1. Integral Equation Formulation for Electromagnetic
Scattering from Small Particles
積分方程方法在小粒子的電磁波散射的應用
TAM, Ho Yin
譚浩賢
A Thesis Submitted in Partial Fulfilment
of the Requirements for the Degree of
Master of Philosophy
in
Physics
The Chinese University of Hong Kong
October 2014

2. Thesis Assessment Committee
Professor CHING, Shuk Chi Emily (Chair)
Professor YOUNG, Kenneth (Thesis Supervisor)
Professor LEUNG, Pui Tang (Thesis Co-supervisor)
Professor WANG, Jianfang (Committee Member)
Professor POON, Wing On Andrew (External Examiner)

3. Abstract
Applications in nanophotonics rely on three important properties when op-
tical light is incident on a metallic nanoparticle (size < 100 nm): (a) it is
possible to have a greatly enhanced intensity inside and just outside the par-
ticle; (b) the intensity-enhanced regions have much smaller dimensions (tens
of nm) than the wavelength (about 400 to 700 nm); and (c) the enhance-
ment is resonant (i.e., strongly wavelength-dependent), with the resonance
sensitive to the size and shape of the particle. The last property allows the
resonance to be tuned according to need.
In fact, the dielectric constant (complex and strongly frequency-dependent)
has a complicated interplay with the geometry of the particle(s), making
it difficult to obtain a good understanding of the problem; many results
are obtained only by brute-force numerical calculations, validated against
measurements typically made only on the far field. The present work exam-
ines different theoretical methodologies, especially those based on an integral
equations, establishes a numerical method of choice, and develops a quasi-
analytic approximation that provides physical insight.
The T-Matrix method relies on an integral equation that enjoys the “leap-
frog” property: the internal field is related directly to the far field, bypassing
the external near field. This property is exhibited in a novel derivation, and
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4. is also used to obtain accurate results for small to medium nanoparticles
under moderate truncation of the infinite set of linear equations. Because
computation is not intensive, convergence is easily achieved, and the results
can serve as benchmarks.
For the Finite Difference Time Domain (FDTD) method, commercial
packages are readily available. Though supposedly exact for sufficiently small
grid size, FDTD suffers from several possible problems: (a) it is computation-
ally intensive, even with several tricks to restrict the computational domain;
and (b) consequently it is not possible in practice to use a sufficiently small
grid size, leading to significant numerical errors in certain regions, some of
which may not be widely appreciated.
Both the T-Matrix method and FDTD, being brute-force numerical al-
gorithms, are also difficult to understand in a physical and qualitative way.
The integral equation method, in particular an analytic approximation
based thereon, allows these problems to be assessed and in some cases over-
come. When the size of particles and the spatial region of interest are much
smaller than the wavelength, and the skin depth is also much longer than
the wavelength, the situation can be treated in a quasi-static approximation.
In this case, the integral equation formalism yields a simple formula for the
internal field, with the dependence on wavelength, size and shape separately
exhibited.
The analytic approximation is evaluated against accurate numerical re-
sults, and then employed to understand the optical response of gold nanopar-
ticles, especially the resonance peak of different shapes: spheroidal and cylin-
drical, leading to an improved physical picture.
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5. 摘摘摘要要要
納米光學的應用取決於金屬納米粒子（尺寸小於 100 nm）在可見光
照射下的三個重要特性： （一）粒子內與外圍的電場可能會大大增強；
（二）該強度增強區域的尺寸（數十 nm）遠小於波長（約 400 至 700
nm）；和 （三）該增強是諧振的（即與波長密切相關），且對粒子的大
小和形狀非常敏感。最後一個特性使諧振可按需要調整。
實際上，介電常數（含實數和虛數部分，且對頻率非常依賴）與粒子的
幾何特性存在交叉關係，引致情況複雜，我們難以對納米粒子的光學問題
得出透徹的了解；許多結果都只以蠻力式的數值方法計算，再與實驗測出
的遠場結果比較驗證。此論文檢視不同的理論方法各自的優勢和限制，特
別注意以積分方程為基礎的一些方法，並發展出一個顯示出物理圖像的準
解析近似方法。
T−矩陣方法建基於積分方程和其所具有的「跨越式」特性：內場直接
與外部遠場連繫，而不涉及外部近場。本論文以此屬性提出一種嶄新的推
導，且當應用於小或中度大小的粒子時，將無限線性方程組適度截尾，即
得出準確的數值解。由於此計算負荷較小，較易收斂，其結果相當準確，
可作為基準。
時域有限差分法（FDTD）為另一常用的方法，且有現成的商用軟件。
雖然對於足夠小的網格尺寸，此方法原則上可得出準確解，可是仍有幾種
可能出現的問題： （一）它的計算量龐大，即使經過幾種技巧來限制計算
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6. 區域大小後亦如是； （二）因此，實際上不可能使用足夠小的網格尺寸。
在某些區域這可能會導致顯著的數值誤差，而其中一些誤差可能沒有受到
足夠的注意。
無論是T−矩陣法和時域有限差分，均屬蠻力式數值算法，其結果也很
難以物理式或定性的方法理解。
積分方程法，特別是基於此的一個解析近似，使這些問題在某些情況下
得以克服。當粒子大小和感興趣的空間區域大小遠小於波長，並且趨膚深
度遠長於波長時，此種情況可以準靜態近似處理。在這種情況下，積分方
程方法可以給出簡單的公式表示內場，且該公式中的波長、粒子大小、形
狀與場的關聯能分別展現。
該解析近似將與準確的數值解進行對比，然後用於理解金納米粒子的光
學反應，特別是不同形狀粒子（橢圓和柱體）的諧振峰值，並得出一個較
佳的物理圖像。
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7. Acknowledgement
I would like to express my greatest thanks to my supervisors, Professor Ken-
neth Young and Professor P. T. Leung for their patient guidance on my re-
search. Their insight on physics and philosophy on research have broadened
my horizon.
I also thank Professor J. F. Wang and his research group, especially Dr.
H. Chen, L. Shao, and Q. Ruan, for their kindly cooperation on FDTD
computation in this thesis.
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8. Contents
1 Introduction 1
1.1 Surface plasmonic resonance of noble metals . . . . . . . . . . 2
1.2 Localized surface plasmon on metallic nanoparticles . . . . . . 2
1.3 Motivation of this thesis . . . . . . . . . . . . . . . . . . . . . 4
2 Overview of Thesis 6
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Key properties . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Organization of the thesis . . . . . . . . . . . . . . . . . . . . 31
3 Integral Equation 35
3.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Integral equation solution to vector wave equation . . . . . . . 40
3.3 Schematic expression of integral equation . . . . . . . . . . . . 44
4 T-Matrix for Scalar Fields 46
4.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Formulation in terms of surface integral . . . . . . . . . . . . . 47
4.3 Expansion in complete sets . . . . . . . . . . . . . . . . . . . . 50
4.4 Alternate form . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.5 Results for incident plane wave . . . . . . . . . . . . . . . . . 56
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9. 4.6 Proof of the assertion used in the alternate form . . . . . . . . 62
4.7 Supplementary tables . . . . . . . . . . . . . . . . . . . . . . . 65
5 T-Matrix for Vector Fields 71
5.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 T-Matrix method for vector wave equation . . . . . . . . . . . 72
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6 Finite-Difference Time-Domain
Method (FDTD) 101
6.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2 Dielectric function . . . . . . . . . . . . . . . . . . . . . . . . 107
6.3 Setup of computational domain . . . . . . . . . . . . . . . . . 111
6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.5 Comparing the computational effort . . . . . . . . . . . . . . . 137
6.6 Conclusion for FDTD . . . . . . . . . . . . . . . . . . . . . . . 141
7 Analytic Approximation 142
7.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.2 Systematic approximation . . . . . . . . . . . . . . . . . . . . 144
7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8 External Field 164
8.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
8.2 Integral form . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.3 Multipole Expansion . . . . . . . . . . . . . . . . . . . . . . . 171
8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
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10. 8.5 Supplement: The derivation of surface integral form . . . . . . 183
9 Conclusion 186
A Proof of the Condition for Uniform Field 190
B Solution to Laplace Equation in Ellipsoidal Coordinates (For
Ellipsoidal Object) 195
B.1 Ellipsoidal coordinates . . . . . . . . . . . . . . . . . . . . . . 195
B.2 Laplace equation in ellipsoidal coordinates . . . . . . . . . . . 200
B.3 Electric potential for applied electric field along z-axis in el-
lipsoidal coordinates . . . . . . . . . . . . . . . . . . . . . . . 201
B.4 External and internal electric potential in ellipsoidal coordinates201
B.5 Internal field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
B.6 Spheroidal particles . . . . . . . . . . . . . . . . . . . . . . . . 206
C 1D Perfectly-Matched-Layer (PML) for Normal Incidence 207
Bibliography 210
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