Analysis of scattering of plane electromagnetic waves by stratified spheres of radially uniaxial anisotropic materials

13 2.3 Harmonic expansion of plane wave and formulation of Debye potentials 26 2.4 Formulation of boundary conditions and derivation of scattering co-efficients.. Second, for the case of a

Liu Huizhe(B Eng., National University of Singapore)

A Thesis Submittedfor the Joint Degree of Doctor of Philosophy

by National University of Singapore and ´Ecole Sup´erieure d’´Electricit´e

2011

First of all, I would like to express my heartfelt gratitude to my main supervisor,Prof Koen Mouthaan Without his kindness in monitoring my progress andhis patience in proofreading my papers and thesis to perfection, I would not havecompleted this work in time Many thanks also go to my previous main supervisor,Prof Joshua Le-Wei Li, who left for UESTC, for initiating this interesting projectand giving me the opportunity to travel to France I am deeply grateful to mysupervisor, Prof Sa¨ıd Zouhdi, for his hospitality during my research attachment inLGEP-Sup´elec and the financial support to attend a series of enriching conferencesand PhD schools in Europe and Africa I am greatly indebted to my supervisor,Prof Leong Mook Seng, for stimulating discussions and his unfailing support.

I am grateful to the National University of Singapore for the research arship provided throughout my candidature I am also grateful to the Frenchgovernment, in particular, the CROUS in France and the French Embassy in Sin-gapore, for the financial support during my stay in France

schol-I also owe my gratitude to teachers and friends schol-I have met in Singpore, in

i

particular, Prof Chen Xudong, Prof Qiu Cheng-Wei, Dr Zhong Yu, Dr Xiao

Ke, Mr Wan Chao, Mr Han Tiancheng, Dr Hu Li from NUS and Dr BorisLuk’yanchuk from DSI for valuable discussions; Miss Zheng Xuhui, Miss Rao Hailu,Miss Ding Pingping, Miss Ye Xiuzhu, Dr Wu Yuming, Ms Liang Dandan, Ms.Wang Xuan, Ms Li Yanan, Dr Li Yanan, Mr Tang Kai, Mr Ye Huapeng, Dr.Liu Xiaofei, Dr Fei Ting for their support to me in one way or another; lab officers

Mr Jack Ng Chin Hock, Mdm Guo Lin, Mr Sing Cheng Hiong for their kindassistance with lab facilities I am also thankful to friends I have met in France,

in particular, Prof Alain Bossavit, Prof Olivier Dubrunfaut, Mr Roger Pereira,

Mr Lotfi Beghou and Prof Alexey P Vinogradov for engaging discussions; Mr.Laurent Santandrea for his kind assistance in coursework and in printing my firstconference poster; Dr Yu Peiqing, Dr Zhu Yu, Dr Liu Bing, Dr Liu Xiaofeng,

Dr Song Li, Mr Tang Qingshan, Dr Hicham Belyamoun, Mr Hassan Hariri,Prof Ahachad Mohammed, Mr Yves Bernard for making my attachment inFrance a memorable one I am also grateful to my thesis committee members,Prof Chen Xudong, Prof Dominique Lesselier and Dr Boris Luk’yanchuk fortheir valuable suggestions to improve my thesis

I would like to express my deepest gratitude to my beloved parents for alwaysbeing supportive of my academic pursuit and for taking care of the preparation of

my wedding ceremony so that I could concentrate on thesis writing Last but notleast, I owe plenty to my beloved husband Mr Zhang Xiaomeng for his patience inediting papers and thesis, discussing ideas and practising presentations with me

Acknowledgments i

1.1 Background 1

1.2 Original contributions and structure of the thesis 9

iii

2 Preliminaries 11

2.1 Problem set-up 11

2.2 Fields and potentials in anisotropic media 13

2.3 Harmonic expansion of plane wave and formulation of Debye potentials 26 2.4 Formulation of boundary conditions and derivation of scattering co-efficients 33

2.5 Conclusions 41

3 Transparent spherical particles with radial anisotropy 42 3.1 Introduction 42

3.2 Theoretical analysis 45

3.2.1 Theoretical background 45

3.2.2 Derivation of transparency relation 48

3.3 Numerical analysis 51

3.3.1 Far-field analysis 51

3.3.2 Near-field analysis 55

3.4 Conclusions 60

4 Plasmon-resonant spherical particles with radial anisotropy 62

4.1 Introduction 62

4.2 Theoretical Formulation 65

4.3 Numerical analysis 72

4.3.1 Effect of anisotropy on proportion of permissible regions 72

4.3.2 Effects of particle sizes, modes, dissipative losses and

config-urations on resonance trace 74

4.4 Conclusions 78

A.1 Scattered electric fields 104

A.2 Far field approximations 106

B Bessel Functions 111

B.1 Cylindrical Bessel functions 111

B.2 Spherical Bessel functions 112

B.3 Riccati-Bessel functions 113

The aim of this thesis is to characterize electrically-small spheres with radialanisotropy and stratified structure for either minimizing or maximizing scatter-ing by plane waves based on the extended Mie scattering model This theory isapplicable to the case of single-layer spheres as well The original contributions

of the thesis are twofold First, for the case of a single-layer sphere with radialanisotropy, a transparency condition comprising of radial and tangential permit-tivity is analytically established As such, particles with carefully engineered radialanisotropy are transparent without using any coating and are ideal for applicationswith space constraint and stringent transparency criteria Second, for the case of acoated sphere with radial anisotropy, a resonance condition is analytically derived.The derived condition provides a guideline for choosing material and structural pa-rameters in the design of plasmon-resonant particles at given frequencies of interestfor applications in sensing For both cases, the effective permittivity of the scat-terers of interest is derived in comparison with pertinent expressions of scatteringcoefficients for isotropic spheres Physical insights are also provided In addition,full-wave numerical analysis is presented to validate the proposed conditions

vii

1.1 Illustration of a metallic nanoparticle (a) with localized surface monic resonance (b) and (c) [1] 7

plas-2.1 Configuration of scattering of an incident plane wave by a stratifiedradially uniaxial anisotropic spherical scatterer 12

3.1 Configuration of scattering of an incident plane wave by a radiallyuniaxial anisotropic spherical scatterer 46

3.2 (a) Normalized total scattering cross section and (b) contribution of

the first-order TM scattering coefficient, with respect to ε r /ε0, with

ε t = 3ε0, µ = µ0, and a = λ0/100 . 52

3.3 (a) Normalized total scattering cross section and (b) contributions

of several TM and TE scattering coefficients, with respect to ε r /ε0,

with ε t = 3ε0, µ = µ0, and a = λ0/5. 54

viii

3.4 (a) Normalized total scattering cross section and (b) contributions

of several TM and TE scattering coefficients, with respect to ε t /ε0,

with ε r = 3ε0, µ = µ0, and a = λ0/5 . 563.5 Contour plots of magnitude of radial components of scattered elec-

tric fields in the x-z plane for (a) an isotropic sphere with ε = 3ε0,

µ = µ0, and a = λ0/5; (b) the same as (a), except for a coating of

a c = 1.49a and ε c = 0.5ε0; (c) the same as (a), except that εt = 3ε0and εr = 0.25ε0; and (d) the same as (a), except that εt = 0.62ε0and εr = 3ε0 58

3.6 Distribution of magnitude of time-average Poynting vector and

as-sociated power flow lines in the E plane for the same four cases as

in Fig 3.5 59

4.1 Configuration of scattering of an incident plane wave by a coatedradially uniaxial anisotropic spherical scatterer 65

4.2 Contour plot of resonance condition over ε t,1 –ε t,2 (ε r,1 –ε r,2) space:

colored region corresponds to 0 < a2/a1 < 1; blank region

corre-sponds to forbidden region 70

4.3 Proportion of permissible area in the 4th quadrant of contour map

against ε t,2 /ε r,2 with ε t,1 /ε r,1 = 1, ε m = 1.7689ε0 and (a)|ε t,2,max | =

|ε 1,max | = 15ε0; (b) |ε t,2,max | = |ε 1,max | = 100ε0 80

4.4 Proportion of permissible area in the 2nd quadrant of contour map

against ε t,1 /ε r,1 with ε t,2 /ε r,2 = 1, ε m = 1.7689ε0 and (a)|ε t,1,max | =

|ε 2,max | = 15ε0; (b) |ε t,1,max | = |ε 2,max | = 100ε0 814.5 Radial ratios at resonance over a range of normalized shell radial

permittivities, at fixed core permittivity of ε2 = −5ε m , for n =

1, 2, 3, and 4, respectively . 82

4.6 Density distribution of the magnitude of the first-order scattering

coefficient in the (ε r,1 /ε m , a2/a1) space for a coated sphere at

res-onance with core permittivity ε2 = −5ε m for different sizes (a)

k m a1 = 0.1; (b) k m a1 = 0.3; (c) k m a1 = 0.5; (d) k m a1 = 0.8 . 83

4.7 Density distribution of the magnitude of the second-order

scatter-ing coefficient in the (ε r,1 /ε m , a2/a1) space for a coated sphere at

resonance with core permittivity ε2 = −5ε m for different sizes (a)

k m a1 = 0.5; (b) k m a1 = 0.8; (c) k m a1 = 1.0; (d) k m a1 = 1.2 . 84

4.8 Density distribution of the magnitude of the first-order

scatter-ing coefficient in the (ε r,1 /ε m , a2/a1) space for a coated sphere

at resonance with electrical size k m a1 = 0.5 for different losses in the core (a) ε2 = (−5 + 0.001)ε m ; (b) ε2 = (−5 + 0.01)ε m; (c)

ε2 = (−5 + 0.1)ε m ; (d) ε2 = (−5 + 0.3)ε m 85

4.9 Density distribution of the magnitude of (a) the first-order tering coefficient; (b) the second-order scattering coefficient in the

scat-(ε r,1 /ε m , a2/a1) space for a coated sphere at resonance with fixed

core permittivity ε2 =−1.4ε m and electrical size k m a1 = 0.5 . 86

4.10 Same as Fig 4.9 except in the (ε r,2 /ε m , a2/a1) space and with fixed

shell permittivity ε1 =−7ε m 87

4.11 Same as Fig 4.9 except in the (ε r,2 /ε m , a2/a1) space and with fixed

shell permittivity ε1 =−0.6ε m 88

LSPR localized surface plasmon resonanceSLEM surface localized electromagnetic mode

TM transverse magnetic

TE transverse electric

NP nanoparticle

RCS radar cross section

SCS scattering cross section

xii

1 The content in Ch 3 is published as

(a) H.-Z Liu, L.-W Li, M S Leong, S Zouhdi, “Transparent uniaxial

anisotropic spherical particles designed using radial anisotropy”, Phys.

Rev E 84, 016605, 2011 (Fig 5 is selected for display on the journal

website as part of the “Kaleidoscope”, http://pre.aps.org/)

(b) H.-Z Liu, L.-W Li, S Zouhdi, M S Leong, “General formulation of

permittivity tensor for minimal backscattering from uniaxial anisotropic

spheres,” META’10, 2nd International Conference on Metamaterials,

Photonic crystals and Plasmonics, Cairo, Egypt, 22-25 February 2010.

2 The content in Ch 4 is published as

(a) H.-Z Liu, K Mouthaan, M S Leong, S Zouhdi, “Maximizing

scat-tering by coated spheres with radial anisotropy,” submitted to Appl.

Phys A.

(b) H.-Z Liu, K Mouthaan, M S Leong, S Zouhdi, “Resonance

condi-tion for coated spheres with radial anisotropy,” PIERS 2011, Progress In

Electromagnetics Research Symposium, Suzhou, China, 12-16

Septem-ber 2011

In addition, there are two coauthored papers not covered in this thesis:

(a) E O Liznev, A V Dorofeenko, H.-Z Liu, A.P Vinogradov, S Zouhdi,

“Epsilon-near-zero material as a unique solution to three different

ap-proaches to cloaking,” Appl Phys A 100, 2, pp 321-325, 2010.

(b) A P Vinogradov, A V Dorofeenko, E O Liznev, H.-Z Liu, and S.

Zouhdi, “Employing Epsilon-near-zero Material in Cloaking”, PIERS

2009, Progress in Electromagnetics Research Symposium, Moscow,

Rus-sia, 18-21 August 2009

xiii

In this thesis the interaction between plane electromagnetic (EM) waves and ified radially uniaxial anisotropic spheres is analytically studied based on theMie theory In particular, minimizing and maximizing scattering are considered,i.e., transparency and resonance Special attention is paid to the role of radialanisotropy in controlling the scattering intensity This introductory chapter coversthe background and the structure of the thesis

Scattering takes place when propagating waves or particles encounter obstaclesalong their paths Electromagnetic scattering, being one of the most commonlyencountered forms of scattering, can be broadly divided into two types, namely,elastic and inelastic scattering Elastic scattering includes Mie scattering [2] andRayleigh scattering [3] where the kinetic energy of the incident wave is conserved

1

in the absence of dissipation On the other hand, inelastic scattering is mon in molecular collisions where the kinetic energy of the incident particles isnot conserved and a frequency shift takes place Examples of inelastic scatter-ing include Brillioun scattering, Raman scattering, inelastic X-ray scattering andCompton scattering In this work, we are concerned with elastic scattering, to bemore specific, Mie scattering extended to the complex media, i.e., radially uniaxialanisotropic media.

com-The Mie scattering model is an analytical tool for solving Maxwell’s tions while dealing with electromagnetic scattering by spherical objects In thesmall-radii limit, Mie scattering is typically approximated by Rayleigh scattering.Although in this work we are mainly concerned with electrically small particles, thefull-wave Mie theory is adopted instead of the Rayleigh approximation This is at-tributed to the breakdown of the Rayleigh approximation in the case of anomalousscattering, to be discussed in greater depth later on

equa-Spherical-shaped scatterers are chosen for two reasons First, the boundarycondition is easy to deal with since the spherical surface is closed and of finiteextent Furthermore, the sphere, being a canonical shape, has exact analyticalsolutions given by the Mie theory Analytical solutions are developed in this worksince they may reveal more physical insights into the problems compared withnumerical methods For irregular or more complex shapes, one may resort tonumerical methods, such as finite difference time domain method (FDTD), finiteelement method (FEM), and method of moment (MoM), which are outside thescope of this work

In general, materials can be described as being isotropic, anisotropic, isotropic, bianisotropic or idealized [4] For isotropic media, the constitutive pa-rameters are scalar For anisotropic media, at least one of the constitutive pa-rameters is in tensorial form, for instance, uniaxial, biaxial, gyroelectric and gyro-magnetic media For biisotropic and bianisotropic media, cross coupling betweenelectric and magnetic fields exists One example of biisotropic media is chiral me-dia [5] Examples of idealized media are perfect electric conductors (PEC) andperfect magnetic conductors (PMC) In this work, we are particularly interested

bi-in radially uniaxial anisotropic media defined bi-in the spherical coordbi-inates Suchmedia are also termed media with radial anisotropy where the permittivity and/orpermeability in the radial direction and that in the direction transverse to the ra-dial direction have differing values Media with radial anisotropy is chosen in thisstudy because they offer one more degree of freedom for exploration over isotropicmedia and are the simplest type of anisotropic media in the context of the sphericalcoordinate system

Having provided some basic understanding on scattering and materials, weare ready to embark on the transparency problems To begin with, we present themyriad of existing techniques for scattering reduction and introduce the rationale

of this work

In the past, stealth technology is employed to minimize scattering and hideaircrafts from radar detection with the employment of radar absorbing paint, non-metallic air frame and modified vehicle shape In recent years, amidst the intenseinterest in metamaterials, the possibility of devising an ideal cloak is conceived

An ideal cloak should be macroscopic and not limited to sub-wavelength sizes ornear field regions, be independent of the object to be cloaked, have minimized ab-sorption and scattering, be passive and broadband [6] Several design mechanismshave been proposed towards this end, namely, coordinate transformation [7–9],dipolar scattering cancellation [10], anomalous localized resonance [11], hard andsoft surfaces [12], tunneling light transmissions [13], active sources [14], and invis-ible fish-scale structure [15] In this section, we highlight two of the techniques,namely, coordinate transformation and scattering cancellation technique.

The advantages and drawbacks of the transformation cloak are illustrated asfollows Theoretically speaking, the transformation cloak is closest to the criteria of

an ideal cloak The basic idea is to control the path of electromagnetic waves by thecareful engineering of permittivity and permeability distributions However, withregard to practical realization, the materials employed in the transformation cloakare too complex First, the transformation cloak possesses a detrimental shortcom-ing, namely singularities of material properties at the inner boundary of the cylin-

drical cloak (ε θ tends to infinity) Simplifications [16] and modifications [17–19]have been made to Pendry’s recipe in order to make material properties more real-istic Furthermore, for both cylindrical and spherical transformation cloaks, zerovalues of constitutive parameters are present at the inner boundary A remedy can

be found in the design of epsilon-near-zero materials in [20] Moreover, the terials employed in constructing the transformation cloak are radius-dependent

ma-To solve this problem, Huang et al proposed a concentric layered structure ofalternating homogeneous isotropic materials for the cylindrical cloak [21] Qiu et

al extended the concept to the spherical cloak [22] For the cylindrical case, thematerials can be approximated as being non-magnetic However, for the sphericalcase, anisotropy in both permittivity and permeability has to be simultaneouslypresent It can be concluded that the materials associated with construction ofthe transformation cloak, especially for the spherical case, involve considerablecomplexity.

We turn our attention to another invisibility technique, namely, scatteringcancellation, which was initially raised by Kerker [23] This technique has beendemonstrated to work well for coated scatterers which are electrically small in size.Small scatterers can be regarded as oscillating dipoles With careful manipulation

of the structural and material parameters of the scatterer, the dipole momentsinduced in the core and shell layers become oppositely directed and cancel eachother As a result, the overall dipole moment becomes zero and no scattering isobserved The scattering cancellation technique has been applied not only to thecoated spheres [10, 24] or cylinders [25] for transparency at a particular frequencypoint, but also to multilayered spheres [26] and cylinders [27] for multi-frequencytransparency This technique is based on the small-radii assumption that the radius

of the particle is much smaller than the incident wavelength As the particlesize increases, not only the dipolar scattering but also higher order multipolarscattering becomes prominent This technique can thus be extended to attenuate

the dominant n-th order scattering mode and hence can significantly reduce the

total scattering As discussed, the scattering cancellation technique can be utilizedfor the design of coated and multilayer structures for minimizing scattering

The scattering cancellation technique is adopted in this work for its ease ofimplementation In our case, we simplify the structure to a single-layer sphereand add one more degree of freedom to the material parameters by making use ofthe radially uniaxial anisotropic materials We attempt to design a single spherewhich is transparent with the introduction of radial anisotropy In addition, wewill examine the contributions of tangential and radial components of permittivity

to total scattering We want to investigate if they can be decoupled and canceleach other just like the cancellation of opposite dipole moments in core and shelllayers observed in the coated design [10]

So far, we have briefly elaborated on our approach to scattering minimization

In the following paragraphs, we will look into the topic of scattering enhancement

As we mentioned earlier, scattering by electrically small particles is typicallyrepresented by Rayleigh scattering which is a first-order approximation of the Mietheory Rayleigh scattering is characterized by a scattering intensity inversely pro-portional to the fourth power of the incident wavelength It gives an explanation

to the blueness of the sky where the blue component, having a shorter wavelengththan the red one, is scattered the most In Rayleigh scattering, the weak dipo-lar scattering prevails over higher order scattering However, Rayleigh scatteringfails to describe one particular case, namely, resonant scattering With resonantscattering, electrically small particles exhibit sharp giant extinction peaks with in-verse hierarchy where the quadrupolar is higher than the dipolar peak The hugescattering intensity is usually only found for particles much greater in size Fur-thermore, complex near field distributions are also observed The mechanism of

+ + +

+ + +

tion, the n-th order component of incident wave is totally converted into scattered

energy In this work, we are particularly interested in scattering enhancementassociated with resonant scattering

Surface plasmons are collective oscillations of electron gas which can be ported at the interface between two media whose real parts of dielectric functionsare of opposite signs As illustrated in Fig 1.1, when the frequency of incidentwave equals one of the surface mode of the metallic nanoparticle, the electrons areshifted with respect to the ion lattice The polarization charges on opposite surfaceelements apply a restoring force on the electron gas Consequently, collective oscil-lations of the electron gas take place, leading to localized surface plasmon resonance(LSPR) Negative dielectric functions can be found in conductors, such as metals,below their plasma frequencies as described by the Drude model Pertaining tooptical frequencies, commonly used metals for sustaining surface plasmons are gold

sup-and silver The main application lies in sensing [28] For instance, the resonancepeaks are highly sensitive to the permittivity of the surrounding medium Oncethe concentration of chemicals in the surrounding medium changes, the frequency

of LSPR also changes and can be detected

Resonant scattering has been analytically studied for both single-layer [29] andcoated [30] spheres Tribelsky et al formulated the permittivity of a small particlethat can support LSPR and derived the theoretical dissipation limit to distinguishanomalous scattering from Rayleigh scattering [29] Alu and Engheta [30] derivedthe resonance condition for coated isotropic spheres and investigated properties ofcomposite media embedded with such resonant inclusions With emerging appli-cations of anisotropic materials in optical devices, LSPR by metallic-anisotropictwo-layer spheres are investigated [31,32] However, in earlier works, the structuralparameters and anisotropy are predefined and the resonant frequency is located

by performing a frequency sweep For some applications, the working frequency ispredefined Therefore, in our work, we attempt to design resonant structures atgiven frequencies of interest Material parameters can be derived from frequenciesvia pertinent dielectric functions

1.2 Original contributions and structure of the

thesis

The original contributions of the thesis encompass two aspects: design of parent single-layer sphere and plasmon-resonant coated sphere, both with radialanisotropy The former is introduced in chapter 3 and the latter in chapter 4 Inchapter 2, preliminaries are given The thesis is concluded in chapter 5

trans-In chapter 2, theoretical foundation is laid with reference to scattering bygeneralized multilayer spheres with radial anisotropy The analysis starts fromfull-wave Mie theory Fields in all regions are formulated in terms of transversemagnetic and transverse electric Debye potentials Furthermore, scattering andtransmission coefficients in all regions are formulated based on Cramer’s rule Inaddition, the transfer matrix method is presented as an alternate approach forsolving scattering coefficients

Chapter 3 investigates the transparency condition for single-layer sphere withradial anisotropy Scattering coefficients are derived based on Mie theory Withthe application of the small-radii assumption, closed-form expressions of scatter-ing coefficients are obtained By nullifying the scattering coefficients and settingthe numerator to zero, the transparency condition is derived Subsequently, withsome manipulations, the coupling effect between tangential and radial components

of permittivity is investigated By making comparisons with pertinent Rayleighresults, an expression of equivalent permittivity for sphere with radial anisotropy

is derived Numerical analysis is performed to validate the derived transparency

condition by examining the scattering cross section and the first few orders of tering coefficient The size tolerance of the transparency condition is also studied.Moreover, the transparency performance of the proposed anisotropy sphere is com-pared against earlier coated design in terms of the magnitudes of electric fields andthe distributions of time-averaged Poynting vectors in the near field.

scat-Chapter 4 looks into the resonance condition for coated spheres with radialanisotropy Starting from Mie theory, pertinent scattering coefficients are obtained

By using the small-radii approximation and the low-dissipation approximation,closed form expressions of scattering coefficients are derived Through comparison

with the expression for a single sphere, the effective permittivity for the n-th order resonant mode is derived By setting the scattering coefficient to one, the n-th order

resonance condition is derived The resonance condition is presented in a contourmap in the permittivity space Numerical analysis is performed to study the effects

of anisotropy on the size of permissible regions in a contour map Furthermore,the applicability of the resonance condition is validated by inspecting the densityplots of the magnitudes of scattering coefficients in the material-structural spacefor various configurations, particle sizes, orders of resonance and losses

Last but not least, a summary of the thesis is provided in chapter 5 tions of this work and proposed future work are discussed

In this chapter, we present the formulation of electromagnetic fields in all regions

in the presence of a stratified spherical scatterer with radial anisotropy, which laysthe theoretical foundation for solving transparency and resonance problems to bediscussed in later chapters

Based on full-wave Mie scattering theory, field solutions for scattering fromuniaxial anisotropic spheres can be analytically derived by a myriad of techniques,such as the expansion of spherical vector wave functions [33] in the cartesian coor-dinates; the introduction of Debye potentials [24, 34] or novel potentials [35, 36] inthe spherical coordinates In particular, Debye potentials are utilized in this work

2.1 Problem set-up

The configuration of the problem is depicted in Fig 2.1 An arbitrary-sized

scat-terer is centered at the origin O The scatscat-terer comprises of N layers of concentric

11

m !m

E k

x

z

Figure 2.1: Configuration of scattering of an incident plane wave by a stratifiedradially uniaxial anisotropic spherical scatterer

spherical shells and core with radial anisotropy The permittivity and permeability

in any layer j (where j = 1, 2, , N , from the outer-most to the core region) can

be represented in tensorial form as (ε j , µ j) To be more specific, we have

where I = brbr + bθbθ + bϕbϕ is the identity matrix, (br, bθ, bϕ) are unit vectors in the

spherical coordinate system Spherical coordinates are adopted in this work for the

convenience in describing scattering problems by spheres ε r,j (µ r,j) is the radial

component of permittivity (permeability) along the br direction, ε t,j (µ t,j) is thetangential component of permittivity (permeability) along the bθ and b ϕ directions

in layer j Hence, the material properties in the stratified sphere are termed “radial

anisotropy” The scatterer is embedded in an isotropic host medium characterized

by scalar permittivity and permeability (ε m , µ m ) An x-polarized and z-traveling

plane wave is incident on the scatterer

Within any region, the materials involved are homogeneous and linear mogeneity suggests that the material properties remain constant everywhere Lin-earity signifies that material properties are independent of the absolute values ofthe electric and magnetic fields, i.e |E| and |H|.

Ho-2.2 Fields and potentials in anisotropic media

Maxwell’s equations, together with appropriate constitutive relations and ary conditions, yield unique solutions to electromagnetic problems In this section,

bound-we attempt to formulate fields in any isolated layer j Boundary conditions will

be dealt with in a later section

To begin with, we shall rewrite vector Maxwell’s equations in scalar differentialform in order to facilitate the introduction of scalar Debye potentials

In the time domain, the four Maxwell’s equations, namely, Faraday’s Law,Ampere’s Law, Gauss’s Law, Gauss’s Law for magnetism, in vector differential

form are given by

In this work, all regions under investigation are presumed to be source-free

with J = 0 and ρ = 0 However, sources do exist in space outside these regions in

order to produce fields in regions of interest Furthermore, all fields are presumed

to take up the harmonic time dependence e −iωt which is suppressed in subsequentderivations with no loss of generality Hence, Maxwell’s equations can be expressed

in phasor form in the spectral domain as

where A( ·) is an arbitrary vector symbol representing E(·), H(·), D(·), B(·) or

J ( ·) A (r) is a complex vector, also named a phasor.

Material properties enter Maxwell’s equations via constitutive relations Forradially uniaxial anisotropic media, constitutive relations are given by

By substituting Eqs (2.1a)–(2.1b) into Eqs (2.4a)–(2.4b), Maxwell’s equations,

i.e Eqs (2.3a)–(2.3d), in any layer j can be rewritten as

With reference to the standard expansion of the curl of an arbitrary vector inthe spherical coordinates, the vector equation representation of Faraday’s law, i.e

Eq (2.5a), can be separated into three scalar equations given by

Similarly, the vector equation representation of Ampere’s law, i.e Eq (2.5b), can

r sin θ

[

∂H j r

with respect to the radial direction This representation is consistent with thesystem of equations (2.6) derived earlier, which is further illustrated as follows.

By taking the TM wave as an example, we substitute Eq (2.7a) into Eqs (2.6e)and (2.6f) and arrive at

With reference to Eq (2.11), it is safe to conclude that Eq (2.12), as well as

Eq (2.6a), is also fulfilled by the TM solution With above examination, we haveshown that the TM solution comply with Maxwell’s equations By following the

same argument, the compliance of the complementary TE solution can also beverified.

Since TM and TE waves are linearly independent, Maxwell’s equations can bedecoupled into two sets of equations One set contains solely TM waves and theother TE waves Therefore, it will be sufficient to study in detail the TM case.The results for the TE case can be inferred by using duality

We now construct the set of Maxwell’s equations with TM fields only Uponapplication of Eqs (2.7a)–(2.7b) to Eqs (2.6a)–(2.6c), TM representations of Fara-day’s law can be expressed as

Having arrived at Maxwell’s equations for TM waves, we are ready to introducethe concept of scalar TM and TE Debye potentials, ΠT M j and ΠT E j [34] Debyepotentials are a particularly useful tool for investigating scattering by sphericalobjects For TM waves, the radial component of magnetic fields is zero, whereas the

transverse components of electric fields, i.e E θ j,T M and E ϕ j,T M, can be representedwith reference to Eq (2.13a) in terms of the gradient of a scalar variable as

j represents the TM Debye potential

in layer j, Eqs (2.14a)–(2.14b) can be rewritten as

After applying duality: ε ↔ µ, E → H, H → −E, to Eqs (2.15a)–(2.15e),

we can obtain the corresponding TE field components TM and TE waves can becombined to yield total fields for every component of electric and magnetic fields

at any point in layer j as follows

So far, we have formulated fields in layer j in terms of Debye potentials Next,

we will construct and solve wave equations in order to derive Debye potentials.The coefficients involved in the expressions of Debye potentials can be solved viamatching boundary conditions, discussed in a later section

By substituting Eqs (2.15c)–(2.15d) into (2.13d), we obtain the wave equationfor TM waves as

variable as

rΠ T M j = rf T M (r)g(θ)h(ϕ). (2.18)Applying Eq (2.18) to the TM wave equation, i.e Eq (2.17), and multiplying

both sides by r sin2θ and dividing both sides by Eq (2.18), we have

(

sin θ dg

dθ

)+ 1

The first term on the left-hand side of the above equation contains variable r only.

On the other hand, the last two terms contain variable θ only and can be equated

As such, Eq (2.19) is separated into three independent ordinary differentialequations, i.e Eqs (2.20a)–(2.20c) Next, we proceed to solve the three equations.

Eq (2.20a) can be rearranged as

d2h

dϕ2 + l2h = 0,

which is a well-known equation whose general solution is a linear combination of

harmonic functions cos (lϕ) and sin (lϕ) with l taking up integer values In order

to obtain single-valued h over the range of 0 to 2π on ϕ, the solution has to take

the form of

The first- and second-order partial derivatives with respect to angle θ can be rewritten with respect to cos θ as

dg

dg dcos θ

dcos θ dθ

dθ

(

dg dcos θ

dcos θ dθ

dcos θ dθ

)

dcos θ dθ

=

(

d2g dcos2θ

dcos θ

dg dcos θ

d dcos θ

dcos θ dθ

)

dcos θ dθ

2g dcos2θ sin

2θ + dg dcos θ

(

− dsin θ dcos θ

)(− sin θ)

2g dcos2θ

(

1− cos2θ)

dcos θ cos θ.

Substituting newly derived partial differentials to Eq (2.20b), it follows

P n l (cos θ), Q l n (cos θ), with reference to Eq (C.2b) in Appendix C For fields to

be finite for any value of θ, including 0 and π, where Q (l) n (cos θ) possesses

singularities, the solution takes the form of

Dividing both sides by r12, we have

n,j +12)th order J ( ·) and Y (·) represent standing

waves; H(1)(·) represents an outgoing traveling wave; H(2)(·) represents an inward

traveling wave Depending on the situation, the solution can be a linear tion of either the former two or the latter two functions Therefore, we have

combina-rf T M ∼ r√1

k t,j r B v T M n,j+12 (k t,j r) ,

rf T M ∼ ˆ B v T M (k t,j r) , (2.21c)

n,j )th order, as defined in Sec B.3 of Appendix B.

Multiplying together Eqs (2.21a)–(2.21c) and taking the summation over all

possible values of n and l, we obtain a general solution to the TM wave equation

cos(lϕ) sin(lϕ)

cos(lϕ) sin(lϕ)

At this point, we have derived expressions of Debye potentials in any layer j with

undetermined coefficients We will examine the incident wave and subsequentlydetermine pertinent coefficients through matching boundary conditions along thespherical surfaces

2.3 Harmonic expansion of plane wave and

for-mulation of Debye potentials

An x-polarized monochromatic plane wave with unit amplitude travels along the z direction in a boundless, homogeneous, isotropic medium characterized by (ε m , µ m)

in the presence of a stratified sphere located at the origin We are going to take alook at fields associated with undisturbed plane wave alone We leave the interac-tion between the plane wave and stratified sphere to a later section

The plane wave is described by

x =bxe ik m z =bxe ik m r cos θ , (2.23)

in a rectangular system of coordinates The time dependence e −iωt is suppressedwith no essential loss of generality

By substituting the expression of the electric field Eq (2.23), into the free time-harmonic Faraday’s law Eq (2.3a), the co-existing magnetic field can bederived as

E y inc = E z inc = H x inc = H z inc = 0.

In dealing with scattering by a spherical object, a spherical coordinate system

is favored Hence, we will convert expressions of electric and magnetic fields from