconditions. The boundary-value problem for the composite particle depictedin Fig. 2.8 has the following formulation.

Given the external excitation Ee,He as an entire solution to the Maxwellequations, find the scattered field Es, Hs and the internal fields Ei,1, H i,1

and Ei,2, H i,2 satisfying the Maxwell equations


∇× Ei,1 = jk0µi,1H i,1 , ∇× H i,1 = −jk0εi,1Ei,1 in Di,1 ,(2.152)

and



n1 × Ei,1 − n1 × Es = n1 × Ee ,

n1 × H i,1 − n1 × Hs = n1 × He (2.154)

on S1,

n2 × Ei,2 − n2 × Es = n2 × Ee ,

n2 × H i,2 − n2 × Hs = n2 × He (2.155)


on S2, and

n12 × Ei,1 + n21 × Ei,2 = 0 ,

n12 × H i,1 + n21 × H i,2 = 0 (2.156)

on S12, and the Silver–Muller radiation condition for the scattered field (2.3).The Stratton–Chu representation theorem for the incident and scattered

fields in Di, where Di = Di,1∪Di,2∪S12 = R3−Ds, together with the boundaryconditions (2.154) and (2.155), lead to the general null-field equation

∇×∫

S1

[ei,1 (r′) − ee (r′)] g (ks, r, r′) dS (r′)

+j

k0εs∇×∇×

∫

S1

[hi,1 (r′) − he (r′)] g (ks, r, r′) dS (r′)

+∇×∫

S2

[ei,2 (r′′) − ee (r′′)] g (ks, r, r′′) dS (r′′) (2.157)

+j

k0εs∇×∇×

∫

S2

[hi,2 (r′′) − he (r′′)] g (ks, r, r′′) dS (r′′) = 0 , r ∈ Di .

Passing from the origin O to the origin O1, using the identities

g (ks, r, r′) = g (ks, r1, r′1) ,

g (ks, r, r′′) = g (ks, r1, r′′1) ,

restricting r1 to lie on a sphere enclosed in Di,1 and taking into account theintegral form of the boundary conditions (2.156)

∫

S12

ei,1 ·(

N3ν

M3ν

)

dS +∫

S12

ei,2 ·(

N3ν

M3ν

)

dS = 0 ,

∫

S12

hi,1 ·(

M3ν

N3ν

)

dS +∫

S12

hi,2 ·(

M3ν

N3ν

)

dS = 0 ,

yields

jk2s

π

∫

S1c

[ei,1 (r′1) − ee (r′

1)] ·(

N3ν(ksr

′1)

M3ν(ksr

′1)

)

+ j√

µs

εs[hi,1 (r′

1) − he (r′1)] ·(

M3ν(ksr

′1)

N3ν(ksr

′1)

)

dS (r′1)

+jk2

s

π

∫

S2c

[ei,2 (r′′2) − ee (r′′

1)] ·(

N3ν (ksr

′′1)

M3ν(ksr

′′1)

)

(2.158)

+ j√

µs

εs[hi,2 (r′′

2) − he (r′′1)] ·(

M3ν(ksr

′′1)

N3ν(ksr

′′1)

)

dS (r′′2) = 0 , ν = 1, 2, . . .


r199

r1O1

O2

P1

M2

D i,2

S2c

Sr

Fig. 2.9. Illustration of the auxiliary sphere in the exterior of S2c

The physical fact that the tangential fields are continuous across the inter-secting surface is the key to the structure of the null-field equations, and itis not used explicitly latter on. To simplify the null-field equations (2.158)we use the Stratton–Chu representation theorem for the incident field in theinterior and exterior of the closed surface S2c

(−Ee(r1)

0

)

= ∇×∫

S2c

ee (r′′1) g (ks, r1, r

′′1) dS(r′′

1)

+j

k0εs∇×∇×

∫

S2c

he (r′′1) g (ks, r1, r

′′1) dS(r′′

1) ,

(r1 ∈ Di,2

r1 ∈ R3 − Di,2

)

.

Choosing an auxiliary sphere in the exterior of S2c as in Fig. 2.9, and restrict-ing r1 to lie on this sphere, we derive

jk2s

π

∫

S2c

[

ee (r′′1) ·(

N3ν (ksr

′′1)

M3ν (ksr

′′1)

)

+ j√

µs

εshe (r′′

1) ·(

M3ν (ksr

′′1)

N3ν (ksr

′′1)

)]

dS (r′′1) = 0 , ν = 1, 2, ... ,

whence, using the identities ee(r′) = ee(r′1) and he(r′) = he(r′

1), we obtain

jk2s

π

∫

S1c

[ei,1 (r′1) − ee (r′)] ·

(N3

ν(ksr′1)

M3ν(ksr

′1)

)

+ j√

µs

εs[hi,1 (r′

1) − he (r′)] ·(

M3ν(ksr

′1)

N3ν(ksr

′1)

)

dS (r′1)

+jk2

s

π

∫

S2c

[

ei,2 (r′′2) ·(

N3ν (ksr

′′1)

M3ν(ksr

′′1)

)

(2.159)

+ j√

µs

εshi,2 (r′′

2) ·(

M3ν(ksr

′′1)

N3ν(ksr

′′1)

)]

dS (r′′2) = 0 , ν = 1, 2, . . .


Similarly, passing from origin O to origin O2 and restricting r2 to lie on asphere enclosed in Di,2, gives

jk2s

π

∫

S1c

[

ei,1 (r′1) ·(

N3ν(ksr

′2)

M3ν(ksr

′2)

)

+ j√

µs

εshi,1 (r′

1) ·(

M3ν(ksr

′2)

N3ν(ksr

′2)

)]

dS (r′1)

+jk2

s

2

∫

S2c

[ei,2 (r′′2) − ee (r′′)] ·

(N3

ν (ksr′′2)

M3ν(ksr

′′2)

)

(2.160)

+ j√

µs

εs[hi,2 (r′′

2) − he (r′′)] ·(

M3ν(ksr

′′2)

N3ν(ksr

′′2)

)

dS (r′′2) = 0 , ν = 1, 2, . . .

The surface fields ei,1, hi,1 and ei,2, hi,2 are approximated by finite expansionsin terms of regular vector spherical wave functions as in (2.132) and (2.133)respectively. Inserting these expansions into the null-field equations (2.159)and (2.160), yields the system of matrix equations

Q311 (ks, 1, ki,1, 1)i1 + Q31

2 (ks, 1, ki,2, 2)i2 = Q311 (ks, 1, ks, 0)e ,

Q311 (ks, 2, ki,1, 1)i1 + Q31

2 (ks, 2, ki,2, 2)i2 = Q312 (ks, 2, ks, 0)e , (2.161)

where the significance of the vectors i1, i2 and e is as in Sect. 2.6. The Qmatrices involving vector spherical wave functions with arguments referringto different origins are given by (2.84)–(2.87).

To compute the scattered-field expansion centered at the origin O we usethe Stratton–Chu representation theorem for the scattered field Es in Ds. Theexpressions of the scattered field coefficients are given by (2.136) with S1c andS2c in place of S1 and S2, respectively, and in matrix form, we have

s = Q111 (ks, 0, ki,1, 1)i1 + Q11

2 (ks, 0, ki,2, 2)i2 . (2.162)

The null-field equations (2.161) and the scattered field equation (2.162) com-pletely describe the T -matrix calculation and correspond to equations (2.134)and (2.137) of Sect. 2.6.

2.7.2 Formulation for a Particle with N Constituents

For a composite particle with N constituents, the system of matrix equationsconsists in the null-field equations in the interior of Slc

N∑

p=1

Q31p (ks, l, ki,p, p)ip = Q31

l (ks, l, ks, 0)e , for l = 1, 2, . . . ,N (2.163)


s =N∑

l=1

Q11l (ks, 0, ki,l, l)il . (2.164)


Considering the global matrix A with block-matrix components

Alp = Q31p (ks, l, ki,p, p) , l, p = 1, 2, . . . ,N , (2.165)

we express the solution to the system of matrix equations (2.163) as

il =

( N∑

p=1

AlpQ31p (ks, p, ks, 0)

)

e , l = 1, 2, . . . ,N ,

where A stay for A−1, and Alp, l, p = 1, 2, . . . ,N , are the block-matrix compo-nents of A. In view of (2.164), the transition matrix of the composite particlebecomes

T =N∑

l=1

N∑

p=1

Q11l (ks, 0, ki,l, l)AlpQ31

p (ks, p, ks, 0) .

For an axisymmetric composite particle, the transition matrix is block-diagonal with each block matrix corresponding to a different azimuthal modem. Since there is no coupling between the different m indices, each block ma-trix can be computed separately. To derive the dimension of the transitionmatrix, we consider an axisymmetric particle and set

Nmax(l) =

Nrank(l) ,Nrank(l) − |m| + 1 ,

m = 0m = 0 ,

where Nrank(l) is the maximum expansion order of the region l and m is theazimuthal mode. Then, we have

dim(Q31

p

)= 2Nmax(p) × 2Nmax ,

dim(Alp)

= 2Nmax(l) × 2Nmax(p) ,

dim(Q11

l

)= 2Nmax × 2Nmax(l),

and therefore

dim (T ) = 2Nmax × 2Nmax .

Thus, Nmax gives the dimension of the transition matrix and

Nmax =

Nrank ,Nrank − |m| + 1 ,

m = 0,m = 0,

where Nrank is determined by the size parameter of the composite particle.It should be mentioned that in our computer code, Nrank and Nrank(l), l =1, 2, . . . ,N , are independent parameters controlling the T -matrix calculations.

The contributions of Zheng [276,277], Zheng and Strom [279], and Zhengand Shao [278] demonstrated the applicability of the null-field method toparticles with complex geometries and optical properties. The above formalismcan be extended to nonaxisymmetric particles in which case, the domain ofanalysis can be discretized into many volume elements with different shapesand optical constants.


2.7.3 Formulation with Discrete Sources

For a composite particle with two constituents, the null-field equations for-mulated in terms of distributed vector spherical wave functions

jk2s

π

∫

S1c

[ei,1 (r′1) − ee (r′)] ·

(N 3

ν (ksr′1)

M3ν(ksr

′1)

)

+ [ei,1 (r′1) − ee (r′)] ·

(N 3

ν (ksr′1)

M3ν(ksr

′1)

)

dS (r′1)

+jk2

s

π

∫

S2c

[

ei,2 (r′′2) ·(

N 3ν (ksr

′′1)

M3ν(ksr

′′1)

)

(2.166)

+ j√

µs

εshi,2 (r′′

2) ·(M3

ν(ksr′′1)

N 3ν (ksr

′′1)

)]

dS (r′′2) = 0 , ν = 1, 2, . . .

and

jk2s

π

∫

S1c

[

ei,1 (r′1) ·(

N 3ν (ksr

′2)

M3ν(ksr

′2)

)

+ j√

µs

εshi,1 (r′

1) ·(M3

ν(ksr′2)

N 3ν (ksr

′2)

)]

dS (r′1)

+jk2

s

2

∫

S2c

[ei,2 (r′′2) − ee (r′′)] ·

(N 3

ν (ksr′′2)

M3ν(ksr

′′2)

)

(2.167)

+ j√

µs

εs[hi,2 (r′′

2) − he (r′′)] ·(M3

ν(ksr′′2)

N 3ν (ksr

′′2)

)

dS (r′′2) = 0 , ν = 1, 2, . . .

are equivalent to the general null-field equation (2.157). The distributed vectorspherical wave functions in (2.166)–(2.167) are defined as

M1,3mn(kr1) = M1,3

m,|m|+l [k(r1 − z1,nez)] ,

N 1,3mn(kr1) = N1,3

m,|m|+l [k(r1 − z1,nez)] ,

and

M1,3mn(kr2) = M1,3

m,|m|+l [k(r2 − z2,nez)] ,

N 1,3mn(kr2) = N1,3

m,|m|+l [k(r2 − z2,nez)] ,

where z1,n∞n=1 is a dense set of points situated on the z-axis and in theinterior of S1c, while z2,n∞n=1 is a dense set of points situated on the z-axis and in the interior of S2c. In the null-field method with discrete sources,


the approximate surface fields eNi,1, hN

i,1 and eNi,2, hN

i,2 are expressed as linearcombinations of distributed vector spherical wave functions as in (2.132) and(2.133) but with M1

µ(ki,1r′1), N 1

µ(ki,1r′1) in place of M1

µ(ki,1r′1), N1

µ(ki,1r′1)

and M1µ(ki,2r

′′2), N 1

µ(ki,2r′′2) in place of M1

µ(ki,2r′′2), N1

µ(ki,2r′′2), respectively.

For a composite particle with N constituents, the block-matrix elementsAlp are given by (2.165) with Q

31

p in place of Q31p , and the expressions of

the elements of the Q31

p matrix are given by (2.84)–(2.87) with the localizedvector spherical wave functions replaced by the distributed vector sphericalwave functions. The transition matrix is

T =N∑

l=1

N∑

p=1

Q11

l (ks, 0, ki,l, l)AlpQ31

p (ks, p, ks, 0) ,

where the Q11

l matrices contains as rows the vectors M1ν , N1

ν and as columns

the vectors M1µ, N 1

µ , while the Q31

p matrices contains as rows and columnsthe vectors M3

ν , N 3ν and M1

µ, N1µ, respectively.

For axisymmetric particles, the z-axis of the particle coordinate system isthe axis of rotation and the use of distributed vector spherical wave functionsimproves the numerical stability of the T -matrix calculations, especially forhighly elongated and flattened composite particles [51].

2.8 Complex Particles

The transition matrix for complex particles consisting of a number of arbi-trary constituents can be derived by combining the results established in theprevious sections. As an example, we consider the particle geometry depictedin Fig. 2.10. The scatterer is a composite particle with two inhomogeneousconstituents and each inhomogeneous part is a layered particle. We denoteby i1 and i1 the vectors containing the expansion coefficients of the internalfields in Di,1, by i2 and i2 the vectors containing the expansion coefficientsof the internal fields in Di,2 and by i3 and i4 the vectors containing the ex-pansion coefficients of the internal fields in Di,3 and Di,4, respectively. i1 andi2 refer to the expansion coefficients corresponding to radiating vector spher-ical wave functions, while i1, i2, i3 and i4 refer to the expansion coefficientscorresponding to regular vector spherical wave functions.

Taking into account the results established for composite particles, we seethat the null-field equations in the interior of S1c = S1∪S12 and S2c = S2∪S12

are

Q331 (ks, 1, ki,1, 1)i1 + Q31

1 (ks, 1, ki,1, 1)i1

+Q332 (ks, 1, ki,2, 2)i2 + Q31

2 (ks, 1, ki,2, 2)i2 = Q311 (ks, 1, ks, 0)e ,

and

2.8 Complex Particles 147

O4

O2

Di,1

Di,2

S2

Ds

O

O1

O3S1

Di,3

Di,4

S3

S4

S12i1, i1

i2, i2

i3

i4

Fig. 2.10. Geometry of a complex particle

Q331 (ks, 2, ki,1, 1)i1 + Q31

1 (ks, 2, ki,1, 1)i1

+Q332 (ks, 2, ki,2, 2)i2 + Q31

2 (ks, 2, ki,2, 2)i2 = Q312 (ks, 2, ks, 0)e ,

respectively. Using the results established for inhomogeneous particles, wededuce that the null-field equations in the exterior of S1c and the interior ofS3 are

−i1 + Q113 (ki,1, 1, ki,3, 3)i3 = 0 ,

−Q311 (ki,1, 3, ki,1, 1)i1 + Q31

3 (ki,1, 3, ki,3, 3)i3 = 0 ,

while the null-field equations in the exterior of S2c and the interior of S4 takethe forms

−i2 + Q114 (ki,2, 2, ki,4, 4)i4 = 0 ,

−Q312 (ki,2, 4, ki,2, 2)i2 + Q31

4 (ki,2, 4, ki,4, 4)i4 = 0 .

To derive the coefficients of the scattered-field expansion centered at O weproceed as in the case of composite particles and obtain

s = Q131 (ks, 0, ki,1, 1)i1 + Q11

1 (ks, 0, ki,1, 1)i1

+Q132 (ks, 0, ki,2, 2)i2 + Q11

2 (ks, 0, ki,2, 2)i2 .

The above equations form a system of seven equations with seven unknowns.Eliminating the vectors corresponding to the internal fields it is possible toobtain a relation between the scattered and incident field coefficients and thus,to derive the expression of the transition matrix.

The T -matrix description is also suitable for scattering configurations withseveral layered particles, Fig. 2.11, or inhomogeneous particles with severalenclosures, Fig. 2.12. In the first case, we compute the transition matrices


O1

Di,11

Ds

S11

O2

S21Di,21

O

S12 S22

Di,22Di,12

Fig. 2.11. Geometry of two layered particles

O3

D i,1

Ds

S1

O2

O

S3S2

D i,3D i,2

Fig. 2.12. Geometry of an inhomogeneous particle with two enclosures

of each layered particle with the formalism presented in Sect. 2.5, and thenuse the multiple scattering formalism to derive the transition matrix of theensemble. In the second case, we compute the transition matrix of the two-particle system, and then calculate the transition matrix of the inhomogeneousparticle using the formalism presented in Sect. 2.4. In fact, we can considerany combination of separate and consecutively enclosing surfaces and for eachcase immediately write down the expression of the transition matrix accordingto the prescriptions given in the earlier sections.

2.9 Effective Medium Model

The optical properties of heterogeneous materials are of considerable inter-est in atmospheric science, astronomy and optical particle sizing. Randomdispersions of inclusions in a homogeneous host medium can be equivalentlyconsidered as effectively homogeneous after using various homogenization for-malisms [36,210]. The effective permittivity of a heterogeneous material relates

2.9 Effective Medium Model 149

the average electric displacement to the average electric field [137],

〈D〉 = εeff 〈E〉 ,

where 〈D〉 =∫

DDdV , 〈E〉 =

∫D

E dV and D is the domain occupied bythe heterogeneous material. The effective-medium theories operate with var-ious assumptions which determine their domain of applicability. The usualderivations of the effective-medium approximations are based on electrosta-tic considerations and assume that the inclusions are much smaller than thewavelength and that the interactions between the inclusions can be neglected.The average electric displacement is expressed in terms of the average electricand polarization fields

〈D〉 = εs 〈E〉 + 〈P 〉 ,

while the average polarization field 〈P 〉 is related to the dipole moment of asingle inclusion p by the relation

〈P 〉 = n0p ,

where εs is the relative permittivity of the host medium and n0 is the numberof particles per unit volume. Defining the exciting field Eexc as the local fieldwhich exists within a fictitious cavity that has the shape of an inclusion

Eexc = 〈E〉 +1

3εs〈P 〉

and assuming that the dipole moment can be expressed in terms of the excitingfield as

p = αEexc

with α being the polarizability scalar, yields

εeff = εs3εs + 2n0α

3εs − n0α. (2.168)

The main issue in the effective-medium approximations is to relate the po-larizability α to the relative permittivity εr, where εr = εi/εs, and εi is therelative permittivity of the inclusion. For spherical particles and at frequenciesfor which the inclusions can be considered very small, the Maxwell–Garnettformula uses the relation

α =43πR3εs

3 (εr − 1)εr + 2

, (2.169)

which yields

εeff = εs

1 + 2c εr−1εr+2

1 − c εr−1εr+2

, (2.170)


where R is the radius of the spherical particles and c is the fractional vol-ume (or the fractional concentration) of the particles, c = (4/3)πR3n0. Notethat size-dependent Maxwell–Garnett formulas incorporating finite-size ef-fects have been derived by using the Lorenz–Mie theory [52, 57], the volumeintegral equation method [129] and variational approaches [211]. However, inmicroscopic view, waves propagating in the composite do undergo multiplecorrelated scattering between the particles. When the concentration of theinclusions is high and the size of the inclusion becomes comparable to thewavelength, more complete models which takes into account the inclusionsize, statistical correlations, and multiple-scattering effects, are needed.

In this section, we consider the scattering of a vector plane wave incidenton a half-space with randomly distributed particles. Our treatment followsthe procedure described by Varadan et al. [236], Varadan and Varadan [235],and Bringi et al. [28] and concerns with the analysis of the coherent scat-tered field and in particular, of the effective propagation constants in the caseof normal incidence. This approach is based on the T -matrix method andon spherical statistics, even though the particles may be nonspherical. Lax’squasi-crystalline approximation [138] is employed to truncate the hierarchy ofequations relating the different orders of correlations between the particles,and the hole correction and the Percus–Yevick [186] approximation are usedto model the pair distribution function. The integral equation of the quasi-crystalline approximation gives rise to the generalized Lorentz–Lorenz law andthe generalized Ewald–Oseen extinction theorem. The generalized Lorentz–Lorenz law consists of a homogeneous system of equations and the resultingdispersion relation gives the effective propagation constant for the coherentwave. The generalized Ewald–Oseen extinction theorem is an inhomogeneousequation that relates the transmitted coherent field to the amplitude, polar-ization, and direction of propagation of the incident field.

Other contributions which are related to the present analysis includes thework of Foldy [69], Waterman and Truell [258], Fikioris and Waterman [65],Twerski [230, 231], Tsang and Kong [223, 224] and Tsang et al. [227]. Thecase of a plane electromagnetic wave obliquely incident on a half-space withdensely distributed particles and the computation of the incoherent field withthe distorted Born approximation have been considered Tsang and Kong [225,226].

2.9.1 T -matrix Formulation

The scattering geometry is shown in Fig. 2.13. A linearly polarized plane waveis incident on a half-space with N identical particles centered at r0l, l =1, 2, . . . ,N , and each particle is assumed to have an imaginary circumscribingspherical shell of radius R. The particles have the same orientation and wechoose αl = βl = γl = 0 for l = 1, 2, . . . ,N . The relative permittivity andrelative permeability of the homogeneous particles are εi and µi, and εs andµs are the material constants of the background medium (which also occupies


ON

Op

O

O1

O l

r0l

x

y

z

ke

eβeα

ε i, µi εs, µs

Rr0p

rlp

P

rp rlr

Fig. 2.13. Geometry of identical, homogeneous particles distributed in the half-space z ≥ 0

the domain z < 0). The vector plane wave is of unit amplitude:

Ee(r) = epolejke·r ,

and propagates along the z-axis of the global coordinate system, ke = ksez.Our analysis is entirely based on the superposition T -matrix method which

expresses the scattered field Es in terms of the fields Es,l scattered from eachindividual particle l (cf. (2.147)),

Es(r) =N∑

l=1

Es,l (rl)

=N∑

l=1

∑

µ

fl,µM3µ (ksrl) + gl,µN3

µ (ksrl) . (2.171)

Taking into account that the particle coordinate systems have the same spa-tial orientation (no rotations are involved), we rewrite the expression of thescattered field coefficients sl, sl = [fl,µ, gl,µ]T, given by (2.142) as

sl = T

⎡

⎣ejksz0le +N∑

p=l

T 31 (ksrlp) sp

⎤

⎦ , (2.172)

where e is the vector of the incident field coefficients in the global coordinatesystem and T is the transition matrix of a particle.


Before going any further we make some remarks on statistically averagedwave fields. We define the probability density function of finding the firstparticle at r01, the second particle at r02, and so forth by p(r01, r02, . . . , r0N ).The probability density function can be written as [236]

p (r01, r02, . . . , r0N ) = p (r0l) p (r01, r02, . . . ,′ , . . . , r0N | r0l)

= p (r0l, r0p) p (r01, r02, . . . ,′ , . . . ,′ , . . . , r0N | r0l, r0p) ,

where p(r0l) is the probability density of finding the particle l at r0l, p(r0l, r0p)is the joint density function of the particles l and p and the ′ superscriptindicates that the term is absent. From Bayes’ rule we have

p (r0l, r0p) = p (r0l) p (r0p | r0l) ,

where p(r0p | r0l) is the conditional probability of finding the particle p atr0p if the particle l is known to be at r0l. The conditional densities with oneand two particles held fixed are related by the relation

p (r01, r02, . . . ,′ , . . . , r0N | r0l) = p (r0p | r0l)

×p (r01, r02, . . . ,′ , . . . ,′ , . . . , r0N | r0l, r0p) .

The configurational average of a statistical quantity f is defined as

Ef =∫

D

· · ·∫

D

fp (r01, r02, . . . , r0N )

×dV (r01) dV (r02) , . . . ,dV (r0N ) ,

while the conditional averages with one and two particles held fixed are

Elf =∫

D

· · ·∫

D

fp (r01, r02, . . . ,′ , . . . , r0N | r0l)

×dV (r01) dV (r02) , . . . ,′ , . . . ,dV (r0N )

and

Elpf =∫

D

· · ·∫

D

fp (r01, r02, . . . ,′ , . . . ,′ , . . . , r0N | r0l, r0p)

×dV (r01) dV (r02) , . . . ,′ , . . . ,′ , . . . ,dV (r0N ) ,

respectively.Averaging (2.172) over the positions of all particles excepting the lth, we

obtain the conditional configurational average of the scattered field coefficients

El sl = T

⎡

⎣ejksz0le +N∑

p=l

1V

∫

Dp

T 31 (ksrlp) Elp sp g (rlp) dV (r0p)

⎤

⎦ ,

(2.173)


where g is the pair distribution function of a statistically homogeneousmedium,

p (r0p | r0l) =1V

g (r0p − r0l) =1V

g (rlp)

and V is the volume accessible to the particles. In our analysis we assume thatboth N and V are very large, but the number of particles per unit volume n0,

n0 = N/V,

is finite. The integration domain Dp is the half-space z0p ≥ 0, excluding aspherical volume of radius 2R centered at r0l, i.e., |r0p−r0l| ≥ 2R. The aboveequation indicates that the conditional average with one particle fixed Elslis given in terms of the conditional average with two particles fixed Elpsp.In order to close the system we use the quasi-crystalline approximation

Elp sp = Ep sp for l = p , (2.174)

which implies that there is no correlation between the lth and pth particlesother than there should be no interpenetration of any two particles.

To solve (2.173) we seek the plane wave solution

El sl = sejKsz0l , (2.175)

where s is the unknown vector of the scattered field coefficients and the ef-fective wave vector Ke is assumed to be parallel to the wave vector of theincident field, i.e., Ke = Ksez. Substituting (2.174) and (2.175) into (2.173),and taking into account that for identical particles the sum

∑Np=l can be

replaced by N − 1 and for a sufficiently large N , n0 ≈ (N − 1)/V , we obtain

sejKsz0l = T

[

ejksz0le + n0

∫

Dp

T 31 (ksrlp) g (rlp) sejKsz0pdV (r0p)

]

.

(2.176)The integral in (2.176) can be written as (excepting the unknown vector s)

Jl = J1,l + J2,l

where

J1,l =∫

Dp

T 31 (ksrlp) ejKsz0pdV (r0p) ,

J2,l =∫

Dp

T 31 (ksrlp) [g (rlp) − 1] ejKsz0pdV (r0p) ,

and the rest of our analysis concerns with the computation of the matrix termsJ1,l and J2,l.


In view of (2.144), J1,l can be expressed as

J1,l =∫

Dp

[A3

−mn,−m′n′ (ksrlp) B3−mn,−m′n′ (ksrlp)

B3−mn,−m′n′ (ksrlp) A3

−mn,−m′n′ (ksrlp)

]

ejKsz0pdV (r0p) ,

whence, taking into account the series representations for the translation co-efficients (cf. (B.72) and (B.73))

A3−mn,−m′n′(ksrlp) =

2jn′−n

√nn′(n + 1) (n′ + 1)

×∑

n′′

jn′′a1 (−m,−m′ | n′′, n, n′) u3

m′−mn′′ (ksrlp) ,

B3−mn,−m′n′(ksrlp) =

2jn′−n

√nn′(n + 1) (n′ + 1)

×∑

n′′

jn′′b1 (−m,−m′ | n′′, n, n′) u3

m′−mn′′ (ksrlp)

with (cf. (B.68) and (B.69))

a1 (−m,−m′ | n′′, n, n′) =∫ π

0

[mm′π|m|

n (β) π|m′|n′ (β) + τ |m|

n (β) τ|m′|n′ (β)

]

×P|m′−m|n′′ (cos β) sinβdβ ,

b1 (−m,−m′ | n′′, n, n′) = −∫ π

0

[mπ|m|

n (β) τ|m′|n′ (β) + m′τ |m|

n (β) π|m′|n′ (β)

]

×P|m′−m|n′′ (cos β) sin βdβ ,

we see that the calculation of J1,l requires the computation of the integral

I1mm′n′′ =

∫

Dp

ejKsz0pu3m′−mn′′ (ksrlp) dV (r0p) .

The procedure to calculate the integral I1mm′n′′ is described in Appendix C

and we have (cf. (C.2)),

I1mm′n′′ =

16πR3

(ksR)2 − (KsR)2jn

′′√

2n′′ + 12

ejKsz0lδmm′Fn′′ (Ks, ks, R)

+2π

k2s (Ks − ks)

jn′′+1

√2n′′ + 1

2ejksz0lδmm′


with

Fn′′ (Ks, ks, R) = (ksR)(h

(1)n′′ (2ksR)

)′jn′′ (2KsR)

− (KsR) h(1)n′′ (2ksR) (jn′′ (2KsR))′ .

Relying on this result we find that the matrix J1,l can be decomposed intothe following manner:

J1,l = J1,RejKsz0l + J1,zejksz0l . (2.177)

The elements of the matrix J1,R,

J1,R =

[(J1,R)11−mn,−m′n′ (J1,R)12−mn,−m′n′

(J1,R)21−mn,−m′n′ (J1,R)22−mn,−m′n′

]

,

are given by

(J1,R)11−mn,−m′n′ = (J1,R)22−mn,−m′n′

=32πR3

(ksR)2 − (KsR)2jn

′−n

√nn′(n + 1) (n′ + 1)

R11mnn′δmm′ ,

(J1,R)12−mn,−m′n′ = (J1,R)21−mn,−m′n′

= − 32πR3

(ksR)2 − (KsR)2jn

′−n

√nn′(n + 1) (n′ + 1)

R12mnn′δmm′ ,

where R11mnn′ and R12

mnn′ are defined as

R11,12mnn′ =

∞∑

n′′=0

(−1)n′′√

2n′′ + 12

Fn′′ (Ks, ks, R) I1,2mnn′n′′ ,

and

I1mnn′n′′ =

∫ π

0

[m2π|m|

n (β) π|m|n′ (β) + τ |m|

n (β) τ|m|n′ (β)

]Pn′′ (cos β) sin βdβ ,

I2mnn′n′′ = m

∫ π

0

[π|m|

n (β) τ|m|n′ (β) + τ |m|

n (β) π|m|n′ (β)

]Pn′′ (cos β) sinβdβ .

(2.178)

The matrix J1,z is of the form:

J1,z =

[(J1,z)

11−mn,−m′n′ (J1,z)

12−mn,−m′n′

(J1,z)21−mn,−m′n′ (J1,z)

22−mn,−m′n′

]

,


and the matrix elements are given by

(J1,z)11−mn,−m′n′ = (J1,z)

22−mn,−m′n′

=4π

k2s (Ks − ks)

jn′−n+1

√nn′(n + 1) (n′ + 1)

S1mnn′δmm′ ,

(J1,z)12−mn,−m′n′ = (J1,z)

21−mn,−m′n′

= − 4π

k2s (Ks − ks)

jn′−n+1

√nn′(n + 1) (n′ + 1)

S2mnn′δmm′ ,

where the expressions of S1mnn′ and S2

mnn′ are similar to those of R11mnn′ and

R12mnn′ excepting the factor Fn′′ ,

S1,2mnn′ =

∞∑

n′′=0

(−1)n′′√

2n′′ + 12

I1,2mnn′n′′ .

For m = 1 we have

S11nn′ = S1

−1nn′ = snn′ , (2.179)

S21nn′ = −S2

−1nn′ = −snn′ , (2.180)

where the coefficient snn′ is given by (cf. (C.6))

snn′ =(−1)n+n′

4

√n(n + 1) (2n + 1)

√n′(n′ + 1) (2n′ + 1) . (2.181)

To derive the expression of J2,l, we need to compute the integral

I2mm′n′′ =

∫

Dp

[g (rlp) − 1] ejKsz0pu3m′−mn′′ (ksrlp) dV (r0p)

and the result is (cf. (C.3))

I2mm′n′′ = 32πR3jn′′

√2n′′ + 1

2ejKsz0lδmm′Gn′′ (Ks, ks, R) ,

where

Gn′′ (Ks, ks, R) =∫ ∞

1

[g (2Rx) − 1] h(1)n′′ (2ksRx) jn′′ (2KsRx)x2dx .

The matrix term J2,l then becomes

J2,l = J2,RejKsz0l (2.182)


and the elements of the matrix J2,R,

J2,R =

[(J2,R)11−mn,−m′n′ (J2,R)12−mn,−m′n′

(J2,R)21−mn,−m′n′ (J2,R)22−mn,−m′n′

]

are given by

(J2,R)11−mn,−m′n′ = (J2,R)22−mn,−m′n′

= 64πR3 jn′−n

√nn′(n + 1) (n′ + 1)

R21mnn′δmm′ ,

(J2,R)12−mn,−m′n′ = (J2,R)21−mn,−m′n′

= −64πR3 jn′−n

√nn′(n + 1) (n′ + 1)

R22mnn′δmm′ ,

where R21mnn′ and R22

mnn′ are defined as

R21,22mnn′ =

∞∑

n′′=0

(−1)n′′√

2n′′ + 12

Gn′′ (Ks, ks, R) I1,2mnn′n′′ .

Taking into account the expressions of the integral terms I1,2mnn′n′′ (cf. (2.178))

it is apparent that

R11mnn′ = R11

−mnn′ , R12mnn′ = −R12

−mnn′ (2.183)

andR21

mnn′ = R21−mnn′ , R22

mnn′ = −R22−mnn′ . (2.184)

Substituting (2.177) and (2.182) into (2.176) gives two types of terms. Onetype of terms has a exp(jksz0l) dependence and corresponds to waves travelingwith the wave number of the incident wave, while the other type of termshas a exp(jKsz0l) dependence and corresponds to waves traveling with thewave number of the effective medium. The terms with wave number ks shouldbalance each other giving the generalized Ewald–Oseen extinction theorem,

n0J1,zs + e = 0 , (2.185)

while balancing the terms with wave number Ks gives the generalized Lorentz–Lorenz law

[I − n0T (J1,R + J2,R)] s = 0 . (2.186)

For an ensemble of particles with different orientations we proceed analo-gously. If the configurational distribution and the orientation distributionare assumed to be statistically independent, the total probability density


function can be expressed as a product of two functions, one of whichp(r01, r02, . . . , r0N ) describes the configurational distribution and the other ofwhich p(Ω1, Ω2, . . . , ΩN ) describes the distribution of particles orientations,i.e.,

p (r01, r02, . . . , r0N ;Ω1, Ω2, . . . , ΩN )= p (r01, r02, . . . , r0N ) p (Ω1, Ω2, . . . , ΩN ) ,

where Ωl is the set of Euler angles specifying the orientation of the lth particleand Ωl = (αl, βl, γl). Each of these probability density functions is normalizedto unity and the problem of computing the configurational and orientation av-erages are separated. Assuming that the particle orientations are statisticallyindependent

p (Ω1, Ω2, . . . , ΩN ) = p (Ω1) p (Ω2) ...p (ΩN ) ,

and that the individual orientation distribution is uniform

p (Ωl) =1

8π2, l = 1, 2, . . . ,N ,

we average (2.172) over all configurations for which the lth particle is heldfixed and over all orientations. We obtain

〈El sl〉 = ejksz0l

⟨T⟩

e +N∑

p=l

1V

∫

Dp

⟨TT 31 (ksrlp) Ep (sp)

⟩g (rlp) dV (r0p) ,

where 〈.〉 denotes the orientation average

〈El sl〉 =1

(8π2)N

∫

Ω

· · ·∫

Ω

[∫

D

· · ·∫

D

slp (r01, r02, . . . ,′ , . . . , r0N | r0l)

×dV (r01) dV (r02) , . . . ,′ , . . . ,dV (r0N )]dΩ1dΩ2, . . . ,dΩN ,

sl = sl(Ω1, Ω2, . . . , ΩN ), Ω is the unit sphere, dΩl = sin βldβldαldγl and⟨T⟩

=1

8π2

∫

Ω

T (αl, βl, γl) dΩl

is the orientation-averaged transition matrix. Making the assumption⟨TT 31 (ksrlp) Ep (sp)

⟩=⟨T⟩T 31 (ksrlp) 〈Ep (sp)〉 ,

which implies that Epsp depends only on Ωp [227], we obtain the generalizedLorentz–Lorenz law in terms of the orientation-averaged transition matrix〈T 〉 instead of the transition matrix T of a particle in a fixed orientation.We note that the submatrices of the orientation-averaged transition matrixare diagonal and their elements do not depend on the azimuthal indices (seeSect. 1.5).


2.9.2 Generalized Lorentz–Lorenz Law

The generalized Lorentz–Lorenz law is a system of homogeneous equations forthe unknown vector s. For a nontrivial solution, the determinant must vanishgiving an equation for the effective wave number Ks. For spherical particles oraxisymmetric particles with the axis of symmetry directed along the z-axis,the transition matrix is diagonal with respect to the azimuthal indices m andm′. Because the nonvanishing incident field coefficients corresponds to m = −1and m = 1, the above matrix equations involve only these azimuthal modes.Taking into account the symmetry relations (2.25), (2.183) and (2.184), weobtain the following equation for the effective wave number Ks

det

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

I−

⎡

⎢⎢⎢⎣

T 111n′′,1n 0 T 12

1n′′,1n 00 T 11

1n′′,1n 0 −T 121n′′,1n

T 211n′′,1n 0 T 22

1n′′,1n 00 −T 21

1n′′,1n 0 T 221n′′,1n

⎤

⎥⎥⎥⎦

×

⎡

⎢⎢⎢⎣

Ann′ 0 Bnn′ 00 Ann′ 0 −Bnn′

Bnn′ 0 Ann′ 00 −Bnn′ 0 Ann′

⎤

⎥⎥⎥⎦

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

= 0 ,

where

Ann′ = n0

[(J1,R)11−1n,−1n′ + (J2,R)11−1n,−1n′

],

Bnn′ = n0

[(J1,R)12−1n,−1n′ + (J2,R)12−1n,−1n′

]. (2.187)

The dimension of the problem can be reduced by using the properties ofdeterminants and the final result is

det

I−[

T 111n′′,1n T 12

1n′′,1n

T 211n′′,1n T 22

1n′′,1n

][Ann′ Bnn′

Bnn′ Ann′

]

= 0 . (2.188)

In the low frequency limit, the size of the particle is considered to be verysmall compared to the incident wavelength. It is then sufficient to take onlythe lowest-order coefficients in the fields expansions, and in this limit, theelements of the T matrix can be obtained in closed form for simple shapes.For spherical particles, we have

T 1111,11 = T 1

1 =j

45(εr − 1) x5 + O

(x7)

,

T 2211,11 = T 2

1 =2j3

εr − 1εr + 2

x3 +2j5

(εr − 1) (εr − 2)(εr + 2)2

x5 + O(x6)

,


where x = ksR is the size parameter. In (2.187) we proceed analogously andconsidering the leading terms in the expansions of the spherical Bessel andHankel functions, we obtain

A11 = a0 +12a2 ,

B11 =32a1,

where

a0 =3jc

(ksR)[(ksR)2 − (KsR)2

] +(1 − c)4

(1 + 2c)2,

a1 =3jc (KsR)

(ksR)2[(ksR)2 − (KsR)2

] ,

a2 =3jc (KsR)2

(ksR)3[(ksR)2 − (KsR)2

] .

As a result, the long-wavelength equation for the effective wave number takesthe form

(1 − T 11

11,11A11

) (1 − T 22

11,11A11

)− T 11

11,11T2211,11B

211 = 0 .

Further, neglecting T 1111,11, yields

1 − T 2211,11A11 = 0 ,

and consequently, (Ks

ks

)2

=2χ + 6jc2χ − 3jc

(2.189)

with

χ = (ksR)3[

(1 − c)4

(1 + 2c)2− 1

T 2211,11

]

.

For an ensemble of spheres of radius R and small fractional concentrations c,

χ ≈ j32

εr + 2εr − 1

,

and we obtain the dispersion relation of the Maxwell–Garnett form (2.170). Athigher frequencies, the root of the equation (2.188) is searched in the complexplane using Muller’s method. The initial guess is provided by (2.189) at lowvalues of ksR and these are used systematically to obtain convergence of rootsat increasingly higher values of ksR.


2.9.3 Generalized Ewald–Oseen Extinction Theorem

After computing the singular solution Ks, the elements of the s vector canbe expressed in terms of an arbitrary constant, and this constant will bedetermined from the Ewald-Oseen extinction theorem. Certainly, this strategytacitly assumes that the system of equations (2.185) reduces to a single scalarequation and to prove this assertion we introduce the notations

s =

⎡

⎢⎢⎣

f1n′

f−1n′

g1n′

g−1n′

⎤

⎥⎥⎦ , e =

⎡

⎢⎢⎣

a1n

a−1n

b1n

b−1n

⎤

⎥⎥⎦ ,

and (cf. (2.179) and (2.180))

J1,z =

⎡

⎢⎢⎣

tnn′ 0 tnn′ 00 tnn′ 0 −tnn′

tnn′ 0 tnn′ 00 −tnn′ 0 tnn′

⎤

⎥⎥⎦ ,

where

tnn′ =4π

k2s (Ks − ks)

jn′−n+1

√nn′(n + 1) (n′ + 1)

snn′ .

Using the fact that for a vector plane wave polarized along the x-axis, theincident field coefficients are given by

a1n = −a−1n = jn−1√

2n + 1 ,

b1n = b−1n = jn−1√

2n + 1 ,

we see that the matrix equation (2.185) is solved by

f1n′ = −f−1n′ ,

g1n′ = g−1n′ ,

and the coefficients f1n′ and g1n′ are the solutions of the inhomogeneoussystem of equations

n0

∞∑

n′=0

tnn′ (f1n′ + g1n′) = −jn−1√

2n + 1 , n = 1, 2, . . .

Taking into account the expression of snn′ given by (2.181), we deduce thatthe above set of inhomogeneous equations reduces to a single scalar equation

πn0

k2s (Ks − ks)

∞∑

n′=0

(−j)n′ √2n′ + 1 (f1n′ + g1n′) = 1 (2.190)


and the assertion is proved. If the incident wave is polarized along the y-axis,we have

f1n = f−1n ,

g1n = −g−1n ,

while the scalar equation takes the form

πjn0

k2s (Ks − ks)

∞∑

n′=0

(−j)n′ √2n′ + 1 (f1n′ + g1n′) = 1 . (2.191)

Thus, the procedure is to first find the singular solution of the Lorentz–Lorenzlaw together with the dispersion relation for the effective wave number Ks,and then to determine the arbitrary constant from the scalar equations (2.190)or (2.191). After the conditional average of the scattered field coefficients hasbeen evaluated, the coherent reflected field can be computed by taking theconfigurational average of (2.171), i.e.,

E Es(r) =∫

...

∫Es(r)p (r01, r02, . . . , r0N )

×dV (r01) dV (r02) , . . . ,dV (r0N )

=N∑

l=1

∫

Dl

El Es,l (rl) p (r0l) dV (r0l)

= n0

∫

Dl

El Es,l (rl)dV (r0l) ,

where the integration domain Dl is the half-space z0l ≥ 0 and p(r0l) = 1/V .We obtain

E Es(r) = − πn0

k2s (Ks + ks)

e−jksez·r∞∑

n=1

jn√

2n + 1 (f1n − g1n)ex

for an incident vector plane wave polarized along the x-axis, and

E Es(r) = − πjn0

k2s (Ks + ks)

e−jksez·r∞∑

n=1

jn√

2n + 1 (f1n − g1n) ey

for an incident vector plane wave polarized along the y-axis. The above rela-tions show that the coherent reflected field is in the backward direction andcontains no depolarization.

2.9.4 Pair Distribution Functions

We conclude this section with some remarks on pair distribution functionsthat have been used in multiple scattering problems. For the hole-correction


approximation, the pair distribution function is given by g(r) = 0, for r <2R, and g(r) = 1, for r ≥ 2R. The hole-correction approximation assumesthat the particles do not interpenetrate each other and considers uniformdistribution outside the hole. When the fractional volume of the particlesc is appreciable, the hole-correction approximation is not correct and thePercus–Yevick pair function is frequently employed. This pair distributionfunction is the solution of the Ornstein–Zernike equation for the total anddirect correlation functions of two particles h(r) and d(r), respectively, [228].In general, the total correlation function

h(r) = g(r) − 1 ,

can be expressed in terms of its Fourier transform H as

h(r) =1

(2π)3

∫ ∞

−∞H (p) ejp·rdV (p) .

Passing to spherical coordinates, using the spherical wave expansion of the“plane waves” exp (jp·r) and assuming spherical symmetry for H, i.e., H(p) =H(p), we obtain

h (r) =1

2π2

∫ ∞

0

H(p)sin (pr)

prp2dp

and further

h (2Rx) =1

16πR3

∫ ∞

0

H( u

2R

) sin (ux)ux

u2du , (2.192)

where

x =r

2R, u = 2Rp .

The Ornstein–Zernike equation with Percus–Yevick approximation has aclosed-form solution for the case of a hard-sphere potential. In the Fouriertransform domain, the solution to the Ornstein–Zernike equation is

H(p) =D(p)

1 − n0D(p),

where the Fourier transform D of the direct correlation function is given by

D(p) =24c

n0

[α + β + δ

u2cos u − α + 2β + 4δ

u3sinu

− 2 (β + 6δ)u4

cos u +2β

u4+

24δ

u5sinu +

24δ

u6(cos u − 1)

]

with

α =(1 + 2c)2

(1 − c)4, β = −6c

(1 + c/2)2

(1 − c)4, δ =

12αc .

Thus, the Fourier transform H(p) can be calculated readily while the totalcorrelation function is obtained from (2.192).


2.10 Particle on or near an Infinite Surface

Computation of light scattering from particles deposited upon a surface is ofgreat interest in the simulation, development and calibration of surface scan-ners for wafer inspection [214]. More recent applications include laser clean-ing [149], scanning near-field optical microscopy (SNOM) [114] and plasmonresonances effects in surface-enhanced Raman spectroscopy (SERS) [30]. Sev-eral studies have addressed this scattering problem using different methods.Simplified theoretical models have been developed on the basis of Lorenz-Mietheory and Fresnel surface reflection [15,240–242]. A coupled-dipole algorithmhas been employed by Taubenblatt and Tran [221] and Nebeker et al. [178]using a three dimensional array of dipoles to model a feature shape and theSommerfeld integrals to describe the interaction between a dipole and a sur-face. The theoretical aspects of the coupled-dipole model has been fully out-lined by Schmehl [206]. A model based on the discrete source method hasbeen given by Eremin and Orlov [59,60], whereas the transmission conditionsat the interface are satisfied analytically and the fields of discrete sources arederived by using the Green tensor for a plane surface. More details on com-putational methods and experimental results can be found in a book editedby Moreno and Gonzales [173].

Similar scattering problems have been solved by Kristensson and Strom[125], and Hackmann and Sammelmann [90] in the framework of the null-fieldmethod. Acoustic scattering from a buried inhomogeneity has been consideredby Kristensson and Strom on the assumption that the free-field T -matrix ofthe particle modifies the free-field T -operator of the arbitrary surface. Byprojecting the free-field T -operator of the surface onto a spherical basis, aninfinite system of linear equations for the free-field T -matrix of the particlehas been derived. In contrast, Hackmann and Sammelmann assumed thatthe free-field T -operator of the surface modifies the free-field T -matrix ofthe particle, and in this case, the free-field T -matrix of the particle has beenprojected onto a rectangular basis and an integral equation for the spectralamplitudes of the fields has been obtained.

2.10.1 Particle on or near a Plane Surface

In this section we extend the results of Bobbert and Vlieger [15] to the caseof axisymmetric particles situated on or near a plane surface. The scatteringproblem is a multiple particles problem and the solution method is the separa-tion of variables technique. To model the scattering problem in the frameworkof the separation of variables technique one must address how the radiationinteracts with the particle. The incident field strikes the particle either di-rectly or after interacting with the surface, while the fields emanating fromthe particle may also reflect off the surface and interact with the particle again.The transition matrix relating the incident and scattered field coefficients iscomputed in the framework of the null-field method and the reflection matrix

2.10 Particle on or near an Infinite Surface 165

O x

z

ke

β0z0

Σ

mrs

mr

Fig. 2.14. Geometry of an axisymmetric particle situated near a plane surface. Theexternal excitation is a vector plane wave propagating in the ambient medium

characterizing the reflection of the scattered field by the surface is computedby using the integral representation for the vector spherical wave functions.

The geometry of the scattering problem is shown in Fig. 2.14. An axisym-metric particle is situated in the neighborhood of a plane surface Σ, so thatits axis of symmetry is normal to the plane surface. The z-axis of the parti-cle coordinate system Oxyz is directed along the axis of symmetry and theorigin O is situated at the distance z0 below the plane surface. The incidentradiation is a linearly polarized vector plane wave propagating in the ambientmedium (the medium below the surface Σ)

Ee(r) = (Ee0,βeβ + Ee0,αeα) ejke·r , (2.193)

and the wave vector ke, ke = ksek, is assumed to be in the xz-plane and toenclose the angle β0 with the z-axis.

The incident wave strikes the particle either directly or after interactingwith the surface. The direct and the reflected incident fields are expanded interms of regular vector spherical wave functions

Ee(r) =∞∑

n1=1

n1∑

m=−n1

a0mn1

M1mn1

(ksr) + b0mn1

N1mn1

(ksr) (2.194)

and

ERe (r) =

∞∑

n1=1

n1∑

m=−n1

aRmn1

M1mn1

(ksr) + bRmn1

N1mn1

(ksr) , (2.195)


respectively, and we define the total expansion coefficients amn1 and bmn1 bythe relations

amn1 = a0mn1

+ aRmn1

,

bmn1 = b0mn1

+ bRmn1

.

The coefficients a0mn1

and b0mn1

are the expansion coefficients of a vector planewave traveling in the (β0, 0) direction,

a0mn1

= − 4jn1

√2n1 (n1 + 1)

[jmπ|m|

n1(β0)Ee0,β + τ |m|

n1(β0)Ee0,α

],

b0mn1

= − 4jn1+1

√2n1 (n1 + 1)

[τ |m|n1

(β0)Ee0,β − jmπ|m|n1

(β0)Ee0,α

],

while the coefficients aRmn1

and bRmn1

are the expansion coefficients of a vectorplane wave traveling in the (π−β0, 0) direction. The part of the incident fieldthat reflects off the surface will undergo a Fresnel reflection and it will beout of phase by an amount of exp(2jksz0 cos β0). This phase factor arises fromthe phase difference between the plane wave and its reflected wave in O. Theresulting expressions for aR

mn1and bR

mn1are

aRmn1

= − 4jn1

√2n1 (n1 + 1)

[jmπ|m|

n1(π − β0)ER

e0,β + τ |m|n1

(π − β0)ERe0,α

],

bRmn1

= − 4jn1+1

√2n1 (n1 + 1)

[τ |m|n1

(π − β0)ERe0,β − jmπ|m|

n1(π − β0)ER

e0,α

],

where

ERe0,β = r‖(β0)e2jksz0 cos β0Ee0,β ,

ERe0,α = r⊥(β0)e2jksz0 cos β0Ee0,α ,

with r‖(β0) and r⊥(β0) being the Fresnel reflection coefficients for parallel andperpendicular polarizations, respectively. The Fresnel reflection coefficientsare given by

r‖(β0) =mrs cos β0 − cos β

mrs cos β0 + cos β,

r⊥(β0) =cos β0 − mrs cos β

cos β0 + mrs cos β,

where mrs is the relative refractive index of the substrate with respect to theambient medium and β is the angle of refraction (Fig. 2.15). The angles ofincidence and refraction are related to each other by Snell’s law


O x

z

k e

β0

Σ

β0 keR

k eT

Eeβ ER

eβ

ETeββ

mrs

z0

Fig. 2.15. Reflection and refraction of a vector plane wave (propagating in theambient medium) at the interface Σ

sin β =1

mrssin β0 ,

cos β = ±√

1 − sin2 β

and for real values of mrs, the sign of the square root is plus, while for complexvalues of mrs, the sign of the square root is chosen such that Immrs cos β > 0.This choice guarantees that the amplitude of the refracted wave propagatingin the positive direction of the z-axis would tend to zero with increasing thedistance z.

The scattered field is expanded in terms of radiating vector spherical wavefunctions

Es(r) =∞∑

n=1

n∑

m=−n

fmnM3mn(ksr) + gmnN3

mn(ksr) (2.196)

and the rest of our analysis concerns with the calculation of the expansion co-efficients fmn and gmn. In addition to the fields described by (2.194)–(2.196), afourth field exists in the ambient medium. This field is a result of the scatteredfield reflecting off the surface and striking the particle. It can be expressed as

ERs (r) =

∞∑

n=1

n∑

m=−n

fmnM3,Rmn(ksr) + gmnN3,R

mn(ksr) , (2.197)

where M3,Rmn(ksr) and N3,R

mn(ksr) are the radiating vector spherical wave func-tions reflected by the surface. Accordingly to Videen [240–242], the field ER

s

will be designated as the interacting field. For r inside a sphere enclosed inthe particle and a given azimuthal mode m, we anticipate an expansion of thereflected vector spherical wave functions of the form


(M3,R

mn(ksr)N3,R

mn(ksr)

)

=∞∑

n1=1

(αmnn1

γmnn1

)

M1mn1

(ksr) +

(βmnn1

δmnn1

)

N1mn1

(ksr) .

(2.198)

Inserting (2.198) into (2.197), we derive a series representation for the inter-acting field in terms of regular vector spherical wave functions

ERs (r) =

∞∑

n1=1

n1∑

m=−n1

fRmn1

M1mn1

(ksr) + gRmn1

N1mn1

(ksr)] , (2.199)

where(

fRmn1

gRmn1

)

=∞∑

n=1

(αmnn1

βmnn1

)

fmn +

(γmnn1

δmnn1

)

gmn . (2.200)

In the null-field method, the scattered field coefficients are related to theexpansion coefficients of the fields striking the particle by the transition matrixT . For an axisymmetric particle, the equations become uncoupled, permittinga separate solution for each azimuthal mode. Thus, for a fixed azimuthalmode m, we truncate the expansions given by (2.194)–(2.196) and (2.199),and derive the following matrix equation:

[fmn

gmn

]= [Tmn,mn1 ]

([amn1

bmn1

]+[

fRmn1

gRmn1

]), (2.201)

where n and n1 range from 1 to Nrank, and m ranges from −Mrank to Mrank,with Nrank and Mrank being the maximum expansion and azimuthal orders,respectively. The expansion coefficients of the interacting field are related tothe scattered field coefficients by a so called reflection matrix:

[fR

mn1

gRmn1

]= [Amn1n]

[fmn

gmn

], (2.202)

where, in view of (2.200),

[Amn1n] =[

αmnn1 γmnn1

βmnn1 δmnn1

].

Now it is apparent that the scattered field coefficients fmn and gmn can beobtained by combining the matrix equations (2.201) and (2.202), and theresult is [270]

(I − [Tmn,mn1 ] [Amn1n])[

fmn

gmn

]= [Tmn,mn1 ]

[amn1

bmn1

]. (2.203)

To derive the expression of the reflection matrix we use the integral represen-tations for the radiating vector spherical wave functions


(M3

mn(ksr)N3

mn(ksr)

)

= − 12πjn+1

1√

2n(n + 1)

∫ 2π

0

∫ π/2−j∞

0

[(mπ

|m|n (β)

τ|m|n (β)

)

eβ

+j

(τ|m|n (β)

mπ|m|n (β)

)

eα

]

ejmαejk(β,α)·r sinβdβ dα , (2.204)

where (ks, β, α) are the spherical coordinates of the wave vector k, and(ek,eβ ,eα) are the spherical unit vectors of k. Each reflected plane wavein (2.204) will contain a Fresnel reflection term and a phase term equiva-lent to exp(2jksz0 cos β). The reflected vector spherical wave functions can beexpressed as(

M3,Rmn(ksr)

N3,Rmn(ksr)

)

= − 12πjn+1

1√

2n(n + 1)

∫ 2π

0

∫ π/2−j∞

0

[(mπ

|m|n (β)

τ|m|n (β)

)

r‖(β)eβR

+ j

(τ|m|n (β)

mπ|m|n (β)

)

r⊥(β)eαR

]

ejmαe2jksz0 cos βejkR(βR,αR)·r

× sin β dβ dα ,

where βR = π − β, αR = α, (ks, βR, αR) are the spherical coordinates of thereflected wave vector kR, and (ekR,eβR,eαR) are the spherical unit vectorsof kR. For r inside a sphere enclosed in the particle, we expand each planewave in terms of regular vector spherical wave functions

(eβR

eαR

)

ejkR·r = −∞∑

n1=1

n1∑

m1=−n1

4jn1

√2n1 (n1 + 1)

[(jm1π

|m1|n1 (π − β)

τ|m1|n1 (π − β)

)

× M1m1n1

(ksr) +

⎛

⎝jτ |m1|

n1 (π − β)

m1π|m1|n1 (π − β)

⎞

⎠N1m1n1

(ksr)

⎤

⎦ e−jm1α

and obtain the following expressions for the elements of the reflection matrix:

αmnn1 =2jn1−n

√nn1(n + 1)(n1 + 1)

∫ π/2−j∞

0

[m2π|m|

n (β)π|m|n1

(π − β)r‖(β)

+ τ |m|n (β)τ |m|

n1(π − β)r⊥(β)

]e2jksz0 cos β sinβ dβ , (2.205)

βmnn1 =2jn1−n

√nn1(n + 1)(n1 + 1)

∫ π/2−j∞

0

m[π|m|

n (β)τ |m|n1

(π − β)r‖(β)

+ τ |m|n (β)π|m|

n1(π − β)r⊥(β)

]e2jksz0 cos β sin β dβ , (2.206)


γmnn1 =2jn1−n

√nn1(n + 1)(n1 + 1)

∫ π/2−j∞

0

m[τ |m|n (β)π|m|

n1(π − β)r‖(β)

+ π|m|n (β)τ |m|

n1(π − β)r⊥(β)

]e2jksz0 cos β sinβ dβ , (2.207)

δmnn1 =2jn1−n

√nn1(n + 1)(n1 + 1)

∫ π/2−j∞

0

[τ |m|n (β)τ |m|

n1(π − β)r‖(β)

+ m2π|m|n (β)π|m|

n1(π − β)r⊥(β)

]e2jksz0 cos β sin β dβ , (2.208)

An approximate expression for the reflection matrix can be derived if weassume that the interacting radiation strikes the surface at normal incidence.Assuming r(0) = r⊥(β) = −r‖(β), changing the variable from β to βR = π−β,and using the relations

π|m|n (π − βR) = (−1)n−|m|π|m|

n (βR) ,

τ |m|n (π − βR) = (−1)n−|m|+1τ |m|

n (βR) ,

yields the following simplified integral representations for the reflected vectorspherical wave functions:(

M3,Rmn(ksr)

N3,Rmn(ksr)

)

= − (−1)n−|m|r(0)2πjn+1

1√

2n(n + 1)

×∫ 2π

0

∫ π

π/2+j∞

[(−mπ

|m|n (βR)

τ|m|n (βR)

)

eβR+ j

(−τ

|m|n (βR)

mπ|m|n (βR)

)

eαR

]

×ejmαRe−2jksz0 cos βRejkR(βR,αR)·r sin βR dβR dαR .

To compute M3,Rmn and N3,R

mn we introduce the image coordinate systemO′x′y′z′ by shifting the original coordinate system a distance 2z0 along thepositive z-axis. The geometry of the image coordinate system is shown inFig. 2.16. Taking into account that kR · r′ = kR · r − 2ksz0 cos βR, wherer′ = (x′, y′, z′), we identify in the resulting equation the integral representa-tions for the radiating vector spherical wave functions in the half-space z < 0:

(M3,R

mn(ksr)N3,R

mn(ksr)

)

= (−1)n−|m|r(0)

(−M3

mn(ksr′)

N3mn(ksr

′)

)

.

In this case the interacting field is the image of the scattered field and theexpansion (2.198) can be derived by using the addition theorem for vectorspherical wave functions. The elements of the reflection matrix are the trans-lation coefficients, and as a result, the amount of computer time required tosolve the scattering problem is significantly reduced. In this regard it should


O

x

z

Σ

z0

x

z

O

z0r

r

P

Fig. 2.16. Image coordinate system

be mentioned that the formalism using the approximate expression for thereflection matrix has been employed by Videen [240–242].

In most practical situations we are interested in the analysis of the scat-tered field in the far-field region and below the plane surface, i.e., for θ > π/2.In this region we have two contributions to the scattered field: the directelectric far-field pattern Es∞(θ, ϕ),

Es∞(θ, ϕ) =1ks

∞∑

n=1

n∑

m=−n

(−j)n+1 [fmnmmn(θ, ϕ) + jgmnnmn(θ, ϕ)] (2.209)

and the interacting electric far-field pattern ERs∞(θ, ϕ),

ERs∞(θ, ϕ) =

1ks

∞∑

n=1

n∑

m=−n

(−j)n+1 [fmnmR

mn(θ, ϕ) + jgmnnRmn(θ, ϕ)

],

(2.210)

where mmn and nmn are the vector spherical harmonics, and mRmn and nR

mn

are the reflected vector spherical harmonics,

mRmn(θ, ϕ) =

1√

2n(n + 1)e−2jksz0 cos θ

×[jmπ|m|

n (π − θ)r‖(π − θ)eθ − τ |m|n (π − θ)r⊥(π − θ)eϕ

]ejmϕ ,

nRmn(θ, ϕ) =

1√

2n(n + 1)e−2jksz0 cos θ

×[τ |m|n (π − θ)r‖(π − θ)eθ + jmπ|m|

n (π − θ) r⊥(π − θ)eϕ

]ejmϕ .


Thus, the solution of the scattering problem in the framework of the separationof variables method involves the following steps:

1. Calculation of the T matrix relating the expansion coefficients of the fieldsstriking the particle to the scattered field coefficients.

2. Calculation of the reflection matrix A characterizing the reflection of vec-tor spherical wave functions by the surface.

3. Computation of an approximate solution by solving the matrix equation(2.203).

4. Computation of the far-field pattern by using (2.209) and (2.210).

In practice, we must compute the integrals in (2.205)–(2.208), which are ofthe form

I =∫ π/2−j∞

0

f (cos β) e2jq cos β sin βdβ .

Changing variables from β to x = −2jq(cos β − 1), we have

I =e2jq

2jq

∫ ∞

0

f

(1 − x

2jq

)e−xdx

and integrals of this type can be computed efficiently by using the Laguerrepolynomials [15].

Scanning near-field optical microscopy [202,203] requires a rigorous analy-sis of the evanescent scattering by small particles near the surface of a dielec-tric prism [31, 142, 199, 252]. Scattering of evanescent waves can be analyzedby extending our formalism to the case of an incident plane wave propagatingin the substrate (Fig. 2.17).

For the incident vector plane wave given by (2.193), the transmitted (orthe refracted) vector plane wave is

EeT(r) =(ET

e0,βeβT + ETe0,αeαT

)ejkeT ·r ,

where

ETe0,β = t‖(β0)ejksz0(cos β−mrs cos β0)Ee0,β ,

ETe0,α = t⊥(β0)ejksz0(cos β−mrs cos β0)Ee0,α ,

β0 is the incident angle and (ekT,eβT,eαT) are the spherical unit vectors ofthe transmitted wave vector keT. The Fresnel transmission coefficients aregiven by

t‖(β0) =2mrs cos β0

cos β0 + mrs cos β,

t⊥(β0) =2mrs cos β0

mrs cos β0 + cos β,


O x

z

k e

β 0

z 0

Σ

mrs

mr

k eTβ

Fig. 2.17. Geometry of an axisymmetric particle situated near a plane surface. Theexternal excitation is a vector plane wave propagating in the substrate

while the angle of refraction is computed by using Snell’s law:

sin β = mrs sin β0 ,

cos β = ±√

1 − sin2 β .

Evanescent waves appear for real mrs and incident angles β0 > β0c, whereβ0c = arcsin(1/mrs). In this case, sin β > 1 and cosβ is purely imaginary. Fornegative values of z, we have

exp (jkeT · r) = exp (−jksz cos β + jksx sin β) = exp (jks |z| cos β + jksx sin β) ,

and we choose the sign of the square root such that Imcos β > 0. Thischoice guarantees that the amplitude of the refracted wave propagating in thenegative direction of the z-axis decreases with increasing the distance |z|. Theexpansion coefficients of the transmitted wave are

aTmn1

= − 4jn1

√2n1 (n1 + 1)

[jmπ|m|

n1(π − β)ET

e0,β + τ |m|n1

(π − β)ETe0,α

],

bTmn1

= − 4jn1+1

√2n1 (n1 + 1)

[τ |m|n1

(π − β)ETe0,β − jmπ|m|

n1(π − β)ET

e0,α

],

and we see that our previous analysis remains unchanged if we replace thetotal expansion coefficients amn1 and bmn1 , by the expansion coefficients ofthe transmitted wave aT

mn1and bT

mn1.

2.10.2 Particle on or near an Arbitrary Surface

In the precedent analysis, we considered scattering by a particle situatednear a plane surface. The scattering problem of a particle situated in the


neighborhood of an arbitrary infinite surface can be solved by using theT -operator formalism. Specifically, we are interested to compute the vectorspherical wave expansions of the reflected fields ER

e , M3,Rmn and N3,R

mn in thecase of an infinite surface. For this purpose, we consider the scattering prob-lem of an infinite surface illuminated by an arbitrary incident field and followthe analysis of Kristensson [124].

We briefly recall the definitions and the basic properties of scalar andvector plane waves [14]. The scalar plane wave is defined by

χ(r,K±) = exp(jK± · r) = exp[j (Kxx + Kyy ± Kzz)] ,

where

K± = Kxex + Kyey ± Kzez .

Using the notation kT =√

K2x + K2

y , Kz can be expressed as Kz =√

k2 − k2T,

where the square root is always chosen to have a positive imaginary part. Forreal k, Kz is given by Kz =

√k2 − k2

T, if kT ≤ k, and by Kz =j√

k2T − k2 if

kT > k. The case kT ≤ k corresponds to harmonic propagating waves, whilethe case kT > k corresponds to evanescent waves. The vector plane wavesM(r,K±) and N(r,K±) are defined in terms of scalar plane wave as

M (r,K±) =1kT

∇× [ezχ (r,K±)] = jKT

kTχ (r,K±) ,

N (r,K±) =1k∇× M (r,K±) = −

(K±k

× KT

kT

)χ (r,K±) ,

where KT is the transverse component of the wave vector K± and is given by

KT = K± × ez = Kyex − Kxey .

The orthogonality relations∫

Σ

M (r,K±) · M(−r,K ′

±)dxdy = 4π2e±j(Kz−K′

z)zδ(Kx − K ′

x,Ky − K ′y

)

∫

Σ

N (r,K±) · N(−r,K ′

±)

dxdy = 4π2 K± · K ′±

kk′ e±j(Kz−K′z)z

×δ(Kx − K ′

x,Ky − K ′y

),

∫

Σ

M (r,K±) · N(−r,K ′

±)

dxdy =∫

Σ

N (r,K±) · M(−r,K ′

±)dxdy=0,

hold on a plane z = const, where

M (−r,K±) =1kT

∇× [ezχ (−r,K±)] = −jKT

kTχ (−r,K±) ,

N (−r,K±) =1k∇× M (−r,K±) = −

(K±k

× KT

kT

)χ (−r,K±) .


O x

z

keβ0

mrs

Sz = z >

z = z<

Di

Ds

Fig. 2.18. Geometry of an infinite surface

Alternative expressions for the vector plane waves are

M (r,K±) = −jeαχ (r,K±) ,

N (r,K±) = −eβχ (r,K±) , (2.211)

and

M (−r,K±) = jeαχ (−r,K±) ,

N (−r,K±) = −eβχ (−r,K±) , (2.212)

where (ek,eβ ,eα) are the spherical unit vectors of the wave vector K±.We consider now the scattering geometry depicted in Fig. 2.18. The inci-

dent field Ee is a vector plane wave or any radiating vector spherical wavefunctions. To solve the scattering problem in the framework of the T -operatormethod we apply the Huygens principle to a surface consisting of a finite partof S and a lower half-sphere. Letting the radius of the sphere tend to infinityand assuming that the integrals over S exist and the integrals over the lowerhalf-sphere vanish (radiation conditions), we obtain

E(r)0

= Ee(r)+

∫

S

[n (r′) × Ei (r′)] ·

[∇′ × G (ks, r

′, r)]

(2.213)

+ [n (r′) × (∇′ × Ei (r′))] · G (ks, r′, r)

dS (r′) ,

r ∈ Ds

r ∈ Di

,

where E(r) = Ee(r) + Es(r) is the total field in the domain Ds andG(ks, r

′, r) is the free space dyadic Green function of wave number ks. Thedyadic Green function can be expanded in terms of vector plane waves,[14,124]


G(ks, r′, r) =

jks

8π2

∫

R2

[M(−r′,Ks

+

)M(r,Ks

+

)

+ N(−r′,Ks

+

)N(r,Ks

+

)] dKsx dKs

y

ksKsz

for z > z′, and

G(ks, r′, r) =

jks

8π2

∫

R2

[M(−r′,Ks

−)

M(r,Ks

−)

+ N(−r′,Ks

−)

N(r,Ks

−)] dKs

x dKsy

ksKsz

for z < z′, where the superscript s indicates that the vector plane wavescorrespond to the domain Ds.

The incident field is a prescribed field, whose sources are assumed to besituated in Ds. This means that in any case, the sources are below the fictitiousplane z = z> and we can represent the external excitation as an integral overup-going vector plane waves

Ee(r) =∫

R2

[A(Ks

+)M(r,Ks+) + B(Ks

+)N(r,Ks+)] dKs

x dKsy

ksKsz

.

For r ∈ Di and z > z>, we use the plane wave expansion of the Green dyadand the orthogonality properties of the vector plane waves to obtain

A(Ks

+

)= − jk2

s

8π2

∫

S

[n(r′) × Ei(r′)] · N

(−r′,Ks

+

)(2.214)

+1ks

[n (r′) × (∇′ × Ei (r′))] · M(−r′,Ks

+

)

dS (r′) ,

B(Ks

+

)= − jk2

s

8π2

∫

S

[n(r′) × Ei(r′)] · M

(−r′,Ks

+

)(2.215)

+1ks

[n (r′) × (∇′ × Ei (r′))] · N(−r′,Ks

+

)

dS (r′) .

Taking into account that the tangential vector plane waves n×M(·,K i+) and

n×N(·,K i+), form a complete system of vector function on S, we represent

the surface fields n × Ei and n × (∇× Ei) as integrals over up-going vectorplane waves, i.e.,

n (r′) × Ei (r′) =∫

R2

[C(K i

+

)c(K i

+

)n (r′) × M

(r′,K i

+

)(2.216)

+ D(K i

+

)d(K i

+

)n (r′) × N

(r′,K i

+

)]dK i

x dK iy


and

n (r′) × [∇′ × Ei (r′)] = ki

∫

R2

[C(K i

+

)c(K i

+

)n (r′) × N

(r′,K i

+

)

+ D(K i

+

)d(K i

+

)n (r′) × M

(r′,K i

+

)]dK i

x dK iy ,

(2.217)

respectively. The plane wave transmission coefficients in (2.216) and (2.217)are given by

c(K i

+

)=

2Ksz

Ksz + K i

z

e−j(Kiz−Ks

z)z′,

d(Ki

+

)=

2mrsKsz

m2rsK

sz + K i

z

e−j(Kiz−Ks

z)z′, (2.218)

where mrs = ki/ks is the relative refractive index of the domain Di with respectto the ambient medium, and the wave vectors Ks

+ and K i+ (in (2.218)) are

related to each other by Snell’s law

kT = kTs = kTi =√

(Ksx)2 + (Ks

y)2 =√

(K ix)2 + (K i

y)2 .

Inserting (2.216) and (2.217) into (2.214) and (2.215) yields[

A(Ks

+

)

B(Ks

+

)

]

=∫

R2Q31(Ks

+,K i+

)[

C(K i

+

)

D(K i

+

)

]

dK ix dK i

y , (2.219)

where

Q31(Ks

+,K i+

)=

[(Q31(Ks

+,K i+

))11 (Q31(Ks

+,K i+

))12

(Q31(Ks

+,K i+

))21 (Q31(Ks

+,K i+

))22

]

and

(Q31(Ks

+,K i+

))11= − jk2

s

8π2

∫

S

c(K i+)[

n (r′) × M(r′,K i

+

)]· N(−r′,Ks

+)

+ mrs

[n (r′) × N

(r′,K i

+

)]· M(−r′,Ks

+)

dS (r′) ,

(Q31(Ks

+,K i+

))12= − jk2

s

8π2

∫

S

d(K i+)[

n (r′) × N(r′,K i

+

)]· N(−r′,Ks

+)

+ mrs

[n (r′) × M

(r′,K i

+

)]· M(−r′,Ks

+)

dS (r′) ,

(Q31(Ks

+,K i+

))21= − jk2

s

8π2

∫

S

c(K i+)[

n (r′)×M(r′,K i

+

)]· M(−r′,Ks

+)

+ mrs

[n (r′) × N

(r′,K i

+

)]· N(−r′,Ks

+)

dS (r′) ,


(Q31(Ks

+,K i+

))22= − jk2

s

8π2

∫

S

d(K i+)[

n (r′) × N(r′,K i

+

)]· M(−r′,Ks

+)

+ mrs

[n (r′) × M

(r′,K i

+

)]· N(−r′,Ks

+)

dS (r′) .

We note that (Q31)αβ , α, β = 1, 2, are not functions in the usual sense andrelations involving Q31 should be understood in a distributional sense. For aplane surface we have

(Q31(Ks

+,K i+

))11plane

=(Q31(Ks

+,K i+

))22plane

= ksKszδ(K i

x − Ksx,K i

y − Ksy

),

(Q31(Ks

+,K i+

))11plane

=(Q31(Ks

+,Ki+

))22plane

= 0

and the system of integral equations (2.219) simplifies to a system of algebraicequations

A(Ks

+

)= ksK

szC(K i

+

),

B(Ks

+

)= ksK

szD(K i

+

).

For solving the general case of an arbitrary surface, one has to invert a systemof two-dimensional integral transforms and this is a formidable analytic andnumerical problem. We can formally assume that the inverse transform existsand represent the solution as[

C(K i

+

)

D(K i

+

)

]

=∫

R2

[Q31(Ks

+,K i+

)]−1

[A(Ks

+

)

B(Ks

+

)

]

dKsx dKs

y . (2.220)

To compute the scattered field we consider (2.213) for r ∈ Ds, expand thefree space dyadic Green function in terms of vector plane waves by assumingz < z<, and obtain

Es(r) =∫

R2

[F (Ks

−)M(r,Ks−) + G(Ks

−)N(r,Ks−)] dKs

x dKsy

ksKsz

, (2.221)

where the amplitudes F (Ks−) and G(Ks

−) are given by

F(Ks

−)

=jk2

s

8π2

∫

S

[n(r′) × Ei(r′)] · N

(−r′,Ks

−)

+1ks

[n (r′) × (∇′ × Ei (r′))] · M(−r′,Ks

−)

dS (r′) ,

G(Ks

−)

=jk2

s

8π2

∫

S

[n(r′) × Ei(r′)] · M

(−r′,Ks

−)

+1ks

[n (r′) × (∇′ × Ei (r′))] · N(−r′,Ks

−)

dS (r′) .


Inserting the integral representations for n × Ei and n × (∇ × Ei) into theabove equations yields a relation between the amplitudes of the field in thedomain Ds, F (Ks

−) and G(Ks−), and the amplitudes of the field in the domain

Di, C(K i+) and D(K i

+):[

F(Ks

−)

G(Ks

−)

]

=∫

R2Q11(Ks

−,K i+

)[

C(K i

+

)

D(K i

+

)

]

dK ix dK i

y , (2.222)

where

Q11(Ks

−,K i+

)=

[(Q11(Ks

−,K i+

))11 (Q11(Ks

−,K i+

))12

(Q11(Ks

−,K i+

))21 (Q11(Ks

−,K i+

))22

]

and

(Q11(Ks

−,K i+

))11=

jk2s

8π2

∫

S

c(K i+)[

n (r′) × M(r′,K i

+

)]· N(−r′,Ks

−)

+ mrs

[n (r′) × N

(r′,K i

+

)]· M(−r′,Ks

−)

dS (r′) ,

(Q11(Ks

−,K i+

))12=

jk2s

8π2

∫

S

d(K i+)[

n (r′) × N(r′,K i

+

)]· N(−r′,Ks

−)

+ mrs

[n (r′) × M

(r′,K i

+

)]· M(−r′,Ks

−)

dS (r′) ,

(Q11(Ks

−,K i+

))21=

jk2s

8π2

∫

S

c(K i+)[

n (r′) × M(r′,K i

+

)]· M(−r′,Ks

−)

+ mrs

[n (r′) × N

(r′,K i

+

)]· N(−r′,Ks

−)

dS (r′) ,

(Q11(Ks

−,K i+

))22=

jk2s

8π2

∫

S

d(K i+)[

n (r′) × N(r′,K i

+

)]· M(−r′,Ks

−)

+ mrs

[n (r′) × M

(r′,K i

+

)]· N(−r′,Ks

−)

dS (r′) .

Combining (2.220) and (2.222) we are led to[

F(Ks

−)

G(Ks

−)

]

=∫

R2

∫

R2Q11(Ks

−,K i+

) [Q31(K

s

+,K i+

)]−1

×

⎡

⎣A(K

s

+

)

B(K

s

+

)

⎤

⎦dKsx dKs

y dK ix dK i

y

=∫

R2T(Ks

−, Ks

+

)⎡

⎣A(K

s

+

)

B(K

s

+

)

⎤

⎦dKsx dKs

y ,

where T (Ks−, K

s

+) is the T -operator of the infinite surface and


T(Ks

−, Ks

+

)=∫

R2Q11(Ks

−,K i+

) [Q31(K

s

+,K i+

)]−1

dK ix dK i

y .

(2.223)For a plane surface, the above relations simplify considerably and we have

(Q11(Ks

−,K i+

))11plane

= ksKsza(Ks

−)δ(K i

x − Ksx,K i

y − Ksy

),

(Q11(Ks

−,K i+

))22plane

= ksKszb(Ks

−)δ(K i

x − Ksx,K i

y − Ksy

),

and(Q11(Ks

−,K i+

))11plane

=(Q11(Ks

−,K i+

))22plane

= 0 ,

where a(Ks−) and b(Ks

−) are the reflection coefficients for a plane interface,

a(Ks

−)

=Ks

z − K iz

Ksz + Ki

z

e2jKszz′

,

b(Ks

−)

=m2

rsKsz − K i

z

m2rsK

sz + K i

z

e2jKszz′

. (2.224)

Thus

F(Ks

−)

= a(Ks

−)A(Ks

+

),

G(Ks

−)

= b(Ks

−)B(Ks

+

),

and(T(Ks

−, Ks

+

))11

plane= a(Ks

−)δ(Ks

x − Ksx, Ks

y − Ksy

),

(T(Ks

−, Ks

+

))22

plane= b(Ks

−)δ(Ks

x − Ksx, Ks

y − Ksy

),

where the wave vectors Ks

+, Ks− and K i

+ are related to each other by Snell’slaw. Actually, if K

s

+ is an arbitrary incident wave vector, then Ks− and K i

+

represents the reflected and the transmitted wave vectors, respectively. SettingKs

z = ks cos β0 and K iz = ki cos β, we see that the coefficients a(Ks

−) andb(Ks

−) given by (2.224) are the Fresnel reflection coefficients for perpendicularand parallel polarizations, respectively.

The final step of our analysis is the expansion of the scattered field givenby (2.221) in terms of regular vector spherical wave functions. Transformingthe integration over the rectangular components of the wave vector into anintegration over polar angles and using (2.211), gives

Es(r) = −∫ 2π

0

∫ π

π/2+j∞[G (β, α) eβ + jF (β, α) eα] ejKs

−·r sin β dβ dα ,


where (ks, β, α) are the spherical coordinates of the wave vector Ks−. Further,

expanding the vector plane waves eβ exp (jKs− · r) and eα exp (jKs

− · r) interms of vector spherical wave functions yields

Es(r) =∞∑

n=1

n∑

m=−n

fRmnM1

mn (ksr) + gRmnN1

mn (ksr) , (2.225)

where

fRmn =

4jn+1

√2n (n + 1)

∫ 2π

0

∫ π

π/2+j∞

[G (β, α) mπ|m|

n (β) + F (β, α) τ |m|n (β)

]

×e−jmα sin βdβ dα ,

gRmn =

4jn+1

√2n (n + 1)

∫ 2π

0

∫ π

π/2+j∞

[G (β, α) τ |m|

n (β) + F (β, α) mπ|m|n (β)

]

×e−jmα sin βdβ dα .

Thus, the derivation of the vector spherical wave expansion of the electro-magnetic field scattered by an arbitrary infinite surface involves the followingsteps:

1. Representation of the incident field as an integral over vector plane waves.2. Calculation of the spectral amplitudes of the scattered field by using the

T -operator equation (2.223).3. Computation of the vector spherical wave expansion of the scattered field

by using (2.225).

Returning to the scattering problem of a particle near an arbitrary infinitesurface, we see that our analysis is complete if, according to Step 1, we areable to represent the plane electromagnetic wave Ee and the radiating vectorspherical wave functions M3

mn and N3mn as integrals over vector plane waves.

For Ee, we have

Ee(r) =∫

R2

[Ae(Ks

+)M(r,Ks+) + Be(Ks

+)N(r,Ks+)] dKs

x dKsy

ksKsz

with

Ae(Ks+) = jEe0,αksK

sz δ(Ks

x − Kex,Ks

y − Key) ,

Be(Ks+) = −Ee0,βksK

sz δ(Ks

x − Kex,Ks

y − Key) ,

and Ke+ = ke, while for M3

mn and N3mn, we obtain

(M3

mn (ksr)N3

mn (ksr)

)

=∫

R2

[(As(Ks

+)Bs(Ks

+)

)

M(r,Ks+) +

(Bs(Ks

+)As(Ks

+)

)

N(r,Ks+)

]

×dKs

x dKsy

ksKsz


with

As(Ks+) =

12πjn+1

1√

2n(n + 1)τ |m|n (β)ejmα ,

Bs(Ks+) =

12πjn+1

1√

2n(n + 1)mπ|m|

n (β)ejmα .

The T -operator solution is a formal solution since we assumed the invertibilityof the integral equation (2.219). The usefulness of the formalism in numericalapplications depends upon the possibility of discretizing (2.219) in a suitableway. This can be done for a class of simple infinite surfaces as for example planesurfaces or surfaces with periodic roughness and small amplitude roughness.It should be noted that for plane surfaces we obtain exactly the results derivedin our earlier analysis.

3

Simulation Results

In this chapter we present computer simulation results for the scattering prob-lems considered in Chap. 2. For the numerical analysis, we use our own soft-ware package and some existing electromagnetic scattering programs. After aconcise description of the T -matrix code, we present the theoretical bases ofthe electromagnetics programs which we used to verify the accuracy of the newimplementation. We then present simulation results for homogeneous, axisym-metric and nonaxisymmetric particles, inhomogeneous, layered and compositeparticles, and clusters of particles. The last sections deal with scattering by aparticle on or near a plane surface and the computation of the effective wavenumber of a half-space with randomly distributed particles.

3.1 T -matrix Program

A Fortran computer program has been written to solve various scatteringproblems in the framework of the null-field method. This section gives a shortdescription of the code, while more details concerning the significance of theinput and output parameters are given in the documentation on CD-ROM.

The main program TMATRIX.f90 calls a T -matrix routine for solving aspecific scattering problem. These routines computes the T -matrix of:

– Homogeneous, dielectric (isotropic, chiral) and perfectly conducting, ax-isymmetric particles (TAXSYM.f90),

– Homogeneous, dielectric (isotropic, uniaxial anisotropic, chiral) and per-fectly conducting, nonaxisymmetric particles (TNONAXSYM.f90)

– Axisymmetric, composite particles (TCOMP.f90)– Axisymmetric, layered particles (TLAY.f90)– An inhomogeneous, dielectric, axisymmetric particle with an arbitrarily

shaped inclusion (TINHOM.f90)– An inhomogeneous, dielectric sphere with a spherical inclusion (TIN-

HOM2SPH.f90)

184 3 Simulation Results

– An inhomogeneous, dielectric sphere with an arbitrarily shaped inclusion(TINHOMSPH.f90)

– An inhomogeneous, dielectric sphere with multiple spherical inclusions(TINHOMSPHREC.f90)

– Clusters of arbitrarily shaped particles (TMULT.f90)– Two homogeneous, dielectric spheres (TMULT2SPH.f90)– Clusters of homogeneous, dielectric spheres (TMULTSPH.f90 and TMULT-

SPHREC.f90)– Concentrically layered sphere (TSPHERE.f90)– A homogeneous, dielectric or perfectly conducting, axisymmetric particle

on or near a plane surface (TPARTSUB.f90)

Three other routine are invoked by the main program

– SCT.f90 computes the scattering characteristics of a particle using thepreviously calculated T matrix

– SCTAVRGSPH.f90 computes the scattering characteristics of an ensembleof polydisperse, homogeneous spherical particles

– EFMED.f90 computes the effective wave number of a half-space with ran-domly distributed particles

Essentially, the code performs a convergence test and computes the T -matrixand the scattering characteristics of particles with uniform orientation distri-bution functions.

An important part of the T -matrix calculation is the convergence proce-dure over the maximum expansion order Nrank, maximum azimuthal orderMrank and the number of integration points Nint. In fact, Nrank, Mrank andNint are input parameters and their optimal values must be found by the user.This is accomplished by repeated convergence tests based on the analysis ofthe differential scattering cross-section as discussed in Sect. 2.1.

The scattering characteristics depend on the type of the orientation distrib-ution function. By convention, the uniform distribution function is called com-plete if the Euler angles αp, βp and γp are uniformly distributed in the intervals(0, 360), (0, 180) and (0, 360), respectively. The normalization constant is4π for axisymmetric particles and 8π2 for nonaxisymmetric particles. The uni-form distribution function is called incomplete if the Euler angles αp, βp andγp are uniformly distributed in the intervals (αp min, αp max), (βp min, βp max),and (γp min, γp max), respectively. For axisymmetric particles, the orientationalaverage is performed over αp and βp, and the normalization constant is

(αp max − αp min) (cos βp min − cos βp max) ,

while for nonaxisymmetric particles, the orientational average is performedover αp, βp and γp, and the normalization constant is

(αp max − αp min) (cos βp min − cos βp max) (γp max − γp min) .

The scattering characteristics computed by the code are summarized later.

3.1 T -matrix Program 185

3.1.1 Complete Uniform Distribution Function

For the complete uniform distribution function, the external excitation is avector plane wave propagating along the Z-axis of the global coordinate sys-tem and the scattering plane is the XZ-plane. The code computes the follow-ing orientation-averaged quantities:

– The scattering matrix 〈F 〉 at a set of Nθ,RND scattering angles– The extinction matrix 〈K〉– The scattering and extinction cross-sections 〈Cscat〉 and 〈Cext〉 and– The asymmetry parameter 〈cos Θ〉.

The scattering angles, at which the scattering matrix is evaluated, are uni-formly spaced in the interval (θmin,RND, θmax,RND). The elements of the scat-tering matrix are expressed in terms of the ten average quantities

⟨|Sθβ |2

⟩,⟨|Sθα|2

⟩,⟨|Sϕβ |2

⟩,⟨|Sϕα|2

⟩,⟨SθβS∗

ϕα

⟩,

⟨SθαS∗

ϕβ

⟩, 〈SθβS∗

θα〉 ,⟨SθβS∗

ϕβ

⟩,⟨SθαS∗

ϕα

⟩,⟨SϕβS∗

ϕα

⟩

for macroscopically isotropic media, and the six average quantities⟨|Sθβ |2

⟩,⟨|Sθα|2

⟩,⟨|Sϕβ |2

⟩,⟨|Sϕα|2

⟩,⟨SθβS∗

ϕα

⟩,⟨SθαS∗

ϕβ

⟩

for macroscopically isotropic and mirror-symmetric media. 〈SpqS∗p1q1

〉 arecomputed at Nθ,GS scattering angles, which are uniformly spaced in the in-terval (0, 180). The scattering matrix is calculated at the same sample an-gles and polynomial interpolation is used to evaluate the scattering matrixat any polar angle θ in the range (θmin,RND, θmax RND). The average quanti-ties 〈SpqS

∗p1q1

〉 can be computed by using a numerical procedure or the an-alytical orientation-averaging approach described in Sect. 1.5. The numericalorientation-averaging procedure chooses the angles αp, βp and γp to samplethe intervals (0, 360), (0, 180) and (0, 360), respectively. The prescrip-tion for choosing the angles is to

– Uniformly sample in αp

– Uniformly sample in cos βp or nonuniformly sample in βp

– Uniformly sample in γp

The integration over αp and γp are performed with Simpson’s rule, and thenumber of integration points Nα and Nγ must be odd numbers. The integra-tion over βp can also be performed with Simpson’s rule, and in this specificcase, the algorithm samples uniformly in cosβp, and the number of integra-tion points Nβ is an odd number. Alternatively, Gauss–Legendre quadraturemethod can be used for averaging over βp, and Nβ can be any integer number.For the analytical orientation-averaging approach, the maximum expansionand azimuthal orders Nrank and Mrank (specifying the dimensions of the T


matrix) can be reduced. In this case, a convergence test over the extinctionand scattering cross-sections gives the effective values N eff

rank and M effrank.

The orientation-averaged extinction matrix 〈K〉 is computed by using(1.125) and (1.126), and note that for macroscopically isotropic and mirror-symmetric media the off-diagonal elements are zero and the diagonal elementsare equal to the orientation-averaged extinction cross-section per particle.

For macroscopically isotropic and mirror-symmetric media, the orientation-averaged scattering and extinction cross-sections 〈Cscat〉 = 〈Cscat〉I and〈Cext〉 = 〈Cext〉I are calculated by using (1.124) and (1.122), respectively,while for macroscopically isotropic media, the code additionally computes〈Cscat〉V accordingly to (1.133), and 〈Cext〉V as 〈Cext〉V = 〈K14〉.

The asymmetry parameter 〈cos Θ〉 is determined by angular integrationover the scattering angle θ. The number of integration points is Nθ,GS andSimpson’s rule is used for calculation. For macroscopically isotropic andmirror-symmetric media, the asymmetry parameter 〈cos Θ〉 = 〈cos Θ〉I is cal-culated by using (1.134), while for macroscopically isotropic media, the codesupplementarily computes 〈cos Θ〉V accordingly to (1.135).

The physical correctness of the computed results is tested by using theinequalities (1.138) given by Hovenier and van der Mee [103]. The messagethat the test is not satisfied means that the computed results may be wrong.

The code also computes the average differential scattering cross-sections〈σdp〉 and 〈σds〉 for a specific incident polarization state and at a set of Nθ,GS

scattering angles (cf. (1.136) and (1.137)). These scattering angles are uni-formly spaced in the interval (0, 180) and coincide with the sample anglesat which the average quantities 〈SpqS

∗p1q1

〉 are computed. The polarizationstate of the incident vector plane wave is specified by the complex amplitudesEe0,β and Ee0,α, and the differential scattering cross-sections are calculatedfor the complex polarization unit vector

epol =1

√|Ee0,β |2 + |Ee0,α|2

(Ee0,βeβ + Ee0,αeα) . (3.1)

Note that Nθ,GS is the number of scattering angles at which 〈SpqS∗p1q1

〉, 〈σdp〉and 〈σds〉 are computed, and also gives the number of integration points forcalculating 〈Cscat〉V, 〈cos Θ〉I and 〈cos Θ〉V. Because the integrals are com-puted with Simpson’s rule, Nθ,GS must be an odd number.

3.1.2 Incomplete Uniform Distribution Function

For the incomplete uniform distribution function, we use a global coordinatesystem to specify both the direction of propagation and the states of polar-ization of the incident and scattered waves, and the particle orientation. Aspecial orientation with a constant orientation angle can be specified by set-ting Nδ = 1 and δmin = δmax, where δ stands for αp, βp and γp. For example,

3.1 T -matrix Program 187

uniform particle orientation distributions around the Z-axis can be specifiedby setting Nβ = 1 and βp min = βp max.

The scattering characteristics are averaged over the particle orientation byusing a numerical procedure. The prescriptions for choosing the sample anglesand the significance of the parameters are as in Sect. 3.1.1. For each particleorientation we compute the following quantities:

– The phase matrix Z at Nϕ scattering planes– The extinction matrix K for a plane wave incidence– The scattering and extinction cross-sections Cscat and Cext for incident

parallel and perpendicular linear polarizations– The mean direction of propagation of the scattered field g for incident

parallel and perpendicular linear polarizations

The azimuthal angles describing the positions of the scattering planes at whichthe phase matrix is computed are ϕ(1), ϕ(2), . . . , ϕ(Nϕ). In each scatteringplane i, i = 1, 2, . . . , Nϕ, the number of zenith angles is Nθ(i), while the zenithangle varies between θmin(i) and θmax(i). For each particle orientation, thephase and extinction matrices Z and K are computed by using (1.77) and(1.79), respectively, while the scattering and extinction cross-sections Cscat

and Cext are calculated accordingly to (1.101) and (1.100), respectively.The mean direction of propagation of the scattered field g is evaluated

by angular integration over the scattering angles θ and ϕ (cf. (1.87)), andthe numbers of integration points are Nθ,asym and Nϕ,asym. The integrationover ϕ is performed with Simpson’s rule, while the integration over θ can beperformed with Simpson’s rule or Gauss–Legendre quadrature method. Wenote that the optical cross-sections and the mean direction of propagation ofthe scattered field are computed for linearly polarized incident waves (vectorplane waves and Gaussian beams) by choosing αpol = 0 and αpol = 90 (cf.(1.18)).

The code also calculates the average differential scattering cross-sectionsσdp and σds in the azimuthal plane ϕGS and at a set of Nθ,GS scatteringangles. The average differential scattering cross-sections can be computed forscattering angles ranging from 0 to 180 in the azimuthal plane ϕGS and from180 to 0 in the azimuthal plane ϕGS +180, or for scattering angles rangingfrom 0 to 180 in the azimuthal plane ϕGS. The calculations are performedfor elliptically polarized vector plane waves (characterized by the complexpolarization unit vector epol as in (3.1)) and linearly polarized Gaussian beams(characterized by the polarization angle αpol).

For spherical particles, the code chooses a single orientation αp min =αp max = 0, βp min = βp max = 0 and γp min = γp max = 0, and setsMrank = Nrank.

The codes perform calculations with double- or extended-precision float-ing point variables. The extended-precision code is slower than the double-precision version by a factor of 5–6, but allow scattering computationsfor substantially larger particles. It should also be mentioned that the


CPU time consumption rapidly increases with increasing particle size andasphericity.

The program is written in a modular form, so that modifications, if re-quired, should be fairly straightforward. In this context, the integration rou-tines, the routines for computing spherical and associated Legendre functionsor the routines for solving linear systems of equations can be replaced by moreefficient routines.

The main “drawback” of the code is that the computer time requirementsmight by higher in comparison to other more optimized T -matrix programs.Our intention was to cover a large class of electromagnetic scattering problemand therefore we sacrifice the speed in favor of the flexibility and code modu-larization. The code shares several modules which are of general use and arenot devoted to a specific application.

3.2 Electromagnetics Programs

The on-line directories created by Wriedt [266] and Flatau [67] provide links toseveral electromagnetics programs. In order to validate our software packagewe consider several computer programs relying on different methods. Thissection briefly reviews these methods without entering into deep details. Moreinformation can be found in the literature cited.

3.2.1 T -matrix Programs

Computer programs using the null-field method to calculate the electromag-netic scattering by homogeneous, axisymmetric particles have been given byBarber and Hill [8]. These programs compute:

– The angular scattering over a designated scattering plane or in all direc-tions for a particle in a fixed orientation

– The angular scattering and the optical cross-sections for an ensemble ofparticles randomly oriented in a 2D plane or in 3D

– The scattering matrix elements for an ensemble of particles randomly ori-ented in 3D

– The normalized scattering cross-sections versus size parameter for a par-ticle in fixed or random orientation

– The internal intensity distribution in the equatorial plane of a spheroidalparticle

We would like to mention that the general program structure and some pro-gramming solutions given by Barber and Hill have been adopted in our com-puter code.

The T -matrix codes developed by Mishchenko and Travis [167] andMishchenko et al. [168, 169] are efficient numerical tools for computing elec-tromagnetic scattering by homogeneous, axisymmetric particles with size

3.2 Electromagnetics Programs 189

significantly larger than a wavelength. The following computer programs areavailable on the World Wide Web at www.giss.nasa.gov/∼crimim:

– A Fortran code for computing the amplitude and phase matrices for ahomogeneous, axisymmetric particle in an arbitrary orientation

– A Fortran code for computing the far-field scattering and absorption char-acteristics of a polydisperse ensemble of randomly oriented, homogeneous,axisymmetric particles

The last code is suitable for practical applications requiring the knowl-edge of size-, shape-, and orientation-averaged quantities such as the op-tical cross-sections and phase and scattering matrix elements. The imple-mentation of the analytical orientation-averaging procedure makes this codethe fastest computer program for randomly oriented, axisymmetric parti-cles. The following computer programs are also provided at the w sitewww.giss.nasa.gov/∼crimim:

– A Lorenz–Mie code for computing the scattering characteristics of an en-semble of polydisperse, homogeneous, spherical particles and

– The Fortran code SCSMTM for computing the T -matrix of a spherecluster and the orientation-averaged scattering matrix and optical cross-sections [153]

3.2.2 MMP Program

MMP is a Fortran code developed by Hafner and Bomholt [94] which usesthe multiple multipole method for solving arbitrary 3D electromagnetic prob-lems. In the multiple multipole method, the electromagnetic fields are ex-pressed as linear combinations of spherical wave fields corresponding tomultipole sources. By locating these sources away from the boundary, themultipole expansions are smooth on the surface and the boundary singu-larities are avoided. The multipoles describing the internal field are locatedoutside the particle, while the multipoles describing the scattered field are po-sitioned inside the particle. Not only spherical multipoles can be used for fieldexpansions; other discrete sources as for instance distributed vector sphericalwave functions, vector Mie potentials or magnetic and electric dipoles canbe employed. Therefore, other names for similar concepts have been given,e.g., method of auxiliary sources [275], discrete sources method [49, 62], fic-titious sources method [140, 158], or Yasuura method [113]. An overview ofthe discrete sources method has been given by Wriedt [267] and a review ofthe latest literature in this field has been published by Fairweather et al. [63].By convention, we categorize the multiple multipole method as a discretesources method with multiple vector spherical wave functions. Below we sum-marize the basic concepts of the discrete sources method for the transmissionboundary-value problem.

Let Ψ3µ,Φ3

µ be a system of radiating solutions to the Maxwell equationsin Ds with the properties ∇×Ψ3

µ = ksΦ3µ and ∇×Φ3

µ = ksΨ3µ. Analogously,


let Ψ1µ,Φ1

µ be a system of regular solutions to the Maxwell equations inDi satisfying ∇ × Ψ1

µ = kiΦ1µ and ∇ × Φ1

µ = kiΨ1µ. For the transmission

boundary-value problem, the system of vector functions⎧⎨

⎩

⎛

⎝n × Ψ3,1

µ

−j√

εs,iµs,i

n × Φ3,1µ

⎞

⎠ ,

⎛

⎝n × Φ3,1

µ

−j√

εs,iµs,i

n × Ψ3,1µ

⎞

⎠ , µ = 1, 2, . . .

⎫⎬

⎭ (3.2)

is assumed to be complete and linearly independent in L2tan(S), where L2

tan(S)is the product space L2

tan(S) = L2tan(S) × L2

tan(S) endowed with the scalarproduct

⟨(x1

x2

),

(y1

y2

)⟩

2,S

=⟨x1,y1

⟩2,S

+⟨x1,y1

⟩2,S

and L2tan(S) is the space of square integrable tangential fields on the surface

S. Approximate solutions to the scattered and internal fields are sought aslinear combinations of basis functions (3.2),

(EN

s,i(r)HN

s,i(r)

)

=N∑

µ=1

as,iNµ

⎛

⎝Ψ3,1

µ (r)

−j√

εs,iµs,i

Φ3,1µ (r)

⎞

⎠+ bs,iNµ

⎛

⎝Φ3,1

µ (r)

−j√

εs,iµs,i

Ψ3,1µ (r)

⎞

⎠ ,

r ∈ Ds,i .

The approximate electric and magnetic fields ENs and HN

s are expressed interms of discrete sources fields with singularities distributed in Di, and there-fore EN

s and HNs are analytic in Ds. Analogously, the approximate electric

and magnetic fields ENi and HN

i are expressed in terms of discrete sourcesfields distributed in Ds, and EN

i and HNi are analytic in Di. Then, using the

Stratton–Chu representation theorem and the continuity conditions on theparticle surface, we obtain the estimate

∥∥Es − EN

s

∥∥∞,Gs

+∥∥Hs − HN

s

∥∥∞,Gs

+∥∥Ei − EN

i

∥∥∞,Gi

+∥∥H i − HN

i

∥∥∞,Gi

≤ C(∥∥n × EN

s + n × Ee − n × ENi

∥∥

2,S

+∥∥n × HN

s + n × He − n × HNi

∥∥

2,S

)(3.3)

in any closed sets Gs ⊂ Ds and Gi ⊂ Di, where ‖.‖∞,G stands for the supre-mum norm, ‖a‖∞,G = maxr∈G |a(r)|. The estimate (3.3) reflects the ba-sic principle of the discrete sources method: an approximate solution to thetransmission boundary-value problem minimizes the residual electric and mag-netic fields on the particle surface. Due to the completeness property of the


system of vector functions (3.2), the expansion coefficients as,iNµ and bs,iN

µ ,µ = 1, 2, . . . , N , can be determined by solving the minimization problem,

[as,iN

µ

bs,iNµ

]

= arg min∥∥n × EN

s + n × Ee − n × ENi

∥∥2

2,S

+∥∥n × HN

s + n × He − n × HNi

∥∥2

2,S

.

This minimization problem leads to a system of normal equations for theexpansion coefficients. Experience has shown that this technique often givesinaccurate and numerically unstable results with highly oscillating error dis-tributions along the boundary surface. Accurate and stable results can beobtained by using the point matching method with an overdetermined sys-tem of equations, that is, the boundary conditions are fulfilled at a set ofsurface points, while the number of surface points exceeds the number of un-knowns (more than twice). The resulting system of equations is solved in theleast-squares sense by using, for example, the QR-factorization.

The main problem in the multiple multipole method is the choice of thenumber, position and the order of multipoles, and the distribution of match-ing points. Some useful criteria can be summarized as follows. Firstly, thematching points on the boundary should be set with enough density, and theintervals between adjacent pairs of matching points must be far less than thewavelength, for suppressing errors of expansion on the boundary. Secondly,the poles should be neither too near nor too far from the boundary, since theformer requirement may avoid alternative errors corresponding to nonphys-ical rough solution on that interval; and a well-conditioned matrix will beformulated with the latter requirement. In view of the quasi-local behavior ofa multipole expansion, only a restricted area around its origin is influencedby this pole, so that a reasonable scheme is to let the area of influence of eachpole fit a portion of the boundary, but the whole boundary must be coveredby the areas of influence of all the poles. Finally, the highest order of a pole islimited by the density of matching points covered by its area of influence. Itshould be noted that the MMP code contains routines for automatically opti-mizing the number, location and order of multipoles. More detailed overviewson methodology, code and simulation technique can be found in the works ofLudwig [147,148], Hafner [91] and Bomholt [18]. The attractiveness and sim-plicity of the physical idea of the MMP and the public availability of the codeby Hafner [92,93] have resulted in widespread applications of this technique.

In addition to the MMP code, the DSM (discrete sources method) codedeveloped by Eremin and Orlov [61] will be used for computer simulations.This DSM code is devoted to the analysis of homogeneous, axisymmetric par-ticles using distributed vector spherical wave functions. For highly elongatedparticles, the sources are distributed along the axis of symmetry of the par-ticle, while for highly flattened particles, the sources are distributed in thecomplex plane (see Appendix B).


3.2.3 DDSCAT Program

DDSCAT is a freely available software package which applies the discrete di-pole approximation (DDA) to calculate scattering and absorption of electro-magnetic waves by particles with arbitrary geometries and complex refractiveindices [55]. The discrete dipole approximation model the particle as an arrayof polarizable points and DDSCAT allows accurate calculations for particleswith size parameter ksa < 15, provided the refractive index mr is not largecompared to unity, |mr − 1| < 1. The discrete dipole approximation (some-times referred to as the coupled dipole method (CDM)) was apparently firstproposed by Purcell and Pennypacker [197]. The theory was reviewed anddeveloped further by Draine [53], Draine and Goodman [56] and Draine andFlatau [54]. An improvement of this method was given by Piller and Mar-tin [194] applying concepts from sampling theory, while Varadan et al. [237],Lakhtakia [130] and Piller [192] extended the discrete dipole approximation toanisotropic, bi-anisotropic and high-permittivity materials, respectively. Sincethe discrete dipole approximation can be derived from the volume integralequation we follow the analysis of Lakhtakia and Mulholland [131] and reviewthe basic concepts of the volume integral equation method and the discretedipole approximation.

The volume integral equation method relies on the integral representationfor the electric field

E (r) = Ee (r) + k2s

∫

Di

G (ks, r, r′) ·[m2

r (r′) − 1]E (r′) dV (r′) ,

r ∈ Ds ∪ Di .

The domain Di is discretized into simply connected elements (cells) Di,m,m = 1, 2, . . . , Ncells, and Di = ∪Ncells

m=1 Di,m. Each element Di,m is modeled asbeing homogeneous such that mr(r) = mr,m for r ∈ Di,m, even though thisassumption can lead to an artificial material discontinuity across the interfaceof two adjacent elements. As a consequence of the discretization process, theelement equation reads as

E (r) = Ee (r) + k2s

(m2

r,m − 1) ∫

Di,m

G (ks, r, r′) · E (r′) dV (r′)

+k2s

Ncells∑

n=1,n =m

(m2

r,n − 1)∫

Di,n

G (ks, r, r′) · E (r′) dV (r′) , r ∈ Di,m .

Due to the singularity at r′ = r it is necessary to approximate the integral

Em (r) =∫

Di,m

G (ks, r, r′) · E (r′) dV (r′) , r ∈ Di,m

analytically in a small volume around the singularity. With Dr being a sphereof radius r around the singularity, we transform the integral as follows:


Em (r) =∫

Di,m−Dr

G (ks, r, r′) · E (r′) dV (r′)

+∫

Dr

G (ks, r, r′) · [E (r′) − E (r)] dV (r′)

+∫

Dr

[G (ks, r, r′) − Gs (ks, r, r′)

]· E (r) dV (r′)

+∫

Dr

Gs (ks, r, r′) · E (r) dV (r′) , (3.4)

where Gs is an auxiliary dyad, and

Gs (ks, r, r′) =1k2s

∇′ ⊗∇′(

14π |r′−r|

).

The first term in (3.4) is a nonsingular integral that can be computed numer-ically. When the integration volume Di,m is spherical and sufficiently small,we can set Di,m = Dr and the first term is zero. For the third term, we usethe coordinate-free representations

G (ks, r, r′) =

[

1 +j

ksR− 1

(ksR)2

]

I

−[

1 +3j

ksR− 3

(ksR)2

]

eR ⊗ eR

g (R)

and

Gs (ks, r, r′) =1

(ksR)2(I − 3eR ⊗ eR

) 14πR

,

where R = r − r′, R = |R|, eR = R/R and g(R) = exp(jksR)/(4πR), andobtain

∫

Dr

[G (ks, r, r′) − Gs (ks, r, r′)

]· E (r) dV (r′) = ME (r)

with

M =2

3k2s

[(1 − jksr) ejksr − 1

].

For the last term, the dyadic identity∫

D

∇′ ⊗ a dV =∫

S

n ⊗ a dS

with a = (1/k2s )∇′(1/(4π|r′−r|)), yields


∫

Dr

Gs (ks, r, r′) · E (r) dV (r′) = − 1k2s

∫

Sr

n (r′)

(r′−r

4π |r′−r|3

)

· E (r) dS (r′)

= − 1k2s

LE (r) ,

where

L =13

.

The usual tradition is to neglect the second term assuming that the field isconstant or claiming that as the integration volume becomes small the inte-grand vanishes. As pointed out by Draine and Goodman [56], vanishing of thesecond term is of the same order as with the third term and cannot be omit-ted. Using the second-order Taylor expansion for each Cartesian componentEk of E,

Ek (r′) = Ek (r) + (r′−r) · ∇Ek (r) +12

(r′−r) · [(r′−r) · ∇]∇Ek (r) ,

and the vector Helmholtz equation

E + m2r,mk2

s E = 0 , in Di,m

we obtain [185]∫

Dr

G (ks, r, r′) · [E (r′) − E (r)] dV (r′) = M1E (r)

with

M1 =m2

r,m

k2s

[1 − jksr −

715

(ksr)2 +

215

j (ksr)3

]ejksr − 1

.

The final result is

Em (r) =(

M + M1 −1k2s

L

)E (r) +

∫

Di,m−Dr

G (ks, r, r′) · E (r′) dV (r′)

and the element equation read as[1 − k2

s

(m2

r,m − 1)(

M + M1 −1k2s

L

)]E (r)

−k2s

(m2

r,m − 1) ∫

Di,m−Dr

G (ks, r, r′) · E (r′) dV (r′)

= Ee (r) + k2s

Ncells∑

n=1,n =m

(m2

r,n − 1) ∫

Di,n

G (ks, r, r′) · E (r′) dV (r′) ,

r ∈ Di,m . (3.5)


Within each volume element, the total field can be approximated by

E (r) ≈Nbasis∑

k=1

EkmW k (r) , r ∈ Di,m ,

where W k are referred to as the shape or basis functions. Frequently, node-based shape functions are used as basis functions, but edge-based shape func-tions, using an expansion of a vector field in terms of values associated withelement faces, are better suited for simulating electromagnetic fields at cornersand edges. A detailed mathematical analysis of edge-based shape functionsfor various two- and three-dimensional elements has been given by Graglia etal. [86]. Linear as well as higher order curvilinear elements have been discussedby Mur and deHoop [176], Lee et al. [139], Wang and Ida [249], Crowley etal. [42] and Sertel and Volakis [209]. In general, the edge-based shape func-tions can be categorized as either interpolatory or hierarchical. An elementis referred to as hierarchical if the vector basis functions forming the nth or-der element are a subset of the vector basis functions forming the (n + 1)thorder element. This property allows to use lowest order elements in regionswhere the field is expected to vary slowly and higher order elements in re-gions where rapid field variation is anticipated. For a tetrahedral element, ahierarchical edge-based element up to and including second order has beenintroduced by Webb and Forghani [259]. Additional to node- and edge-basedshape functions, low order vector spherical wave functions have been em-ployed by Peltoniemi [185]. In our analysis we assume that the electromag-netic field is constant over each element and consider the simplest type ofbasis functions, the constant pulse functions: W k(r) = ek for r ∈ Di,m, andW k(r) = 0 for r /∈ Di,m, where k = x, y, z and (ex,ey,ez) are the Cartesianunit vectors. This approach is called collocation method, since a certain valueof the electromagnetic field is assigned to the centroid rm of each element,i.e., E(r) ≈ E(rm) for r ∈ Di,m. Substituting this expansion into (3.5) andsetting Di,m = Dr (which gives

∫Di,m−Dr

G · E dV = 0), yields the system oflinear equations

Ee (rm) =Ncells∑

n=1

Amn · E (rn) , m = 1, 2, . . . , Ncells , (3.6)

where the dyadic kernel is given by

Amn =[1 − k2

s

(m2

r,m − 1)(

M + M1 −1k2s

L

)]Iδmn

−k2s

(m2

r,n − 1)(1 − δmn) VnG (ks, rm, rn) ,

and Vn is the volume of the n element.In computing the self-term Em we extracted from the original integrand a

singular function which approximates the integrand in the neighborhood of thesingularity. However, there are other efficient schemes to accurately compute


singular integrals [3]. For example, in the framework of the volume integralequation method, Sertel and Volakis [209] transferred one of the derivative ofthe scalar Green function to the testing function using the divergence theo-rem. The resulting integrals are evaluated by the annihilation method, whichconverts first-order singular integrals into nonsingular integrals by using anappropriate parametric transformation (change of variables).

Whereas the volume integral equation method deals with the total field,the discrete dipole approximation exploits the concept of the exciting field,each volume element being explicitly modeled as an electric dipole moment.In this regard, we consider the field that excites the volume element Di,m

Eexc (rm) = Ee (rm)+k2s

Ncells∑

n=1,n =m

(m2

r,n − 1)VnG (ks, rm, rn) ·E (rn) (3.7)

and rewrite (3.5) as[1 − k2

s

(m2

r,m − 1)(

M + M1 −1k2s

L

)]E (rm) = Eexc (rm) . (3.8)

It is clear from (3.7) that the exciting field consists of the incident field and thefield contributions of all other volume elements Di,n with n = m. The heartof the discrete dipole approximation is (3.7), while (3.8) is used to express thetotal field in terms of the exciting field. In dyadic notations, (3.8) read as

Amm · E (rm) = Eexc (rm) ,

and the desired relation is

E (rm) = A−1

mm · Eexc (rm) . (3.9)

Inserting (3.9) into (3.7), we obtain the expression of the field that excitesthe volume element Di,m in terms of the exciting fields of all other volumeelements Di,n

Eexc (rm) = Ee (rm) + k2s

Ncells∑

n=1,n =m

(m2

r,n − 1)Vn

×[G (ks, rm, rn) · A−1

nn

]· Eexc (rn) .

Defining the polarizability dyad by

tn = εs

(m2

r,n − 1)VnA

−1

nn

and taking into account that ks = k0√

εs, yields

Ee (rm) =Ncells∑

n=1

Bmn · Eexc (rn) , m = 1, 2, . . . , Ncells , (3.10)


where the dyadic kernel is

Bmn = Iδmn − k20 (1 − δmn) G (ks, rm, rn) · tn .

In terms of the dipole moment pm

pm = εs

(m2

r,m − 1)VmE (rm) ,

(3.9) can be written as

pm = tm · Eexc (rm)

and the effect of the exciting field can be interpreted as inducing a dipolemoment pm in each volume element Di,m. This is the origin of the term“discrete dipole approximation.”

Parenthetically we note that the discrete dipole approximation can beused to derive the size-dependent Maxwell–Garnett formula. For a collectionof identical spheres of volume V and relative refractive index mr, the polariz-ability dyad is

tm = αI ,

and the dipole moment satisfies the equation

pm = αEexc (rm) ,

where the polarizability scalar α is given by

α = εs

(m2

r − 1)V

[1 − k2

s

(m2

r − 1)(

M + M1 −1k2s

L

)]−1

. (3.11)

The size-dependent Maxwell–Garnett formula can now be obtained by com-bining (3.11) and (2.168) [129].

The maximum linear cross-sectional extent 2rm of Di,m is such that

rmmr,m

λ< 0.1 ,

that is, the dimension rm is no more than a tenth of the wavelength in thevolume element Di,m. This condition ensures that the spatial variations ofthe electromagnetic field inside each volume element are small enough, sothat each volume element can be thought of as a dipole scatterer. Conse-quently, for large scale simulations, the systems of equations (3.6) and (3.10)are excessively large and iterative solvers as the conjugate or bi-conjugategradient method are employed. The coupling submatrices in (3.6) and (3.10)depend on the absolute distance among elements, and the overall matricesare block Toplitz. Thus, (3.6) and (3.10) can be cast in circulant form, andthe fast Fourier transform can be employed to carry out the matrix–vectorproducts [68,80].


3.2.4 CST Microwave Studio Program

CST Microwave Studio is a commercial software package which can be used foranalyzing the electromagnetic scattering by particles with complex geometriesand optical properties. Essentially, the electromagnetic simulator is based onthe finite integration technique (FIT) developed by Weiland [260]. The finiteintegration technique is simple in concept and provides a discrete reformu-lation of Maxwell’s equations in their integral from. Following the analysisof Clemens and Weiland [38] we recall the basic properties of the discreterepresentation of Maxwell’s equation by the finite integration technique.

The first discretization step of the finite integration method consists inthe restriction of the electromagnetic field problem, which represent an openboundary problem, to a simply connected and bounded space domain Ω con-taining the region of interest. The next step consists in the decomposition ofthe computational domain into a finite number of volume elements (cells).This decomposition yields the volume element complex G, which serves ascomputational grid. Assuming that Ω is brick-shaped, we have the volumeelement complex

G = Dijk/Dijk = [xi, xi+1] × [yj , yj+1] × [zk, zk+1]

i = 1, 2, . . . , Nx − 1, j = 1, 2, . . . , Ny − 1, k = 1, 2, . . . , Nz − 1 ,

and note that each edge and facet of the volume elements are associated witha direction.

Next, we consider the integral form of the Maxwell equations, i.e., forFaraday’s induction law and Maxwell–Ampere law we integrate the differentialequations on an open surface and apply the Stokes theorem, while for Gauss’electric and magnetic field laws we integrate the differential equations on abounded domain and use the Gauss theorem.

The Faraday law in integral form∮

∂S

E · ds = −∫

S

∂B

∂t· dS

with S being an arbitrary open surface contained in Ω, can be rewritten for thefacet Sz,ijk of the volume element Dijk as the ordinary differential equation

ex,ijk + ey,i+1jk − ex,ij+1k − ey,ijk = − ddt

bz,ijk ,

where

ex,ijk =∫ (xi+1,yj ,zk)

(xi,yj ,zk)

E · ds

is the electric voltage along the edge Lx,ijk of the facet Sz,ijk, and


X

Y

Z ex,ijk

ex,ij+1k

ey,ijk

ey,i+1jk

bz,ijk

bx,ijk bx,i+1jk

by,ijk

by,ij+1

bz,ijk-1

Fig. 3.1. Electric voltages and magnetic fluxes on a volume element

bz,ijk =∫

Sz,ijk

B · dS

represents the magnetic flux through the facet Sz,ijk. Note that the orientationof the element edges influences the signs in the differential equation (Fig. 3.1).The Gauss magnetic field law in integral form

∫

S

B · dS = 0

with S being an arbitrary closed surface contained in Ω, yields for the bound-ary surface of the volume element Dijk,

−bx,ijk + bx,i+1jk − by,ijk + by,ij+1k − bz,ijk + bz,ijk+1 = 0 .

The discretization of the remaining two Maxwell equations requires the in-troduction of a second volume element complex G, which is the dual of theprimary volume element complex G. For the Cartesian tensor product grid G,the dual grid G is defined by taking the foci of the volume elements of G asgridpoints for the volume elements of G. With this definition there is a one-to-one relation between the element edges of G cutting through the elementsurfaces of G and conversely. For the dual volume elements, the Maxwell–Ampere law

∮

∂S

H · ds =∫

S

(∂D

∂t+ J

)· dS ,

and the Gauss electric field law∫

S

D · dS =∫

D

ρdV , S = ∂D


are discretized in an analogous manner. The differential equations are for-mulated in terms of the magnetic grid voltages hα,ijk along the dual edgesLα,ijk, the dielectric fluxes dα,ijk and the conductive currents jα,ijk throughthe dual facets Sα,ijk, α = x, y, z, and the electric charge qijk in the dualvolume element Dijk (qijk =

∫Dijk

ρdV ).

Collecting the equations of all element surfaces of the complex pair G, Gand introducing the integral voltage-vectors e and h, the flux state-vectors band d, the conductive current-vector j, and the electric charge-vector q, wederive a set of discrete matrix equations, the so-called Maxwell grid equations

Ce = − ddt

b , Ch =ddt

d + j

and

Sb = 0 , Sd = q .

The transformation into frequency domain for the Maxwell grid equationswith e(t) = Reee−jωt, yields

Ce = jωb , Ch = −jωd + j .

The discrete curl-matrices C and C, and the discrete divergence matrices Sand S are defined on the grids G and G, respectively, and depend only onthe grid topology. The integral voltage- and flux state-variables allocated onthe two different volume element complexes are related to each other by thediscrete material matrix relations

d = M εe + p , j = Mκe , h = Mνb − m ,

where M ε is the permittivity matrix, Mκ is the matrix of conductivities,Mν is the matrix of reluctivities, and p and m are the electric and magneticpolarization vectors, respectively. The discrete grid topology matrices havethe same effect as the vector operators curl and div. For instance, the discreteanalog of the equation ∇ · ∇× = 0 (div curl = 0) is SC = 0 and SC =0. Transposition of these equations together with the relation between thediscrete curl-matrices C = C

T, yields the discrete equations CST = 0 and

CST

= 0, both corresponding to the equation ∇ × ∇ = 0 (curl grad = 0).Basic algebraic properties of the Maxwell grid equations also allow to proveconservation properties with respect to energy and charges.

The finite integration technique suffers somewhat from a deficiency in be-ing able to model very complicated cavities including curved boundaries withhigh precision, but the usage of the perfect boundary approximation eliminatethis deficiency [123].

3.3 Homogeneous, Axisymmetric and Nonaxisymmetric Particles 201

3.3 Homogeneous, Axisymmetricand Nonaxisymmetric Particles

In the following analysis we show results computed with the TAXSYM andTNONAXSYM routines. The flow diagram of these routines is shown inFig. 3.2. The input data file provides the variables specifying the optical prop-erties, geometry, type of discrete sources and error tolerances for the conver-gence tests over Nrank, Mrank and Nint. The model parameters control theinterpolation and integration processes, and the solution of the linear systemof equations. The current version of the TAXSYM code is directly applica-ble to spheroids, cylinders and rounded oblate cylinders. Nonaxisymmetricgeometries currently supported include ellipsoids, quadratic prisms and reg-ular polyhedral prisms. The user should be able to write new routines togenerate particles with other shapes. The codes can also read particle geom-etry information from files instead of automatically generating one of thegeometries listed above.

3.3.1 Axisymmetric Particles

The scattering characteristics of axisymmetric particles can be computed withlocalized or distributed sources. Specifically, localized sources are used to an-alyze the electromagnetic scattering by particles which are not too extremein terms of size parameter and aspect ratio, while distributed sources areemployed to compute the T matrix of large particles with extreme geome-tries. For highly elongated particles, the sources are distributed along the axisof symmetry, while for flattened particles, the sources are distributed in thecomplex plane.

T-Matrix Routine

T Matrix Scattering Characteristics

Convergence Test Results

Input Data Model Control Parameters Particle Geometry

Fig. 3.2. Flow diagram of TAXSYM and TNONAXSYM routines


x

z

Y

Z

X

β

α

O

a

b

Fig. 3.3. Geometry of a prolate spheroid

Localized Sources

In our first example, we consider prolate spheroids in random and fixedorientation, and compare the results obtained with the TAXSYM code tothe solutions computed with the codes developed by Mishchenko [167–169].The orientation of the axisymmetric particle with respect to the global coor-dinate system is specified by the Euler angles of rotation αp and βp, and theincident field is a linearly polarized plane wave propagating along the Z-axis(Fig. 3.3). The rotational semi-axis (along the axis of symmetry) is ksa = 10,the horizontal semi-axis is ksb = 5, and the relative refractive index of thespheroid is mr = 1.5. The maximum expansion order and the number of in-tegration points are Nrank = 17 and Nint = 100, respectively. In Figs. 3.4 and3.5 we plot some elements of the scattering matrix for a randomly orientedspheroid. The agreement between the curves is acceptable. For a fixed orien-tation of the prolate spheroid, we list in Tables 3.1 and 3.2 the phase matrixelements Z11 and Z44, and Z21 and Z42, respectively. The Euler angles ofrotation are αp = βp = 45, and the matrix elements are computed in theazimuthal planes ϕ = 45 and ϕ = 225 at three zenith angles: 30, 90,and 150. The relative error is around 10% for the lowest matrix element andremains below 1% for other elements.

In the next example, we show results computed for a perfectly conductingspheroid of size parameter ksa = 10, aspect ratio a/b = 2, and Euler angles ofrotation αp = βp = 45. The perfectly conducting spheroid is simulated fromthe dielectric spheroid by using a very high value of the relative refractiveindex (mr = 1.e+30), and the version of the code devoted to the analysis ofperfectly conducting particles is taken as reference. For this application, the


10−4

10−3

10−2

10−1

100

101

102

Sca

tterin

g M

atrix

Ele

men

ts

0 30 60 90 120 150 180Scattering Angle (deg)

F11 - TAXSYMF11 - code of MishchenkoF22 - TAXSYMF22 - code of Mishchenko

Fig. 3.4. Scattering matrix elements F11 and F22 of a dielectric prolate spheroid

−0.3

−0.2

−0.1

0.0

0.1

0.2

Sca

tterin

g M

atrix

Ele

men

ts


F21 - TAXSYMF21 - code of MishchenkoF43 - TAXSYMF42 - code of Mishchenko

Fig. 3.5. Scattering matrix elements F21 and F43 of a dielectric prolate spheroid

maximum expansion order is Nrank = 18, while the number of integrationpoints is Nint = 200. The normalized differential scattering cross-sectionspresented in Fig. 3.6 are similar for both methods.

To verify the accuracy of the code for isotropic, chiral particles, we considera spherical particle of size parameter ksa = 10. The refractive index of theparticle is mr = 1.5 and the chirality parameter is βki = 0.1, where ki = mrks.Calculations are performed for Nrank = 18 and Nint = 200. Figure 3.7 com-pares the normalized differential scattering cross-sections computed with the


Table 3.1. Phase matrix elements 11 and 44 computed with (a) the TAXSYMroutine and (b) the code of Mishchenko

ϕ θ Z11 (a) Z11 (b) Z44 (a) Z44 (b)

45 30 4.154e−01 4.152e−01 3.962e−01 3.961e−01

45 90 9.136e−01 9.142e−01 5.453e−01 5.459e−01

45 150 5.491e−02 5.489e−02 2.414e−03 2.420e−03

225 30 8.442e−01 8.439e−01 8.406e−01 8.402e−01

225 90 5.331e−02 5.329e−02 1.493e−04 1.360e−04

225 150 3.807e−02 3.805e−02 −1.400e−02 −1.402e−02

Table 3.2. Phase matrix elements 21 and 42 computed with (a) the TAXSYMroutine and (b) the code of Mishchenko

ϕ θ Z21 (a) Z21 (b) Z42 (a) Z42 (b)

45 30 2.134e−02 2.134e−02 1.229e−01 1.229e−01

45 90 3.016e−01 3.015e−01 6.681e−01 6.685e−01

45 150 −2.655e−03 −2.699e−03 −5.479e−02 −5.477e−02

225 30 6.677e−02 6.689e−02 −4.190e−02 −4.161e−02

225 90 −2.910e−02 −2.908e−02 −4.466e−02 −4.466e−02

225 150 −3.042e−02 −3.039e−02 1.810e−02 1.810e−02

10−3

10−2

10−1

100

DS

CS


TAXSYM - perfect conductor - parallelTAXSYM - perfect conductor - perpendicularTAXSYM - dielectric - parallelTAXSYM - dielectric - perpendicular

Fig. 3.6. Normalized differential scattering cross-sections of a perfectly conductingprolate spheroid


10−6

10−5

10−4

10−3

10−2

10−1

100

DS

CS


TAXSYM - polarization angle = 0˚TAXSYM - polarization angle = 90˚code of Bohren - polarization angle = 0˚code of Bohren - polarization angle = 90˚

Fig. 3.7. Normalized differential scattering cross-sections of an isotropic chiralsphere

TAXSYM routine and the program developed by Bohren [16]. This programwas coded by Ute Comberg and is available from www.T-matrix.de. The scat-tering characteristics are computed in the azimuthal plane ϕ = 0 and for twopolarizations of the incident wave.

In the next example, we present computer simulations for Gaussian beamscattering. The particle is a prolate spheroid with semi-axes a = 2.0 µm andb = 1.0 µm, and relative refractive index mr = 1.5. The wavelength of theincident radiation is λ = 0.628 µm, and the orientation of the spheroid isspecified by the Euler angles αp = 0 and βp = 90. The maximum expansionand azimuthal orders are Nrank = 30 and Mrank = 12, respectively, while thenumber of integration points is Nint = 300. The variation of the differentialscattering cross-sections with the axial position z0 of the particle is shownin Fig. 3.8. The waist radius of the Gaussian beam is w0 = 10 µm, and theoff-axis coordinates are x0 = y0 = 0. The beam parameter is s = 1/(ksw0) =0.01, and for this value of s, the localized beam model gives accurate results.The scattering cross-section decreases from Cscat = 16.788 to Cscat = 14.520when z0 increases from 0 to 200 µm. Figure 3.9 illustrates the influence of theoff-axis coordinate x0 on the angular scattering for w0 = 10 µm and y0 =z0 = 0. In this case, the scattering cross-section decreases more rapidly andattains the value Cscat = 2.326 for x0 = 10 µm. The results of the differentialscattering cross-sections for different values of the beam waist radius are shownin Fig. 3.10. For w0 = 5 µm, the scattering cross-section is Cscat = 15.654,while for w0 = 2.5 µm, Cscat = 12.280.


10−4

10−3

10−2

10−1

100

101

102

DS

CS

- p

aral

lel


kz = 0kz = 2000

Fig. 3.8. Variation of the normalized differential scattering cross-sections with theaxial position of a prolate spheroid illuminated by a Gaussian beam

10−5

10−4

10−3

10−2

10−1

100

101

102

DS

CS

- p

aral

lel


kx = 0kx = 50kx = 100

Fig. 3.9. Variation of the normalized differential scattering cross-sections with theoff-axis coordinate of a prolate spheroid illuminated by a Gaussian beam

Distributed Sources

While localized sources are used for not extremely aspherical particles, dis-tributed sources are suitable for analyzing particles with extreme geometries,i.e., particles whose shape differs significantly from a sphere. Extremely de-formed particles are encountered in various scientific disciplines as for instance


10−4

10−3

10−2

10−1

100

101

102

DS

CS

- p

aral

lel


kw = 100kw = 50kw = 25

Fig. 3.10. Variation of the normalized differential scattering cross-sections with thebeam waist radius

Fig. 3.11. Particles with extreme geometries: prolate spheroid, fibre, oblate cylinderand Cassini particle

astrophysics, atmospheric science and optical particle sizing. For example,light scattering by finite fibres is needed in optical characterization of asbestosor other mineral fibres, while flat particles are encountered as aluminium ormica flakes in coatings.

Figure 3.11 summarizes the particle shapes considered in our exemplarysimulation results. The spheroid is a relatively simple shape but convergenceproblems occur for large size parameters and high aspect ratios. The finite fibreis a more extreme shape because the flank is even and without convexities.This shape is modeled by a rounded prolate cylinder, i.e., by a cylinder withtwo half-spheres at the ends. In polar coordinates, a rounded prolate cylinderas shown in Fig. 3.12 is described by

r (θ) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(a − b) cos θ +√

b2 − (a − b)2 sin2 θ ,

b/ sin θ ,

−(a − b) cos θ +√

b2 − (a − b)2 sin2 θ ,

θ < θ0,

θ0 ≤ θ ≤ π − θ0,

π − θ0 < θ < π,


x

z

O

2b

2a

z1

z2

zN

Fig. 3.12. Geometry of a prolate cylinder and the distribution of the discrete sourceson the axis of symmetry

where θ0 = arctan[b/(a − b)]. The rounded oblate cylinder is constructedquite similar, i.e., the flank is rounded and top and bottom are flat. An oblatecylinder as shown in Fig. 3.13 is described in polar coordinates by

r (θ) =

⎧⎪⎪⎨

⎪⎪⎩

a/ cos θ ,

(b − a) sin θ +√

a2 − (b − a)2 cos2 θ ,

−a/ cos θ ,

θ < θ0

θ0 ≤ θ ≤ π − θ0

π − θ0 < θ < π

,

where θ0 = arctan[(b − a)/a]. Cassini particles are a real challenge for lightscattering simulations because the generatrix contains concavities on its topand bottom. The Cassini ovals can be described in polar coordinates by theequation

r (θ) =√

a2 − 2a2 sin2 θ +√

b4 − 4a4 sin2 θ + 4a4 sin4 θ

and the shape depends on the ratio b/a. If a < b the curve is an oval loop,for a = b the result is a lemniscate, and for a > b the curve consists of twoseparate loops. If a is chosen slightly smaller than b we obtain a concave,bone-like shape, and this concavity becomes deeper as a approaches to b.

For the prolate particles considered in our simulations, the sources are dis-tributed on the axis of symmetry as in Fig. 3.12, while for the oblate particles,the sources are distributed in the complex plane as in Fig. 3.13. The wave-length of the incident radiation is λ = 0.6328 µm, the relative refractive index


x

z

O

2b

2a

ImZ

ReZ

z1z2zN

O

Fig. 3.13. Geometry of an oblate cylinder and the distribution of the discretesources in the complex plane

Table 3.3. Surface parameters of particles with extreme geometries

Particle type a (µm) b (µm) αp () βp ()

Prolate spheroid 8.5 0.85 0 0

Fibre 3 0.06 0 0

Oblate cylinder 0.03 3 0 0

Cassini particle 1.1 1.125 0 45

Table 3.4. Maximum expansion and azimuthal orders for particles with extremegeometries

Particle type Nrank Nint

Prolate spheroid 100 1000

Fibre 50 3000

Oblate cylinder 36 5000

Cassini particle 28 1000

is mr = 1.5 and the scattering characteristics are computed in the azimuthalplane ϕ = 0. The parameters describing the geometry and orientation ofthe particles are given in Table 3.3, while the parameters controlling the con-vergence process are listed in Table 3.4. Note that the Cassini particle has adiameter of about 3.15 µm and an aspect ratio of about 1/4.


In Figs. 3.14–3.17 we plot the normalized differential scattering cross-sections together with the results computed with the discrete sources methodfor parallel and perpendicular polarizations, and for the case of normal inci-dence. It is apparent that the agreement between the curves is acceptable.

We conclude this section with an extensive validation test for an oblatecylinder of radius ksb = 15, length 2ksa = 7.5 and relative refractive index

106

105

104

103

102

101

100

101

102

DS

CS


TAXSYM - parallelTAXSYM - perpendicularDSM - parallelDSM - perpendicular

Fig. 3.14. Normalized differential scattering cross-sections of a prolate spheroidwith a = 8.5 µm and b = 0.85 µm. The curves are computed with the TAXSYMroutine and the discrete sources method (DSM)

109

108

107

106

105

104

103

102

101

DS

CS



Fig. 3.15. Normalized differential scattering cross-sections of a fibre particle witha = 3.0 µm and b = 0.06 µm. The curves are computed with the TAXSYM routineand the discrete sources method (DSM)


10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

102

103

DS

CS

−180 −120 −60 0 60 120 180Scattering Angle (deg)

TAXSYm - parallelTAXSYM - perpendicularDSM - parallelDSM - perpendicular

Fig. 3.16. Normalized differential scattering cross-sections of an oblate cylinderwith a = 0.03 µm and b = 3.0 µm. The curves are computed with the TAXSYMroutine and the discrete sources method (DSM)

10−5

10−4

10−3

10−2

10−1

100

101

102

DS

CS

−180 −120 −60 0 60 120 180Scattering Angle (deg)


Fig. 3.17. Normalized differential scattering cross-sections of a Cassini particle witha = 1.1 and b = 1.125. The curves are computed with the TAXSYM routine andthe discrete sources method (DSM)

mr = 1.5. The T -matrix calculations are performed with Nrank = 30 sourcesdistributed in the complex plane and by using Nint = 300 integration pointson the generatrix curve. The scattering characteristics plotted in Fig. 3.18 arecalculated for the Euler angles of rotation αp = βp = 0 and for the caseof normal incidence. Also shown are the results computed with the discrete


10−710−610−510−410−310−210−1100101102103104

DSC

S


TAXSYM - parallelTAXSYM - perpendicularDSM - parallelDSM - perpendicularMMP - parallelMMP - perpendicularDDA - parallelDDA - perpendicularCST - parallelCST - perpendicular

Fig. 3.18. Normalized differential scattering cross-sections of an oblate cylinderwith ksb = 15 and 2ksa = 7.5. The curves are computed with the TAXSYM routine,discrete sources method (DSM), multiple multipole method (MMP), discrete dipoleapproximation (DDA) and finite integration technique (CST)

10−5

10−4

10−3

10−2

10−1

100

DS

CS


TNONAXSYM - parallelTNONAXSYM - perpendicularDDA - parallelDDA - perpendicular

Fig. 3.19. Normalized differential scattering cross-sections of a dielectric cube com-puted with the TNONAXSYM routine and the discrete dipole approximation (DDA)

sources method, multiple multipole method, discrete dipole approximationand finite integration technique. No substantial differences between the curvesare noted. Further scattering patterns for finite fibres having a large aspectratio have been published by Pulbere and Wriedt [196].

3.3.2 Nonaxisymmetric Particles

In Fig. 3.19, we compare the results obtained with the TNONAXSYM routineto the discrete dipole approximation solutions. Calculations are performed for


y

z

x

O2l

a

Fig. 3.20. Geometry of a hexagonal prism

a dielectric cube of length 2ksa = 10 and relative refractive index mr = 1.5.The incident wave propagates along the Z-axis of the global coordinate systemand the Euler orientation angles are αp = βp = γp = 0. Convergence isachieved for Nrank = 14 and Mrank = 12, while the numbers of integrationpoints on each square surface are Nint1 = Nint2 = 24.

As a second example, we consider a hexagonal prism (Fig. 3.20) of length2l = 1.154 µm, hexagon side a = 1.154 µm and relative refractive indexmr = 1.5. The wavelength of the incident radiation is taken to be λ =0.628 µm, and the orientation of the hexagonal prism is specified by the Eulerangles αp = βp = 0 and γp = 90. For this application, the maximumexpansion and azimuthal orders are Nrank = 22 and Mrank = 20, respec-tively. Figure 3.21 compares results obtained with the TNONAXSYM routineand the finite integration technique. The agreement between the curves isacceptable.

Figure 3.22 illustrates the normalized differential scattering cross-sectionsfor a positive uniaxial anisotropic cube of length 2a = 0.3 µm, and relativerefractive indices mrx = mry = 1.5 and mrz = 1.7. The principal coordinatesystem coincides with the particle coordinate system, i.e., αpr = βpr = 0.Calculations are performed at a wavelength of λ = 0.3 µm and for the caseof normal incidence. The parameters controlling the T -matrix calculationsare Nrank = 18 and Mrank = 18, while the numbers of integration points oneach cube face are Nint1 = Nint2 = 50. The behavior of the far-field patternsobtained with the TNONAXSYM routine, discrete dipole approximation andfinite integration technique is quite similar.

Figure 3.23 shows plots of the differential scattering cross-sections versusthe scattering angle for a negative uniaxial anisotropic ellipsoid of semi-axislengths a = 0.3 µm, b = 0.2 µm and c = 0.1 µm, and relative refractive indicesmrx = mry = 1.5 and mrz = 1.3. The orientation of the principal coordinatesystem is given by the Euler angles αpr = βpr = 0 and the wavelength of


10−5

10−4

10−3

10−2

10−1

100

101

DS

CS


TNONAXSYM - parallelTNONAXSYM - perpendicularCST - parallelCST - perpendicular

Fig. 3.21. Normalized differential scattering cross-sections of a dielectric hexagonalprism computed with the TNONAXSYM routine and the finite integration technique(CST)

10−5

10−4

10−3

10−2

10−1

100

DSC

S


TNONAXSYM - parallelTNONAXSYM - perpendicularDDA - parallelDDA - perpendicularCST - parallelCST - perpendicular

Fig. 3.22. Normalized differential scattering cross-sections of a uniaxial anisotropiccube computed with the TNONAXSYM routine, discrete dipole approximation(DDA) and finite integration technique (CST)

the incident radiation is λ = 0.3 µm. The maximum expansion and azimuthalorders are Nrank = 11 and Mrank = 11, respectively, and the numbers ofintegration points in the zenith and azimuthal directions are Nint1 = 100 and


10−4

10−3

10−2

10−1

100

DSC

S


TNONAXSYM - phi = 0˚TNONAXSYM - phi = 90˚CST - phi = 0˚CST - phi = 90˚

Fig. 3.23. Normalized differential scattering cross-sections of a uniaxial anisotropicellipsoid computed with the TNONAXSYM routine and the finite integration tech-nique (CST)

Nint2 = 100, respectively. The curves show that the differential scatteringcross-sections are reasonably well reproduced by the TNONAXSYM routine.

Next we present calculations for a dielectric, a perfectly conducting anda chiral cube of length 2ksa = 10. The refractive index of the dielectriccube is mr = 1.5 and the chirality parameter is βki = 0.1. The scatteringcharacteristics are computed by using the symmetry properties of the tran-sition matrix. For particles with a plane of symmetry perpendicular to theaxis of rotation (mirror symmetric particles with the surface parameteriza-tion r(θ, ϕ) = r(π − θ, ϕ)) the T matrix can be computed by integrating θover the interval [0, π/2], while for particles with azimuthal symmetry (parti-cles with the surface parameterization r(θ, ϕ) = r(θ, ϕ+2π/N), where N ≥ 2)the T matrix can be computed by integrating ϕ over the interval [0, 2π/N ].For T -matrix calculations without mirror and azimuthal symmetries, thenumbers of integration points on each square surface are Nint1 = Nint2 = 30.For calculations using azimuthal symmetry, the numbers of integration pointson the top and bottom quarter-square surfaces are Nint1 = Nint2 = 20, whilefor calculations using mirror symmetry, the numbers of integration points onthe lateral half-square surface are Nint1 = Nint2 = 20. The integration sur-faces are shown in Fig. 3.24, and the differential scattering cross-sections areplotted in Figs. 3.25–3.27. The results obtained using different techniques aregenerally close to each other.


y

z

x

O

2a

Fig. 3.24. Integration surfaces for a cube

10−4

10−3

10−2

10−1

100

101

DSC

S


parallelperpendicularparallel - azimuthal symmetryperpendicular - azimuthal symmetryparallel - azimuthal and mirror symmetryperpendicular - azimuthal and mirror symmetry

Fig. 3.25. Normalized differential scattering cross-sections of a dielectric cube usingthe symmetry properties of the transition matrix

3.3.3 Triangular Surface Patch Model

Some nonaxisymmetric particle shapes such as ellipsoids, quadratic prismsand regular polyhedral prisms are directly included in the Fortran program


10−4

10−3

10−2

10−1

100

101

DSC

S


parallel - azimuthal symmetryperpendicular - azimuthal symmetryparallelperpendicular

Fig. 3.26. Normalized differential scattering cross-sections of a chiral cube usingthe symmetry properties of the transition matrix

10−5

10−4

10−3

10−2

10−1

100

101

DSC

S

0 60 120 180 240 300 360

Scattering Angle (deg)

parallel - azimuthal and mirror symmetryperpendicular - azimuthal and mirror symmetryparallelperpendicular

Fig. 3.27. Normalized differential scattering cross-sections of a perfectly conductingcube using the symmetry properties of the transition matrix

as it is provided on the CD-ROM with the book. To handle arbitrary particlegeometries, the program can also read particle shape data from an input file.

The particle shape data is based on a surface description using a triangularsurface patch model. There are various 3D object file formats suitable for ameshed particle shape. We decided for the Wavefront .obj file format but the


program will only support the polygonal format subset and not the free-formgeometry (also included in the .obj file format). In this case, a triangular sur-face patch model of the particle surface has to be generated by an adequatesoftware. For shapes given by an implicit equation, the HyperFun polygo-nizer [106,183] supporting high-level language functional representations is apossible candidate for surface mesh generation. Function representation is ageneralization of traditional implicit surfaces and constructive solid geometry,which allows the construction of complex shapes such as isosurfaces of real-valued functions [184]. The HyperFun polygonizer generates a VRML outputof a sufficiently regular triangular patch model, which is then converted to the.obj file format by using the 3D Exploration program.

To compute the T -matrix elements by surface integrals we employ a mod-ified midpoint or centroid quadrature. The integral over each surface patchis approximated by multiplying the value of the integrand at the centroid bythe patch area [79]

∫

S

fdS ≈∑

i

f(vi,c)area[vi,1, vi,2, vi,3] , (3.12)

where, vi,1, vi,2, vi,3 are the vertices spanning a triangle and vi,c denotes themass center of the triangle [vi,1, vi,2, vi,3]

vi,c =13

3∑

j=1

vi,j . (3.13)

For a sufficiently large number of surface patches, the use of this centroidintegration is satisfactorily accurate. In convergence checks versus the numberof triangular faces, we found that this centroid quadrature is quite stableand the computational results are not much influenced by the number ofintegration elements.

As an example, we consider a sphere which has been cut at a quarter of itsdiameter on the z-axis as shown in Fig. 3.28. The cut sphere has been meshed

Fig. 3.28. Geometry of a cut sphere with 10,132 faces


10−4

10−3

10−2

10−1

100

101

102

103

104

DSC

S


cut sphere - parallelcut sphere - perpendicularsphere - parallelsphere - perpendicular

Fig. 3.29. Normalized differential scattering cross-sections of a sphere and a cutsphere

Fig. 3.30. Geometry of a cube shaped particle with 31,284 faces

with 10132 faces, the Mie parameter computed from the radius is 10, therefractive index is 1.5 and the direction of the incident plane wave is alongthe z-axis. In Fig. 3.29 the scattering pattern is plotted together with thescattering pattern of a sphere with the same parameters. For better visibilitythe curves are shifted and it is apparent that there are pronounced differencesin the scattering diagrams of a sphere and a cut sphere.

There are different ways to compute scattering by arbitrary particle shapesreconstructed from a number of measured data points. Next we present anexample using compactly supported radial basis functions (CS-RBF) for scat-tered data points processing. For this application we used the software toolkitby Kojekine et al. [121] available from www.karlson.ru. The data points rep-resent a cube shaped particle as depicted in Fig. 3.30 and a number of 378


Fig. 3.31. Geometry of a cube shaped particle with 17,680 faces approximated byCS-RBF

10−4

10−3

10−2

10−1

100

101

102

103

104

DSC

S


cubecube cs-rbf

Fig. 3.32. Normalized differential scattering cross-sections for parallel polarizationof a cube and a reconstructed cube

surface points were used in shape reconstruction. The result of the shape re-construction can be seen in Fig. 3.31, while in Figs. 3.32 and 3.33 the scatteringdiagrams of the original and the reconstructed cubes are plotted. There areonly minor differences in the backscattering region. The particle shapes havebeen discretized to a large number of surface patches by using a divide by fourscheme and the Rational Reducer Professional software for grid reduction, sothat there are finally 17,680 triangular faces with the reconstructed cube.

Scattering results for nonaxisymmetric particles using the superellipsoidas a model particle shape have been given by Wriedt [268]. The programSScaTT (Superellipsoid Scattering Tool) includes a small graphical user in-terface to generate various superellipsoid particle shape modes and is available


10−4

10−3

10−2

10−1

100

101

102

103

104

DS

CS

0 30 60 90 120 150 180


cube

cube cs-rbf

Fig. 3.33. Normalized differential scattering cross-sections for perpendicular polar-ization of a cube and a reconstructed cube

T-Matrix Routine

T Matrix Scattering Characteristics

Convergence Test Results

Input Data Model Control Parameters

T matrix of theinclusion / particles

Fig. 3.34. Flow diagram of the TINHOM routine

on CD-ROM with this book. The effect of particle surface discretization on thecomputed scattering patterns has been analyzed by Hellmers and Wriedt [99].

3.4 Inhomogeneous Particles

This section discusses numerical and practical aspects of T -matrix calcula-tions for inhomogeneous particles. The flow diagram of the TINHOM rou-tine is shown in Fig. 3.34. The program supports scattering computation foraxisymmetric and dielectric host particles with real refractive index. Themain feature of this routine is that the T -matrix of the inclusion is providedas input parameter. The inclusion can be a homogeneous, axisymmetric or


r

x

z

O

x1

z1a1

b1

β1

Fig. 3.35. Geometry of an inhomogeneous sphere with a spheroidal inclusion

nonaxisymmetric particle, a composite or a layered scatterer and an aggre-gate. This choice enables the analysis of particles with complex structure andenhances the flexibility of the program. For computing the T -matrix of theinclusion, the refractive index of the ambient medium must be identical withthe relative refractive index of host particle.

We consider an inhomogeneous sphere of radius ksr = 10 as shown inFig. 3.35. The inhomogeneity is a prolate spheroid of semi-axes ksa1 = 5 andksb1 = 5. The relative refractive indices of the sphere and the spheroid withrespect to the ambient medium are mr = 1.2 and mr1 = 1.5, respectively,and the Euler angles specifying the orientation of the prolate spheroid withrespect to the particle coordinate system are αp1 = βp1 = 45. The differentialscattering cross-sections are calculated for the case of normal incidence, andthe maximum expansion and azimuthal orders for computing the inclusion Tmatrix are Nrank = 10 and Mrank = 5, respectively. In Fig. 3.36, we comparethe T -matrix results to those obtained with the multiple multipole method.The agreement between the scattering curves is acceptable.

In Fig. 3.37, we show an inhomogeneous sphere with a spherical inclusion.The radii of the host sphere and the inclusion are R = 1.0 µm and r = 0.5 µm,respectively, while the relative refractive indices with respect to the ambi-ent medium are mr = 1.33 and mr1 = 1.5, respectively. The inhomogeneity isplaced at the distance z1 = 0.25 µm with respect to the particle coordinate sys-tem and the wavelength of the incident radiation is chosen as λ = 0.6328 µm.


10−6

10−5

10−4

10−3

10−2

10−1

100

DS

CS

0 60 120 180 240 300 360


TINHOM - parallelTINHOM - perpendicularMMP - parallelMMP - perpendicular

Fig. 3.36. Normalized differential scattering cross-sections of an inhomogeneoussphere with a spheroidal inclusion. The results are computed with the TINHOMroutine and the multiple multipole method (MMP)

z1

O1

O x

z

R

r

Fig. 3.37. Geometry of an inhomogeneous sphere with a spherical inclusion


10−5

10−4

10−3

10−2

10−1

100

101

102

DS

CS

0 60 120 180 240 300 360


TINHOM2SPH - parallelTINHOM2SPH - perpendicularcode of Videen - parallelcode of Videen - perpendicular

Fig. 3.38. Normalized differential scattering cross-sections of an inhomogeneoussphere with a spherical inclusion

The results of the differential scattering cross-sections computed with the TIN-HOM2SPH routine and the code developed by Videen et al. [243] are shown inFig. 3.38. The plots demonstrate a complete agreement between the scatteringcurves. We note that the FORTRAN code developed by Videen et al. [243](see also, [180]), uses the Lorenz–Mie theory to represent the electromagneticfields. To exploit the orthogonality of the vector spherical wave functions, thetranslation addition theorem is used to re-expand the field scattered by theinhomogeneity about the origin of the host sphere.

In order to provide a numerical test of the accuracy of the recursive T -matrix algorithm described in Sect. 2.6, we consider an inhomogeneous spherewith multiple spherical inclusions. The radius of the spherical inclusions isksr = 0.3, and the maximum expansion and azimuthal orders for comput-ing the inclusion T -matrix are Nrank = Mrank = 3. The radius of the hostsphere is ksR = ksRcs(1) = 3.5 and 200 inhomogeneities are generated byusing the sequential addition method [47]. The relative refractive indices withrespect to the ambient medium are mr = 1.2 and mr1 = 1.5. According tothe guidelines of the recursive T -matrix algorithm, two auxiliary surfaces ofradii ksRcs(2) = 2.7 and ksRcs(3) = 2.2 are considered in the interior of thehost particle. The values of the maximum expansion and azimuthal ordersfor each auxiliary surface are given in Table 3.5. The matrix inversion is per-formed with the bi-conjugate gradient method instead of the LU-factorizationmethod. The results presented in Fig. 3.39 correspond to the direct methodand the recursive algorithm with one and two auxiliary surfaces. The agree-ment between the scattering curves is satisfactory, and note that the recursivealgorithm with two auxiliary surfaces is faster by a factor of 2 than the directmethod.


Table 3.5. Maximum expansion and azimuthal orders for the auxiliary surfaces

Auxiliary surface Nrank Mrank

1 8 5

2 6 4

3 6 4

10−5

10−4

10−3

10−2

10−1

DS

CS

0 30 60 90 120 150 180


parallel - direct methodparallel - RTA with 1 surfaceparallel - RTA with 2 surfacesperpendicular - direct methodperpendicular - RTA with 1 surfaceperpendicular - RTA with 2 surfaces

Fig. 3.39. Normalized differential scattering cross-sections of an inhomogeneoussphere with multiple spherical inclusions

3.5 Layered Particles

In the following examples we consider electromagnetic scattering by axisym-metric, layered particles. An axisymmetric, layered particle is an inhomoge-neous particle consisting of several consecutively enclosing and rotationallysymmetric surfaces with a common axis of symmetry. The scattering compu-tations are performed with the TLAY routine and the set of routines devotedto the analysis of inhomogeneous particles. Particle geometries currently sup-ported by the TLAY routine include layered spheroids and cylinders. We notethat localized and distributed sources can be used for T -matrix calculations,and in contrast to the TINHOM routine, the relative refractive index of thehost particle can be a complex quantity (absorbing dielectric medium).

Figure 3.40 shows a layered spheroid with ksa = 10, ksb = 8, ksa1 = 8,ksb1 = 3 and ksz1 = 3. The relative refractive indices of the host spheroidand the inhomogeneity with respect to the ambient medium are mr = 1.2and mr1 = 1.5, respectively. Numerical results are presented as plots of thedifferential scattering cross-sections for the inhomogeneous spheroid in fixedor random orientation. The scattering characteristics are computed with lo-calized and distributed sources, whereas the distributed sources are placed


a

b

b1

a1

z1

O1

O

z

Fig. 3.40. Geometry of a layered spheroid

Table 3.6. Parameters of calculation for an oblate layered spheroid

Type of sources Nrank-host particle Nrank-inclusion Nint

Localized 17 12 500

Distributed 14 8 2,000

along the symmetry axis of the prolate spheroids. The maximum expansionorders are Nrank = 20 for the host spheroid and Nrank = 13 for the inclusion.The global number of integration points is Nint = 200 for localized sources,and Nint = 1000 for distributed sources. The TINHOM routine uses as in-put parameter the inclusion T -matrix, which is characterized by Nrank = 13and Mrank = 5. The results presented in Figs. 3.41 and 3.42 show a completeagreement between the different methods.

Scattering by oblate spheroids can be computed with localized and dis-crete sources distributed in the complex plane. Figure 3.43 compares resultsobtained with localized and distributed sources for an oblate layered spheroidwith ksa = 5, ksb = 10, ksa1 = 3 and ksb1 = 5. The relative refractive indicesare mr = 1.2 and mr1 = 1.5, and the parameters of calculation are given inTable 3.6.


10−6

10−5

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10−3

10−2

10−1

100

101

DS

CS

0 60 120 180 240 300 360


TLAY - localized sources - parallelTLAY - localized sources - perpendicularTLAY - distributed sources - parallelTLAY - distributed sources - perpendicularTINHOM - parallelTINHOM - perpendicular

Fig. 3.41. Normalized differential scattering cross-sections of a layered spheroid

10−3

10−2

10−1

100

101

<D

SC

S>


TLAY - localized sources - parallelTLAY - localized sources - perpendicularTLAY - distributed sources - parallelTLAY - distributed sources - perpendicularTINHOM - parallelTINHOM - perpendicular

Fig. 3.42. Averaged differential scattering cross-sections of a layered spheroid

For highly elongated particles, distributed sources are required to achieveconvergence. The results plotted in Fig. 3.44 correspond to a layered cylinderwith ksL = 10, ksr = 2, ksL1 = 5 and ksr1 = 1 (Fig. 3.45). The relativerefractive indices of the host particle and the inclusions are mr = 1.2 andmr1 = 1.4, respectively. The maximum expansion orders are Nrank = 14 for


10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

DS

CS


parallel - distributed sourcesparallel - localized sourcesperpendicular - distributed sourcesperpendicular - localized sources

Fig. 3.43. Normalized differential scattering cross-sections of an oblate layeredspheroid

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

DS

CS


parallelperpendicular

Fig. 3.44. Normalized differential scattering cross-sections of a layered cylinder

the host particle and Nrank = 8 for the inclusion, while the global number ofintegration points is Nint = 1,000.

In the next example, the host particle is a sphere of radius ksR = 10, theinclusion is a prolate spheroid with ksa1 = 8 and ksb1 = 3, and the relativerefractive indices are mr = 1.2 and mr1 = 1.5. Figures 3.46 and 3.47 compareresults obtained with the routines TLAY, TINHOMSPH and TINHOM. Theagreement between the scattering curves is very good.


L

r

x

z

L 1

r1

Fig. 3.45. Geometry of a layered cylinder

10−5

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10−1

100

101

102

DSC

S


TLAY - parallelTLAY - perpendicularTINHOMSPH - parallelTINHOMSPH - perpendicularTINHOM - parallelTINHOM - perpendicular

Fig. 3.46. Normalized differential scattering cross-sections of a layered particleconsisting of a host sphere and a spheroidal inclusion

In Fig. 3.48, we show results for a concentrically layered sphere consisting ofthree layers of radii ksr1 = 10, ksr2 = 7 and ksr3 = 4. The relative refractiveindices with respect to the ambient medium are mr1 = 1.2 + 0.2j, mr2 =1.5 + 0.1j and mr3 = 1.8 + 0.3j. The scattering curves obtained with theTSPHERE and TLAY routines are close to each other.


10−4

10−3

10−2

10−1

100

101

102

<D

SC

S>


TLAY - parallelTLAY - perpendicularTINHOMSPH - parallelTINHOMSPH - perpendicularTINHOM - parallelTINHOM - perpendicular

Fig. 3.47. Averaged differential scattering cross-sections of a layered particle con-sisting of a host sphere and a spheroidal inclusion

10−5

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10−3

10−2

10−1

100

101

DS

CS


TLAY - parallelTLAY - perpendicularTSPHERE - parallelTSPHERE - perpendicular

Fig. 3.48. Normalized differential scattering cross-sections of a layered sphere

3.6 Multiple Particles

In the following analysis we investigate electromagnetic scattering by a sys-tem of particles. The basic routine for analyzing this type of scattering prob-lem is the TMULT routine. As for inhomogeneous scatterers, the individ-ual T -matrices of the particles are input parameters of the code and theymay correspond to homogeneous, axisymmetric or nonaxisymmetric particles


X

Z

n1

n2

O2 x2

z2

a1

b1

β2

x1

z1

O1 x20

z20

a2

b2

Fig. 3.49. Geometry of a system of two prolate spheroids

and inhomogeneous, composite or layered particles. The flow diagram of theTMULT routine is as in Fig. 3.34.

Figure 3.49 illustrates a system of two spheroids of semi-axes ksa1 =ksa2 = 4 and ksb1 = ksb2 = 2, and relative refractive indices mr1 = mr2 = 1.5.The Cartesian coordinates of the center of the second spheroid are ksx20 =ksz20 = 4

√2 and y20 = 0, while the Euler angles specifying the orientation of

the spheroids are chosen as αp1 = βp1 = 0 and αp2 = βp2 = 45. The firstcomputational step involves the calculation of the individual T -matrix of aspheroid by using the TAXSYM code, and for this calculation, the maximumexpansion and azimuthal orders are Nrank = 10 and Mrank = 4, respectively.The individual T -matrices are then used to compute the T -matrix of thespheroids with respect to the origin O1, and the dimension of the system T -matrix is given by Nrank = 20 and Mrank = 18. In Fig. 3.50, we show thedifferential scattering cross-sections computed with the TMULT routine andthe multiple multipole method. The scattering curves are in good agreement.

As a second example, we consider a system of five spherical particles illu-minated by a plane wave propagating along the Z-axis of the global coordinatesystem. The spheres are identical and have a radius of ksr = 2 and a relativerefractive index of mr = 1.5, while the length specifying the position of thespheres is ksl = 6 (Fig. 3.51). The system T -matrix is computed with respectto the origin O and is characterized by Nrank = 18 and Mrank = 16. Thescattering characteristics are computed with the TMULTSPH routine, andnumerical results are again presented in the form of the differential scatteringcross-sections. The curves plotted in Fig. 3.52 correspond to a fixed orientation


10−5

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10−2

10−1

100

101

DS

CS


TMULT - parallelTMULT - perpendicularMMP - parallelMMP - perpendicular

Fig. 3.50. Normalized differential scattering cross-sections of a system of two prolatespheroids computed with the TMULT routine and the multiple multipole method(MMP)

X

Y

Z

l

l l

l

r

Fig. 3.51. Geometry of a system of five spheres

of the system of spheres (αp = βp = γp = 0) and are similar for the T -matrixand the multiple multipole solutions. For a random orientation, we computethe elements of the scattering matrix, and the results are shown in Figs. 3.53and 3.54 together with the results computed with the SCSMTM code devel-


10−5

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10−2

10−1

100

101

DSC

S


TMULTSPH - parallelTMULSPH - perpendicularMMP - parallelMMP - perpendicular

Fig. 3.52. Normalized differential scattering cross-sections of a system of fivespheres computed with the TMULT routine and the multiple multipole method(MMP)

10−3.0

10−2.0

10−1.0

100.0

101.0

Scat

terin

g M

atrix

Ele

men

ts


F11 - TMULTSPHF11 - SCSMTMF22 - TMULTSPHF22 - SCSMTM

Fig. 3.53. Scattering matrix elements F11 and F22 of a system of five spherescomputed with the TMULT routine and the code SCSMTM of Mackowski andMishchenko [153]

oped by Mackowski and Mishchenko [153]. The curves are generally close toeach other.

Figure 3.55 illustrates a system of four identical spheres that can be usedto compare different routines. The radius of each spherical particle is ksr = 3and the relative refractive index is mr = 1.5. In the global coordinate system


−0.03

−0.02

−0.01

0.00

0.01

Scat

terin

g M

atrix

Ele

men

ts


F21 - TMULTSPHF21 - SCSMTMF43 - TMULTSPHF43 - SCSMTM

Fig. 3.54. Scattering matrix elements F21 and F43 of a system of five spherescomputed with the TMULT routine and the code SCSMTM of Mackowski andMishchenko [153]

OXY Z, two local coordinate systems O1x1y1z1 and O2x2y2z2 are defined,and the positions of the origins O1 and O2 are characterized by x01 = y01 =z01 = L and x02 = y02 = z02 = −L, respectively, where ksL = 6.5. Ineach local coordinate system, a system of two spheres is considered, and thedistance between the sphere centers is 2ksl = 3.5, while the Euler angles spec-ifying the orientation of the two-spheres system are αp1 = βp1 = 45 andαp2 = βp2 = 30. The Cartesian coordinates of the spheres are computedwith respect to the global coordinate system and the TMULTSPH routineis used to compute the scattering characteristics. The dimension of the T -matrix of the four-sphere system is given by Nrank = 21 and Mrank = 19. Asecond technique for analyzing this scattering geometry involves the compu-tation of the T -matrix of the two-sphere system by using the TMULT2SPHroutine. In this case, Nrank = 16 and Mrank = 5, and the T -matrix of the two-sphere system serves as input parameter for the TMULT routine. Figures 3.56and 3.57 illustrate the differential scattering cross-sections for a fixed (αp =βp = γp = 0) and a random orientation of the system of spheres. The far-fieldpatterns are reproduced very accurately by both methods.

To demonstrate the capabilities of the TMULTSPH routine we presentsome exemplary results for polydisperse aggregates. Monodisperse aggre-gates of small spherical particles are characterized by the number of primaryspherule N , the fractal dimension Df , the fractal pre-factor kf , the radius ofgyration Rg, and the radius of the primary spherules rp. These morphologicalparameters are related in the form of a scaling law


O

O2

O1

X

Z

x1

x2

z1

z2

β1

α 1

2lx1

z1

y1

r

O1

L

L

−L

−L

Fig. 3.55. Geometry of a system of four spheres

10−6

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10−3

10−2

10−1

100

DS

CS


TMATMULT - parallelTMATMULT - perpendicularTMATMULTSPH - parallelTMATMULTSPH - perpendicular

Fig. 3.56. Normalized differential scattering cross-sections of a system of fourspheres


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10−2

10−1

100

<D

SC

S>


TMATMULT - parallelTMATMULT - perpendicularTMATMULTSPH - parallelTMATMULTSPH - perpendicular

Fig. 3.57. Averaged differential scattering cross-sections of a system of four spheres

N = kf

(Rg

rp

)Df

, (3.14)

where the radius of gyration is determined from the position of each primaryspherule ri to the geometrical center of the cluster r0,

R2g =

1N

N∑

i=1

|ri − r0|2

and r0 = (1/N)∑N

i=1 ri. Equation (3.14) is important because the values ofparameters are linked to real physical processes. If Df ≈ 1.8 the processbelongs to a diffusion limited aggregation and if Df ≈ 2.1 it belongs toa reaction limited aggregation. For simulating diffusion limited aggregation,we developed a Fortran program and used the Cartesian coordinates of thespherule as input parameters for the TMULTSPH routine. This program isbased on the fast algorithm described by Filippov et al. [66] and generatesaggregates by using two different methods. In the first method, each newprimary spherule will be stuck on the mother aggregate after touching oneof the existing spherules. This method is known as the diffusion limited al-gorithm (DLA) and is used for aggregates not larger than about N = 150spherules. With the second method, complete small aggregates are stuck onthe mother aggregate. The small aggregates are generated with the DLA al-gorithm and consists of about 20 or more primary spherules. This method iscalled diffusion limited cluster cluster aggregation (DLCCA) and gives morerealistic aggregates. An example of a monodisperse aggregate representing


Fig. 3.58. Monodisperse aggregate with Df = 1.8, N = 130 and rp = 10 nm

Fig. 3.59. Polydisperse aggregate with Df = 1.8, N = 203 and rp between 10 and20 nm

a soot particle from a combustion processes is shown in Fig. 3.58, while apolydisperse aggregate is shown in Fig. 3.59. For the monodisperse aggregate,the scattering characteristics are shown in Fig. 3.60, and the essential parame-ters controlling the convergence process are Nrank = 8 and Mrank = 6 for thesystem T -matrix, and Nrank = 4 and Mrank = 3 for the primary spherules. Thedifferential scattering cross-sections of the polydisperse aggregate are plottedin Fig. 3.61, and the parameters of calculation for the system T -matrix in-crease to Nrank = 12 and Mrank = 10. The program described above has beenused by Riefler et al. [204] to characterize soot from a flame by analyzing themeasured scattering patterns.


10−6

10−5

10−4

10−3

10−2

10−1

DS

CS



Fig. 3.60. Normalized differential scattering cross-sections of a monodisperseaggregate

10−9

10−8

10−7

10−6

10−5

10−4

10−3

DS

CS



Fig. 3.61. Normalized differential scattering cross-sections of a polydisperseaggregate

3.7 Composite Particles

Electromagnetic scattering by axisymmetric, composite particles can be com-puted with the TCOMP routine. An axisymmetric, composite particle consistsof several nonenclosing, rotationally symmetric regions with a common axisof symmetry. In contrast to the TMULT routine, TCOMP is based on a for-malism which avoid the use of any local origin translation. The scattering


z1

Ll

r

z3

x

z

Fig. 3.62. Geometry of a composite particle consisting of three identical cylinders

characteristics can be computed with localized or distributed sources, and theprogram supports calculations for dielectric particles. Particle geometries in-cluded in the library are half-spheroids with offset origins and three cylinders.

In Fig. 3.62, we show a composite particle consisting of three identicalcylinders of radius ksr = 2 and length ksl = 4. The axial positions of thefirst and third cylinder are given by ksz1 = 4 and ksz3 = −4, and the relativerefractive indices are chosen as mr1 = mr3 = 1.5 and mr2 = 1.0. Becausemr2 = 1.0, the scattering problem is equivalent to the multiple scatteringproblem of two identical cylinders. On the other hand, the composite particlecan be regarded as an inhomogeneous cylinder with a cylindrical inclusion.Therefore, the scattering characteristics are computed with three independentroutines: TCOMP, TMULT, and TLAY. Localized and distributed sources areused for T -matrix calculations with the TCOMP routine, while distributedsources are used for calculations with the TLAY routine. Figure 3.63 shows thedifferential scattering cross-sections for a fixed orientation of the compositeparticle (αp = βp = 45), while Fig. 3.64 illustrates the numerical results fora random orientation. The behavior of the far-field patterns are quite similar.

In order to demonstrate the capability of the code to compute the elec-tromagnetic scattering by composite particles with elongated and flattenedregions, we consider the geometry depicted in Fig. 3.65. For this application,the sources are distributed on the real and imaginary axes. The parametersspecifying the particle shape are: ksr = 1, ksR = 3, ksl = 6 and ksL = 7,while the relative refractive indices are identical mr1 = mr2 = mr3 = 1.5.For the prolate cylinders, Nrank = 10 sources are distributed along the axisof symmetry, and for the oblate cylinder, Nrank = 10 sources are distributed


10−6

10−5

10−4

10−3

10−2

10−1

100

101

DS

CS


TCOMP - localized sources - parallelTCOMP - localized sources - perpendicularTCOMP - distributed sources - parallelTCOMP - distributed sources - perpendicularTMULT - parallelTMULT - perpendicularTLAY - parallelTLAY - perpendicular

Fig. 3.63. Normalized differential scattering cross-sections of a composite particleconsisting of three identical cylinders

10−4

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10−2

10−1

100

101

<DSC

S>


TCOMP - localized sources - parallelTCOMP - localized sources - perpendicularTCOMP - distributed sources - parallelTCOMP - distributed sources - perpendicularTMULT - parallelTMULT - perpendicularTLAY - parallelTLAY - perpendicular

Fig. 3.64. Averaged differential scattering cross-sections of a composite particleconsisting of three identical cylinders

on the imaginary axis. The global number of integration points is Nint = 100.The differential scattering cross-sections are plotted in Fig. 3.66 for the caseof normal incidence.

Scattering by composite spheroids as shown in Fig. 3.67 can be computedwith localized and distributed sources. The results plotted in Fig. 3.68 corre-spond to a composite spheroid with ksa = 20, ksb = 5, ksz1 = ksz2 = 10 and


z1

Ll

r

z3

x

z

R

Fig. 3.65. Geometry of a composite particle consisting of three cylinders

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10−2

10−1

DSC

S


TCOMP - parallelTCOMP - perpendicular

Fig. 3.66. Normalized differential scattering cross-sections of a composite particleconsisting of three cylinders


z1

z2

x

z

a

b

Fig. 3.67. Geometry of a composite spheroid

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10−2

10−1

100

101

DSC

S


parallel - distributed sourcesparallel - localized sourcesperpendicular - distributed sourcesperpendicular - localized sources

Fig. 3.68. Normalized differential scattering cross-sections of a composite spheroid

mr1 = mr2 = 1.2, while the parameters controlling the T -matrix calculationare given in Table 3.7.

3.8 Complex Particles

In the following analysis we consider the inhomogeneous particle depicted inFig. 3.69. The host particle is a sphere with radius R = 1.0 µm and relative

3.8 Complex Particles 243

Table 3.7. Parameters of calculation for a composite spheroid

Type of sources Nrank-half-spheroid Nrank-composite particle Nint

Localized 22 34 1000

Distributed 18 32 1000

O1

O X

Z

O2

x1

x2

z2

z1

mr

mr1,1

mr1,2

mr2,1

mr2,2

Fig. 3.69. Geometry of an inhomogeneous sphere containing a composite and alayered spheroid as separate inclusions

refractive index mr = 1.2, while the wavelength of the incident radiation is λ =0.628 µm. The inhomogeneities are a composite and a layered prolate spheroid.The composite particle consists of two identical half-spheroids with semi-axesa1 = 0.3 µm and b1 = 0.2 µm, and relative refractive indices (with respectto the ambient medium) mr1,1 = 1.5 and mr1,2 = 1.33. The layered particleconsists of two concentric prolate spheroids with semi-axes a2,1 = 0.3 µm,b2,1 = 0.2 µm, and a2,2 = 0.15 µm, b2,2 = 0.1 µm, and relative refractiveindices mr2,1 = 1.5 and mr2,2 = 1.8. The position of the composite particlewith respect to the global coordinate system of the host particle is specified bythe Cartesian coordinates x1 = y1 = z1 = 0.3 µm, while the Euler orientationangles are αp1 = βp1 = 45. For the layered particle, we choose x2 = y2 =z2 = −0.3 µm and αp2 = βp2 = 0.

The results plotted in Fig. 3.70 are computed with the TMULT routineand show the differential scattering cross-sections for the two-spheroid system.The particles are placed in a medium with a refractive index of 1.2, and the


10−6.00

10−5.00

10−4.00

10−3.00

10−2.00

10−1.00

DSC

S


TMULT - parallelTMULT - perpendicular

Fig. 3.70. Normalized differential scattering cross-sections of a composite and alayered particle

10−6

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10−3

10−2

10−1

100

101

102

DSC

S


TINHOM - parallelTINHOM - perpendicular

Fig. 3.71. Normalized differential scattering cross-sections of an inhomogeneoussphere. The inclusion consists of a composite and a layered particle

dimension of the system T -matrix is given by Nrank = 14 and Mrank = 12.The incident wave travel along the Z-axis of the global coordinate systemand the angular scattering is computed in the azimuthal plane ϕ = 0. Thesystem T -matrix serves as input parameter for the TINHOM routine. Theresulting T -matrix is characterized by Nrank = 16 and Mrank = 14, andcorresponds to a sphere containing a composite and a layered spheroid asseparate inclusions. Figure 3.71 illustrates the differential scattering cross-sections for the inhomogeneous sphere in the case of normal incidence.

3.9 Particle on or Near a Plane Surface 245

3.9 Particle on or Near a Plane Surface

In this section, we present scattering results for an axisymmetric particlesituated on or near a plane surface. For this purpose we use the TPARTSUBroutine and a computer program based on the discrete sources method [59,60].

Figures 3.72–3.74 show the differential scattering cross-sections for Fe-,Si- and SiO-spheroids with semi-axes a = 0.05 µm and b = 0.025 µm. Therelative refractive indices are: mr = 1.35 + 1.97j for Fe, mr = 4.37 + 0.08j for

10−6

10−5

10−4

10−3

10−2

10−1

100

101

DSC

S


TPARTSUB - parallelTPARTSUB - perpendicularDSM - parallelDSM - perpendicular

Fig. 3.72. Normalized differential scattering cross-sections of a Fe-spheroid com-puted with the TPARTSUB routine and the discrete sources method (DSM)

10−6

10−5

10−4

10−3

10−2

10−1

100

101

DSC

S



Fig. 3.73. Normalized differential scattering cross-sections of a Si-spheroid com-puted with the TPARTSUB routine and the discrete sources method (DSM)


10−6

10−5

10−4

10−3

10−2

10−1

100

101

DSC

S



Fig. 3.74. Normalized differential scattering cross-sections of a SiO-spheroid com-puted with the TPARTSUB routine and the discrete sources method (DSM)

Si, and mr = 1.67 for SiO. The particles are situated on a silicon substrate,the wavelength of the incident radiation is λ = 0.488 µm, and the incidentangle is β0 = 45. The plotted data show that the T -matrix method leads toaccurate results.

In the next example, we investigate scattering of evanescent waves byparticles situated on a glass prism. We note that evanescent wave scatteringis important in various sensor applications such as the total internal reflectionmicroscopy TIRM [195]. Choosing the wavelength of the external excitationas λ = 0.488 µm and taking into account that the glass prism has a refractiveindex of mrs = 1.5, we deduce that the evanescent waves appear for incidentangles exceeding 41.8. In Figs. 3.75–3.77, we plot the differential scatteringcross-section for Ag-, diamond-, and Si-spheres with a diameter of d = 0.2 µm.The relative refractive indices of Ag- and diamond particles are mr = 0.25 +3.14j and mr = 2.43, respectively. The scattering plane coincides with theincident plane and the angle of incidence is β0 = 60. The plotted data showa good agreement between the discrete sources and the T -matrix solutions.

3.10 Effective Medium Model

The effective wave number of a half-space with randomly distributed spher-oidal particles can be computed with the EFMED routine.

First we consider spherical particles and compare our results to the solu-tions obtained with the Matlab program QCAMIE included in the Electro-


10−5

10−4

10−3

10−2

10−1

100

101

DSC

S



Fig. 3.75. Normalized differential scattering cross-sections of a metallic Ag-spherecomputed with the TPARTSUB routine and the discrete sources method (DSM)

10−5

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10−2

10−1

100

101

DSC

S



Fig. 3.76. Normalized differential scattering cross-sections of a Diamond-spherecomputed with the TPARTSUB routine and the discrete sources method (DSM)

magnetic Wave Matlab Library and available from www.emwave.com. Thislibrary contains several Matlab programs which are based on the theory givenin the book of Tsang et al. [229]. Figures 3.78 and 3.79 show the normalizedphase velocity ks/ReKs and the effective loss tangent 2ImKs/ReKs asfunctions of the size parameter x = ksR, and it is apparent that no substantialdifferences between the curves exist.


10−5

10−4

10−3

10−2

10−1

100

101

DSC

S



Fig. 3.77. Normalized differential scattering cross-sections of a Si-sphere computedwith the TPARTSUB routine and the discrete sources method (DSM)

0.84

0.85

0.86

0.87

0.88

0.89

Pha

se V

eloc

ity

0.0 0.5 1.0 1.5 2.0 2.5

Size Parameter

EFMEDQCAMIE

Fig. 3.78. Normalized phase velocity versus size parameter computed with theEFMED routine and the Matlab program QCAMIE

Next we consider the test examples given by Neo et al. [179] and comparethe T -matrix results to the solutions obtained with the long-wavelength quasi-crystalline approximation

K2s = k2

s

1 +3c εr−1

εr+2

1 − c εr−1εr+2

[

1 + j2x3

3

εr−1εr+2

1 − c εr−1εr+2

(1 − c)4

(1 + 2c)2

]


0.0

0.1

0.2

0.3

0.4

Effe

ctiv

e Lo

ss T

ange

nt

0.0 0.5 1.0 1.5 2.0 2.5Size Parameter

EFMEDQCAMIE

Fig. 3.79. Effective loss tangent versus size parameter computed with the EFMEDroutine and the Matlab program QCAMIE

and the quasi-crystalline approximation with coherent potential [118],

K2s = k2

s

⎧⎨

⎩1 +

c (εr − 1)

1 + k2s (εr−1)3K2

s(1 − c)

×

⎡

⎣1 + j2Ksk

2s R

3

9εr − 1

1 + k2s (εr−1)3K2

s(1 − c)

(1 − c)4

(1 + 2c)2

⎤

⎦

⎫⎬

⎭.

Figure 3.80 shows plots of ImKs/k0 versus concentration for spherical parti-cles of radius a = 3.977 ·10−3 µm and relative refractive index (with respect tothe ambient medium) mr = 1.194. The wave number in free space is k0 = 10,while the refractive index of the ambient medium is m = 1.33.

In Fig. 3.81 we plot ImKs/k0 versus concentration for a = 1.047 ·10−2 µm, mr = 1.789 and m = 1.0. Computed results using the T -matrixmethod agree with the quasi-crystalline approximation. It should be observedthat both the quasi-crystalline approximation and the quasi-crystalline ap-proximation with coherent potential do predict maximum wave attenuationat a certain concentration.

In Figs. 3.82 and 3.83, we show calculations of ReKs and ImKs asfunctions of the size parameter x = ks maxa, b for oblate and prolate spher-oids. The spheroids are assumed to be oriented with their axis of symmetryalong the Z-axis or to be randomly oriented. Since the incident wave is alsoassumed to propagate along the Z-axis, the medium is not anisotropic and ischaracterized by a single wave number Ks. The axial ratios are a/b = 0.66 foroblate spheroids and a/b = 1.5 for prolate spheroids. As before, the fractional


10−5.0

10−4.0

10−3.0

Im(K

/k)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Concentration

EFMED (T-Matrix Multiple Scattering Theory)QCAQCA - CP

Fig. 3.80. ImKs/k0 for a = 3.977 · 10−3 µm, mr = 1.194 and m = 1.33. Theresults are computed with the T -matrix method, quasicrystalline approximation(QCA) and quasicrystalline approximation with coherent potential (QCA-CP)

10−6.0

10−5.0

10−4.0

Im(K

/k)

0.0 0.5 1.0 1.5 2.0 2.5 3.0Concentration

EFMED (T-Matrix Multiple Scattering Theory)QCAQCA - CP

Fig. 3.81. ImKs/k0 for a = 1.047 · 10−2 µm, mr = 1.789 and m = 1.0. Theresults are computed with the T -matrix method, quasicrystalline approximation(QCA) and quasicrystalline approximation with coherent potential (QCA-CP)

concentration is c = 0.2 and the relative refractive index of the particles ismr = 1.789. The significant feature in Fig. 3.82 is the occurrence of a maximumin ReKs at x = 1.6 for spherical particles. In the low-frequency regime up tox = 0.5, the ImKs curves increase rapidly with x and then show a smooth


10.4

10.8

11.2

11.6

12.0

12.4

Re(

K)

0.0 0.4 0.8 1.2 1.6 2.0

Size Parameter

sphereoblate - fixed orientationoblate - random orientationprolate - fixed orientationprolate - random orientation

Fig. 3.82. ReKs versus the size parameter x = ks maxa, b for oblate and pro-late spheroids with a/b = 0.66 and a/b = 1.5, respectively. The calculations areperformed for c = 0.2, mr = 1.789 and m = 1.0

10−5.0

10−4.0

10−3.0

10−2.0

10−1.0

100.0

Im(K

)

0.0 0.4 0.8 1.2 1.6 2.0Size Parameter

sphereoblate - fixed orientationoblate - random orientationprolate - fixed orientationprolate - random orientation

Fig. 3.83. ImKs versus the size parameter x = ks maxa, b for oblate and prolatespheroids. The parameters of calculation are as in Fig. 3.82

increase with x. Note that for a given size parameter x, the radius of a spher-ical particle is larger than the equal-volume-sphere radii of an oblate and aprolate spheroid, and this feature is reflected in the variations of ReKs andImKs.

A

Spherical Functions

In this appendix we recall the basic properties of the solutions to the Besseland Legendre differential equations and discuss some computational aspects.Properties of spherical Bessel and Hankel functions and (associated) Legendrefunctions can be found in [1, 40,215,238].

In a source-free isotropic medium the electric and magnetic fields satisfythe equations

∇×∇× X − k2X = 0,

∇ · X = 0,

where X stands for E or H. Since

∇×∇× X = −∆X + ∇ (∇ · X) ,

we see that X satisfies the vector wave equation

∆X + k2X = 0

and we deduce that each component of X satisfies the scalar wave equationor the Helmholtz equation

∆u + k2u = 0.

The Helmholtz equation written in spherical coordinates as

1r2

∂

∂r

(r2 ∂u

∂r

)+

1r2 sin θ

∂

∂θ

(sin θ

∂u

∂θ

)+

1r2 sin2 θ

∂2u

∂ϕ2+ k2u = 0

is separable, so that upon replacing

u(r) = f1(r)Y (θ, ϕ) ,

254 A Spherical Functions

where (r, θ, ϕ) are the spherical coordinates of the position vector r, we obtain

r2f ′′1 + 2rf ′

1 +[k2r2 − n(n + 1)

]f1 = 0, (A.1)

1sin θ

∂

∂θ

(sin θ

∂Y

∂θ

)+

1sin2 θ

∂2Y

∂ϕ2+ n(n + 1)Y = 0. (A.2)

The above equations are known as the Bessel differential equation and thedifferential equation for the spherical harmonics. Further, setting

Y (θ, ϕ) = f2(θ)f3 (ϕ) ,

we find that

1sin θ

ddθ

(sin θ

df2

dθ

)+[n(n + 1) − m2

sin2 θ

]f2 = 0, (A.3)

f ′′3 + m2f3 = 0. (A.4)

The differential equation (A.3) is known as the associated Legendre equation,while the solution to the differential equation (A.4) is f3(ϕ) = exp(jmϕ).

A.1 Spherical Bessel Functions

With the substitution x = kr, the spherical Bessel differential equation canbe written in the standard form

x2f ′′(x) + 2xf ′(x) +[x2 − n(n + 1)

]f(x) = 0 .

For n = 0, 1, . . . , the functions

jn(x) =∞∑

p=0

(−1)pxn+2p

2pp!(2n + 2p + 1)!!

and

yn(x) = − (2n)!2nn!

∞∑

p=0

(−1)px2p−n−1

2pp! (−2n + 1) (−2n + 3) · · · (−2n + 2p − 1), (A.5)

where 1 · 3 · · · (2n + 2p + 1) = (2n + 2p + 1)!!, are solutions to the sphericalBessel differential equation (the first coefficient in the series (A.5) has to beset equal to one). The functions jn and yn are called the spherical Bessel andNeumann functions of order n, respectively, and the linear combinations

h(1,2)n = jn ± jyn

are known as the spherical Hankel functions of the first and second kind. Fromthe series representations we see that

A.1 Spherical Bessel Functions 255

2n + 1x

zn(x) = zn−1(x) + zn+1(x), (A.6)

(2n + 1) z′n(x) = nzn−1(x) − (n + 1)zn+1(x), (A.7)

andddx

[xzn(x)] = xzn−1(x) − nzn(x), (A.8)

where zn stands for any spherical function. The Wronskian relation for thespherical Bessel and Neumann functions is

jn(x)y′n(x) − j′n(x)yn(x) =

1x2

and for small values of the argument we have

jn(x) =xn

(2n + 1)!!1 + O

(x2)

,

yn(x) = − (2n − 1)!!xn+1

1 + O

(x2)

as x → 0, while for large value of the argument,

jn(x) =1x

cos[x − (n + 1)π

2

]1 + O

(1x

),

yn(x) =1x

sin[x − (n + 1)π

2

]1 + O

(1x

),

h(1)n (x) =

ej[x−(n+1)π/2]

x

1 + O

(1x

),

h(2)n (x) =

e−j[x−(n+1)π/2]

x

1 + O

(1x

)

as x → ∞.Spherical Bessel functions can be expressed in terms of trigonometric func-

tions, and the first few spherical functions have the explicit forms

j0(x) =sinx

x, (A.9)

y0(x) = −cos x

x, (A.10)

j1(x) =sinx

x2− cos x

x, (A.11)

y1(x) = −cos x

x2− sin x

x, (A.12)


and

h(1)0 (x) =

ejx

jx,

h(2)0 (x) = −e−jx

jx.

The spherical Bessel functions are computed by downward recursion. Thisrecursion begins with two successive functions of small values and producesfunctions proportional to the Bessel functions rather than the actual Besselfunctions. The constant of proportionality between the two sets of functions isobtained from the function of order zero j0. Alternatively, the spherical Besselfunctions can be computed with an algorithm involving the auxiliary functionχn [169]

χn(x) =jn(x)

jn−1(x).

In this case, the functions χn are calculated with the downward recurrencerelation

χn(x) =1

2n+1x − χn+1(x)

,

the constant of proportionality is obtained from the function of order one,χ1(x) = 1/x − cot x, and the spherical Bessel functions are computed withthe upward recursion

jn(x) = χn(x)jn−1(x)

starting at j1. The spherical Neumann functions are calculated by upwardrecursion starting with the functions of order zero and one, y0 and y1, respec-tively.

A.2 Legendre Functions

With the substitution x = cos θ, the associated Legendre equation trans-forms to

(1 − x2

)f ′′(x) − 2xf ′ (x) +

[n(n + 1) − m2

1 − x2

]f (x) = 0. (A.13)

This equation is characterized by regular singularities at the points x = ±1and at infinity. For m = 0, there are two linearly independent solutions to theLegendre differential equation and these solutions can be expressed as powerseries about the origin x = 0. In general, these series do not converge for

A.2 Legendre Functions 257

x = ±1, but if n is a positive integer, one of the series breaks off after a finitenumber of terms and has a finite value at the poles. These polynomial solutionsare called Legendre polynomials and are denoted by Pn(x). For m = 0, thesolutions to (A.13) which are finite at the poles x = ±1 are the associatedLegendre functions. If m and n are integers, the associated Legendre functionsare defined as

Pmn (x) =

(1 − x2

)m/2 dmPn(x)dxm

.

Useful recurrence relations satisfied by Pmn are

(2n + 1) xPmn (x) = (n − m + 1) Pm

n+1(x) + (n + m) Pmn−1(x),

(A.14)

(2n + 1)√

1 − x2Pmn (x) = (n + m − 1) (n + m) Pm−1

n−1 (x) (A.15)

− (n − m + 1) (n − m + 2) Pm−1n+1 (x),

(2n + 1)√

1 − x2Pm−1n (x) = Pm

n+1(x) − Pmn−1 (cos θ) , (A.16)

(2n + 1)(1 − x2

) ddx

Pmn (x) = (n + 1) (n + m) Pm

n−1 (x)

−n (n − m + 1) Pmn+1(x), (A.17)

(1 − x2

) ddx

Pmn (x) = (n + m)Pm

n−1 (x) − nxPmn (x). (A.18)

The angular functions πmn and τm

n are related to the associated Legendrefunctions by the relations

πmn (θ) =

Pmn (cos θ)sin θ

, (A.19)

τmn (θ) =

ddθ

Pmn (cos θ). (A.20)

For positive values of m and for θ → 0 or θ → π,

Pmn (cos θ) → (n + m)!

(n − m)!

(2

2n + 1

)m

Jm

(2n + 1

2θ

)

and

mπmn (θ) → 1

2(n + m)!

(n − m)!(m − 1)!

(θ

2

)m−1

,

τmn (θ) → 1

2(n + m)!

(n − m)!(m − 1)!

(θ

2

)m−1

,


while for n 1 and away from θ = 0 and θ = π, we have

Pmn (cos θ) →

(2π

) 12

(sin θ)−12 nm− 1

2 cos(

2n + 12

θ +mπ

2− π

4

)

and

πmn (θ) →

(2π

) 12

(sin θ)−32 nm− 1

2 cos(

2n + 12

θ +mπ

2− π

4

),

τmn (θ) →

(2π

) 12

(sin θ)−12 nm+ 1

2 sin(

2n + 12

θ +mπ

2− π

4

).

In our analysis we use the associated Legendre functions with positive valuesof the index m. For m ≥ 0, the normalized associated Legendre functions aregiven by

Pmn (cos θ) = cmnPm

n (cos θ), (A.21)

where cmn is a normalization constant and

cmn =

√2n + 1

2· (n − m)!(n + m)!

.

Similarly, the normalized angular functions πmn and τm

n are related to theangular functions πm

n and τmn by the relations

πmn (θ) = cmnπm

n (θ), (A.22)

τmn (θ) = cmnτm

n (θ) . (A.23)

An algorithm for computing the normalized associated Legendre functionsPm

n involves the following steps.

(1) For m = 0, compute Pn+1 by using the recurrence relation

Pn+1(cos θ) =1

n + 1

√(2n + 1)(2n + 3) cos θPn(cos θ)

− n

n + 1

√2n + 32n − 1

Pn−1(cos θ) , n ≥ 1

with the starting values

P0(cos θ) =√

22

,

P1(cos θ) =

√32

cos θ.

A.2 Legendre Functions 259

(2) For m ≥ 1, compute Pmn+1 by using the recurrence relation

Pmn+1(cos θ) =

√(2n + 1)(2n + 3)

(n + 1 − m)(n + 1 + m)cos θPm

n (cos θ)

−√

(2n + 3)(n − m)(n + m)(2n − 1)(n + 1 − m)(n + 1 + m)

Pmn−1(cos θ), n ≥ m

with the initial values

Pm−1m (cos θ) = 0,

Pmm (cos θ) =

√2m + 12 (2m)!

(2m − 1)!! sinm θ.

Considering the angular functions πmn , we see that π0

n diverges at θ = 0 andθ = π. Because in our applications, the product mπm

n appears explicitly, we setπ0

n = 0 for n ≥ 0. For m ≥ 1, the angular functions πmn are computed by using

(A.19), (A.21) and (A.22), and the recurrence relations for the normalizedassociated Legendre functions.

The angular functions τmn can be calculated with the following algorithm.

(1) For m = 0, compute

dd cos θ

Pn (cos θ) = P ′n (cos θ)

by using the recurrence relation

P ′n(cos θ) = n

√2n + 12n − 1

Pn−1(cos θ) +

√2n + 12n − 1

cos θP ′n−1(cos θ)

with the starting value

P ′0 (cos θ) = 0,

and set [144]

τ0n(θ) = − sin θP ′

n (cos θ) , n ≥ 0 .

(2) For m ≥ 1, compute τmn with the recurrence relation

τmn (θ) = n cos θπm

n (θ) − (n + m)

√(2n + 1)(n − m)(2n − 1)(n + m)

πmn−1(θ), n ≥ m.


The following orthogonality relations are important for solving scatteringproblems:

∫ π

0

Pmn (cos θ)Pm

n′ (cos θ) sin θ dθ = δnn′ ,

∫ π

0

[τmn (θ)τm

n′ (θ) + m2πmn (θ)πm

n′(θ)]sin θ dθ = n(n + 1)δnn′ ,

∫ π

0

[πmn (θ)τm

n′ (θ) + τmn (θ)πm

n′(θ)] sin θ dθ = 0.

For negative arguments we have the symmetry relation

Pmn (− cos θ) = (−1)n−mPm

n (cos θ)

and consequently,

πmn (π − θ) = (−1)n−mπm

n (θ), (A.24)

τmn (π − θ) = (−1)n−m+1τm

n (θ). (A.25)

B

Wave Functions

In this appendix, we summarize the basic properties of the characteristic so-lutions to the scalar and vector wave equations. We recall the integral andorthogonality relations, and the addition theorems under coordinate rotationsand translations.

B.1 Scalar Wave Functions

The spherical wave functions form a set of characteristic solutions to theHelmholtz equation and are given by [215]

u1,3mn (kr) = z1,3

n (kr) P |m|n (cos θ) ejmϕ,

where n = 0, 1, . . . ,, m = −n, . . . , n, (r, θ, ϕ) are the spherical coordinates ofthe position vector r, z1

n designates the spherical Bessel functions jn, z3n stands

for the spherical Hankel functions of the first-order h(1)n , and P

|m|n denotes the

normalized associated Legendre functions. u1mn is an entire solution to the

Helmholtz equation (solution which are defined in all of R3) and u3mn is a

radiating solution to the Helmholtz equation in R3 − 0.The Green function or the fundamental solution is defined as

g (k, r, r′) =ejk|r−r′|

4π |r − r′| , r = r′

and straightforward differentiation shows that for fixed r′ ∈ R3, the funda-mental solution satisfies the Helmholtz equation in R3 −r′. The expansionof the Green function in terms of spherical wave functions is given by

g (k, r, r′) =jk2π

∞∑

n=0

n∑

m=−n

u3−mn (kr′) u1

mn (kr) ,

u1−mn (kr′) u3

mn (kr) ,

r < r′

r > r′.

262 B Wave Functions

The spherical harmonics are defined as

Ymn (θ, ϕ) = P |m|n (cos θ) ejmϕ

and they satisfy the orthogonality relation∫ 2π

0

∫ π

0

Ymn (θ, ϕ) Ym′n′ (θ, ϕ) sin θ dθ dϕ = 2πδm,−m′δnn′ .

The spherical wave functions can be expressed as integrals over plane waves[26,70,215]

u1mn(kr) =

14πjn

∫ 2π

0

∫ π

0

Ymn (β, α) ejk(β,α)·r sinβ dβ dα (B.1)

and

u3mn(kr) =

12πjn

∫ 2π

0

∫ π2 −j∞

0

Ymn (β, α) ejk(β,α)·r sinβ dβ dα, (B.2)

where (k, β, α) are the spherical coordinates of the wave vector k. We notethat (B.2) is valid for z > 0 since only then the integral converges.

The spherical wave expansion of the plane wave exp(jk ·r) is of significantimportance in electromagnetic scattering and is given by [215]

P (k, β, α, r) = ejk(β,α)·r =∞∑

n=0

n∑

m=−n

2jnY−mn (β, α) u1mn(kr). (B.3)

The quasi-plane waves have been introduced by Devaney [46], and for z > 0,they are defined as

Q(k, β, α, r) =∫ 2π

0

∫ π2 −j∞

0

∆(β, α, β′, α′)ejk′(β′,α′)·r sin β′ dβ′ dα′,

where

∆(β, α, β′, α′) =12π

∞∑

n=0

n∑

m=−n

Y−mn (β, α) Ymn (β′, α′) .

If β and β′ belong to [0, π], the expression of ∆ simplifies to

∆(β, α, β′, α′) = δ(cos β′ − cos β)δ(α′ − α).

As in (B.3), the expansion of quasi-plane waves in terms of radiating sphericalwave functions is given by

Q(k, β, α, r) =∞∑

n=0

n∑

m=−n

jnY−mn (β, α) u3mn(kr). (B.4)

B.1 Scalar Wave Functions 263

The function ∆ allows to formally analytically continue a function f(β, α),defined for real β, onto the complex values of β. In particular we see that

f(β, α) =∫ 2π

0

∫ π

0

f (β′, α′) ∆(β, α, β′, α′) sin β′ dβ′ dα′,

where β can assume all values lying on the contour [0, π2 − j∞), and therefore

∫ 2π

0

∫ π2 −j∞

0

f(β, α)ejk(β,α)·r sinβ dβ dα

=∫ 2π

0

∫ π

0

f(β, α)Q(k, β, α, r) sin β dβ dα, (B.5)

where we have used the definition of Q. In view of (B.2) and (B.5), we deducethat the integral representation for the radiating spherical wave functions isgiven by

u3mn(kr) =

12πjn

∫ 2π

0

∫ π

0

Ymn (β, α) Q(k, β, α, r) sin β dβ dα. (B.6)

In the far-field region, a plane wave can be expressed as a superposition ofincoming and outgoing spherical waves. To derive this expression, we startwith the series representation (B.3) written as

P (k, β, α, r) = ejk(β,α)·r =∞∑

n=0

n∑

m=−n

2jnjn(kr)Y−mn (β, α) Ymn(θ, ϕ),

use the asymptotic behavior of the spherical Bessel functions for large valuesof the argument

jn (kr) =1kr

sin(kr − nπ

2

)

=1

2jkr

[ej(kr−nπ

2 ) − e−j(kr−nπ2 )], kr → ∞

take into account the completeness relations for spherical harmonics,

δ (er − ek) = δ(ϕ − α)δ(cos θ − cos β)

=12π

∞∑

n=0

n∑

m=−n

Y−mn (β, α) Ymn (θ, ϕ) ,

δ (er + ek) = δ [(π + ϕ) − α] δ [cos (π − θ) − cos β]

=12π

∞∑

n=0

n∑

m=−n

(−1)nY−mn (β, α) Ymn (θ, ϕ) ,


where er and ek are the unit vectors in the directions of r and k, respectively,and obtain

ejk·r =2π

jkr

[δ (er − ek) ejkr − δ (er + ek) e−jkr

], kr → ∞. (B.7)

The vector spherical harmonics are defined as [175,228,229]

mmn(θ, ϕ) =1

√2n(n + 1)

[jmπ|m|

n (θ)eθ − τ |m|n (θ)eϕ

]ejmϕ, (B.8)

nmn(θ, ϕ) =1

√2n(n + 1)

[τ |m|n (θ)eθ + jmπ|m|

n eϕ

]ejmϕ, (B.9)

where (er,eθ,eϕ) are the spherical unit vectors of the position vector r, andwe have er ×mmn = nmn and er ×nmn = −mmn. We omit the third vectorspherical harmonics

pmn(θ, ϕ) =1

√2n(n + 1)

Ymn(θ, ϕ)er

since it will be not encountered in our analysis. The orthogonality relationsfor vector spherical harmonics are

∫ 2π

0

∫ π

0

mmn(θ, ϕ) · mm′n′(θ, ϕ) sin θ dθ dϕ

=∫ 2π

0

∫ π

0

nmn(θ, ϕ) · nm′n′(θ, ϕ) sin θ dθ dϕ = πδm,−m′δnn′ , (B.10)

and ∫ 2π

0

∫ π

0

mmn(θ, ϕ) · nm′n′(θ, ϕ) sin θdθ dϕ = 0 . (B.11)

The system of vector spherical harmonics is orthogonal and complete inL2

tan(Ω) (the space of square integrable tangential fields defined on the unitsphere Ω) and in terms of the scalar product in L2

tan(Ω) we have

∫ 2π

0

∫ π

0

mmn(θ, ϕ) · m∗m′n′(θ, ϕ) sin θ dθ dϕ

=∫ 2π

0

∫ π

0

nmn(θ, ϕ) · n∗m′n′(θ, ϕ) sin θ dθ dϕ = πδmm′δnn′ (B.12)

and ∫ 2π

0

∫ π

0

mmn(θ, ϕ) · n∗m′n′(θ, ϕ) sin θ dθ dϕ = 0. (B.13)

B.2 Vector Wave Functions 265

We define the vector spherical harmonics of left- and right-handed type as

lmn(θ, ϕ) =1√2

[mmn(θ, ϕ) + jnmn(θ, ϕ)] (B.14)

=j

2√

n(n + 1)

[mπ|m|

n (θ) + τ |m|n (θ)

](eθ + jeϕ)ejmϕ

and

rmn(θ, ϕ) =1√2

[mmn(θ, ϕ) − jnmn(θ, ϕ)] (B.15)

=j

2√

n(n + 1)

[mπ|m|

n (θ) − τ |m|n (θ)

](eθ − jeϕ)ejmϕ,

respectively, and it is straightforward to verify the orthogonality relations

∫ 2π

0

∫ π

0

lmn(θ, ϕ) · l∗m′n′(θ, ϕ) sin θ dθ dϕ

=∫ 2π

0

∫ π

0

rmn(θ, ϕ) · r∗m′n′(θ, ϕ) sin θ dθ dϕ = πδmm′δnn′ (B.16)

and∫ 2π

0

∫ π

0

lmn(θ, ϕ) · r∗m′n′(θ, ϕ) sin θ dθ dϕ = 0. (B.17)

Since lmn and rmn are linear combinations of mmn and nmn, we deduce thatthe system of vector spherical harmonics of left- and right-handed type is alsoorthogonal and complete in L2

tan(Ω).

B.2 Vector Wave Functions

The independent solutions to the vector wave equations can be constructedas [215]

M1,3mn(kr) =

1√

2n(n + 1)∇u1,3

mn(kr) × r,

N1,3mn(kr) =

1k∇× M1,3

mn(kr),

where n = 1, 2, ..., and m = −n, ..., n. The explicit expressions of the vectorspherical wave functions are given by


M1,3mn(kr) =

1√

2n(n + 1)z1,3

n (kr)[jmπ|m|

n (θ)eθ − τ |m|n (θ)eϕ

]ejmϕ,

N1,3mn(kr) =

1√

2n(n + 1)

n(n + 1)

z1,3n (kr)

krP |m|

n (cos θ)er

+

[krz1,3

n (kr)]′

kr

[τ |m|n (θ)eθ + jmπ|m|

n (θ)eϕ

]

ejmϕ,

where

[krz1,3

n (kr)]′

=d

d (kr)[krz1,3

n (kr)].

The superscript ‘1’ stands for the regular vector spherical wave functions whilethe superscript ‘3’ stands for the radiating vector spherical wave functions. Itis useful to note that for n = m = 0, we have M1,3

00 = N1,300 = 0. M1

mn, N1mn

is an entire solution to the Maxwell equations and M3mn, N3

mn is a radiatingsolution to the Maxwell equations in R3 − 0.

The vector spherical wave functions can be expressed in terms of vectorspherical harmonics as follows:

M1,3mn(kr) = z1,3

n (kr)mmn(θ, ϕ),

N1,3mn(kr) =

√n(n + 1)

2z1,3

n (kr)kr

Ymn (θ, ϕ) er +

[krz1,3

n (kr)]′

krnmn(θ, ϕ),

In the far-field region, the asymptotic behavior of the spherical Hankel func-tions for large value of the argument yields the following representations forthe radiating vector spherical wave functions [40]:

M3mn(kr) =

ejkr

kr

(−j)n+1

mmn(θ, ϕ) + O

(1r

), r → ∞,

N3mn(kr) =

ejkr

kr

j (−j)n+1

nmn(θ, ϕ) + O

(1r

), r → ∞.

The orthogonality relations on a sphere Sc, of radius R, are∫

Sc

[er × Mmn(kr)] · [er × Mm′n′(kr)] dS(r) = πR2z2n(kR)δm,−m′δnn′ ,

∫

Sc

[er × Nmn(kr)] · [er × Nm′n′(kr)] dS(r) = πR2

[kRzn(kR)]′

kR

2

×δm,−m′δnn′ ,


and∫

Sc

[er × Mmn(kr)] · [er × Nm′n′(kr)] dS(r) = 0,

and we also have∫

Sc

[er × Mmn(kr)] · Nm′n′(kr)dS (r)

= −∫

Sc

[er × Nmn(kr)] · Mm′n′(kr)dS (r)

= πR2zn(kR)[kRzn(kR)]′

kRδm,−m′δnn′ (B.18)

and∫

Sc

[er × Mmn(kr)] · Mm′n′(kr)dS (r)

=∫

Sc

[er × Nmn(kr)] · Nm′n′(kr)dS (r) = 0. (B.19)

The spherical vector wave expansion of the dyad gI is of basic importance inour analysis and is given by [175,228,229]

g(k, r, r′)I

=jkπ

∞∑

n=1

n∑

m=−n

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

[M3

−mn(kr′)M1mn(kr) + N3

−mn(kr′)N1mn(kr)

]

+Irrotational terms , r < r′

[M1

−mn(kr′)M3mn(kr) + N1

−mn(kr′)N3mn(kr)

]

+Irrotational terms , r > r′

(B.20)

Using the calculation rules for dyadic functions and the identity ag = a · gI,we find the following simple but useful expansions

∇× [a(r′)g(k, r, r′)]

=jk2

π

∞∑

n=1

n∑

m=−n

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

[a(r′) · M3

−mn(kr′)]N1

mn(kr)

+[a(r′) · N3

−mn(kr′)]M1

mn(kr)

, r < r′

[a(r′) · M1

−mn(kr′)]N3

mn(kr)

+[a(r′) · N1

−mn(kr′)]M3

mn(kr)

, r > r′

(B.21)


and

∇×∇× [a(r′)g(k, r, r′)]

=jk3

π

∞∑

n=1

n∑

m=−n

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

[a(r′) · M3

−mn(kr′)]M1

mn(kr)

+[a(r′) · N3

−mn(kr′)]N1

mn(kr)

, r < r′

[a(r′) · M1

−mn(kr′)]M3

mn(kr)

+[a(r′) · N1

−mn(kr′)]N3

mn(kr)

, r > r′

(B.22)

Relying on these expansions and using the Stratton–Chu representation the-orem, orthogonality relations of vector spherical wave functions on arbitrarilyclosed surfaces can be derived. Let Di be a bounded domain with bound-ary S and exterior Ds, and let n be the unit normal vector to S directedinto Ds. The wave number in the domain Ds is denoted by ks, while thewave number in the domain Di is denoted by ki. For r ∈ Di, application ofStratton–Chu representation theorem to the vector fields Es(r) = M3

mn(ksr)and Hs(r) = −j

√εs/µsN

3mn(ksr) gives

∫

S

[(n × M3

mn

)·(

M3−m′n′

N3−m′n′

)+(n × N3

mn

)·(

N3−m′n′

M3−m′n′

)]dS = 0,

(B.23)while for r ∈ Ds, yields

jk2s

π

∫

S

[(n × M3

mn

)·(

M1−m′n′

N1−m′n′

)+(n × N3

mn

)·(

N1−m′n′

M1−m′n′

)]dS

=(

0δmm′δnn′

). (B.24)

Similarly, for r ∈ Ds, the Stratton–Chu representation theorem applied to thevector fields Ei(r) = M1

mn(kir) and H i(r) = −j√

εi/µiN1mn(kir) leads to

∫

S

[(n × M1

mn

)·(

M1−m′n′

N1−m′n′

)+(n × N1

mn

)·(

N1−m′n′

M1−m′n′

)]dS = 0.

(B.25)The regular and radiating spherical vector wave functions can be expressedas integrals over vector spherical harmonics [26]

M1mn(kr) = − 1

4πjn+1

∫ 2π

0

∫ π

0

(−j)mmn (β, α) ejk(β,α)·r sin β dβ dα,

(B.26)

N1mn(kr) = − 1

4πjn+1

∫ 2π

0

∫ π

0

nmn (β, α) ejk(β,α)·r sin β dβ dα,

(B.27)


and

M3mn(kr) = − 1

2πjn+1

∫ 2π

0

∫ π2 −j∞

0

(−j)mmn (β, α) ejk(β,α)·r sinβ dβ dα,

(B.28)

N3mn(kr) = − 1

2πjn+1

∫ 2π

0

∫ π2 −j∞

0

nmn (β, α) ejk(β,α)·r sin β dβ dα

(B.29)

for z > 0, respectively.The above system of vector functions is also known as the system of local-

ized vector spherical wave functions. Another system of vector functions whichis suitable for analyzing axisymmetric particles with extreme geometries is thesystem of distributed vector spherical wave functions [49]. For an axisymmet-ric particle with the axis of rotation along the z-axis, the distributed vectorspherical wave functions are defined as

M1,3mn(kr) = M1,3

m,|m|+l [k(r − znez)] ,

N 1,3mn(kr) = N1,3

m,|m|+l [k(r − znez)] , (B.30)

where zn∞n=1 is a dense set of points on the z-axis (Fig. B.1), ez is the unitvector in the direction of the z-axis, n = 1, 2, ...,m ∈ Z, and l = 1 if m = 0 andl = 0, if m = 0. M1

mn, N 1mn is an entire solution to the Maxwell equations and

M3mn, N 3

mn is a radiating solution to the Maxwell equations in R3 − znez.In the case of prolate scatterers, the distribution of the poles on the axis ofrotation adequately describes the particle geometry. In contrast, it is clearfrom physical considerations that this arrangement is not suitable for oblatescatterers. In this case, the procedure of analytic continuation of the vectorfields onto the complex plane along the source coordinate zn can be used

Ozn

z1

z2

r

r-znezM

x

y

z

Γ (support of DS)

Fig. B.1. Sources distributed on the z-axis


O z, Re z

ρ

Im z

ΣL

L z

η

Fig. B.2. Illustration of the complex plane. L is the generatrix of the surface andL is the image of L in the complex plane

(Fig. B.2). The complex plane Σ = z = (Re z, Im z)/Re z, Im z ∈ R is thedual of the azimuthal plane ϕ = const., Σ = η = (ρ, z)/ρ ≥ 0, z ∈ R, andis defined by taking the real axis Re z along the z-axis. The vector sphericalwave functions can be expressed in terms of the coordinates of the sourcepoint z ∈ Σ and the field point η ∈ Σ as

M1,3mn(kr) =

1√

2n(n + 1)z1,3

n (kR)

jmπ|m|n

(θ) [

sin(θ − θ)er

+ cos(θ − θ)eθ

]− τ |m|

n

(θ)

eϕ

ejmϕ (B.31)

and

N1,3mn(kr) =

1√

2n(n + 1)

n(n + 1)

z1,3n (kR)

kRP |m|

n (cos θ)

×[cos(θ − θ)er − sin(θ − θ)eθ

]+

[kRz1,3

n (kR)]′

kR

×

τ |m|n

(θ) [

sin(θ − θ)er + cos(θ − θ)eθ

]

+ jmπ|m|n

(θ)

eϕ

ejmϕ, (B.32)

where

R2 = ρ2 + (z − z)2, sin θ =ρ

Rand cos θ =

z − z

R.

B.3 Rotations

We consider two coordinate systems Oxyz and Ox1y1z1 having the same ori-gin. The coordinate system Ox1y1z1 is obtained by rotating the coordinate

B.3 Rotations 271

x

O

z

y

z1β

αγ

x1

y1

θ

θ1

M

r

Fig. B.3. Coordinate rotations

system Oxyz thought the Euler angles (α, β, γ) as shown in Fig. B.3. With(θ, ϕ) and (θ1, ϕ1) being the spherical angles of the same position vector r inthe coordinate systems Oxyz and Ox1y1z1, the addition theorem for sphericalwave functions under coordinate rotations is [58,213,239]

u1,3mn(kr, θ, ϕ) =

n∑

m′=−n

Dnmm′(α, β, γ)u1,3

m′n(kr, θ1, ϕ1), (B.33)

where the Wigner D-functions are defined as [262]

Dnmm′(α, β, γ) = (−1)m+m′

ejmαdnmm′(β)ejm′γ . (B.34)

The functions d are given by

dnmm′(β) = ∆mm′dn

mm′(β), (B.35)

where dnmm′ are the Wigner d-functions and

∆mm′ =

⎧⎪⎪⎨

⎪⎪⎩

1, m ≥ 0 , m′ ≥ 0(−1)m′

, m ≥ 0 , m′ < 0(−1)m , m < 0 , m′ ≥ 0(−1)m+m′

, m < 0 , m′ < 0,

(B.36)

with the property ∆mm′ = ∆m′m. The expression of Wigner d-functions forpositive and negative values of the indices m and m′ is given by

dnmm′(β) =

√(n + m′)!(n − m′)!(n + m)!(n − m)!

∑

σ

(−1)n−m′−σCn−m′−σn+m Cσ

n−m

×(

cosβ

2

)m+m′+2σ (sin

β

2

)2n−m−m′−2σ


and note that the above equation is valid for β < 0, if

cos(

β

2

)=

√1 + cos β

2, sin

(β

2

)= −√

1 − cos β

2

and for β > π, if

cos(

β

2

)= −√

1 + cos β

2, sin

(β

2

)=

√1 − cos β

2.

The Wigner d-functions are real and have the following symmetry proper-ties [58]:

dnm−m′(β) = (−1)n+m′

dnmm′(β + π), (B.37)

dn−mm′(β) = (−1)n+mdn

mm′(β + π), (B.38)

dn−m−m′(β) = (−1)m+m′

dnmm′(β), (B.39)

dnmm′(β) = (−1)m+m′

dnm′m(β), (B.40)

and

dnmm′(−β) = (−1)m+m′

dnmm′(β) = dn

m′m(β). (B.41)

Taking into account the above symmetry relations, we can express the d-functions in terms of the d-functions with positive values of the indices mand m′

dnmm′(β) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

dnmm′(β) , m ≥ 0 , m′ ≥ 0,

(−1)ndnm−m′(β + π) , m ≥ 0 , m′ < 0,

(−1)ndn−mm′(β + π) , m < 0 , m′ ≥ 0,

dn−m−m′(β), m < 0 , m′ < 0.

(B.42)

The orthogonality property of the Wigner d-functions is similar to that of theassociated Legendre functions and is given by

∫ π

0

dnmm′(β)dn′

mm′(β) sin βdβ =2

2n + 1δnn′ . (B.43)

The Wigner d-functions are related to the generalized spherical functionsPn

mm′ by the relation [78]

dnmm′(β) = jm

′−mPnmm′ (cos β) . (B.44)

The generalized spherical functions are complex and have the following sym-metry properties:

B.3 Rotations 273

Pnm−m′(x) = (−1)nPn

mm′(−x),

Pn−mm′(x) = (−1)nPn

mm′(−x),

Pn−m−m′(x) = Pn

mm′(x),

Pnmm′(x) = Pn

m′m(x),

where x = cos β. The orthogonality relation for the generalized spherical func-tions follows from the orthogonality relation for the Wigner d-function:

∫ π

0

Pnmm′ (cos β) Pn′

mm′ (cos β) sin βdβ =(−1)m+m′

2n + 1δnn′ .

In practice, the generalized spherical functions can be found from the recur-rence relation [162]

n√

(n + 1)2 − m2√

(n + 1)2 − m′2Pn+1mm′(x)

= (2n + 1) [n(n + 1)x − mm′] Pnmm′(x)

−(n + 1)√

n2 − m2√

n2 − m′2Pn−1mm′(x)

with the initial values

Pn0−1mm′ (x) = 0,

Pn0mm′(x) =

(−j)|m−m′|2n0

√(2n0)!

(|m − m′|)! (|m + m′|)!

× (1 − x)|m−m′|

2 (1 + x)|m+m′|

2 ,

and n0 = max(|m|, |m′|). From (B.42) and (B.44) we see that it is sufficient tocompute the generalized spherical functions for positive values of the indicesm and m′.

If the Euler angles (α1, β1, γ1) and (α2, β2, γ2) describe two consecutiverotations of a coordinate system and the Euler angles (α, β, γ) describes theresulting rotation, the addition theorem for the D-functions is [169,239]

Dnmm′(α, β, γ) =

n∑

m′′=−n

Dnmm′′(α1, β1, γ1)Dn

m′′m′(α2, β2, γ2) (B.45)

and the unitarity condition read as

n∑

m′′=−n

Dnmm′′(α1, β1, γ1)Dn

m′′m′(−γ1,−β1,−α1) = Dnmm′(0, 0, 0) = δmm′ .

(B.46)


If in (B.45) we set α1 = α2 = 0 and γ1 = γ2 = 0, then β = β1 + β2, and weobtain the addition theorem for the d-functions

dnmm′(β) =

n∑

m′′=−n

dnmm′′(β1)dn

m′′m′(β2).

In particular, when β2 = −β1 we derive the unitarity condition

n∑

m′′=−n

dnmm′′(β1)dn

m′′m′(−β1) =n∑

m′′=−n

(−1)m′+m′′dn

mm′′(β1)dnm′′m′(β1)

=n∑

m′′=−n

dnmm′′(β1)dn

m′m′′(β1)=dnmm′(0)=δmm′ .

The product of two d-functions can be expanded in terms of the Clebsch–Gordan coefficients Cm+m1u

mn,m1n1

dnmm′(β)dn1

m1m′1(β) =

n+n1∑

u=|n−n1|Cm+m1u

mn,m1n1du

m+m1m′+m′1(β) C

m′+m′1u

m′n,m′1n1

, (B.47)

and note that the Clebsch–Gordan coefficients are nonzero only when |n −n1| ≤ u ≤ n + n1. The following symmetry properties of the Clebsch–Gordancoefficients are used in our analysis [169,239]:

Cm+m1umn,m1n1

= (−1)n+n1+uC−m−m1u−mn,−m1n1

, (B.48)

Cm+m1umn,m1n1

= (−1)n+n1+uCm+m1um1n1,mn, (B.49)

Cm+m1umn,m1n1

= (−1)m+n

√2u + 12n1 + 1

C−m1n1mn,−m−m1u, (B.50)

Cm+m1umn,m1n1

= (−1)m1+n+u

√2u + 12n + 1

Cmnm+m1u,−m1n1

. (B.51)

To compute the Clebsch–Gordan coefficients we first define the coefficientsSu

mn,m1n1by the relation [77]

Cm+m1umn,m1n1

= gumn,m1n1

Sumn,m1n1

,

where

gumn,m1n1

=√

2u + 1

√(u + m + m1)!(u − m − m1)!

(n − m)!(n + m)!

×√

(n + n1 − u)! (u + n − n1)! (u − n + n1)!(n1 − m1)!(n1 + m1)! (n + n1 + u + 1)!

B.3 Rotations 275

The S-coefficients obey the three-term downward recurrence relation

Su−1mn,m1n1

= puSumn,m1n1

+ quSu+1mn,m1n1

for u = n + n1, n + n1 − 1, . . . ,max(|m + m1|, |n − n1|) with

pu =(2u + 1) (m − m1)u(u + 1) − (m + m1) [n (n + 1) − n1 (n1 + 1)]

(u + 1) (n + n1 − u + 1) (n + n1 + u + 1),

qu = −u(u + n − n1 + 1) (u − n + n1 + 1)(u + 1) (n + n1 − u + 1)

× (u + m + m1 + 1) (u − m − m1 + 1)n + n1 + u + 1

and the starting values

Sn+n1+1mn,m1n1

= 0,

Sn+n1mn,m1n1

= 1.

The rotation addition theorem for vector spherical wave functions is [213]

M1,3mn (kr, θ, ϕ) =

n∑

m′=−n

Dnmm′(α, β, γ)M1,3

m′n(kr, θ1, ϕ1), (B.52)

N1,3mn (kr, θ, ϕ) =

n∑

m′=−n

Dnmm′(α, β, γ)N1,3

m′n(kr, θ1, ϕ1) , (B.53)

and in matrix form we have[

M1,3mn (kr, θ, ϕ)

N1,3mn (kr, θ, ϕ)

]

= R (α, β, γ)

[M1,3

m′n′(kr, θ1, ϕ1)

N1,3m′n′(kr, θ1, ϕ1)

]

,

where R is the rotation matrix. The rotation matrix has a block-diagonalstructure and is given by

R (α, β, γ) =[

Rmn,m′n′ (α, β, γ) 00 Rmn,m′n′ (α, β, γ)

],

where

Rmn,m′n′ (α, β, γ) = Dnmm′(α, β, γ)δnn′ .

The unitarity condition for the D-functions yields

R (−γ,−β,−α) = R−1 (α, β, γ) (B.54)


and since

Dnm′m (−γ,−β,−α) = Dn∗

mm′(α, β, γ),

we also haveR (−γ,−β,−α) = R† (α, β, γ) , (B.55)

where X† stands for the complex conjugate transpose of the matrix X.Properties of Wigner functions and generalized spherical functions (which

have been introduced in the quantum theory of angular momentum) are alsodiscussed in [27,160].

B.4 Translations

We consider two coordinate systems Oxyz and Ox1y1z1 having identical spa-tial orientations but different origins (Fig. B.4). The vectors r and r1 arethe position vectors of the same field point in the coordinate systems Oxyzand Ox1y1z1, respectively, while the vector r0 connects the origins of bothcoordinate systems and is given by r0 = r − r1.

In general, the addition theorem for spherical wave functions can be writ-ten as [27,70]

umn (kr) =∞∑

n′=0

n′∑

m′=−n′

Cmn,m′n′ (kr0) um′n′ (kr1) .

Integral and series representations for the translation coefficients Cmn,m′n′

can be obtained by using the integral representations for the spherical wave

x

O

z

y

z1

x1

y1

M

r r1

r0O1

Fig. B.4. Coordinate translation

B.4 Translations 277

functions. For regular spherical wave functions, we use the integral represen-tation (B.1), the relation r = r0 + r1, and the spherical wave expansion ofthe plane wave exp(jk · r1) to obtain

u1mn (kr) =

∞∑

n′=0

n′∑

m′=−n′

C1mn,m′n′ (kr0) u1

m′n′ (kr1) (B.56)

with

C1mn,m′n′ (kr0) =

jn′−n

2π

∫ 2π

0

∫ π

0

Ymn (β, α) Y−m′n′ (β, α) ejk(β,α)·r0 sin β dβ dα .

(B.57)Taking into account the spherical wave expansion of the plane wave exp(jk·r0)and integrating over α, we find the series representation

C1mn,m′n′ (kr0) = 2jn

′−n∑

n′′

jn′′a (m,m′ | n′′, n, n′) u1

m−m′n′′ (kr0) (B.58)

with the expansion coefficients

a (m,m′ | n′′, n, n′) =∫ π

0

P |m|n (cos β) P

|m′|n′ (cos β) P

|m−m′|n′′ (cos β) sin βdβ.

We note that the expansion coefficients a(·) are defined by the spherical har-monic expansion theorem [70]

P |m|n (cos β) P

|m′|n′ (cos β) =

∑

n′′

a (m,−m′ | n′′, n, n′) P|m+m′|n′′ (cos β) ,

where the summation over n′′ is finite covering the range |n − n′|, |n − n′| +2, ..., n+n′. These coefficients can be identified with a product of two Wigner3j symbols, which are associated with the coupling of two angular momen-tum eigenvectors. The azimuthal integration in (B.57) can be analyticallyperformed by using the identity

k · r0 = kρ0 sin β cos (α − ϕ0) + kz0 cos β

and the standard integral

∫ 2π

0

ejx cos(α−ϕ0)ej(m−m′)α dα = 2πjm′−mJm′−m(x)e−j(m′−m)ϕ0 ,

where (ρ0, ϕ0, z0) are the cylindrical coordinates of r0, and the result is

C1mn,m′n′ (kr0) = jn

′+m′−n−me−j(m′−m)ϕ0

∫ π

0

Jm′−m (kρ0 sinβ)

×P |m|n (cos β) P

|m′|n′ (cos β) ejkz0 cos β sinβ dβ . (B.59)


If the translation is along the z-axis, the addition theorem involves a singlesummation

u1mn (kr) =

∞∑

n′=0

C1mn,mn′ (kz0) u1

mn′ (kr1)

and the translation coefficients simplify to

C1mn,mn′ (kz0) = jn

′−n

∫ π

0

P |m|n (cos β) P

|m|n′ (cos β) ejkz0 cos β sinβ dβ. (B.60)

For radiating spherical wave functions we consider the integral representation(B.2) and the relation r = r0 + r1. For r1 > r0, we express u3

mn as (cf. (B.5))

u3mn(kr) =

12πjn

∫ 2π

0

∫ π

0

Ymn (β, α) Q(k, β, α, r1)ejk(β,α)·r0 sinβdβdα

and use the spherical wave expansion of the quasi-plane wave Q(k, β, α, r1)to derive

u3mn (kr) =

∞∑

n′=0

n′∑

m′=−n′


m′n′ (kr1) .

For r1 < r0, we represent the radiating spherical wave functions as

u3mn(kr) =

12πjn

∫ 2π

0

∫ π

0

Ymn (β, α) Q(k, β, α, r0)ejk(β,α)·r1 sinβdβdα

and use the spherical wave expansion of the plane wave exp(jk · r1) to obtain

u3mn (kr) =

∞∑

n′=0

n′∑

m′=−n′


m′n′ (kr1)

with

C3mn,m′n′ (kr0)

=jn

′−n

π

∫ 2π

0

∫ π

0

Ymn (β, α) Y−m′n′ (β, α) Q(k, β, α, r0) sin β dβ dα . (B.61)

Inserting the spherical wave expansion of the quasi-plane wave Q(k, β, α, r0)into (B.61) yields

C3mn,m′n′ (kr0) = 2jn

′−n∑

n′′

jn′′a (m,m′ | n′′, n, n′) u3

m−m′n′′ (kr0) . (B.62)

Finally, we note the integral representation for the translation coefficientsC3

mn,m′n′ in the specific case of axial translation:


C3mn,mn′ (kz0) = 2jn

′−n

∫ π2 −j∞

0

P |m|n (cos β) P

|m|n′ (cos β) ejkz0 cos β sinβdβ .

(B.63)Recurrence relations have been derived for the Cmn,mn′ coefficients [150].

The derivation is simple for the case of axial translation and positive m. Usingthe integral representation (B.60) (or (B.63)) and the recurrence relations forthe normalized associated Legendre functions (A.15) and (A.16), give

√(n − m − 1)(n − m + 1)

(2n − 1)(2n + 1)Cmn−1,mn′ (kz0)

+

√(n + m)(n + m + 1)

(2n + 1)(2n + 3)Cmn+1,mn′ (kz0)

=

√(n′ + m − 1)(n′ + m)

(2n′ − 1)(2n′ + 1)Cm−1n,m−1n′−1 (kz0)

+

√(n′ − m + 1)(n′ − m + 2)

(2n′ + 1)(2n′ + 3)Cm−1n,m−1n′+1 (kz0) , (B.64)

while the recurrence relation (A.14), yields√

(n − m + 1)(n + m + 1)(2n + 1)(2n + 3)

Cmn+1,mn′ (kz0)

−√

(n − m)(n + m)(2n − 1)(2n + 1)

Cmn−1,mn′ (kz0)

=

√(n′ − m)(n′ + m)(2n′ − 1)(2n′ + 1)

Cmn,mn′−1 (kz0)

−√

(n′ − m + 1)(n′ + m + 1)(2n′ + 1)(2n′ + 3)

Cmn,mn′+1 (kz0) . (B.65)

We note that the convention Cmn,mn′ = 0 for m > n or m > n′, is assumed inthe above equations. For the recurrence relationships to be of practical use,initial values are needed. This is accomplished by using the integral represen-tations

jn (kz0) =1

jn√

2(2n + 1)

∫ π

0

Pn (cos β) ejkz0 cos β sin β dβ,

h(1)n (kz0) =

1jn

√2

2n + 1

∫ π2 −j∞

0

Pn (cos β) ejkz0 cos β sin β dβ ,


which yields

C100,0n′ (kz0) = (−1)n′ √

2n′ + 1jn′ (kz0) ,

C300,0n′ (kz0) = (−1)n′ √

2n′ + 1h(1)n′ (kz0) .

Using these starting values, the Cmm,mn′ coefficients can be computed fromthe recurrence relation (B.64) with m = n + 1,

Cmm,mn′(kz0) =

√2m + 1

2m

[√(n′ + m − 1) (n′ + m)

(2n′ − 1) (2n′ + 1)Cm−1m−1,m−1n′−1(kz0)

+

√(n′ − m + 1) (n′ − m + 2)

(2n′ + 1) (2n′ + 3)Cm−1m−1,m−1n′+1 (kz0)

]

,

where Cmm,mn′ can be obtained from C00,0n′ for all values of m ≥ 1 andn′ ≥ m, while (B.65) can then be used to compute Cmn,mn′ for n ≥ m + 1.

In general, the translation addition theorem for vector spherical wave func-tions can be written as [43,213]

Mmn (kr) =∞∑

n′=1

n′∑

m′=−n′

Amn,m′n′(kr0)Mm′n′(kr1)

+Bmn,m′n′(kr0)Nm′n′(kr1),

Nmn (kr) =∞∑

n′=1

n′∑

m′=−n′

Bmn,m′n′(kr0)Mm′n′(kr1)

+Amn,m′n′(kr0)Nm′n′(kr1).

As in the scalar case, integral and series representations for the translationcoefficients can be obtained by using the integral representations for the vectorspherical wave functions. First we consider the case of regular vector sphericalwave functions. Using the integral representation (B.26), the relation r =r0 + r1, and the vector spherical wave expansion

(−j)mmn (β, α) ejk(β,α)·r1 =∞∑

n′=1

n′∑

m′=−n′

a1mn,m′n′M1

m′n′ (kr1)

+b1mn,m′n′N1

m′n′ (kr1)


with

a1mn,m′n′ = − 4jn

′+1

√2n′ (n′ + 1)

[mm′π|m|

n (β)π|m′|

n′ (β)

+ τ |m|n (β)τ |m

′|n′ (β)

]e−jm′α,

b1mn,m′n′ = − 4jn

′+1

√2n′ (n′ + 1)

[mπ|m|

n (β)τ |m′|

n′ (β)

+ m′τ |m|n (β)π|m

′|n′ (β)

]e−jm′α,

we obtain

M1mn (kr) =

∞∑

n′=1

n′∑

m′=−n′

A1mn,m′n′(kr0)M1

m′n′(kr1)

+B1mn,m′n′(kr0)N1

m′n′(kr1), (B.66)

N1mn (kr) =

∞∑

n′=1

n′∑

m′=−n′

B1mn,m′n′(kr0)M1

m′n′(kr1)

+A1mn,m′n′(kr0)N1

m′n′(kr1), (B.67)

where

A1mn,m′n′(kr0) =

jn′−n

2π√

nn′(n + 1) (n′ + 1)

∫ 2π

0

∫ π

0

[mm′π|m|

n (β)π|m′|

n′ (β)

+ τ |m|n (β)τ |m

′|n′ (β)

]ej(m−m′)αejk(β,α)·r0 sin β dβdα,

B1mn,m′n′(kr0) =

jn′−n

2π√

nn′(n + 1) (n′ + 1)

∫ 2π

0

∫ π

0

[mπ|m|

n (β)τ |m′|

n′ (β)


′|n′ (β)

]ej(m−m′)αejk(β,α)·r0 sin βdβ dα.


Inserting the spherical wave expansion of the plane wave exp(jk ·r0) and inte-grating over α, we derive the following series representation for the translationcoefficients:


2jn′−n

√nn′(n + 1) (n′ + 1)

∑

n′′

jn′′

×a1 (m,m′ | n′′, n, n′) u1m−m′n′′ (kr0) ,


2jn′−n

√nn′(n + 1) (n′ + 1)

∑

n′′

jn′′

×b1 (m,m′ | n′′, n, n′) u1m−m′n′′ (kr0) ,

where

a1 (m,m′ | n′′, n, n′) =∫ π

0

[mm′π|m|

n (β)π|m′|

n′ (β) + τ |m|n (β)τ |m

′|n′ (β)

]

×P|m−m′|n′′ (cos β) sin β dβ, (B.68)

b1 (m,m′ | n′′, n, n′) =∫ π

0

[mπ|m|

n (β) τ|m′|n′ (β) + m′τ |m|

n (β)π|m′|

n′ (β)]

×P|m−m′|n′′ (cos β) sinβ dβ. (B.69)

We note that the coefficients a1(·) and b1(·) can be expressed in terms of thecoefficients a(·) by making use of the recurrence relations for the associatedLegendre functions. As in the scalar case, the integration with respect to theazimuthal angle α gives


jn′+m′−n−m

√nn′(n + 1) (n′ + 1)

e−j(m′−m)ϕ0

×∫ π

0

Jm′−m(kρ0 sin β)[mm′π|m|

n (β)π|m′|n′ (β)

+ τ |m|n (β)τ |m′|

n′ (β)]ejkz0 cos β sin β dβ


and


jn′+m′−n−m

√nn′(n + 1) (n′ + 1)

e−j(m′−m)ϕ0

×∫ π

0

Jm′−m(kρ0 sin β)[mπ|m|

n (β)τ |m′|n′ (β)

+ m′τ |m|n (β)π|m′|

n′ (β)]ejkz0 cos β sinβdβ.

If the translation is along the z-axis the double summation in (B.66) and(B.67) reduces to a single summation over the index n′, and we have

A1mn,mn′(kz0) =

jn′−n

√nn′(n + 1) (n′ + 1)

∫ π

0

[m2π|m|

n (β)π|m|n′ (β)

+ τ |m|n (β)τ |m|

n′ (β)]ejkz0 cos β sinβ dβ

and

B1mn,mn′(kz0) =

mjn′−n

√nn′(n + 1) (n′ + 1)

∫ π

0

[π|m|

n (β)τ |m|n′ (β)

+ τ |m|n (β)π|m|

n′ (β)]ejkz0 cos β sinβ dβ.

Passing to the radiating vector spherical wave functions we consider the in-tegral representation (B.28) and the relation r = r0 + r1. For r1 > r0, thisrepresentation can written as

M3mn(kr) = − 1

2πjn+1

∫ 2π

0

∫ π

0

(−j)mmn (β, α) Q(k, β, α, r1)ejk(β,α)·r0

× sin β dβ dα,

whence, using the vector spherical wave expansion

(−j)mmn (β, α) Q(k, β, α, r1) =∞∑

n′=1

n′∑

m′=−n′

a3mn,m′n′M3

m′n′ (kr1)

+b3mn,m′n′N1

m′n′ (kr1) ,

with

a3mn,m′n′ =

12a1

mn,m′n′ , b3mn,m′n′ =

12b1mn,m′n′ ,


yields

M3mn (kr) =

∞∑

n′=1

n′∑

m′=−n′


m′n′(kr1)


m′n′(kr1),

N3mn (kr) =

∞∑

n′=1

n′∑

m′=−n′


m′n′(kr1)


m′n′(kr1).

For r1 < r0, we represent the radiating vector spherical wave functions as

M3mn(kr) = − 1

2πjn+1

∫ 2π

0

∫ π

0

(−j)mmn (β, α) Q(k, β, α, r0)ejk(β,α)·r1

× sin β dβ dα,

proceed as in the case of regular vector spherical wave functions, and obtain

M3mn (kr) =

∞∑

n′=1

n′∑

m′=−n′


m′n′(kr1)


m′n′(kr1),

N3mn (kr) =

∞∑

n′=1

n′∑

m′=−n′


m′n′(kr1)


m′n′(kr1),

where


jn′−n

π√

nn′(n + 1) (n′ + 1)

∫ 2π

0

∫ π

0

[mm′π|m|

n (β)π|m′|

n′ (β)(B.70)

+ τ |m|n (β)τ |m

′|n′ (β)

]ej(m−m′)αQ(k, β, α, r0) sin βdβdα,


jn′−n

π√

nn′(n + 1) (n′ + 1)

∫ 2π

0

∫ π

0

[mπ|m|

n (β)τ |m′|

n′ (β) (B.71)


′|n′ (β)

]ej(m−m′)αQ(k, β, α, r0) sin β dβ dα .

Making use of the spherical wave expansion of the quasi-plane waveQ(k, β, α, r0), we derive the series representations



2jn′−n

√nn′(n + 1) (n′ + 1)

∑

n′′

jn′′

×a1 (m,m′ | n′′, n, n′) u3m−m′n′′ (kr0) , (B.72)


2jn′−n

√nn′(n + 1) (n′ + 1)

∑

n′′

jn′′

×b1 (m,m′ | n′′, n, n′) u3m−m′n′′ (kr0) . (B.73)

In (B.70) and (B.71) the integration with respect to the azimuthal angle αcan be analytically performed and in the case of axial translation, the integralrepresentations for the translation coefficients A3

mn,m′n′ and B3mn,m′n′ read as

A3mn,mn′(kz0) =

2jn′−n

√nn′(n + 1) (n′ + 1)

∫ π2 −j∞

0

[m2π|m|

n (β)π|m|n′ (β)

+ τ |m|n (β)τ |m|

n′ (β)]ejkz0 cos β sin β dβ,

and

B3mn,mn′(kz0) =

2mjn′−n

√nn′(n + 1) (n′ + 1)

∫ π2 −j∞

0

[π|m|

n (β)τ |m|n′ (β)

+ τ |m|n (β)π|m|

n′ (β)]ejkz0 cos β sin β dβ.

The vector addition coefficients Amn,m′n′ and Bmn,m′n′ can be related to thescalar addition coefficients Cmn,m′n′ . For axial translations and positive valuesof m, these relations can be obtained by partial integration and by using theintegral representations for the addition coefficients, the associated Legendreequation (A.13) and the recurrence relation (A.17). We obtain [150]

Amn,mn′(kz0) =

√n′(n′ + 1)n(n + 1)

[Cmn,mn′ (kz0)

+kz0

n′ + 1

√(n′ − m + 1)(n′ + m + 1)

(2n′ + 1)(2n′ + 3)Cmn,mn′+1 (kz0)

+kz0

n′

√(n′ − m)(n′ + m)(2n′ + 1)(2n′ − 1)

Cmn,mn′−1 (kz0)

]

and

Bmn,mn′(kz0) = jkz0m

√nn′(n + 1) (n′ + 1)

Cmn,mn′ (kz0) .


For negative values of the index m, the following symmetry relations can beused for practical calculations

A−mn,−mn′(kz0) = Amn,mn′(kz0),

B−mn,−mn′(kz0) = −Bmn,mn′(kz0).

The translation coefficients for axial translation can be obtained from theabove recurrence relations. In general, through rotation of coordinates, thenumerical advantages to a common axis can be exploited and a transformationfrom Oxyz to Ox1y1z1 can be accomplished through three steps [150]:

1. The coordinate system Oxyz is rotated with the Euler angles α = ϕ0,β = θ0 and γ = 0, where (r0, θ0, ϕ0) are the spherical coordinates of theposition vector r0.

2. The rotated coordinate system is axially translated with r0.3. The translated coordinate system is rotated back to the original orienta-

tion with the Euler angles α = 0, β = −θ0 and γ = −ϕ0.

The rotation-axial translation-rotation scheme gives

Xmn,m′n′(kr0) =n∑

m′′=−n

Dnmm′′(ϕ0, θ0, 0)Xm′′n,m′′n′ (kr0)

×Dn′

m′′m′(0,−θ0,−ϕ0),

where X stands for A or B. The translation addition theorem can be writtenin matrix form as

[Mp

mn (kr)Np

mn (kr)

]

= T pq (kr0)

[M q

m′n′ (kr1)N q

m′n′ (kr1)

]

,

where the pair (p, q) takes the values (1, 1), (3, 3), and (3, 1), and

T 11 (kr0) = T 33 (kr0) =

[A1

mn,m′n′ (kr0) B1mn,m′n′ (kr0)

B1mn,m′n′ (kr0) A1

mn,m′n′ (kr0)

]

and

T 31 (kr0) =

[A3

mn,m′n′ (kr0) B3mn,m′n′ (kr0)

B3mn,m′n′ (kr0) A3

mn,m′n′ (kr0)

]

.

We conclude this section by noticing some symmetry properties of the trans-lation coefficients for the inverse transformation:

[Mp

mn (kr1)Np

mn (kr1)

]

= T pq (−kr0)

[M q

m′n′ (kr)N q

m′n′ (kr)

]

.


Using the integral representation for the translation coefficients, making thetransformation ϕ0 → ϕ0 + π, changing the variable of integration from β toπ − β and using the identities (A.24) and (A.25), yields

Amn,m′n′ (−kr0) = (−1)n+n′Amn,m′n′ (kr0) ,

Bmn,m′n′ (−kr0) = (−1)n+n′+1Bmn,m′n′ (kr0) .

Further, since

Amn,m′n′ (kr0) = (−1)n+n′A−m′n′,−mn (kr0) ,

Bmn,m′n′ (kr0) = (−1)n+n′+1B−m′n′,−mn (kr0) ,

we obtain

Amn,m′n′ (−kr0) = A−m′n′,−mn (kr0) , (B.74)

Bmn,m′n′ (−kr0) = B−m′n′,−mn (kr0) . (B.75)

Recurrence relations for the scalar and vector addition theorem has also beengiven by Chew [32,33], Chew and Wang [35] and Kim [117]. The relationshipbetween the coefficients of the vector addition theorem and those of the scalaraddition theorem has been discussed by Bruning and Lo [29], and Chew [32].

C

Computational Aspectsin Effective Medium Theory

In this appendix we compute the basic integrals appearing in the analysis ofelectromagnetic scattering from a half-space of randomly distributed particles.Our derivation follows the procedures described by Varadan et al. [236], Tsangand Kong [223,226], and Tsang et al. [228].

C.1 Computation of the Integral I1mm′n′′

The integral I1mm′n′′ is

I1mm′n′′ =

∫

Dp

ejKe·r0pu3m′−mn′′ (ksrlp) dV (r0p) ,

where Ke = Ksez, and the integration domain Dp is the half-space z0p ≥ 0,excluding a spherical volume of radius 2R centered at r0l. The volume integralcan be transformed into a surface integral by making use of the followingresult. Let u and v be two scalar fields satisfying the Helmholtz equation inthe bounded domain D, with the wave numbers Ks and ks, i.e.,

∆u + K2s u = 0,

∆v + k2s v = 0.

Then, from Green’s theorem we have∫

D

uvdV =1

k2s − K2

s

∫

D

(v∆u − u∆v) dV

=1

k2s − K2

s

∫

S

(v

∂u

∂n− u

∂v

∂n

)dS,

where S is the boundary surface of the domain D, and n is the outward unitnormal vector to S. Using this result we transform the integral I1

mm′n′′ asfollows:

290 C Computational Aspects in Effective Medium Theory

O

Op

Olr0l

z

R

r0p rlpθlp

SR

Sz

S

Ω

2R

∞

∞

Fig. C.1. Integration surfaces SR, S∞ and Sz

I1mm′n′′ =

1k2s − K2

s

∫

SR∪S∞∪Sz

[u3

m′−mn′′ (ksrlp)∂

∂n

(ejKe·r0p

)

− ejKe·r0p∂u3

m′−mn′′

∂n(ksrlp)

]dS (r0p) .

The integral can be decomposed into three integrals

I1mm′n′′ = I1,R

mm′n′′ + I1,∞mm′n′′ + I1,z

mm′n′′ ,

where I1,Rmm′n′′ is the integral over the spherical surface SR of radius 2R cen-

tered at r0l, I1,∞mm′n′′ is the integral over the surface of a half-sphere S∞ with

radius R∞ in the limit R∞ → ∞, and I1,zmm′n′′ is the integral over the xy-

plane Sz (the plane z = 0). The choice of the integration surfaces is shown inFig. C.1.

Using the identity r0p = r0l + rlp and replacing, for convenience, thevariables rlp, θlp and ϕlp by r, θ and ϕ, respectively, we obtain

I1,Rmm′n′′ = − 1

k2s − K2

s

ejKsz0l

×∫

SR

h

(1)n′′ (ksr) P

|m′−m|n′′ (cos θ) ej(m′−m)ϕ ∂

∂r

(ejKe·r)

− ejKe·r ∂

∂r

[h

(1)n′′ (ksr) P

|m′−m|n′′ (cos θ) ej(m′−m)ϕ

]dS(r),

whence, taking into account the series representation

ejKe·r =∞∑

n′=0

2jn′√

2n′ + 12

jn′ (Ksr) Pn′ (cos θ) (C.1)

C.1 Computation of the Integral I1mm′n′′ 291

for Ke · r = Ksr cos θ, the orthogonality of the associated Legendre functionsand the relation Ke · r0l = Ksz0l, we end up with

I1,Rmm′n′′ =

16πR3

(ksR)2 − (KsR)2jn

′′√

2n′′ + 12

ejKsz0lδmm′Fn′′ (Ks, ks, R) ,

where

Fn′′ (Ks, ks, R) = (ksR)(h

(1)n′′ (2ksR)

)′jn′′ (2KsR)

− (KsR)h(1)n′′ (2ksR) (jn′′ (2KsR))′ .

To compute I1,∞mm′n′′ we use the stationary point method. Using the inequality

ImKs > 0 and the asymptotic expressions of the spherical Hankel functionsof the first kind

h(1)n′′ (ksr) =

(−j)n′′+1ejksr

ksr,

and

ddr

[h

(1)n′′ (ksr)

]=

j(−j)n′′+1ejksr

r,

we see that I1,∞mm′n′′ vanishes as R∞ → ∞.

To evaluate I1,zmm′n′′ we pass to cylindrical coordinates, integrate over ϕ,

and obtain

I1,zmm′n′′ = − 2π

k2s − K2

s

δmm′

∫ ∞

0

jKsh

(1)n′′ (ksr) Pn′′ (cos θ)

− ∂

∂z

[h

(1)n′′ (ksr) Pn′′ (cos θ)

]

z=−z0l

ρdρ.

Further, using Kasterin’s representation [228,258]

h(1)n′′ (ksr) Pn′′ (cos θ) = (−j)n′′

Pn′′

(1

jks

∂

∂z

)h

(1)0 (ksr)

with

h(1)0 (ksr) =

ejksr

jksr,

we find that

I1,zmm′n′′ = − 2π

k2s − K2

s

(−j)n′′δmm′

×∫ ∞

0

[(jKs −

∂

∂z

)Pn′′

(1

jks

∂

∂z

)ejks

√ρ2+z2

jks

√ρ2 + z2

]

z=−z0l

ρdρ.


Performing the integration and using the relation

Pn′′

(1

jks

∂

∂z

)(e−jksz

k2s

)= (−1)n′′ 1

k2s

√2n′′ + 1

2e−jksz,

we obtain

I1,zmm′n′′ =

2π

k2s (Ks − ks)

jn′′+1

√2n′′ + 1

2ejksz0lδmm′ .

The final result is

I1mm′n′′ = I1,R

mm′n′′ + I1,zmm′n′′

=16πR3

(ksR)2 − (KsR)2jn

′′√

2n′′ + 12

ejKsz0lδmm′Fn′′ (Ks, ks, R)

+2π

k2s (Ks − ks)

jn′′+1

√2n′′ + 1

2ejksz0lδmm′ . (C.2)

C.2 Computation of the Integral I2mm′n′′

The integral I2mm′n′′ is

I2mm′n′′ =

∫

Dp

[g (rlp) − 1] ejKe·r0pu3m′−mn′′ (ksrlp) dV (r0p) ,

where Ke = Ksez, and the domain of integration contains the half-spacez0p ≥ 0 less a sphere of radius 2R centered at r0l. For a spherical symmetricpair distribution function, g(rlp) = g(rlp), the function g(rlp)−1 tends to zerofor a few 2R-values. Therefore, if the point r0l is at least several diametersdeep in the scattering medium, the volume of integration can be extended toinfinity (Fig. C.2). Taking into account that r0p = r0l +rlp, and replacing thevariables rlp, θlp and ϕlp by r, θ and ϕ, respectively, we obtain

I2mm′n′′ = ejKe·r0l

∫ 2π

0

∫ π

0

∫ ∞

2R

[g(r) − 1] ejKe·rh(1)n′′ (ksr) P

|m′−m|n′′ (cos θ)

×ej(m′−m)ϕr2 sin θ dr dθ dϕ .

Using the plane wave expansion (C.1), the orthogonality relations of the as-sociated Legendre functions and the equation Ke · r0l = Ksz0l, we derive

I2mm′n′′ = 4πjn

′′√

2n′′ + 12

ejKsz0lδmm′

∫ ∞

2R

[g(r) − 1] jn′′ (Ksr) h(1)n′′ (ksr) r2dr.

Changing the variables from r to x = r/(2R), we find that

C.3 Computation of the Terms S11nn′ and S2

1nn′ 293

Op

Ol

R∞

rlpθlp

SR

S∞2R

Fig. C.2. Illustration of the domain of integration bounded by the spherical surfaceSR and a sphere situated in the far-field region

I2mm′n′′ = 32πR3jn

′′√

2n′′ + 12

ejKsz0lδmm′Gn′′ (Ks, ks, R) , (C.3)

where

Gn′′ (Ks, ks, R) =∫ ∞

1

[g (2Rx) − 1] h(1)n′′ (2ksRx) jn′′ (2KsRx)x2dx.

C.3 Computation of the Terms S11nn′ and S2

1nn′

The terms S11nn′ and S2

1nn′ are given by

S11nn′ =

∞∑

n′′=0

(−1)n′′√

2n′′ + 12

I11nn′n′′ ,

S21nn′ =

∞∑

n′′=0

(−1)n′′√

2n′′ + 12

I21nn′n′′ ,

where

I11nn′n′′ =

∫ π

0

[π1

n(β)π1n′(β) + τ1

n(β)τ1n′(β)

]Pn′′ (cos β) sin β dβ,

I21nn′n′′ =

∫ π

0

[π1

n(β)τ1n′(β) + τ1

n(β)π1n′(β)

]Pn′′ (cos β) sin β dβ.

To compute S11nn′ , we consider the function

fnn′(β) = π1n(β)π1

n′(β) + τ1n(β)τ1

n′(β) (C.4)


for fixed values of the indices n and n′. This function can be expanded interms of Legendre polynomials

fnn′ (β) =∞∑

n′′=0

ann′,n′′Pn′′ (cos β) , (C.5)

where the expansion coefficients are given by

ann′,n′′ =∫ π

0

fnn′(β)Pn′′ (cos β) sin β dβ = I11nn′n′′ .

Using the special value of the Legendre polynomial at β = π,

Pn′′ (cos π) = (−1)n′′√

2n′′ + 12

,

we see that (C.5) leads to

fnn′(π) =∞∑

n′′=0

(−1)n′′√

2n′′ + 12

I11nn′n′′ = S1

1nn′ .

On the other hand, since

π1n(π) =

(−1)n−1

2√

2

√n(n + 1) (2n + 1),

τ1n(π) =

(−1)n

2√

2

√n(n + 1) (2n + 1).

(C.4) implies that

fnn′(π) = snn′ =(−1)n+n′

4

√n(n + 1) (2n + 1)

√n′(n′ + 1) (2n′ + 1) (C.6)

and therefore,

S11nn′ = snn′ .

Analogously, we find that

S21nn′ = −snn′ .

D

Completenessof Vector Spherical Wave Functions

The completeness and linear independence of the systems of radiating andregular vector spherical wave functions on closed surfaces have been estab-lished by Doicu et al. [49]. In this appendix we prove the completeness andlinear independence of the regular and radiating vector spherical wave func-tions on two enclosing surfaces. A similar result has been given by Aydin andHizal [5] for both the regular and radiating vector spherical wave functionswith nonzero divergence, i.e., the solutions to the vector Helmholtz equation.

Before formulating the boundary-value problem for the Maxwell equa-tions we introduce some normed spaces which are relevant in electromagneticscattering theory. With S being the boundary of a domain Di we denoteby [39,40]

Ctan(S) = a/a ∈ C(S),n · a = 0

the space of all continuous tangential fields and by

C0,αtan(S) =

a/a ∈ C0,α(S),n · a = 0

, 0 < α ≤ 1 ,

the space of all uniformly Holder continuous tangential fields equipped withthe supremum norm and the Holder norm, respectively. The space of uniformlyHolder continuous tangential fields with uniformly Holder continuous surfacedivergence (∇s) is defined as

C0,αtan,d(S) =

a/a ∈ C0,α

tan(S),∇s · a ∈ C0,α(S)

, 0 < α ≤ 1 ,

while the space of square integrable tangential fields is introduced as

L2tan(S) =

a/a ∈ L2(S),n · a = 0

.

Now, let S1 and S2 be two closed surfaces of class C2, with S1 enclosing S2

(Fig. D.1). Let the doubly connected domain between S1 and S2 be denotedby Di,1, the domain interior to S2 by Di,2, the domain interior to S1 by Di,

296 D Completeness of Vector Spherical Wave Functions

O

S1Di,1

Ds

S2Di,2

n1

n2

εi,µi

Fig. D.1. The closed surfaces S1 and S2

and the domain exterior to S1 by Ds. We assume that the origin is in Di,2,and denote by n1 and n2 the outward normal unit vectors to S1 and S2,respectively. The wave number in the domain Di,1 is ki = k0

√εiµi, where k0

is the wave number in free space, and εi and µi are the relative permittivityand permeability of the domain Di,1. We introduce the product spaces

Ctan(S1,2) = Ctan(S1) × Ctan(S2),

C0,αtan(S1,2) = C0,α

tan(S1) × C0,αtan(S2) ,

C0,αtan,d(S1,2) = C0,α

tan,d(S1) × C0,αtan,d(S2) ,

L2tan(S1,2) = L2

tan(S1) × L2tan(S2),

and define the scalar product in L2tan(S1,2) by

⟨(x1

x2

),

(y1

y2

)⟩

2,S1,2

= 〈x1,y1〉2,S1+ 〈x2,y2〉2,S2

.

The Maxwell boundary-value problem for the doubly connected domain Di,1

has the following formulation.Find a solution Ei,H i ∈ C1(Di,1) ∩ C(Di,1) to the Maxwell equations in

Di,1

∇× Ei = jk0µiH i , ∇× H i = −jk0εiEi,

satisfying the boundary conditions

n1 × Ei = f1 on S1 ,

n2 × Ei = f2 on S2 ,

where f1 and f2 are given tangential vector fields.Denoting by σ(Di,1) the spectrum of eigenvalues of the Maxwell boundary-

value problem, we assume that for f1 ∈ C0,αtan,d(S1) and f2 ∈ C0,α

tan,d(S2), both

D Completeness of Vector Spherical Wave Functions 297

Ei and H i belong to C0,α(Di,1), and for ki /∈ σ(Di,1), the Maxwell boundary-value problem possesses an unique solution.

Theorem 1. Consider Di,1 a bounded domain of class C2 with boundaries S1

and S2. Define the vector potentials

A1(r) =∫

S1

a1(r′)g(ki, r, r′)dS(r′) ,

A2(r) =∫

S2

a2(r′)g(ki, r, r′)dS(r′)

for a1 ∈ L2tan(S1) and a2 ∈ L2

tan(S2), assume ki /∈ σ(Di,1) and

∇×∇× A1 + ∇×∇× A2 = 0

in Ds and Di,2. Then a1 ∼ 0 on S1 (a1 vanishes almost everywhere on S1),and a2 ∼ 0 on S2.

Proof. Defining the electromagnetic fields

E =j

k0εi(∇×∇× A1 + ∇×∇× A2) ,

H = − jk0µi

∇× E = ∇× A1 + ∇× A2,

passing to the boundary along a normal direction and using the jump relationsfor the curl of a vector potential with square integrable density [49], we findthat

0 = limh→0+

‖n1 ×H (· + hn1 (·))

−

n1 ×[∇×

∫

S1

a1(r′)g(ki, ·, r′)dS(r′)]

+12a1

− n1 ×[∇×

∫

S2

a2(r′)g(ki, ·, r′)dS(r′)]∥∥∥∥

2,S1

and

0 = limh→0+

‖n2 ×H (· − hn2 (·))

−

n2 ×[∇×

∫

S2

a2(r′)g(ki, ·, r′)dS(r′)]− 1

2a2

− n2 ×[∇×

∫

S1

a1(r′)g(ki, ·, r′)dS(r′)]∥∥∥∥

2,S2

.


Defining the operators

(M11a) (r) = n1 (r) ×[∇×

∫

S1

a(r′)g(ki, r, r′)dS(r′)]

, r ∈ S1,

(M12a) (r) = n1(r) ×[∇×

∫

S2


, r ∈ S1 ,

and

(M21a) (r) = n2(r) ×[∇×

∫

S1


, r ∈ S2 ,

(M22a) (r) = n2(r) ×[∇×

∫

S2


, r ∈ S2 ,

we obtain(

12I + M11

)a1 + M12a2 = 0,

almost everywhere on S1, and

−M21a1 +(

12I −M22

)a2 = 0,

almost everywhere on S2. Note that M11 and M22 are the singular magneticoperators on the surfaces S1 and S2, respectively, while M12 and M21 arenonsingular operators. In compact operator form, we have

a + Ka = 0,

where

a =(

a1

a2

)and K =

[2M11 2M12

−2M21 −2M22

].

The above integral equation is a Fredholm integral equation of the second kind,and according to Mikhlin [161] we find that a ∼ a0 ∈ Ctan(S1,2). Noticing thatthe operator K map Ctan(S1,2) into C

0,αtan(S1,2) and C

0,αtan(S1,2) into C

0,αtan,d(S1,2),

we see that a ∼ a0 ∈ C0,αtan,d(S1,2), where a0 =

(a10

a20

). The properties of the

vector potentials with uniformly Holder continuous densities, then yields E ,H ∈ C0,α(Di,1). The jump relations for the double curl of the vector potentialsA1 and A2 [49], give n1 × E− = 0 on S1, and n2 × E+ = 0 on S2. Therefore,E and H solve the homogeneous Maxwell boundary-value problem, and sinceki /∈ σ(Di,1), it follows that E = H = 0 in Di,1. Finally, from the jumprelations (for continuous densities)

n1 ×H+ − n1 ×H− = a10 = 0,


and

n2 ×H+ − n2 ×H− = a20 = 0,

we obtain a1 ∼ 0 on S1, and a2 ∼ 0 on S2. The following theorems state the completeness and linear independence of

the system of regular and radiating spherical vector wave functions on twoenclosing surfaces.

Theorem 2. Let S1 and S2 be two closed surfaces of class C2, with S1 en-closing S2, and let n1 and n2 be the outward normal unit vectors to S1 andS2, respectively. The system of vector functions(

n1 × M3mn

n2 × M3mn

)

,

(n1 × N3

mn

n2 × N3mn

)

,

(n1 × M1

mn

n2 × M1mn

)

,

(n1 × N1

mn

n2 × N1mn

)

,

n = 1, 2, . . . , m = −n, . . . , n/ki /∈ σ (Di,1)

is complete in L2tan(S1,2).

Proof. It is sufficient to show that for a =(

a1

a2

)∈ L2

tan(S1,2), the set of

closure relations∫

S1

a∗1 ·(n1 × M3

mn

)dS +

∫

S2

a∗2 ·(n2 × M3

mn

)dS = 0,

∫

S1

a∗1 ·(n1 × N3

mn

)dS +

∫

S2

a∗2 ·(n2 × N3

mn

)dS = 0,

and∫

S1

a∗1 ·(n1 × M1

mn

)dS +

∫

S2

a∗2 ·(n2 × M1

mn

)dS = 0,

∫

S1

a∗1 ·(n1 × N1

mn

)dS +

∫

S2

a∗2 ·(n2 × N1

mn

)dS = 0

for n = 1, 2, . . ., and m = −n, . . . , n, yields a1 ∼ 0 on S1, and a2 ∼ 0 on S2.As in the proof of Theorem 1, we consider the vector potentials A1 and A2

with densities a′1 = n1 × a∗

1, and a′2 = n2 × a∗

2, respectively, and define thevector field

E =j

k0εi(∇×∇× A1 + ∇×∇× A2).

Restricting r to lies inside a sphere enclosed in S2 and using the vector spheri-cal wave expansion of the dyad gI, we see that the first set of closure relationsgives E = 0 in Di,2. Analogously, but restricting r to lies outside a sphere en-closing S1, we deduce that the second set of closure relations yields E = 0 inDs. Theorem 1 can now be used to conclude.


Theorem 3. Under the same assumptions as in Theorem2, the system ofvector functions(

n1 × M3mn

n2 × M3mn

)

,

(n1 × N3

mn

n2 × N3mn

)

,

(n1 × M1

mn

n2 × M1mn

)

,

(n1 × N1

mn

n2 × N1mn

)

,

n = 1, 2, . . . , m = −n, . . . , n/ki /∈ σ (Di,1)

is linearly independent in L2tan(S1,2).

Proof. We need to show that for any Nrank, the relations

Nrank∑

n=1

n∑

m=−n

αmn

(n1 × M3

mn

n2 × M3mn

)

+ βmn

(n1 × N3

mn

n2 × N3mn

)

+γmn

(n1 × M1

mn

n2 × M1mn

)

+ δmn

(n1 × N1

mn

n2 × N1mn

)

= 0 ,

(on S1

on S2

)

give αmn = βmn = γmn = δmn = 0, for n = 1, 2, . . . , Nrank and m =−n, . . . , n. Defining the electromagnetic field

E =Nrank∑

n=1

n∑

m=−n

αmnM3mn + βmnN3

mn + γmnM1mn + δmnN1

mn,

we see that n1 × E = 0 on S1 and n2 × E = 0 on S2. The uniqueness of theMaxwell boundary-value problem then gives E = 0 in Di,1, and since E is ananalytic function we deduce that E = 0 in Di−0. Using the representationsfor the vector spherical wave functions M1,3

mn and N1,3mn in terms of vector

spherical harmonincs mmn and nmn, and the fact that the system of vectorspherical harmonics is orthogonal on the unit sphere, we obtain

αmnh(1)n (kir) + γmnjn (kir) = 0,

βmn

[kirh

(1)n (kir)

]′+ δmn [kirjn (kir)]

′ = 0,

for r > 0. Taking into account the expressions of the spherical Bessel andHankel functions for small value of the argument

jn(x) =xn

(2n + 1)!!1 + O

(x2)

and

h(1)n (x) = −j

(2n − 1)!!xn+1

1 + O

(x2)

,


we multiply the first equation by (kir)n+1 and let kir → 0. We obtain αmn = 0and further γmn = 0. Employing the same arguments for the second equation,we deduce that βmn = 0 and δmn = 0.

In the same manner we can prove the completeness and linear indepen-dence of the system of distributed vector spherical wave functions M1,3

mn andN 1,3

mn.

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Index

absorption cross-section, 51amplitude matrix

definition, 43in terms of T-matrix elements, 59

angular frequency, 2angular functions

computational algorithm, 259definition, 257normalized functions, 258orthogonality relations, 260symmetry relations, 260

anisotropic mediabiaxial, 8negative uniaxial, 8positive uniaxial, 8

asymmetry parameterdefinition, 52orientation-averaged, 80

average quantities of heterogeneousmaterials

electric displacement, 148electric field, 148polarization field, 149

backscattering theorem, 57basis functions

constant pulse-functions, 195edge-based, 195node-based, 195

Bayes’ rule, 152Bessel differential equation, 254boundary conditions for

normal component of electricdisplacement, 4

normal component of magneticinduction, 4

tangential component of electric field,4

tangential component of magneticfield, 4

boundary-value problem forcomposite particles, 140homogeneous, isotropic particles, 85inhomogeneous particles, 106isotropic, chiral particles, 102multiple particles, 124uniaxial anisotropic particles, 104

chirality parameter, 8Clebsch–Gordan coefficients

computational algorithm, 274symmetry relations, 274

coherency extinction matrix, 48coherency phase matrix, 44coherency vector of

incident field, 13scattered field, 44total field, 46

complex amplitude vector, 10complex polarization unit vector, 10complex Poynting’s vector, 4conditional average

of the scattered field coefficients, 152with one and two fixed particles, 152

conditional density withone fixed particle, 152two fixed particles, 152

318 Index

configurational average ofreflected field, 162statistical quantities, 152

constitutive relations foranisotropic media, 7chiral media, 8isotropic media, 6

continuity equation, 2current density, 2

degree of polarization, 15dichroism, 46differential equation for spherical

harmonics, 254differential scattering cross-section

definition, 42for parallel and perpendicular

polarizations, 42normalized, 42orientation-averaged, 81

diffusion limited aggregation, DLA, 236diffusion limited cluster cluster

aggregation, DLCCA, 236dimension of T matrix for

composite particles, 144homogeneous, isotropic particles, 96layered particles, 119multiple particles, 132

dipole moment, 149, 197discrete dipole approximation, DDA,

192discrete sources method, DSM, 189dispersion relation for

anisotropic media, 24isotropic media, 26isotropic, chiral media, 31uniaxial anisotropic media, 25

dyadic Green’s functionasymptotic behavior, 38definition, 37differential equation, 38expansion in terms of vector plane

waves, 175dyadic kernel for

discrete dipole approximation, 197volume integral equation method, 195

effective loss tangent, 247effective permittivity, 149

effective wave number, 157effective wave vector, 153efficiencies for

absorption, 51extinction, 51scattering, 51

electric charge density, 2electric conductivity, 6electric dipole operator, 97electric displacement, 2electric field, 2electric permittivity, 6electric susceptibility, 6electromagnetics programs

CST Microwave Studio, 198discrete dipole approximation, DDA,

192multiple multipole method, MMP,

189T-matrix method, 183, 188

ellipsometric parametersellipticity angle, 11orientation angle, 11vibration ellipse, 11

Euler angles, 15evanescent waves scattering, 172Ewald-Oseen extinction theorem, 161exciting field, 149extinction cross-section

definition, 51in terms of T-matrix elements, 60orientation-averaged, 72

extinction matrixdefinition, 48orientation-averaged, 75

extinction theorem, 41

far-field patternsdefinition, 41in terms of T-matrix elements, 58integral representations, 41

Faraday’s induction lawdifferential form, 2discrete form, 198integral form, 198

finite integration technique, FIT, 198fractal dimension, 234fractal pre-factor, 234

Index 319

Fresnell reflection coefficients, 166Fresnell transmission coefficients, 172

Gauss’ electric field lawdifferential form, 2integral form, 199

Gauss’ magnetic field lawdifferential form, 2discrete form, 199integral form, 199

Gaussian beambeam parameter, 18Davis approximation, 18diffraction length, 18generalized localized approximation,

18waist radius, 18

generalized Lorentz–Lorenz law, 158generalized multiparticle Mie-solution,

137generalized spherical functions

definition, 272orthogonality relation, 273recurrence relations, 273symmetry relations, 272

Green’s functiondefinition, 261differential equation, 36spherical waves expansion, 261

Green’s second vector theorem, 34Green’s second vector-dyadic theorem,

38

Helmholtz equation, 253Huygens principle, 41

image coordinate system, 170interacting field, 167internal fields expansions in

isotropic chiral media, 32uniaxial anisotropic media, 27

joint density function, 152jump relations for

curl of a vector potential, 97double curl of a vector potential, 97

Kasterin’s representation, 291

Legendre differential equation, 254

Legendre functionsassociated, 257computational algorithm, 258normalized functions, 258orthogonality relation, 260polynomials, 257recurrence relations, 257special values, 258symmetry relation, 260

logarithmic derivatives coefficientscomputational algorithm, 101definition, 100

long-wavelength dispersion equation,160

magnetic dipole operator, 97magnetic field, 2magnetic induction, 2magnetic media

diamagnetic, 6paramagnetic, 6

magnetic permeability, 6magnetic susceptibility, 6magnetization vector, 6Maxwell’s equation

discrete form, 200in frequency domain, 2in time domain, 1

Maxwell–Ampere lawdifferential form, 2integral form, 199

Maxwell-Garnett formula, 149, 197mean direction of propagation of the

scattered fielddefinition, 52orientation-averaged, 80

multiple multipole method, MMP, 189

normalized phase velocity, 247null-field equations for

complex particles, 146composite particles with

distributed sources, 145localized sources, 141

homogeneous, isotropic particles withdistributed sources, 90localized sources, 86

inhomogeneous particles withdistributed sources, 120

320 Index

null-field equations for (Continued)localized sources, 107

layered particles, 115multiple particles, 127

null-field method with distributedsources for

composite particles, 145homogeneous, isotropic particles, 89layered particles, 120

Ohm’s law, 6optical theorem, 53Ornstein–Zernike equation, 163orthogonalization method, 92

pair distribution functiondefinition, 152for the hole-correction approximation,

163Percus–Yevick, 163

permeability tensor, 7permittivity tensor, 7phase function

definition, 52normalization condition, 52

phase matrix, 44plane waves

asymptotic representation, 264spherical waves expansion, 262

polarizability dyad, 196polarizability scalar, 149, 197polarization angle, 12polarization unit vector for

circularly polarized waves, 13linearly polarized waves, 12

polarization vector, 6polarized waves

circularly, 13ellipticaly, 12linearly, 12

Poynting’s theorem, 4Poynting’s vector, 4probability density function

configurational, 151of particle orientations, 157

Q matrixfor isotropic, chiral particles, 103for spherical particles, 99

for uniaxial anisotropic particles, 105with respect to a single origin, 88with respect to two origins, 112

quasi-crystalline approximation, 153quasi-plane waves

definition, 262spherical waves expansion, 262

radius of gyration, 234reciprocity

amplitude matrix, 57extinction matrix, 57far-field pattern, 56phase matrix, 57tensor scattering amplitude, 56

recursive aggregate T-matrix algorithm,138

reference framebeam coordinate system, 43global coordinate system, 42

reflected vector plane wave, 165reflected vector spherical harmonics,

171reflected vector spherical wave functions

integral representations, 168, 169series representations, 167simplified integral representations,

170reflection matrix, 168refractive index, 7rotation matrix

definition, 275symmetry relations, 275

scalar addition coefficientscomputational algorithm, 279integral representations, 277, 278series representations, 277, 278

scaling law, 234scattering characteristic, 246

Ag-sphere on plane surface, 246axisymmetric particles, 201Cassini particle, 210chiral cube, 215chiral sphere, 203complex particle, 244composite particle, 238, 239composite spheroid, 240concentrically layered sphere, 229

Index 321

cube shaped particle, 220cut sphere, 218diamond sphere on plane surface, 246dielectric cube, 213Gaussian beam scattering, 205hexagonal prism, 213inhomogeneous sphere, 224layered spheroid, 226monodisperse aggregate, 237oblate cylinder, 210oblate spheroid, 226perfectly conducting cube, 215perfectly conducting spheroid, 202polydisperse aggregate, 237prolate spheroid, 202reconstructed cube, 220Si-sphere on plane surface, 246sphere with spherical inclusion, 222spheroid on plane surface, 245system of particles, 230uniaxial anisotropic cube, 213uniaxial anisotropic ellipsoid, 213

scattering cross-sectiondefinition, 51in terms of T-matrix elements, 61orientation-averaged, 74

scattering matrixdefinition, 67for macroscopically isotropic media,

67for mirror-symmetric media, 68orientation-averaged, 79relation to phase matrix, 68

Silver–Muller radiation condition, 34single scattering albedo, 51Snell’s law, 166spectral amplitudes of

incident field, 176internal field, 176scattered field, 178

speed of light, 7spherical functions

Bessel, 254computational algorithm, 256Hankel, 254Neumann, 254recurrence relations, 254special values, 255Wronskian relation, 255

spherical harmonicsdefinition, 262orthogonality relation, 262

spherical particlesconcentrically layered, 122homogeneous, 99

spherical wave functionsintegral representations, 262radiating, 261regular, 261rotation addition theorem, 271translation addition theorem, 276

spherical waves expansion ofGaussian beams, 19vector plane waves, 16

stationary point method, 55, 291Stokes parameters, 14Stokes rotation matrix, 14Stokes vector

of incident field, 13of scattered field, 45transformation under rotation, 14

Stratton–Chu representation theoremfor

radiating fields, 34regular fields, 35

superposition T-matrix method, 132surface charge density, 4surface current, 3surface fields approximations for

homogeneous, isotropic particles, 86inhomogeneous particles, 109isotropic, chiral particles, 102multiple particles, 128uniaxial anisotropic particles, 105

surface integral equationdirect method, 98indirect method, 98

T matrixconvergence of calculation, 96definition, 58for complex particles, 147for composite particles, 144for concentrically layered spheres, 123for homogeneous, isotropic particles,

89for inhomogeneous particles, 111for isotropic, chiral particles, 103

322 Index

for layered particles, 116for multiple particles, 130for uniaxial anisotropic particles, 105orientation-averaged, 71recurrence relation for layered

particles, 117rotation addition theorem, 70symmetries for special shapes, 93symmetry relation, 65unitarity condition, 64

T operator fora plane surface, 180an arbitrary surface, 179

tensor scattering amplitudedefinition, 43in terms of T-matrix elements, 59

time-averaged powerabsorbed by the particle, 50incident, 50scattered by the particle, 50

time-averaged Poynting vector, 5total electric displacement, 2transmission boundary-value problem,

33transmitted vector plane wave, 172

vector addition coefficientscomputational algorithm, 285integral representations, 281, 284series representations, 282, 284symmetry relations, 286, 287

vector plane wavesalternative representations, 175definition, 174orthogonality relations, 174

vector potential, 97vector spherical harmonics

definition, 264of left- and right- handed type, 265orthogonality relations, 264

vector spherical wave functionsasymptotic representations, 266distributed, 269in terms of vector spherical

harmonics, 266integral representations, 268left- and right-handed type, 32orthogonality relations, 266, 268quasi-spherical

Cartesian components, 29integral representations, 27

radiating, 266regular, 266rotation addition theorem, 275translation addition theorem, 280

vector wave equation, 253volume element complex, 198volume integral equation method, 192volume integral representation, 40

wave number, 7Wigner D-functions

addition theorem, 273definition, 271

Wigner d-functionsaddition theorem, 274definition, 271orthogonality relation, 272symmetry relations, 272unitarity condition, 274

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