Spectral Theory of Localized Resonances and Applications 9819962439, 9789819962433

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Youjun Deng Hongyu Liu 

Spectral Theory of Localized Resonances and Applications

Spectral Theory of Localized Resonances and Applications

Youjun Deng • Hongyu Liu

Spectral Theory of Localized Resonances and Applications

Youjun Deng School of Mathematics and Statistics Central South University Changsha, Hunan, China

Hongyu Liu Department of Mathematics City University of Hong Kong Hong Kong, China

ISBN 978-981-99-6243-3 ISBN 978-981-99-6244-0 https://doi.org/10.1007/978-981-99-6244-0

(eBook)

Jointly published with Science Press, Beijing, China. The print edition is not for sale in China Mainland. Customers from China Mainland please order the print book from: Science Press. © Science Press 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.

Preface

This book is devoted to the spectral theory of localized resonances including surface plasmon/polariton resonances, atypical resonances, anomalous localized resonances and interior transmission resonances. Those resonance phenomena arise in different physical contexts, but share similar features: the resonant fields exhibit highly oscillatory behaviours manifested as the energy blowup and more intriguingly the resonant behaviours are localized in space. They form the fundamental basis for many cutting-edge technologies and applications including invisibility cloaking and super-resolution imaging. We aim to present a systematic and comprehensive treatment on these resonance phenomena and the associated applications in a unified manner from a mathematical and spectral perspective. We include in our study the acoustic, electromagnetic and elastic wave scattering. The governing equations include the Laplacian equation, the Helmholtz equation, the Maxwell system and the Lamé system. The mathematical study of those resonances is both theoretically and practically important, yet is still in its early stage and far from being complete. Nevertheless, there are abundant results in the literature, especially those due to the fruitful collaborations between the research groups of the two authors of this book. Hence, we think it is a good time to summarize these results into this book. On the one hand, the field is still being under extensive and rapid development. The book can serve as a handy reference book for researchers in this field. On the other hand, it can also serve as a textbook or an inspiring source for beginning researchers or postgraduate students who are interested in entering this field. There are many people helping us during the writing of this book. First, we would like to thank Chaohua Duan, Yang Gao, Tao Li, Wanjing Tang and Liyan Zhu, whose help on the typesetting is indispensable. We would also like to thank our collaborators including Habib Ammari, Yat Tin Chow, Xiaoping Fang, Yan Jiang, Hongjie Li, Pierre Millien, Mahesh Sukula, Xianchao Wang, Wei Wu, Kai Zhang

v

vi

Preface

and Guanghui Zheng, and the book is based on our longterm and fruitful research outputs. Finally, but not the least, we would like to thank Huaian Diao, Hongjie Li and Guanghui Zheng, who read the first draft of the book and provided many helpful and constructive comments and suggestions. Youjun Deng would like to acknowledge the research support by NSFC-RGC Joint Research Grant No. 12161160314. Hongyu Liu would like to acknowledge the research support by the Hong Kong RGC General Research Funds (projects 11311122, 12301420 and 11300821), the NSFC/RGC Joint Research Fund (project N_CityU101/21), and the France-Hong Kong ANR/RGC Joint Research Grant, A_CityU203/19. Changsha, China Hong Kong, China

Youjun Deng Hongyu Liu

Contents

1

Introduction and Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Layer Potentials in Electro-Magnetic System . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Layer Potentials in Elastic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Bessel and Neumann Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Maxwell’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Introduction to Plasmonic Resonances . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Drude’s Model for the Electric Permittivity and Magnetic Permeability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Boundary Integral Operators and Resolvent Estimates . . . . . . . 2.1.4 Layer Potential Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Derivation of the Asymptotic Formula . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Numerical Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Elastic Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Layer Potential Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Asymptotics for the Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Far-Field Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Asymptotics for the Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Resolvent Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Polariton Resonance for Elastic Nanoparticles . . . . . . . . . . . . . . . .

3

Anomalous Localized Resonances and Their Cloaking Effect . . . . . . . . . . 3.1 Elastostatic Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Mathematical Setup of Elastostatics Problem . . . . . . . . . . . . . . . . . 3.1.2 Preliminaries on Layer Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Spectral Analysis of N-P Operator in Spherical Geometry . . . 3.1.4 Anomalous Localized Resonances and Their Cloaking Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4 6 9 9 9 12 15 24 25 46 48 48 51 53 57 59 67 69 77 77 77 79 81 90 vii

viii

Contents

3.1.5

Cloaking by Anomalous Localized Resonance on a Coated Structure in Two Dimensional Case . . . . . . . . . . . . . 3.2 Electrostatic Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Layer Potential Formulation and Spectral Theory of a Neumann-Poincaré-Type Operator. . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Analysis of Cloaking Due to Anomalous Localized Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

5

97 115 115 117 121

Localized Resonances for Anisotropic Geometry . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Conductivity Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Some Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Quantitative Analysis of the Electric Field . . . . . . . . . . . . . . . . . . . . 4.1.3 Application to Calderón’s Inverse Inclusion Problem . . . . . . . . 4.2 Helmholtz Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Asymptotic and Quantitative Analysis of the Scattering Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Resonance Analysis of the Exterior Wave Field . . . . . . . . . . . . . . 4.2.3 Resonance Analysis of the Interior Wave Field . . . . . . . . . . . . . . . 4.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129 129 130 135 144 145

Localized Resonances Beyond the Quasi-Static Approximation. . . . . . . . 5.1 Spectral System of Neumann-Poincaré Operators in Helmoholtz System and Its Asymptotic Behavior . . . . . . . . . . . . . . . . . . 5.1.1 Layer Potential and Spectral Properties of Neumann-Poincaré Operator in R3 . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Asymptotic Behavior of Spectral System of Neumann-Poincaré Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Two Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Helmholtz System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Atypical Resonance and ALR Results in Three Dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Spectral System of the N-P Operator and Its Application to Atypical Resonance in R3 . . . . . . . . . . . . . . . . . . . . . 5.2.3 Atypical Resonance and ALR Results in Two Dimensions. . . 5.3 Maxwell’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Integral Formulation of the Maxwell System . . . . . . . . . . . . . . . . . 5.3.2 Spectral Analysis of the Integral Operators . . . . . . . . . . . . . . . . . . . 5.3.3 Atypical Resonance and Its Cloaking Effect . . . . . . . . . . . . . . . . . . 5.3.4 Invisibility Cloaking Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Elastic Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Spectrum System of the Neumann-Poincaré Operator . . . . . . . . 5.4.3 Atypical Resonance Beyond the Quasi-Static Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 CALR Beyond the Quasi-Static Approximation . . . . . . . . . . . . . .

183

150 169 176 181

184 184 187 191 196 200 210 213 219 220 225 233 240 248 251 257 267 273

Contents

6

Interior Transmission Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Scalar Case (Helmholtz Equations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Boundary-Localized Transmission Eigenstates . . . . . . . . . . . . . . . 6.2.2 Super-Resolution Wave Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Numerical Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Pseudo Surface Plasmon Resonances and Potential Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Concluding Remarks and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Vectorial Case (Maxwell Equations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Boundary-Localized Transmission Eigenmodes . . . . . . . . . . . . . . 6.3.3 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Application of Boundary-Localized Transmission Eigenfunctions: Artificial Mirage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

281 281 285 285 311 321 324 325 328 328 329 348 351 356

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

Chapter 1

Introduction and Preliminaries

1.1 Overview Surface plasmon resonance is the resonant oscillation of conduction electrons at the interface between negative and positive permittivity material stimulated by incident light. It is a type of surface wave, guided along the interface in much the same way that light can be guided by an optical fiber. Similar resonance phenomenon occurs at the interface of negative and positive elastic materials and is referred to as the surface polariton resonance. In what follows, we write SPR to signify both surface plasmon resonance and surface polariton resonance, if not explicitly specified. The SPRs are resonance phenomena at the subwavelength scale and enable controlling waves at micro-scale beyond the diffraction limit. In particular, for certain delicately designed core-shell plasmonic/polariton structures, anomalous localized resonances (ALR) can be induced. In addition to the spacelocalized feature, ALR strongly depends on the position of the exciting source and moreover can induce invisibility cloaking effect. Mathematically, analyzing SPRs and ALRs can be boiled down to analyzing certain layer-potential operators sitting on the material interfaces, including in particular the so-called Neumann-Poincaré (NP) operators. SPRs and ALRs are intensively investigated in the quasistatic regime, namely the subwavelength scale, which is physically objectionable. The mathematical reduction to spectral analysis of NP operators enables the natural extension to study resonance phenomena analogous to SPRs but beyond the quasistatic approximation. It turns out that resonances can still occur, though under certain more complicated and restrictive conditions. Moreover, the resonant fields possess distinct features with some of them similar to those of SPRs while some are different. They are referred to as the atypical resonances. Interestingly, ALR as well as cloaking can still be induced by atypical resonant structures. SPRs, ALRs and atypical resonances are associated with metamaterial structures, namely artificially engineered materials with non-naturally occurring properties. Transmission resonance is associated with regular material structures when © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Y. Deng, H. Liu, Spectral Theory of Localized Resonances and Applications, https://doi.org/10.1007/978-981-99-6244-0_1

1

2

1 Introduction and Preliminaries

illuminated by incident waves and invisibility/transparency occurs. Transmission eigenvalue problems arising in acoustic, electromagnetic and elastic scattering are generally non-elliptic and non-self-adjoint, which makes the corresponding spectral analysis highly challenging and intriguing. Recent study uncovers that the transmission eigenfunctions possess rich structures, especially they oscillate severely and the high oscillation tends to localize on the boundary of the underlying domain, which justifies that they form certain localized resonant modes. All the localized resonances discussed above can produce important applications in invisibility cloaking and super-resolution imaging, with some of them having been realized in lab while some being conceptual and visionary. In Chaps. 2–5, we shall deal with various localized resonances associated with metamaterial structures, whereas in Chap. 6, we shall consider the interior transmission resonance. As mentioned above, one of the key tools in our mathematical analysis is certain integral operators, a.k.a. layer potential operators. In the rest of this chapter, we introduce those layer potential operators as preliminaries for our subsequent discussion.

1.2 Layer Potentials in Electro-Magnetic System Let .Gk be the fundamental solution to the PDE operator .Δ + k 2 in .Rd , .d = 2, 3, that is ⎧ i (1) ⎪ ⎪ ⎨ − 4 H0 (k|x|), .Gk (x) = ik|x| ⎪ ⎪ ⎩− e , 4π |x|

d=2 (1.2.1) d=3

where .H0(1) (k|x|) is the Hankel function of first kind of order zero and .i = particular, we set .G0 the fundamental solution to Laplacian, which admits

G0 (x) =

.

⎧ 1 ⎪ ⎪ ⎨ 2π ln |x|,

d=2

⎪ ⎪ ⎩−

d=3

1 , 4π |x|

√ −1. In

(1.2.2)

For any bounded Lipschitz domain .D ⊂ Rd , .d = 2, 3, we denote by .SDk : H −1/2 (∂D) → H 1 (Rd \ ∂D) the single layer potential operator given by  SDk [φ](x) :=

Gk (x − y)φ(y) dsy ,

.

∂D

(1.2.3)

1.2 Layer Potentials in Electro-Magnetic System

3

and .(KDk )∗ : H −1/2 (∂D) → H −1/2 (∂D) the Neumann-Poincaré operator k ∗ .(KD ) [φ](x)

 := p.v. ∂D

∂Gk (x − y) φ(y) dsy , ∂ν

(1.2.4)

where p.v. stands for the Cauchy principle value. In (1.2.4) and also in what follows, unless otherwise specified, .ν signifies the exterior unit normal vector to the boundary of the concerned domain. The following notation will be used throughout this book. For a function u defined on .Rd \∂D, we denote .

u|± (x) := lim u(x ± tν(x)),

x ∈ ∂D,

t→0+

and .

 ∂u  (x) := lim 〉∇u(x ± tν(x)), ν(x)〈, ∂ν ± t→0+

x ∈ ∂D,

if the limits exist. Here and throughout this book, .〉, 〈 denotes the scalar product on Rd . It is known that the single layer potential operator .SDk is continuous across .∂D and satisfies the following trace formula

.

.

 ∂ 1  SDk [φ] = (± I + (KDk )∗ )[φ] on ± ∂ν 2

∂D,

(1.2.5)

∂ stands for the normal derivative and the subscripts .± indicate the limits where . ∂ν from outside and inside of a given inclusion D, respectively. For a density .φ ∈ TH(div, ∂D), we define the vectorial single layer potential associated with the fundamental solution .Gk introduced in (1.2.1) by

 ADk [φ](x) :=

Gk (x − y)φ(y)ds(y),

.

x ∈ R3 .

(1.2.6)

∂D

For a scalar density .ϕ ∈ L2 (∂D), the single layer potential is defined similarly by  SDk [ϕ](x) :=

Gk (x − y)ϕ(y)ds(y),

.

x ∈ R3 .

(1.2.7)

∂D

We will also need the following boundary operators: MDk : L2T (∂D) −→ L2T (∂D) .

φ− ‫ → ׀‬MDk [φ] = ν(x) × ∇ ×



(1.2.8) Gk (x, y)φ(y)ds(y), ∂D

4

1 Introduction and Preliminaries

NDk : TH(curl, ∂D) −→ TH(div, ∂D) φ ‫׀‬−→ NDk [φ] = 2ν(x) × ∇ × ∇  Gk (x, y)ν(y) × φ(y)ds(y), ×

.

(1.2.9)

∂D

LDk : TH(div, ∂D) −→ TH(div, ∂D) .

φ ‫׀‬−→ LDk [φ] = ν(x) × (k 2 ADk [φ](x) + ∇SDk [∇∂D · φ](x)). (1.2.10)

In this book, we shall denote by .AD , .SD , .MD , and .ND the operators .AD0 , 0 0 .M , and .N D D corresponding to .k = 0, respectively. The vectorial single k layer potential .AD [φ] is continuous on .R3 and its curl satisfies the following jump formula:

0 .S , D

 φ ν × ∇ × ADk [φ]± = ∓ + MDk [φ] 2

.

on ∂D,

(1.2.11)

where ∀x ∈ ∂D,

.

 ν(x) × ∇ × ADk [φ]± (x) = lim ν(x) × ∇ × ADk [φ](x ± tν(x)). t→0+

1.3 Layer Potentials in Elastic System Set .C(x) := (Cij kl (x))di,j,k,l=1 , .x ∈ Rd , .d = 2, 3, to be a four-rank tensor defined by Cij kl (x) := λ(x)δij δkl + μ(x)(δik δj l + δil δj k ), x ∈ Rd ,

.

(1.3.1)

where .λ, μ ∈ C are complex-valued functions and referred as Lamé constants, and .δ is the Kronecker delta. In (1.3.1), .C(x) denotes an isotropic elasticity tensor distributed in the space. For a regular elastic material, the Lamé constants should satisfy the following two strong convexity conditions, (i). μ > 0

.

and

(ii). dλ + 2μ > 0.

(1.3.2)

We first introduce the elastostatic operator .Lλ,μ corresponding to the Lamé constants .(λ, μ) defined as Lλ,μ w := μΔw + (λ + μ)∇∇ · w,

.

(1.3.3)

1.3 Layer Potentials in Elastic System

5

for .w ∈ C3 . The traction (the conormal derivative) of .w on the .∂Ω is defined as follows ∂ν w = λ(∇ · w)ν + 2μ(∇ s w)ν,

.

(1.3.4)

From [145], the fundamental solution .Gω = (Gωi,j )di,j =1 for the operator .Lλ,μ + ω2 is given by Gω (x) = Gsω (x) + Gpω (x),

.

x ∈ Rd ,

x /= 0,

(1.3.5)

where Gsω (x) = (ks2 I + ∇ 2 )

.

1 2 ∇ Gks (x), μks2

1 2 ∇ Gkp (x). μkp2

Gpω (x) = −

(1.3.6)

The parameters .ks and .kp satisfy ω2 , μ

ks2 =

.

kp2 =

ω2 . λ + 2μ

(1.3.7)

Then the single layer potential associated with the fundamental solution .Gω is defined as  ω .SΩ [ϕ](x) = Gω (x − y)ϕ(y)ds(y), x ∈ R3 , (1.3.8) ∂Ω

for .ϕ ∈ L2 (∂Ω)3 . On the boundary .∂Ω, the conormal derivative of the single layer potential satisfies the following jump formula  

∗ 1 [ϕ](x) ∂ν SωΩ [ϕ]|± (x) = ± I + KωΩ 2

.

x ∈ ∂Ω,

(1.3.9)

where (KωΩ )∗ [ϕ](x) = p.v.



.

∂Ω

∂ν x Gω (x − y)ϕ(y)ds(y),

(1.3.10)

with .p.v. standing for the Cauchy principal value and the subscript .± indicating the limits from outside and inside .Ω, respectively. The operator .(KωΩ )∗ is called Neumann-Poincaré (N-P) operator.

6

1 Introduction and Preliminaries

1.4 Bessel and Neumann Functions When considering the symmetric material structures, especially the spherical structures, we are able to represent the solutions to the related electro-magnetic system or elastic system with concrete form of serials. Usually the Bessel functions are used. Denote by .Jn and .Yn are Bessel functions of order n of the first and the second kind, respectively. Note that the spherical Bessel functions of the first and second kind are respectively defined by jn (t) =

.

π Jn+1/2 (t), 2t

yn (t) =

π Yn+1/2 (t), 2t

t > 0.

Let .Ynm be spherical harmonic function of degree n and order m. Recall that the Bessel function .Jn (t) and the Neumann function .Yn (t) admit the following asymptotic expansions (see, e.g., [1, 99]): ⎧ t2 ⎪ ⎪ ⎪ 1 − + O(t 4 ), n = 0, t ⪡ 1 ⎨ 4 .Jn (t) =

t n

t 4  ⎪ √ 1 1 t 2 ⎪ ⎪ ⎩ 1− + O 2 , n ≥ 1, t ⪡ n + 1 Γ (n + 1) 2 n+1 2 n (1.4.1) and ⎧ 2 t  ⎪ ln + γ + O(t), n = 0, t ⪡ 1 ⎪ ⎨π 2 .Yn (t) =

n

t 4  ⎪ √ 1 t 2 ⎪ ⎩ − Γ (n) 2 1+ + O 2 , n ≥ 1, t ⪡ n + 1 π t n−1 2 n (1.4.2) where .Γ is the Gamma function and .γ = 0.5772... is the Euler-Mascheroni constant. In occasions, we should consider that both the order n and the variable t are sufficiently large, we then need some more sophisticated asymptotic results. Note that the Bessel function admits the following asymptotic formula (see [143], p. 129):  Jn (t) =

.

  π 2 nπ + n arcsin(n/t) − ) 1 + o(1) , cos( t 2 − n2 − √ 2 4 π t 2 − n2 (1.4.3)

1.4 Bessel and Neumann Functions

7

for .t > n and .n → ∞. Besides, there also admits the following asymptotic formula (see [143], p. 129): Jn (nz) =

.

zn e n

√

1−z2

(2π n)1/2 (1 − z2 )1/4 (1 +

√

1 − z 2 )n

 1 + o(1) ,

0 < z < 1.

Chapter 2

Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

In this chapter, we consider the surface plasmon resonance for electro-magnetic scattering governed by the Maxwell system [12] and the surface polariton resonance for elastic scattering governed by the Lamé system [54]. As mentioned earlier, we consider the two resonance phenomena in the quasi-static regime, i.e. the size of the metamaterial structures are smaller than the operating wavelength. We shall mainly treat the resonance corresponding to a single metamaterial structure, which is usually referred to as a nanoparticle in the literature. This is physically unobjectionable since the collective resonances of multiple nanoparticles form the fundamental basis for many engineering and industrial applications. We aim at rigorously deriving the resonance conditions that couple the geometric and medium configurations of the nanoparticle as well as understanding the qualitative and quantitative behaviours of the resonant fields.

2.1 Maxwell’s Problem We mention that there are a lot of relevant results on plasmonic resonances for nanoparticals within the electro-magnetic system, e.g. [8, 12, 18, 64]. The readers are encouraged to refer to the relevant research for different parameters setup and new insights regarding the plasmonic resonance.

2.1.1 Introduction to Plasmonic Resonances Consider an open connected domain D with .C 1,η boundary in .Rd (d = 2, 3) for some .0 < η < 1. Given a harmonic function .u0 in .Rd , we consider the following

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Y. Deng, H. Liu, Spectral Theory of Localized Resonances and Applications, https://doi.org/10.1007/978-981-99-6244-0_2

9

10

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

transmission problem in .Rd :

.

⎧ ⎨∇ · (εD ∇u) = 0

in Rd ,

⎩u − u = O(|x|1−d ) 0

as |x| → ∞,

(2.1.1)

where .εD = εc χ (D) + εm χ (Rd \D) with .εc , εm being two positive constants, and χ (Ω) is the characteristic function of the domain .Ω = D or .Rd \D. With the help of the single-layer potential, we can rewrite the perturbation .u − u0 , which is due to the inclusion D, as

.

u − u0 = SD [ϕ] ,

.

(2.1.2)

where .ϕ ∈ L2 (∂D) is an unknown density, and .SD [ϕ] is the refraction part of the potential in the presence of the inclusion. The transmission problem (2.1.1) can be rewritten as ⎧  ⎪ Δu = 0 in D (Rd \D) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨u|+ = u|− on ∂D , (2.1.3) .   ⎪ ∂u  ∂u  ⎪ ε = ε on ∂D , m c ⎪ ∂ν ∂ν + − ⎪ ⎪ ⎪ ⎩ u − u0 = O(|x|1−d ) as |x| → ∞ . With the help of the jump condition (1.2.5), solving the above system (2.1.3) can be regarded as solving the density function .ϕ ∈ L2 (∂D) of the following integral equation ∂u0 . = ∂ν



εc + εm ∗ I − KD [ϕ] . 2(εc − εm )

(2.1.4)

With the harmonic property of .u0 , we can write u0 (x) =

.

1 ∂ α u0 (0)xα α! d

(2.1.5)

α∈N

with .α = (α1 , . . . , αd ) ∈ Nd , ∂α = ∂1α1 . . . ∂dαd and .α! = α1 ! . . . αd ! . Consider .ϕ α as the solution of the Neumann-Poincaré operator: ∂x α = . ∂ν



εc + εm ∗ I − KD [ϕ α ] . 2(εc − εm )

(2.1.6)

εc +εm The invertibilities of the operator .( 2(ε I − KD∗ ) from .L2 (∂D) onto .L2 (∂D) c −εm ) 2 2 and from .L0 (∂D) onto .L0 (∂D) are proved, for example, in [15, 78], provided that

2.1 Maxwell’s Problem

11

εc +εm | > 1/2. We can substitute (2.1.5) and (2.1.6) back into (2.1.2) to get | 2(ε c −εm )

.



1

1 ∂ α u0 (0) ∂ α u0 (0)SD [ϕ α ] = G(x − y)ϕ α (y)ds(y) . α! α! ∂D

u − u0 =

.

|α|≥1

|α|≥1

(2.1.7) Using the Taylor expansion, G(x − y) = G(x) − y · ∇G(x) + O(

.

1 ), |x|d

(2.1.8)

which holds for all .x such that .|x| → ∞ while .y is bounded [15], we get the following result by substituting (2.1.8) into (2.1.7) that (u − u0 )(x) = ∇u0 (0) · M(λ, D)∇G(x) + O(

.

1 ) |x|d

as |x| → ∞,

(2.1.9)

where .M = (mij )di,j =1 is the polarization tensor associated with the domain D and the contrast .λ defined by  mij (λ, D) :=

.

∂D

yi (λI − KD∗ )−1 νj (y)ds(y) ,

(2.1.10)

with λ :=

.

εc + εm 2(εc − εm )

(2.1.11)

and  .νj being the j -th component of .ν. Here we have used in (2.1.9) the fact that ∂D ν ds = 0. Typically the constants .εc and .εm are positive in order to make the system (2.1.3) physical. This corresponds to the situation with .|λ| > 12 . However, recent advances in nanotechnology make it possible to produce noble metal nanoparticles with negative permittivities at optical frequencies [75, 118]. Therefore, it is possible that for some frequencies, .λ actually belongs to the spectrum of .KD∗ . If this happens, the following integral equation

.

  0 = λI − KD∗ [ϕ]

.

on ∂D

(2.1.12)

has non-trivial solutions .ϕ ∈ L2 (∂D) and the nanoparticle resonates at those frequencies.

12

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

Therefore, we have to investigate the mapping properties of the NeumannPoincaré operator. Assume that .∂D is of class .C 1,η , .0 < η < 1. It is known that the operator .KD∗ : L2 (∂D) → L2 (∂D) is compact [78], and its spectrum is discrete and accumulates at zero. All the eigenvalues are real and bounded by .1/2. Moreover, .1/2 is always an eigenvalue and its associated eigenspace is of dimension one, which is nothing else but the kernel of the single-layer potential .SD . In two dimensions, it can be proved that if .λi /= 1/2 is an eigenvalue of .KD∗ , then .−λi is an eigenvalue as well. This property is known as the twin spectrum property; see [108]. The Fredholm eigenvalues are the eigenvalues of .KD∗ . It is easy to see, from the properties of .KD∗ , that they are invariant with respect to rigid motions and scaling. They can be explicitly computed for ellipses and spheres. If a and b denote the semi-axis lengths of an ellipse then it can be shown that .±((a − b)/(a + b))i are its Fredholm eigenvalues [79]. For the sphere, they are given by .1/(2(2i + 1)). It is worth noticing that the convergence to zero of Fredholm eigenvalues is exponential for ellipses while it is algebraic for spheres. Equation (2.1.12) corresponds to the case when plasmonic resonance occurs in D; see [70]. Finally, we briefly investigate the eigenvalue of the Neumann-Poincaré operator of multiple particles. Let .D1 and .D2 be two smooth bounded domains such that the distance .dist(D1 , D2 ) between .D1 and .D2 is positive. Let .ν (1) and .ν (2) denote the outward normal vectors at .∂D1 and .∂D2 , respectively. The Neumann-Poincaré operator .K∗D1 ∪D2 associated with .D1 ∪ D2 is given by [11]  ∗ .KD ∪D 1 2

:=

KD∗1 ∂ S ∂ν (2) D1

∂ S ∂ν (1) D2 KD∗2

 .

(2.1.13)

In Sect. 2.1.6 we will be interested in how the eigenvalues of .K∗D1 ∪D2 behave numerically as .dist(D1 , D2 ) → 0.

2.1.2 Drude’s Model for the Electric Permittivity and Magnetic Permeability Let D be a bounded domain in .Rd with .C 1,η boundary for some .0 < η < 1, and let .(εm , μm ) be the pair of electro-magnetic parameters (electric permittivity and magnetic permeability) of .Rd \ D and .(εc , μc ) be that of D. We assume that .εm and .μm are real positive constants. We have εD = εm χ (Rd \ D) + εc χ (D) and

.

μD = μm χ (Rd \ D) + μc χ (D).

2.1 Maxwell’s Problem

13

Suppose that the electric permittivity .εc and the magnetic permeability .μc of the nanoparticle are changing with respect to the operating angular frequency .ω while those of the surrounding medium, .εm , μm , are independent of .ω. Then we can write εc (ω) = ε' (ω) + iε'' (ω), .

(2.1.14)

μc (ω) = μ' (ω) + iμ'' (ω).

Because of causality, the real and imaginary parts of .εc and .μc obey the following Kramer–Kronig relations:  +∞ 1 1 ε'' (s)ds, ε (ω) = − p.v. π ω − s −∞  +∞ 1 1 ε' (s)ds, ε'' (ω) = p.v. π ω − s −∞ .  +∞ 1 1 μ'' (ω) = − p.v. μ' (s)ds, π ω − s −∞  +∞ 1 1 μ' (ω) = p.v. μ'' (s)ds, π −∞ ω − s '

(2.1.15)

where .p.v. denotes the principle value. √ √ In the sequel, we set .kc = ω εc μc and .km = ω εm μm and denote by λε (ω) =

.

εc (ω) + εm , 2(εc (ω) − εm )

λμ (ω) =

μc (ω) + μm . 2(μc (ω) − μm )

(2.1.16)

We have λε (ω) =

.

2 + (ε '' (ω))2 ε' (ω)ε'' (ω) (ε' (ω))2 − εm −i . ' 2 '' 2 ' 2((ε (ω) − εm ) + (ε (ω)) 2((ε (ω) − εm )2 + (ε'' (ω))2

A similar formula holds for .λμ (ω). The electric permittivity .εc (ω) and the magnetic permeability .μc (ω) can be described by the Drude Model; see, for instance, [118]. We have εc (ω) = ε0 (1 −

.

ωp2 ω(ω + iτ −1 )

)

and

μc (ω) = μ0 (1 − F

ω2 ), ω2 − ω02 + iτ −1 ω

14

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

or equivalently, ε' (ω) = ε0

ω2 + τ −2 − ωp2

,

ε'' (ω) = ε0

ωp2 τ −1

ω2 + τ −2 ω(ω2 + τ −2 ) 2 −2 2 2 μ0 (τ ω + (ω − ω0 )((1 − F )ω2 − ω02 ) . ' μ (ω) = , (ω2 − ω02 )2 + τ −2 ω2 −1 μ'' (ω) = 2 μ0 F2 τ2 ω−2 2 , (ω −ω0 ) +τ

,

ω

where .ωp is the plasma frequency of the bulk material, .τ > 0 is the nanoparticle’s bulk electron relaxation rate (.τ −1 is the damping coefficient), F is a filling factor, and .ω0 is a localized plasmon resonant frequency. When ω2 + τ −2 < ωp2

.

and (1 − F )(ω2 − ω02 )2 − F ω02 (ω2 − ω02 ) + τ −2 ω2 < 0,

the real parts of .ε(ω) and .μ(ω) are negative. Typical values are ● ● ● ●

τ = 10−14 s; 15 H z; .ω = 10 −12 F m−1 ; .ε = (1.33)2 ε ; .ε0 = 9 · 10 m 0 15 −1 for a gold nanoparticle; .ωp = 2 · 10 s .

It is interesting to have an idea on the size of .ℑm(λε ) (resp. .ℑm(λμ )) since it will be a lower bound for the distance .dist(λε , σ (KD∗ )) (resp. .dist(λμ , σ (KD∗ ))) between .λε (resp. .λμ ) and the spectrum of the Neumann-Poincaré operator .KD∗ . Finally, we define dielectric and magnetic plasmonic resonances. We say that .ω is a dielectric plasmonic resonance if the real part of .λε is an eigenvalue of ∗ .K . Analogously, we say that .ω is a magnetic plasmonic resonance if the real D part of .λμ is an eigenvalue of .KD∗ . Note that if .ω is a dielectric (resp. magnetic) plasmonic resonance, then the polarization tensor .M(λε (ω), D) defined by (2.1.10) (resp. .M(λμ (ω), D)) blows up. In the case of two particles .D1 and .D2 with the same electro-magnetic parameters, .εc (ω) and .μc (ω), we say that .ω is a dielectric (resp. magnetic) plasmonic resonance, if the real part of .λε is an eigenvalue of .K∗D1 ∪D2 . Analogously, we say that .ω is a magnetic plasmonic resonance if the real part of .λμ is an eigenvalue of ∗ .K D1 ∪D2 . Let the polarization tensor .M(λ, D1 ∪ D2 ) = (mij )di,j =1 be defined by  (1)    ∗ −1 νj mij (λ, D1 ∪ D2 ) := yi (λI − KD ) ds(y) (2) (y) νj ∂D1 1   .    (1) ν + yi (λI − K∗D )−1 j(2) (y) ds(y) , νj ∂D2 2 

(2.1.17)

2.1 Maxwell’s Problem

15

where .ν (l) = (ν1 , . . . , νd ), .l = 1, 2, and .[ ]l ' denotes the .l ' th component. As for single particles, .M(λ(ω), D1 ∪ D2 ) = (mij )di,j =1 blows up for .λ(ω) such that .ω is a dielectric or magnetic plasmonic resonance. (l)

(l)

2.1.3 Boundary Integral Operators and Resolvent Estimates We start by recalling some well-known properties about boundary integral operators and proving a few technical lemmas that will be used for deriving the asymptotic expansions of the electric and magnetic fields in the presence of nanoparticles. As will be shown in Sect. 2.1.6, the plasmonic resonances for multiple identical particles are shifted from those of the single particle as the separating distance between the particles becomes comparable to their size. We first review commonly used function spaces. Let .∇∂D · denote the surface 3 divergence. Denote by .L2T (∂D) := {φ ∈ L2 (∂D) , ν · φ = 0}. Let .H s (∂D) be the usual Sobolev space of order s on .∂D. We also introduce the function spaces   TH(div, ∂D) : = φ ∈ L2T (∂D) : ∇∂D · φ ∈ L2 (∂D) ,   TH(curl, ∂D) : = φ ∈ L2T (∂D) : ∇∂D · (φ × ν) ∈ L2 (∂D) , .

equipped with the norms ‖φ‖TH(div,∂D) = ‖φ‖L2 (∂D) + ‖∇∂D · φ‖L2 (∂D) ,

.

‖φ‖TH(curl,∂D) = ‖φ‖L2 (∂D) + ‖∇∂D · (φ × ν)‖L2 (∂D) . We define the vectorial curl for .ϕ ∈ H 1 (∂D) by .curl∂D ϕ = −ν × ∇∂D ϕ. The following result from [36] will be useful. Proposition 2.1.1 The following Helmholtz decomposition holds: L2T (∂D) = ∇∂D (H 1 (∂D))⊕curl∂D (H 1 (∂D)).

.

(2.1.18)

Next, we recall that, for .k > 0, the fundamental outgoing solution .Gk to the Helmholtz operator .(Δ + k 2 ) in .R3 is given by Gk (x) = −

.

eik|x| . 4π |x|

(2.1.19)

16

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

For a density .φ ∈ TH(div, ∂D), we define the vectorial single layer potential associated with the fundamental solution .Gk introduced in (2.1.19) by  ADk [φ](x) :=

Gk (x − y)φ(y)ds(y),

.

x ∈ R3 .

(2.1.20)

∂D

For a scalar density .ϕ ∈ L2 (∂D), the single layer potential is defined similarly by  k .SD [ϕ](x)

:=

Gk (x − y)ϕ(y)ds(y),

x ∈ R3 .

(2.1.21)

∂D

We will also need the following boundary operators: MDk : L2T (∂D) −→ L2T (∂D) .



φ ‫׀‬−→ MDk [φ] = ν(x) × ∇ ×

(2.1.22) Gk (x, y)φ(y)ds(y), ∂D

NDk : TH(curl, ∂D) −→ TH(div, ∂D) φ ‫׀‬−→ NDk [φ] = 2ν(x) × ∇ × ∇  × Gk (x, y)ν(y) × φ(y)ds(y),

.

(2.1.23)

∂D

LDk : TH(div, ∂D) −→ TH(div, ∂D) .

φ ‫׀‬−→ LDk [φ] = ν(x) × (k 2 ADk [φ](x) + ∇SDk [∇∂D · φ](x)). (2.1.24)

In the following, we denote by .AD , .SD , .MD , and .ND the operators .AD0 , .SD0 , 0 0 .M , and .N D D corresponding to .k = 0, respectively. Let .KD be the .L2 -adjoint of .KD∗ . Since .KD and KD∗ : L2 (∂D) → L2 (∂D)

.

are compact and all the eigenvalues of .KD∗ are real, we have .σ (KD ) = σ (KD∗ ). We start with stating the following jump formula. We refer the reader to Appendix A for its proof. Proposition 2.1.2 Let .φ ∈ L2T (∂D). Then .ADk [φ] is continuous on .R3 and its curl satisfies the following jump formula:  φ ν × ∇ × ADk [φ]± = ∓ + MDk [φ] 2

.

on ∂D,

(2.1.25)

2.1 Maxwell’s Problem

17

where ∀x ∈ ∂D,

.

 ν(x) × ∇ × ADk [φ]± (x) = lim ν(x) × ∇ × ADk [φ](x ± tν(x)). t→0+

Next, we prove the following integral identities. Proposition 2.1.3 We have MD∗ = rMD r,

(2.1.26)

.

where .r is defined by r(φ) = ν × φ,

.

∀φ ∈ L2T (∂D).

(2.1.27)

Moreover, ∇ · ADk [φ] = SDk [∇∂D · φ]

.

in R3 ,

∀φ ∈ T H (div, ∂D) .

∗  ∇∂D · MDk [φ] = −k 2 ν · ADk [φ] − KDk [∇∂B · φ],

.

(2.1.28)

∀φ ∈ T H (div, ∂D) . (2.1.29)

Furthermore, ∇∂D · MD [φ] = −KD∗ [∇∂D · φ],

.

∀φ ∈ T H (div, ∂D) ,

MD∗ [∇∂D φ] = −∇∂D KD [φ],

.

(2.1.30) (2.1.31)

and MD [curl∂D φ] = curl∂D KD [φ],

.

∀φ ∈ T H (curl, ∂D).

(2.1.32)

Proof The proof of (2.1.28) can be found in [44]. We give it here for the sake of completeness. If .φ ∈ T H (div, ∂D), then  k .∇ · AD [φ](x) = ∇x · (Gk (x − y)φ(y)) ds(y), x ∈ R3 \ ∂D. ∂D

Using the fact that ∇x · (Gk (x − y)φ(y)) = φ(y)∇x Gk (x − y) = −φ(y)∇y Gk (x − y),

.

and since ∇y Gk (x − y) = ∇∂D Gk (x − y) +

.

∂Gk (x − y), ∂ν(y)

18

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

we get  ∇ · ADk [φ](x) = −

φ(y) · ∇∂D,y Gk (x − y)ds(y),

.

x ∈ R3 \ ∂D.

∂D

Using the fact that .−∇∂D is the adjoint of .∇∂D · we obtain  ∇ · ADk [φ](x) =

Gk (x − y)∇∂D · φ(y)ds(y),

.

x ∈ R3 \ ∂D.

∂D

Next, since .S k [∇∂D φ] is continuous across .∂D, the above relation can be extended to .R3 and we get (2.1.28). Now, in order to prove (2.1.29), we observe that, for any .φ ∈ T H (div, ∂D), ∇ × ∇ × ADk [φ](x) = k 2 ADk [φ](x) + ∇SDk [∇∂D · φ](x),

.

Using the jump relations on .

x ∈ R3 \ ∂D.

∂SDk we obtain that ∂ν

 ∇∂D · φ ν · ∇ × ∇ × ADk [φ]± = k 2 ν · ADk + (KDk )∗ [∇∂D · φ] ± 2

.

on ∂D.

Recall from [44, p.169] that if .f ∈ C 1 (R3 \ D) ∩ C 0 (R3 \ D),  then .∇∂D · (ν × f) = −ν · (∇ × f). Using the jump formula for .ν × ∇ × ADk [φ]± = MDk [φ] ∓ φ/2, we arrive at (2.1.29). Setting .k = 0 in (2.1.29) gives (2.1.30). Identity (2.1.31) can be deduced from (2.1.30) by duality. Now, we prove (2.1.32). Define .r(a) = ν × a for any smooth vector field .a on 1 .∂D. For .φ ∈ H (∂D), we have MD∗ [∇∂D φ] = −∇∂D KD [φ].

.

Since .MD∗ = rMD r (see [71]) and .curl∂D = −r(∇∂D ), it follows that r (MD [curl∂D φ]) = ∇∂D KD [φ].

.

Composing by .r −1 = −r, we get MD [curl∂D φ] = curl∂D KD [φ],

.

which completes the proof. Lemma 2.1.1 Proof Take .φ

The kernel of the operator .ND in .L2T (∂D) is .∇∂D (H 1 (∂D)). = ∇∂D U with .U ∈ H 1 (∂D). From (2.1.28), it follows that ND [∇∂D U ](x) = −2ν(x) × ∇SD [∇∂D · curl∂D U ].

.

⨆ ⨅

2.1 Maxwell’s Problem

19

Since .∇∂D · curl∂D U = 0, we have .ND [φ] = 0. Now, take .φ ∈ L2T (∂D) such that .ND [φ] = 0. Then, on .∂D, we have 2ν(x) × ∇SD [∇∂D · r(φ)] =2ν(x)  ∂ . × ∇∂D SD [∇∂D · r(φ)] + SD [∇∂D · (r(φ)] ∂ν = − 2curl∂D SD [curl∂D φ]. Since .Ker(curl∂D ) = R (see [36]), we obtain that .S∂D [curl∂D φ] = c ∈ R. Then, curl∂D φ = 0, which implies that .φ ∈ ∇∂D H 1 (∂D) (see again [36]). ⨆ ⨅

.

Proposition 2.1.4 We have the following Calderón type identity: ND MD∗ = MD ND .

.

(2.1.33)

Proof Let .φ ∈ H 1/2 (∂D). We have  

  MD ND [φ] = 2MD r ∇ × ∇ × AD r(φ)

.

and  

  MD ND [φ] = 2MDk r ∇SDk ∇∂D · r(φ) .

.

Since  

∂ SD [∇∂D · r(φ)]ν r ∇SD [∇∂D · r(φ)] =ν × ∇∂D SD [∇∂D · r(φ)] + ∂ν . = − curl∂D SD [∇∂D · r(φ)], we can deduce from (2.1.32) that   MD ND [φ] = −2curl∂D KD SD [∇∂D · r(φ)] .

.

Now, using the fact that .MD∗ = rMD r and that .r −1 = −r, we also have 

 ND MD∗ [φ] = −2r ∇ × ∇ × AD MD r(φ)

.



 ND MD∗ [φ] = −2r ∇SD ∇∂D · MD r(φ) .

.

Moreover, (2.1.30) yields 

 ND MD∗ [φ] = 2r ∇SD KD∗ ∇∂D · r(φ) .

.

20

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

Using Calderón’s identity .SD KD∗ = KD SD and the fact that r(∇KD ) = r(∇∂D KD ) = −curl∂D KD ,

.

it follows that 

 ND MD∗ [φ] = −2curl∂D KD SD ∇∂D · r(φ) ,

.

⨆ ⨅

which completes the proof.

As seen in the Sect. 2.1.1, we have to solve Fredholm type equations involving the resolvent of .KD . We will also need to control the resolvent of .MD . For doing so, the main difficulty is due to the fact that .KD and .MD are not self-adjoint. However, we will make use of a symmetrization technique in order to estimate the norms of the resolvents of .KD and .MD . The following result holds. Proposition 2.1.5 The operator .KD : L2 (∂D) −→ L2 (∂D) satisfies the following resolvent estimate ‖(λI − KD )−1 ‖L2 (∂D) ≤

.

c , dist(λ, σ (KD ))

where .dist(λ, σ (KD )) is the distance between .λ and the spectrum .σ (KD ) of .KD and c is a constant depending only on D. Proof We start from Calderón’s identity ∀φ ∈ L2 (∂D),

.

SD KD [φ] = KD∗ SD [φ].

Since .SD : L2 (∂D) −→ L2 (∂D) is a self-adjoint positive definite invertible operator in dimension three, we can define a new inner product on .L2 (∂D). We denote .H the Hilbert space .L2 (∂D) equipped with the following inner product  2 〈φ, ψ〉H = 〈SD [φ], ψ〉 ∀(φ, ψ) ∈ L2 (∂D) .

.

Since .SD is continuous and invertible, the norm associated with the inner product 〈. , .〉H is equivalent to the .L2 (∂D)-norm. Now, .KD is a self-adjoint compact operator on .H . We can write [69]

.

‖(λI − KD )−1 ‖H ≤

.

1 . dist(λ, σ (KD ))

Switching back to the original norm we get the desired result. Proposition 2.1.6 We have .−σ (MD ) =

σ (KD∗ ) \ { 21 }.

⨆ ⨅

2.1 Maxwell’s Problem

21

Proof First, we note that .−1/2 is not an eigenvalue of .MD ; see [71, 111]. Let λ ∈ σ (MD ). Take .φ ∈ L2T (∂D) such that

.

.

(λI − M ) [φ] = 0

(2.1.34)

Using the Helmholtz decomposition (2.1.18), we write φ = ∇∂D U + curl∂D V .

.

Equation (2.1.34) becomes .

(λI − MD ) (∇∂D U + curl∂D V ) = 0,

(2.1.35)

which yields curl∂D (λI − KD ) V = − (λI − MD ) ∇∂D U.

.

Taking the surface divergence we get λΔ∂D U − ∇∂D · MD [∇∂D U ] = 0,

.

and hence, by using (2.1.30),   λI + KD∗ [Δ∂D U ] = 0.

.

Therefore, either .−λ ∈ σ (KD∗ ) or .Δ∂D U = 0, which implies that U is constant and .∇∂D · φ = 0. Now, we take the surface curl of (2.1.35) to get .

− λΔ∂D V + curl∂D MD [curl∂D V ] = 0.

Using (2.1.32), we obtain Δ∂D (λI − KD ) [V ] = 0.

.

c Then, .λV − KD [V ] = c for some constant .c. By replacing V by .V ' = V + λ−1/2 ' ' and using the fact that .KD [1] = 1/2, we arrive at .λV −KD [V ] = 0. If .λ ∈ / σ (K ), then .φ would be constant, which would yield a contradiction. Now, let .λ ∈ σ (KD ) \ {1/2} and let .ϕ be an eigenvector associated with .λ. From .

(λI − KD ) [ϕ] = 0,

Taking the surface curl and using (2.1.32) gives .

(λI − MD ) [curl∂D ϕ] = 0.

22

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

Either .λ ∈ σ (MD ) or .curl∂D ϕ = 0, which means that .ϕ is constant [36]. Since λ /= 1/2, .ϕ cannot be constant. ⨆ ⨅  1  Lemma 2.1.2 Let .φ ∈ H := curl∂D H (∂D) (.H is the space of divergence free vectors in .L2T ). The following resolvent estimate holds:

.

‖ (λI + MD )−1 [φ]‖H ≤

.

c . dist (λ, σ (KD ))

(2.1.36)

Proof We proceed exactly as in the proof of Proposition 2.1.5. If we denote by 〈., .〉H the usual scalar product on .H, then we introduce a new scalar product defined by

.

∀φ, ψ ∈ H × H,

.

〈φ, ψ〉N = 〈ND [φ], ψ〉H ,

 where .ND H is the operator induced by .ND given in (2.1.23) on .H. Then, we first prove that .H is stable by .ND . If .φ ∈ H, then .ND [φ] ∈ TH(div, ∂D) (see [44]) and, using the fact that for any .f ∈ H (curl, Ω), .∇∂D · (ν × f) = −ν · ∇ × f, we get ∇∂D · ND [φ] = ν · ∇ × ∇SD [∇∂D · (ν × φ)] = 0,

.

which means that .ND [φ] ∈ H. For the sake of simplicity we will denote by .ND the induced operator on .H. It is easy to see that this bilinear operator is well defined, continuous and positive. Then, .ND is self-adjoint [44]. The bilinear form is positive since  〈N [φ], φ〉H = N [φ](x) · φ(x)ds(x), 

∂D



∂D

ν(x) × ∇SD [∇∂D · (ν × φ)] (x) · φ(x)ds(x),

=

−curl∂D SD [curl∂D φ] (x) · φ(x)ds(x),

=

.

∂D



SD [curl∂D φ] (x)curl∂D φ(x)ds(x),

=− ∂D

= −〈SD [curl∂D φ] , curl∂D φ〉L2 (∂D) . If we equip .H with this new scalar product, then we can see by using Proposition 2.1.33 that .MD is self-adjoint and therefore, ∀φ ∈ H,

.

‖ (λI − MD )−1 [φ]‖N ≤

1 ‖φ‖H . dist (λ, σ (MD ))

2.1 Maxwell’s Problem

23

Using the fact that .ND is injective and continuous on .H, we can go back to the original norm to have ‖ (λI − MD )−1 [φ]‖ ≤

∀φ ∈ H,

.

C ‖φ‖H , dist (λ, σ (MD )) ⨆ ⨅

which completes the proof.

Proposition 2.1.7 Let .λ ∈ C \ [− 12 , 12 ]. There exists a positive constant C such that ∀φ ∈ L2T (∂D),

.

‖ (λI − MD )−1 [φ]‖L2 (∂D) ≤ T

C ‖φ‖L2 (∂D) . T dist(λ, σ (MD )) (2.1.37)

 2 Proof Let .ψ, φ ∈ L2T (∂D) be such that .

(λI − MD ) [ψ] = φ.

(2.1.38)

Using Helmholtz decomposition (2.1.18), we can write ψ = ∇∂D U + curl∂D V ,

.

with .U ∈ H 1 (∂D) and .V ∈ H 1/2 (∂D). Taking the surface divergence of (2.1.38), together with using (2.1.30), (2.1.32), and the fact that .∇∂D · curl∂D f = 0, ∀f , yields .

  λI − KD∗ [Δ∂D U ] = ∇∂D · φ,

which can be written as −1  Δ∂D U = λI − KD∗ [∇∂D φ] .

.

(2.1.39)

Now we deal with the curl part. If we apply .ND on (2.1.38) we get by using (2.1.33) together with Lemma 2.1.1 that .

  λI − MD∗ ND curl∂D V = ND φ,

or equivalently, −1  ND curl∂D V = λI − MD∗ ND φ.

.

(2.1.40)

From the Helmholtz decomposition of .φ: .φ = ∇∂D φ1 +curl∂D φ2 , (2.1.40) becomes −1  ND curl∂D V = λI − MD∗ ND [curl∂D φ2 ] .

.

(2.1.41)

24

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

Now, we can work in the function space .H = curl∂D H 1/2 (∂D). We denote by .N D the operator induced by .ND on .H and by .R(N D ) ⊂ H the range of the induced  operator. .MD also induces an operator .M D on .H; see the proof of (2.1.33).   ∗ −1 N  Next, we want to make sure that . λI − M D curl∂D V belongs to .R(ND ) D  so that we can apply .N D ’s left inverse (recall from Lemma 2.1.1 that .ND is  −1   injective). For doing so, we show that the range of .ND is stable by . λI − MD∗ .  Take .f = N D [g] ∈ R(ND ). Then,   ∗ −1 [f] ∈ R(N λI − M D ) ⇔ ∃h ∈ H, D ⇔ ∃h ∈ H, ⇔ ∃h ∈ H, .

⇔ ∃h ∈ H,

  ∗ −1 N  λI − M D [g] = ND [h] D   ∗  N D [g] = λI − MD ND [h]     N D [g] = ND λI − MD [h]    g = λI − M D [h]

( by injectivity of N D)  −1  ⇔ ∃h ∈ H, λI − M [g] = h. D  So we have the stability of .R(N D ) by .MD and    −1  ∗ −1 N  −1 . λI − M N D = λI − MD D D

.

(2.1.42)

Applying this to (2.1.41) we get −1   curl∂D V = λI + M curl∂D φ2 . D

.

Using Lemma 2.1.2 we get the desired result.

⨆ ⨅

2.1.4 Layer Potential Formulation For a given plane wave solution .(Ei , Hi ) to the Maxwell equations  .

∇ × Ei = iωμm Hi in R3 , ∇ × Hi = −iωεm Ei in R3 ,

let .(E, H) be the solution to the following Maxwell equations: ⎧ in R3 \ ∂D, ⎨ ∇ × E = iωμH . ∇ × H = −iωεE in R3 \ ∂D, ⎩ [ν × E] = [ν × H] = 0 on ∂D,

(2.1.43)

2.1 Maxwell’s Problem

25

subject to the Silver-Müller radiation condition: .

√ √ lim |x|( μ(H − Hi ) × xˆ − ε(E − Ei )) = 0,

|x|→∞

where .xˆ = x/|x|. Here, .[ν × E] and .[ν × H] denote the jump of .ν × E and .ν × H along .∂D, namely,   [ν × E] = (ν × E)+ − (ν × E)− ,

.

  [ν × H] = (ν × H)+ − (ν × H)− .

Using the layer potentials defined in Sect. 2.1.3, the solution to (2.1.43) can be represented as:  E(x) =

.

Ei (x) + μm ∇ × SDkm [φ](x) + ∇ × ∇ × ADkm [ψ](x), x ∈ R3 \ D, x ∈ D, μc ∇ × ADkc [φ](x) + ∇ × ∇ × ADkc [ψ](x), (2.1.44)

and H(x) = −

.

 i  ∇ × E (x), ωμ

x ∈ R3 \ ∂D,

(2.1.45)

where the pair .(φ, ψ) ∈ T H (div, ∂D) × T H (div, ∂D) is the unique solution to ⎤ ⎡μ + μ c m LDkc − LDkm I + μc MDkc − μm MDkm ⎥ ⎢ 2  2 2 2 .⎣ ⎦ kc kc2 kc km km kc km km I+ + MD − MD LD − LD 2μc 2μm μc μm     φ ν × Ei  . (2.1.46) = iων × Hi ∂D ψ The invertibility of the system of equations (2.1.46) on .T H (div, ∂D) × T H (div, ∂D) was proved in [121]. Moreover, there exists a constant .C = C(ε, μ, ω) such that   ‖φ‖T H (div,∂D) + ‖ψ‖T H (div,∂D) ≤ C ‖Ei × ν‖T H (div,∂D) + ‖Hi × ν‖T H (div,∂D) . (2.1.47)

.

2.1.5 Derivation of the Asymptotic Formula We will need the following notation. For a multi-index .α ∈ N3 , let .x α = x1α1 x2α2 x3α3 , α1 α2 α3 α .∂ = ∂ 1 ∂2 ∂3 , with .∂j = ∂/∂xj .

26

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

& + z, where .D & is a .C 1,η (.0 < η < 1) domain containing the origin. Let .D = δ D y−z & y) = ψ(y). & y) = φ(y) and .ψ(& & For any .y ∈ ∂D, let & .y = δ ∈ ∂ D. Denote by .φ(& k k We have the following expansions for .MD and .LD . Proposition 2.1.8 Let .φ ∈ L2T (∂D). As .δ → 0, we have MDk [φ](x) = MD φ](& x) + O(δ 2 ). & [&

(2.1.48)

.

Proof Let .x ∈ ∂D, and write & .x = k .MD [φ](δ& x + z)

1 =− 4π δ

Changing .y by & .y = k .MD [φ](δ& x + z)

y−z δ

x−z δ .

We have 

 ∂D



ν D (δ& x + z) × ∇&x ×

eik|δ&x+z−y| φ(y) |δ& x + z − y|

ds(y).

in the integral we get

1 =− 4π δ

  ikδ|&x−&y| e & φ(& y) δ 2 ds(& ν D (δ& x + z) × ∇&x × y). δ|& x −& y| & ∂D



x−z Since .∀x ∈ ∂D, ν D (x) = ν D & ( δ ),

MDk [φ](x) = MBδk [& φ](& x).

.

& it follows that For any & .x ∈ δ D, δk & MD x) = MD φ](& x) + & [& & [φ](&

.

 & ∂D

  νD x) × (∇&x × (ikδ)) + O δ 2 , & (& ⨆ ⨅

which gives the result. Proposition 2.1.9 Let .φ ∈ T H (div, ∂D). For any .y ∈ ∂D, we have

LDkm [φ](y) − LDkc [φ](y) =    & y −& y'  1 2 ' ' & ∇ · φ(& y ) ds(& y ) − kc2 )ν D y) × AB [& φ](& y) + δ(km & (& & 8π ∂ D y −& y' | ∂ D & |&   (2.1.49) + O δ2 .

.

Proof Note that, for .y ∈ ∂D, δk & ADk [φ](y) = δAD y) & [φ](&

.

and ∇∂D SDk [∇∂ D & · φ](y) =

.

1 ∇ & S &δk [∇∂ D φ](& y). & ·& δ ∂D D

2.1 Maxwell’s Problem

27

We can expand δk & AD y) = AD φ](& y) + O(δ). & [& & [φ](&

.

We also have 1 δk ∇∂ D φ](& y) = − & SD & ·& & [∇∂ D 4π   1 1 2 2 ' ' 2 3 ' 3 1 + kδ|& y −& y | − k δ |& × ∇∂ D y −& y | + O(δ |& y −& y| ) & y −& y' | 2 & |& ∂D

.

φ(& y' )ds(& y' ) ∇∂ D & ·& and δk ∇∂ D φ](& y) = − & SD & ·& & [∇∂ D

.



1 ∇ & ·& φ(& y' )ds(& y' ) y −& y' | ∂ D & |& ∂D  k2δ2 ∇∂ D + |& y −& y' |∇∂ D φ(& y' ) + O(δ 3 ). & & ·& 8π & ∂D

1 ∇ & 4π ∂ D

   & ∈ C 1 (R3 \ D)  ∈ C 1 (D), & and & SD & SD Now, since .∀f ∈ L2 (∂ D), & [f ] R3 \D & [f ] D & & we the tangential component of the gradient of .S [f ] is continuous across .∂ D, can state that    & (&  & (& & νD ∀& y ∈ ∂ D, y) × ∇∂ D y) = ν D (& y) × ∇SD y) & (& & [f ] ∂ D & [f ] R3 \D & SD   (& = ν D (& y) × ∇SD & [f ] D & y).

.

Then we can write 1 δk & νD y) ∀& y ∈ ∂ D, y) × ∇∂ D φ](& y) = − ν D & (& & (& & SD & ·& & [∇∂ D 4π   1 ∇ & ·& × ∇∂ D φ(& y' )dσ (& y' ) & '| ∂D & |& y − y & ∂D   & y −& y' k2δ3 ' 3 3 & ∇ · φ(& y )dσ (& y) + O(k δ ) . − & 2 ∂D y −& y' | ∂ D & |&

.

The proof is then complete.

⨆ ⨅

28

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

2.1.5.1

Far-Field Expansion

&β for every .β ∈ N3 by φ β and .ψ Define .&  δ .W & D

& φβ &β ψ



 =

ν(& y) × (& yβ ∂ β Ei (z)) iων(& y) × (& yβ ∂ β H i (z))

 (2.1.50)

with μ

m +μc

δ .W & D

=

2

δkc δkm I + μc MD & − μm MD & kc km LD,δ & − LD,δ &



kc km LD,δ & − LD,δ &

δkc δkm c ω2 ( εm +ε & − εm MD & ) 2 I + εc MD

.

(2.1.51) Using (2.1.44) we have the following expansion for .E(x) for .x far away from .z: ∞ ∞



E(x) = Ei (x) +

.

δ 2+|α|+|β|

|α|=0 |β|=0

(−1)|α| α!β!

  μm ∇∂ α Gkm (x − z) ×

& ∂D

& φ β (& yα & y)ds(& y) 

α

+ ∇ × ∇∂ Gkm (x − z) ×

& ∂D

& y ψ β (& y)ds(& y) . α&

(2.1.52)

& φ β ds(& yα & y).

(2.1.53)

For .β ∈ N3 , define the tensors by  Meα,β :=

.

& ∂D

&β ds(& & yα ψ y)

 Mhα,β :=

and

& ∂D

The following lemma holds. Lemma 2.1.3 For .x ∈ R3 \ D, we have E(x) = Ei (x) +

∞ ∞



.

δ 2+|α|+|β|

|α|=0 |β|=0

(−1)|α|  μm ∇∂ α Gkm (x − z) × Mhα,β α!β!

 + ∇ × ∇∂ α Gkm (x − z) × Meα,β .

(2.1.54)

φ β and Proposition 2.1.10 Let .β ∈ N3 . We can write the following expansions for .& & .ψ β : & φβ =

∞

.

n=0

φ β,n , δ n&

&β = ψ

∞

n=0

&β,n . δnψ

2.1 Maxwell’s Problem

29

& .β, .E, and .H such that Moreover, there exists a .C ≥ 0 depending on .D, ⎿n/2⏌+1 1 dist(λμ , σ (MD & ))  ⎾n/2⏋ 1 , dist(λε , σ (MD & ))  ⎿n/2⏌+1 1 (n+1) ≤C dist(λε , σ (MD & ))  ⎾n/2⏋ 1 . dist(λμ , σ (MD & ))

(n+1) ∀n ∈ N, ‖& φ β,n ‖TH(div,∂ D) & ≤C

.

&β,n ‖ ∀n ∈ N, ‖ψ & TH(div,∂ D)



(2.1.55)

Proof We proceed by induction. Using Propositions 2.1.8 and 2.1.9 we find that   −1 & φ β,0 = (μc − μm )−1 (λμ I + MD ν(& y) × (& yβ ∂ β E(z) , &) .   &β,0 = iω−1 (εc − εm )−1 (λε I + M & )−1 ν(& y) × (& yβ ∂ β H(z) . ψ D

(2.1.56)

Note that .∇∂ D φ β,0 = 0 for .β = 0. Indeed, & ·&    ∇∂ D φ = (μc − μm )−1 (λμ I − KB∗ )−1 ∇∂ D y) × (& yβ ∂ β E(z) , & ·& & · ν(&

.

and     ∇∂ D y) × & yβ ∂ β E(z) =ν(& y) · ∇ × [& yβ E(z)] & · (ν(& .

=0. &β,0 = 0 for .β = 0. Using Proposition 2.1.7, we In the same way we have .∇∂ D &·ψ get the result. &β,1 are φ β,1 and .ψ For the first-orders the equations satisfied by .& 2 &β,0 ] = 0, (μc − μm )(λμ I + MD φ β,1 ] + (kc2 − km )ν(& y) × AD & )[& & [ψ .

&β,1 ] + (k 2 − k 2 )ν(& ω2 (εc − εm )(λε I + MD y) × AD φ β,0 ] = 0. & )[ψ & [& c m

(2.1.57)

30

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

The fact that .AD & is bounded together with Proposition 2.1.7 gives the estimate of & ‖& φ β,1 ‖L2 (∂ D) & and .‖ψ β,1 ‖L2 (∂ D) & . If we take the surface divergence of (2.1.57), we T T get

.

  ∗ 2 &β,0 ] = 0, (μc − μm )(λμ I − KD φ β,1 ] + (kc2 − km )∇∂ D y) × AD & [ψ & ·& & · ν(& & )[∇∂ D .   ∗ &β,1 ] + (k 2 − k 2 )∇ & · ν(& y) × AD φ β,0 ] = 0. ω2 (εc − εm )(λε I − KD & [& &·ψ c m ∂D & )[∇∂ D     &β,0 ] = ν(& Since .∇∂ D y) × AD y) · ∇ × AD φ β,0 ] and .f ‫ →׀‬ν · ∇ × AD & [ψ & [& & [f] & · ν(& & into .L2 (∂ D) & , we can estimate the .L2 norm of .∇∂ D is bounded from .L2T (∂ D) φ β,1 & ·& as follows ' '  −1 '  '  1 ∗ ' ' & . ν(& y) · ∇ × AD & [φ β,0 ] ' & ' μ − μ λμ − KD & c m L2 (∂ D) ≤

c ‖& φ β,0 ‖L2 . T dist(λμ , σ (KD & ))

Since .σ (MD & ) = σ (KD ) (see Proposition 2.1.6) we get the result. The estimate for &β,1 ‖L2 is obtained in the same way. ‖∇∂ D &·ψ &β,n+1 satisfy the following system: φ β,n+1 and .ψ Now, fix .n ∈ N∗ ; .&

.

2 (μc − μm )(λμ I + MD φ β,i+1 ] + (kc2 − km )ν(& y) & )[&   &β,i ] + B & [ψ &β,i ] = 0, × AD & [ψ D .

&β,i+1 ] + (k 2 − k 2 )ν(& ω2 (εc − εm )(λε I + MD y) & )[ψ c m   × AD φ β,i ] + BD φ β,i ] = 0, & [& & [&

where the operator .BD & is defined by & −→TH(div, ∂ D) & TH(div, ∂ D)  .  & y −& y'  1 ∇∂ D y' ) dσ (& y' ). f− ‫→ ׀‬ & · f(& ' y −& y| 8π ∂ D & |& φ β,n+1 , The operator .BD & is bounded, and we can get the norm estimates for .& &β,n+1 , as before. &β,n+1 , .∇ & · & .ψ φ β,n+1 and .∇ & · ψ ⨆ ⨅ ∂D

∂D

2.1 Maxwell’s Problem

31

By Lemma 2.1.3, for .x ∈ R3 \ D,   E(x) = Ei (x) + δ 2 μm ∇Gkm (x − z) × Mh0,0 + ∇ × ∇Gkm (x − z) × Me0,0 3 3  



+δ 3 μm ∇Gkm (x − z) × Mh0,j + ∇ × ∇Gkm (x − z) × Me0,j j =1

.

j =1

3 3 



−δ 3 μm ∇∂j Gkm (x − z) × Mhj,0 + ∇ × ∇∂j Gkm (x − z) × Mej,0 j =1

j =1

+O(δ 4 ). (2.1.58) We start by computing .Mh0,0 :  Mh0,0 =

∂ D˜

 h . M0,0

=

∂ D˜

 Mh0,0 =

∂ D˜

& φ 0 (& y)ds(& y), & φ 0 (& y)∇& yds(& y), & y∇∂B · & φ 0 (& y)ds(& y).

φ given in Proposition 2.1.10 we have Now, using the expansion of .&  Mh0,0 =

.

& ∂D

& y∇∂ D φ 0,0 (& y)ds(& y) + & ·&

 & ∂D

& y∇∂ D φ 0,1 (& y)ds(& y) + O(δ 2 ). & ·&

Recall (2.1.56) for .β = 0: −1 & φ 0,0 = (μc − μm )−1 (λμ I + MD y) × E(z)] [ν(& &) .

&0,0 = iω−1 (εc − εm )−1 (λε I + M & )−1 [ν(& ψ y) × H (z)] . D

We can see, using (2.1.30) and the fact that     i i ∇∂ D y) = ∇∂ D y) = 0, & · E (z) × ν(& & · H (z) × ν(&

.

that &0,0 = 0. ∇∂ D φ 0,0 = ∇∂ D & ·& &·ψ

.

32

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

Now, taking the surface divergence of (2.1.57) for .β = 0, it follows that  ∗ 2 2 & & (μc − μm )(λμ I −KD )[∇ · φ ] + (k − k )∇ · ν(& y) × A [ ψ ] = 0, & & & 0,1 0,0 c m ∂D & D ∂D  2 2 & & · ψ 0,1 ]+(kc − km )∇∂ D y) × AD & [φ 0,0 ] = 0. & · ν(&

.

ω (εc − εm )(λε I 2

∗ − KD & & )[∇∂ D

(2.1.59) Since .∇∂ D & · (ν × ·) = ν · (∇ × ·) we need to study the quantities ν · ∇ × AD φ 0,0 ] & [&

.

and &0,0 ]. ν · ∇ × AD & [ψ

.

The following lemma holds. Lemma 2.1.4 We have ⎧  −1 1 ∗ ⎪ ⎪ ∇SD [ν · Ei (z)] & λμ I + KD ⎪ & ⎪ μc − μm ⎪ ⎪ ⎨ & in R3 \ D, .∇ × AD φ 0,0 ] =  −1 & [& μm 1 ⎪ i ∗ ⎪ λ E (z) + ∇S I + K ⎪ & μ & D ⎪ D ⎪ μ μ2c − μm μc ⎪ ⎩ c i & [ν · E (z)]in D,

(2.1.60)

and ⎧  −1 ⎪ i ∗ ⎪ λ ∇S I + K [ν · Hi (z)] ⎪ & ε D & ω(εc −εm ) ⎪ D ⎪ ⎪ ⎨ in R3 \ D, & &0,0 ] =  −1 .∇ × AD & [ψ iεm i ∗ i ⎪ ⎪ ∇S λ I + K H (z) + & ε ⎪ & D D ⎪ ωεc ω(εc2 − εm εc ) ⎪ ⎪ ⎩ & [ν · Hi (z)]in D. (2.1.61) Proof We only prove (2.1.60). We shall consider the solution to the following system ⎧ Δu = 0 in R3 , ⎪ ⎪ ⎨ & on ∂ D, ν · ∇u|− = ν · ∇u|+ . i & ⎪ μc ν × ∇u|− − μm ν × ∇u|+ = ν × E (z) on ∂ D, ⎪ ⎩ −1 |x| → ∞. u = O(|x| )

(2.1.62)

2.1 Maxwell’s Problem

33

We can see that both the left-hand side and the right-hand side of (2.1.60) are divergence free. We want to prove that they are both equal to the field .∇u in .R3 . First we check that they satisfy the jump relations. We already have the continuity of the normal part of the curl of a vectorial single layer potential [44]. Recall that −1 & φ 0,0 = (μc − μm )−1 (λμ I + MD y) × Ei (z)]. & ) [ν(&

.

Then, ν × ∇ × AD [& φ 0,0 ]|± =

.

1 μc − μm

  −1 I ∓ + MD λI + MD [ν(& y) × Ei (z)], & & 2

so we have μc ν × ∇ × AD [& φ 0,0 ]|− − μm ν × ∇ × AD [& φ 0,0 ]|+ = ν(& y) × Ei (z).

.

The continuity of the tangential derivative of a scalar single layer potential gives  μc ν ×

.

   −1 1 i 1  ∗ i = μm ν × ∇SD [ν · E (z)] E (z) & λμ I + KD & − μc − μm μc   −1 μm  ∗ i ∇SD [ν · E (z)] , + 2 & λμ I + KD & + μc − μm μc

and the jump of the normal derivative of a scalar single layer potential can be written as follows    −1  −1 I  ∗ i ∗ ∗ λ + K I + K [ν ·Ei (z)], .ν ·∇SD [ν ·E (z)] = ±  & λμ I + KD μ & & & D D ± 2 which gives the correct jump relation for the normal derivative. The only problem left is to prove the uniqueness of the system. Now let .& u be the & Note that solution to (2.1.62) with the term .ν × Ei (z) replaced by vector .0 on .∂ D. .μc ν × ∇& u|− = μm ν × ∇& u|+ is equivalent to μc

.

∂& u  ∂& u   = μm  , ∂T − ∂T +

& Then by choosing any test function in where T is any tangential direction on .∂ D. 1 & & Thus, .H (∂ D) and integrating by parts we can get .μc& u|− = μm& u|+ on .∂ D. 



 ∂& u  ∂& u  u|− = 0, u|+ + μc  & .0 ≤ μ|∇& u| dx = − μm  & ∂ν − ∂ν + & & ∂D R3 ∂D 2

which proves .& u = 0 and completes the proof.

⨆ ⨅

34

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

It is worth mentioning that it was proved in [71] that 1 −1 i i ∇ × AD & ) [ν × E (z)] = E (z) & ( I + MD 2

.

& in D,

which, by taking .μm = 0 (or let .μc = ∞), can be seen as the extreme case in (2.1.60). Now that we have a better understanding of .ν × ∇ × AD [& φ 0,0 ], by Lemma 2.1.4, & up to constants such that we can introduce the unique solutions .ue , uh ∈ H 1 (D) e &0,0 ] with .ue , .uh satisfying .∇u = ∇ × AD φ 0,0 ], .∇uh = ∇ × AD & [& & [ψ  .

& in D, Δue = 0 e & & ν · ∇u |− = ν · (∇ × AD & [φ 0,0 ]) on ∂ D,

(2.1.63)

& in D, Δuh = 0 & & [ ψ ]) on ∂ D. ν · ∇uh |− = ν · (∇ × AD & 0,0

(2.1.64)

and  .

The expressions of .∇ue and .∇uh are given by Lemma 2.1.4. Now, by using &0,1 : φ 0,1 and .ψ Eq. (2.1.59), we can compute the surface divergence of .&  h  2 k 2 − km ∗ −1 ∂u  (λμ I − KD ∇∂ D φ 0,1 = c  , & ·& &) ∂ν − μc − μm  e  2 2 &0,1 = kc − km (λε I − K ∗ )−1 ∂u  . ∇∂ D &·ψ & D ∂ν − ω2 (εc − εm )

.

Then we have the following lemma. Lemma 2.1.5 Let .v e be the solution to ⎧ e Δv = 0 ⎪ ⎪ ⎪ ⎨ v e |+ − v e |− = 0    . ∂v e  ∂v e  e · ν ε = (ε − ε )∇u − ε ⎪    c m c m ⎪ ∂ν − ∂ν + ⎪ − ⎩ e v →0

& x ∈ R3 \ ∂ D, & x ∈ ∂ D, & x ∈ ∂ D,

(2.1.65)

|x| → ∞,

and let .v h be the solution to ⎧ h Δv = 0 ⎪ ⎪ ⎪ ⎨ v h |+ − v h |− = 0    . ∂v h  ∂v h  h · ν μ − μ = (μ − μ )∇u ⎪    c c m m ⎪ ∂ν + ∂ν − ⎪ − ⎩ h v →0

& x ∈ R3 \ ∂ D, & x ∈ ∂ D, & x ∈ ∂ D, |x| → ∞.

(2.1.66)

2.1 Maxwell’s Problem

35

Then the following asymptotic expansions hold: Me0,0 = δ

2 − k2 km c ω2 εm

Mh0,0 = δ

2 − k2 km c μm

.

 & D

∇(ue + v e ) + O(δ 2 ),



& D

∇(uh + v h ) + O(δ 2 ).

Proof By Proposition 2.1.10, we have  h .M0,0

=

& ∂D

& φ 0 ds(& y) = δ



= −δ =δ

& ∂D

 & ∂D

& φ 0,1 ds(& y) + O(δ 2 )

& y∇∂ D φ 0,1 ds(& y) + O(δ 2 ) & ·&

2 − k2 km c μc − μm

 & ∂D

∗ −1 & y(λμ I − KD &)

 ∂uh    y) + O(δ 2 ).  ds(& ∂ν −

Using the fact that λμ =

.

μm 1 + , 2 μc − μm

& we get that for .f ∈ L2 (∂ D),     I μc − μm  ∗ ∗ λμ I − KD .f = & [f ] + − + KD [f ] . μm 2 Then,  2 − k2   I ∂uh  km c ∗ ∗ −1 & & y(− + KD y y) +  ds(& & )(λμ I − KD &) − ∂ν 2 μm & & ∂D ∂D  ∂uh     y) + O(δ 2 ).  ds(& ∂ν −

Mh0,0 = δ

.

An integration by parts gives  .

∂uh  & y y) =  ds(& & ∂ν − ∂D

 & D

∇uh dx.

We now take a look at the transmission problem (2.1.66) solved by .v h . Using the jump relation of the normal derivative of the scalar single layer potential we find

36

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

  ∂uh ∗ [f ] = that, writing .v h = SD gives & [f ] with f being such that . λμ I − K ∂ν  I ∂v h  ∗ . − + KD [f ] =  , & 2 ∂ν − and hence,  h   ∂v h  I ∗ ∗ −1 ∂u   . (− + KD  = & )(λμ I − KD &) 2 ∂ν − ∂ν −

.

Integrating by parts we get Mh0,0 = δ

.

2 − k2  km c μm

 & D

 ∇uh dx +

& D

 ∇v h dx + O(δ 2 ).

The evaluation for .Me0,0 can be done in exactly the same way.

⨆ ⨅

Lemma 2.1.6 We have e .M α,β

 i  = ∇(xα xβ ) × ∂ β Hi (z) + iω(εc − εm ) ωεm D &   &β,0 ]) + O(δ), . ∇ × (xα ∇ × AD & [ψ

(2.1.67)

 1  ∇(xα xβ ) × ∂ β Ei (z) − (μc − μm ) μm D &   ∇ × (xα ∇ × AD φ β,0 ]) + O(δ). & [&

(2.1.68)

& D

Mhα,β =

& D

In particular, we have Mej,0 =

.

Mhj,0 Me0,j

Mh0,j

i & εc − εm |D|ej × Hi (z) − ej × ωεm εm

 & D

∇uh + O(δ), .

(2.1.69)

 1 & μc − μm i = |D|ej × E (z) − ej × ∇ue + O(δ), . (2.1.70) μm μm & D  i & εc − εm &j,0 ] + O(δ), . = |D|ej × ∂j Hi (z) − ∇SD & [∇∂ D &·ψ ωεm εm & D (2.1.71)  1 & μc − μm = |D|ej × ∂j Ei (z) − ∇SD φ j,0 ] + O(δ), & [∇∂ D & ·& μm μm & D (2.1.72)

where .(e1 , e2 , e3 ) is an orthonormal basis of .R3 .

2.1 Maxwell’s Problem

37

Proof We shall only consider .Mhα,β . .Meα,β can be calculated in exactly the same way. We have Mhα,β = Mh,0 α,β + O(δ),

.

where .Mh,0 α,β is given by  Mh,0 α,β =

.

& ∂D

& φ β,0 ds(& yα & y).

μm 1 & + we have that for any .f ∈ L2T (∂ D), 2 μc − μm

Since .λμ =

.

  λμ I + MD & [f] −



μm I + MD f. & [f] = 2 μc − μm

By applying Lemma 2.1.10, it follows that Mh,0 α,β =

.

1 μm

 & ∂D

1 − μm

& yα ν(& y) × (& yβ ∂ β Ei (z))ds(& y)  & ∂D

I −1 & yα ( + MD y) × & yβ ∂ β Ei (z)]ds(& y). & )(λμ I + MD & ) [ν(& 2

Using the jump relations on .MD & and the fact that & φ β,0 =

.

 −1 1 λμ I + MD [ν(& y) × & yβ ∂ β Ei (z)], & μc − μm

we can write Mh,0 α,β =

.

1 μm

 & ∂D

& yα ν(& y) × (& yβ ∂ β Ei (z))ds(& y) μc − μm − μm

 & ∂D

 & yα ν(& y) × ∇ × SD φ β,0 ]− ds(& y). & [&

The curl theorem yields Mh,0 α,β =

.

1 μm

 & D

∇(xα xβ ) × ∂ β Ei (z)dx −

μc − μm μm

 & D

∇ × (xα ∇ × SD φ β,0 ])dx, & [&

and thus (2.1.68) holds. By using the definition of .ue and .uh we get the case where .|α| = 1, .|β| = 0. ⨆ ⨅

38

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

Denote by .G(x, z) the matrix valued function (Dyadic Green function) G(x, z) = εm (Gkm (x − z)I +

.

1 2 Dx Gkm (x − z)). 2 km

It can be seen that .G(x, z) satisfies ∇x ×

.

1 ∇x × G(x, z) − ω2 μm G(x, z) = −δz I. εm

We can also easily check that ∇ × G(x, z) = εm ∇ × (Gkm (x − z)I ) = εm ∇Gkm (x − z) × I.

.

Theorem 2.1.1 Define the polarization tensors  M :=

.

e

∗ −1 & y(λε I −KD y) & ) [ν]ds(& & ∂D

 and

M := h

& ∂D

∗ −1 & y(λμ I −KD y). & ) [ν]ds(&

(2.1.73)

Then the following far-field expansion holds: E(x)−Ei (x) = −δ 3 ω2 μm G(x, z)M e Ei (z)−δ 3

.

iωμm ∇ ×G(x, z)M h Hi (z)+O(δ 4 ). εm (2.1.74)

Before we proceed, we stress that the polarization tensors .M e , .M h defined above are matrix with each entry .meij and .mhij , .i, j = 1, 2, 3, defined by (2.1.10) with .λ = λε and .λ = λμ , respectively. They are different from the vector valued tensors we defined in Eq. (2.1.53). Proof We shall give the analysis term by term in (2.1.58). It is easy to check that 3

.

ej × ∂j Ei (z) = iωμm Hi (z)

j =1

and

3

ej × ∂j Hi (z) = −iωεm Ei (z)

j =1

and 3

.

j =1

∇∂j Gkm (x − z) × ej × Ei (z) = ω2 μm G(x, z)Ei (z).

2.1 Maxwell’s Problem

39

Then by Lemma 2.1.6 it follows that ∇×

3

.

∇∂j Gkm (x − z) × Mej,0 =

j =1

ω2 μm ∇ × G(x, z)

   i & i (z) − εc − εm ∇uh + O(δ), |D|H εm ωεm & D

and μm

3

.

∇∂j Gkm (x − z) × Mhj,0



& i (z) − (μc − μm ) = ω μm G(x, z) |D|E 2

j =1

 & D

∇ue



+ O(δ). Furthermore, we obtain from Lemma 2.1.10 that ⎤ ⎡ 3 3  −1

 

1 ∗ ⎣ λμ I − KD ∇∂ D y) × (& yj ∂j Ei (z)) ⎦ , . ∇∂ D φ j,0 = & · ν(& & ·& & μc − μm j =1

3

∇∂ D φ j,0 & ·&

j =1

⎤ 3  −1

  1 ∗ ⎣ λμ I − KD ν(& y) · ∇ × (& yj ∂j Ei (z)) ⎦ , = & μc − μm ⎡

j =1

j =1

which gives 3

.

j =1

−1 iωμm  ∗ λμ I − KD [ν · Hi (z)]. ∇∂ D φ j,0 = − & ·& & μc − μm

Similarly, we have 3

.

&j,0 = − ∇∂ D &·ψ

j =1

−1 εm  ∗ λμ I − KD [ν · Ei (z)]. & εc − εm

Recall from (2.1.71) that Me0,j =

.

i & εc − εm |D|ej × ∂j Hi (z) − ωεm εm

 & D

&j,0 ] + O(δ). ∇SD & [∇∂ D &·ψ

40

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

Summing over j gives ∇ × ∇Gkm (x − z) ×

3

j =1

i & |D|ej × ∂j Hi (z) ωεm 

= ∇ × ∇Gkm (x − z) × .

i & |D|∇z × Hi (z) ωεm



& i (z) = −∇ × ∇Gkm (x − z) × |D|E & i (z) = −∇ × ∇ × G(x, z)|D|E & i (z). = ω2 μm G(x, z)|D|E Hence, we can deduce that ∇ × ∇Gkm (x − z) ×

3

.



Me0,j

j =1

& i (z) + = ω μm G(x, z) |D|E





2

∇SD & [ν · H (z)] + O(δ). i

& D

A similar computation yields μm ∇Gkm (x − z) ×

3

.

Mh0,j =

j =1



& i (z) + iωμm ∇Gkm (x − z) × |D|H

 & D

 ∗ −1 i ∇SD & (λμ I − KD & ) [ν · H (z)] + O(δ),

and therefore, μm ∇Gkm (x − z) ×

3

.

Mh0,j =

j =1

   μm ∗ −1 i & i (z) + iω ∇ × G(x − z) |D|H ∇SD & (λμ I − KD & ) [ν · H (z)] + O(δ). εm & D Moreover, Lemma 2.1.5 gives ∇

.

× ∇Gkm (x − z) × Me0,0

μm ∇Gkm (x − z) × Mh0,0

μm 2 =δ (k − kc2 )G(x, z) εm m



(k 2 − kc2 ) ∇ × G(x, z) =δ m εm

& D

∇(ue + v e ) + O(δ 2 )

 & D

∇(uh + v h ) + O(δ 2 ).

2.1 Maxwell’s Problem

41

Combining the previous asymptotic expansions we arrive at   1 2 G(x, z) μm (km − kc2 ) ∇(ue + v e ) εm & D    2 2 ∗ −1 i ∇ue + km ∇SD + (μc − μm )km & (λε I − KD & ) [ν · E (z)]

E(x) − Ei (x) = δ 3

.

& D

+ δ3





& D

1 2 ∇ × G(x, z) (km − kc2 ) ∇(uh + v h ) + ω2 μm (εc − εm ) εm & D   ∗ −1 i 4 + iωμm ∇SD & (λμ I − KD & ) [ν · H (z)] + O(δ ). & D

 & D

∇uh

(2.1.75) ⨆ ⨅

The proof is then complete.

We shall analyze further (2.1.75). Recall that, from the proof of Lemma 2.1.5, we have    e  εm e e ∗ −1 ∂u  & y(λε I − KD ) . ∇(u + v )dx =  ds(x) & εc − εm ∂ D ∂ν − & & D and  .

& D

∇(uh + v h )dx =

μm μc − μm

 & ∂D

∗ −1 & y(λμ I − KD &)

 ∂uh     ds(x). ∂ν −

Noticing that 2 2 μm (km − kc2 ) = (μc − μm )km

.

μm εm − μc εc , (μc − μm )(εc − εm )

we get  2 .μm (km

− kc2 )

 ∇(u + v e

& D

2 (μc − μm )km



e

2 ) + (μc − μm )km

μm εm − μc εc (μc − μm )(εc − εm )

 & ∂D

& D

∇ue =  ∂ue     ∂ν −  ∂ue  & + y  . & ∂ν − ∂D

∗ −1 & y(λε I − KD &)

Moreover, for any f , we have  −1 μm εm − μc εc ∗ λε I − KD [f ] + f = & (μc − μm )(εc − εm )  −1 −1   μm εm − μc εc ∗ ∗ ∗ λε I − KD λ [f ] + λ I − K I − K [f ]. ε ε & & D (μc − μm )(εc − εm ) .

42

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

so that .

 −1 μm εm − μc εc ∗ ∗ ∗ −1 λε I − KD [f ] + f = (λμ I + KD & & )(λε I − KD & ) [f ] (μc − μm )(εc − εm )

We can then write   2 2 e e 2 .μm (km − kc ) ∇(u + v ) + (μc − μm )km ∇ue = & D

 2 − (μc − μm )km

& ∂D

& D

∗ ∗ −1 & y(λμ I + KD & )(λε I − KD &)

 ∂ue     . ∂ν −

Recall that by definition, .

∂ue  φ 0,0 ].  = ν · ∇ × AD & [& ∂ν −

Then, by using Lemma 2.1.4, we obtain ν ·∇ ×AD φ 0,0 ] = & [&

.

  −1 μm 1  ∗ ν ·Ei (z)+ 2 ν ·∇SD [ν ·Ei (z)] , & λμ I + KD & − μc μc − μm μc

which together with the jump relations for the normal derivative of the scalar layer potential yields  2 μm (km − kc2 )

.

−

μm 2 k μc m



 & D

2 ∇(ue + v e ) + (μc − μm )km

μc − μm 2 km − μc

 & ∂D

& D

∇ue =

∗ ∗ −1 i & y(λμ I + KD & )(λε I − KD & ) [ν · E (z)]

1 ∗ ∗ −1 i ∗ ∗ −1 & y(λμ I +KD & )(λμ I +KD & ) [ν ·E (z)]. & )(λε I −KD & ) (− I +KD 2 & ∂D

εc 1 , then we can write If we set .λε = − + 2 εc − εm .

−

−1  1 μm  ∗ i ∗ − I + K λε I − KD & [ν · E (z)] & D 2 μc −1  εc μm  ∗ ∗ i − λε I − KD I + KD =− & & [ν · E (z)], μc εc − εm

2.1 Maxwell’s Problem

43

or equivalently, .

−

−1  1 μm  ∗ ∗ i − λε I − KD I + K & & [ν · E (z)] = D μc 2  −1 εc μm μm ∗ λε I − KD ν · Ei (z) + [ν · Ei (z)]. − & μc μc (εc − εm )

Then, since .

−

 −1 μc − μm  ∗ ∗ λμ I + KD λε I − KD [ν · Ei (z)] = & & μc −1  μc − μm μc − μm ∗ ν · Ei (z) − (λμ − λε ) λε I − KD [ν · Ei (z)], & μc μc

we can write  −1 μc − μm  ∗ ∗ λμ I + KD λ I − K [ν · Ei (z)] ε & & D μc −1  1 μm  ∗ ∗ i − λε I − KD − I + K & & [ν · E (z)] D μc 2   −1 εc μm μc − μm ∗ (λμ − λε ) λε I − KD [ν · Ei (z)]. = ν · Ei (z) + − & μc (εc − εm ) μc .

−

A direct computation gives .

μc − μm εc μm 1 − (λμ − λε ) = + λε , μc (εc − εm ) μc 2

and therefore,  2 μm (km − kc2 )



.

B

 2 km

& yν · E

i

∂B

2 ∇(ue + v e ) + (μc − μm )km



2 (z)ds(& y) − km

1 + λε 2

 ∂B

∇ue = B

 −1 & y λε I − KB∗ [ν · Ei (z)]ds(& y).

A similar computation yields  2 (km − kc2 )

.

 iωμm

 & D

& yν·H (z)ds(& y)−iωμm i

& ∂D

∇(uh + v h ) + ω2 μm (εc − εm ) 

1 + λμ 2

 & ∂D

& D

∇uh =

−1  ∗ & y λε I − KD [ν·Hi (z)]ds(& y). &

44

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

Now it remains to compute the last term in (2.1.75) which is  2 km

.

& D

∗ −1 i ∇SD y & (λε I − KD & ) [ν · E (z)]d&



2 = km

  −1 ∂  ∗ & λε I − KD y SD [ν · Ei (z)]ds(& y). & & − ∂ν & ∂D

 εm ∂ 1  + Writing that .λε = together with the fact that . SD = & − 2 ε + ε ∂ν c m  1 ∗ − I + KD & , we obtain 2 .

−1  −1 ∂ εm  ∗ ∗ SD λε I − KD [ν·Ei (z)]. [ν·Ei (z)] = −ν·Ei (z)+ & λε I − KD & & εc + εm ∂ν

Hence,  2 .km

& D

∇SD & (λε I

∗ −1 − KD & ) [ν

 · E (z)]d& y= i

2 −km

& ∂D

& yν · Ei (z)]ds(& y)

  −1  1 2 ∗ & λε − + km y λε I − KD [ν · Ei (z)]ds(& y). & 2 ∂D &

Similarly, we have  iωμm

.

& D

∗ −1 i ∇SD & (λμ I − KD & ) [ν · H (z)] = iωμm

 & ∂D

& yν · Hi (z)ds(& y)

  −1  1 ∗ & + iωμm λμ − y λε I − KD [ν · Hi (z)]ds(& y). & 2 ∂D &

Finally, we arrive at  E(x) − Ei (x) = −δ 3 ω2 μm G(x, z)

.

∂B

iωμm − δ3 ∇ × G(x, z) εm

& y(λε I − KB∗ )−1 [ν · Ei (z)]



∂B

& y(λμ I − KB∗ )−1 [ν · Hi (z)] + O(δ 4 ).

When a plasmonic resonance occurs, the term .λε =

εc +εm 2(εc −εm )

can have a real part

that is lower and become close to an eigenvalue of the operator .KB∗ . Using Lemma 2.1.10 we can easily see that each of the potentials .φ β,n and 1 .ψ β,n are controlled in norm by powers of . dσ , where .dσ is the distance of .λε to ∗ the spectrum .σ (MB ) = −σ (KB ) \ {−1/2}. So the asymptotic development given by Theorem 2.1.1 is valid when .δ/dσ 0 and

.

3λ + 2μ > 0.

(2.2.4)

& may have multiple connected components, corresponding to It is noted that .D the case that there are multiple nanoparticles. Associated with the elastic medium configuration described above in .R3 , the elastic wave scattering is governed by the following Lamé system, ∇ · C(x)∇ s u + ω2 u = f in R3 ,

.

(2.2.5)

1 (R3 )3 , .C(x) = & where .u ∈ Hloc λδij δkl + & μ(δik δj l + δj k δil ) and the operator .∇ s u is defined by

∇ s u = (∇u + ∇u⏉ )/2

.

2.2 Elastic Problem

51

with .⏉ denoting the transpose. Here we suppose that the source term .f ∈ H −1 (R3 )3 is compactly supported in .R3 \ D. To complete the description of the elastic system, we impose the following radiating condition for .u(x) in (2.2.5) as .|x| → +∞, (∇ × ∇ × u)(x) × .

x − ikT ∇ × u(x) =O(|x|−2 ), |x|

x · [∇(∇ · u)](x) − ikL ∇u(x) =O(|x|−2 ), |x|

(2.2.6)

√ √ √ where .kT := ω/ μ, .kL := ω/ λ + 2μ and .i := −1 is the imaginary unit.

2.2.1 Layer Potential Techniques In order to represent the solution to the Lamé system, we shall make use of the layer potential technique. Note that some elementary notations are introduced in the first chapter. To begin with, we introduce the Lamé operator defined by Lλ,μ u := ∇ · C(x)∇ s u = μΔu(x) + (λ + μ)∇∇ · u(x).

.

Then the Kupradze matrix .Gω , which is the fundamental solution to the PDO (partial differential operator) .Lλ,μ + ω2 , is given as follows (cf. [5]),  1 e cT 2 e I+ D .Gω = − 4π μ|x| 4∂iω2 iω|x|

iω|x| cT

−e |x|

iω|x| cL

 ,

(2.2.7)

where .I is the .3 × 3 identity matrix, .D 2 denotes the standard double differentiation, and cT =

.

√

μ,

cL =

(

λ + 2μ.

(2.2.8)

If .ω = 0, .G0 is the Kelvin matrix of the fundamental solution to the elastostatic system, and it is given by G0 (x) = −

.

γ2 xx⏉ γ1 1 I− , 4π |x|3 4π |x|

(2.2.9)

where the superscript .⏉ stands for the transpose of a matrix and γ1 =

.

1 2



1 1 + μ 2μ + λ

and

γ2 =

1 2



1 1 − . μ 2μ + λ

(2.2.10)

52

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

Based on the above notations, the single and double layer potentials can then be defined by  SωD [ϕ](x) =

Gω (x − y)ϕ(y)ds(y),

.

 ω .DD [ϕ](x)

x ∈ R3 ,

(2.2.11)

x ∈ R3 \∂D,

(2.2.12)

∂D

= ∂D

∂ Gω (x − y)ϕ(y)ds(y), ∂ν y

for any .ϕ ∈ H −1/2 (∂D)3 , where and also in what follows, the conormal derivative on .∂D for a function .u is defined by .

∂u = λ(∇ · u)ν + μ(∇u + ∇u⏉ )ν, ∂ν

(2.2.13)

with .ν signifying the exterior unit normal vector to the boundary .∂D. There hold the following jump relations for the double layer potential and the conormal derivative of the single layer potential on the boundary (cf. [5]), .

 ∂ ω 1 SD [ϕ]|± (x) = ± I + (KωD )∗ [ϕ](x), 2 ∂ν  1 DωD [ϕ]|± (x) = ∓ I + KωD [ϕ](x), 2

.

a.e. x ∈ ∂D, a.e. x ∈ ∂D,

(2.2.14)

(2.2.15)

where .I is the identity operator and .KωD is the operator defined by  KωD [ϕ](x) = p.v.

.

∂D

∂ Gω (x − y)ϕ(y)ds(y), ∂ν y

(2.2.16)

and .(KωD )∗ , also called the Neumann-Poincaré (NP) operator, is given as follows (KωD )∗ [ϕ](x) = p.v.



.

∂D

∂ Gω (x − y)ϕ(y)ds(y), ∂ν x

(2.2.17)

where p.v. means the Cauchy principal value. In what follows, when .ω = 0, we denote .S0D , D0D , K0D , and .(K0D )∗ by .SD , DD , KD , and .K∗D for simplicity. With the above preparations, one has the following integral representation of the solution to the elastic system (2.2.2)–(2.2.6) with .(ϕ, ψ) ∈ H −1/2 (∂D)3 × H −1/2 (∂D)3 (cf. [5]),  u(x) =

.

x ∈ D, SωD1 [ϕ](x), ω2 SD [ψ](x) + F(x), x ∈ R3 \D,

(2.2.18)

2.2 Elastic Problem

53

where  F(x) =

Gω2 (x − y)f(y)dy,

(2.2.19)

with ℜω1 > 0, ℑω1 ≤ 0,

(2.2.20)

.

ω ω1 = √ c

.

R3

and ω2 = ω.

.

It can be verified that Lλ,μ F + ω2 F = f

.

in R3 ,

(2.2.21)

By using the transmission conditions across the boundary .∂D, the solution .u(x) in (2.2.18) should satisfy  .

x ∈ ∂D, SωD1 [ϕ]|− = (SωD2 [ψ](x) + F(x))|+ , ω2 ∂ ∂ ω1 (S [ψ](x) + F(x))| , S [ϕ]| = + x ∈ ∂D. − ∂ν D ∂ν D

(2.2.22)

With the help of the jump relation (2.2.14), the pair .(ϕ, ψ) is the solution to the following system of integral equations,  .

−SωD2 Sω1  I D ω1 ∗  I c − 2 + (KD ) − 2 − (KωD2 )∗



ϕ ψ



 =

 F  ∂F  ∂ν

.

(2.2.23)

∂D

2.2.2 Asymptotics for the Integral Operators In this subsection we derive the asymptotic properties for the operators .SωD and ω ∗ .(K ) involved in the integral reformulation of the elastic system in Sect. 2.1.1. We D suppose that .δ ∈ R+ is sufficiently small, that is, the size of the nanoparticle D is small enough. First of all, we introduce some notations. For a multi-index .α ∈ N3 , let .xα = x1α1 x2α2 x3α3 and .∂ α = ∂1α1 ∂2α2 ∂2α2 , with .∂j = ∂/∂xj . We also define by .ej , 3 & + z, for any .y ∈ ∂D, .j = 1, 2, 3 the unit coordinate vectors in .R . Since .D = δ D & y) = ψ(y), and let & we let & ϕ (& y) = ϕ(y) and .ψ(& .y = (y − z)/δ ∈ ∂ D. Denote by .& & .∂/∂& ν be the conormal derivative on the boundary .∂ D.

54

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

Lemma 2.2.1 Suppose .δ ∈ R+ and .δ ⪡ 1, then there holds the following asymptotic expansion for .x ∈ R3 

+∞ in 1

Gδ (x) = − 4π (n + 2)n! n=0

.

n+1 cTn+2



+∞ 1 in (n − 1) + 4π (n + 2)n!

1

−

cTn+2

n=0

+



1

n+2 cL  1

n+2 cL

δ n |x|n−1 I (2.2.24) δ n |x|n−3 xx⏉ .

Proof By using (2.2.7) and Taylor series expansion with respect to .δ and straightforward computations, one can obtain (2.2.24). The proof is complete. ⨆ ⨅ Proposition 2.2.1 Let .ϕ ∈ H −1/2 (∂D)3 . For .δ ∈ R+ and .δ ⪡ 1, there holds SωD [ϕ](x) = δSD ϕ ](& x) + δ 2 ωRD ϕ ](& x) + δ 3 ω2 ID ϕ ](& x) + O(δ 4 ), & [& & [& & [&

.

(2.2.25)

where  RD ϕ ](& x) = γ3 & [&

.

& ∂D

& ϕ (& y)ds(& y),

(2.2.26)

with −i .γ3 = 12π





2

3 + cL cT3

,

and  ID ϕ ](& x) = & [&

.

& ∂D

Λ(& x −& y)& ϕ (& y)ds(& y),

(2.2.27)

with 1 .Λ = 32π



1 3 + 4 cT4 cL



1 |x|I − 32π



1 1 − 4 cT4 cL



xx⏉ , |x|

and .cT and .cL defined in (2.2.8). Proof Let .x ∈ ∂D and denote & .x = (x − z)/δ. Then one has that  SωD [ϕ](δ& x + z) =

Gω (δ& x + z − y)ϕ(y)ds(y).

.

∂D

(2.2.28)

2.2 Elastic Problem

55

By using .y = δ& y + z and change of variables in the above integral, there holds  SωD [ϕ](δ& x + z) =

.

& ∂D

Gω (δ& x − δ& y)& ϕ (& y)δ 2 ds(& y).

Substituting (2.2.7), the expression for .Gω , into the previous equation yields that  SωD [ϕ](δ& x + z) =

.

& ∂D

δGδω (& x −& y)& ϕ (& y)ds(& y).

Therefore, the following identity is achieved, SωD [ϕ](x) = δSδω ϕ ](& x). & [& D

.

(2.2.29)

& from the expansion (2.2.24) for the When .δ is sufficient small, for any & .x ∈ ∂ D, fundamental solution it follows that SωD [ϕ](x) = δSD ϕ ](& x) + δ 2 ωRD ϕ ](& x) + δ 3 ω2 ID ϕ ](& x) + O(δ 4 ). & [& & [& & [&

.

⨆ ⨅

The proof is complete. Proposition 2.2.2 Let .ϕ ∈ H −1/2 (∂D)3 . For .δ ∈ R+ and .δ ⪡ 1, there holds (KωD )∗ [ϕ](x) = K∗D ϕ ](& x) + δ 2 ω2 PD ϕ ](& x) + O(δ 3 ), & [& & [&

.

(2.2.30)

where  PD ϕ ](& x) = & [&

.

& ∂D

∂ Λ(& x −& y)& ϕ (& y)ds(& y), ∂ν&x

(2.2.31)

with .Λ defined in (2.2.28). Proof Denoting & .x = (x − z)/δ for any .x ∈ ∂D, one can find that (KωD )∗ [ϕ](δ& x + z) = p.v.

.

1 δ

 ∂D

∂ Gω (δ& x + z − y)ϕ(y)ds(y), ∂ν&x

since for any .x ∈ ∂D there holds .ν D (x) = ν D x). Then one has by setting .y = δ& y+z & (& in the previous integral that (KωD )∗ [ϕ](δ& x + z) = p.v.

.

1 δ

 & ∂D

∂ Gω (δ& x − δ& y)& ϕ (& y)δ 2 ds(& y). ∂ν&x

56

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

By substituting (2.2.7) into the last equation and direct calculation, one can obtain that  ∂ ω ∗ Gδω (& x −& y)& ϕ (& y)ds(& y), .(KD ) [ϕ](δ& x + z) = p.v. ∂ν & & x ∂D which gives the following identity ∗ (KωD )∗ [ϕ](x) = (Kδω ϕ ](& x). & ) [& D

.

(2.2.32)

Thus when .δ is sufficient small, from the expansion (2.2.24) for the fundamental solution it follows that (KωD )∗ [ϕ](x) = K∗D ϕ ](& x) + δ 2 ω2 PD ϕ ](& x) + O(δ 3 ), & [& & [&

.

⨆ ⨅

and this completes the proof.

We have derived the asymptotic expansions for the single layer potential operator and the Neumann-Poincaré type operator .(KωD )∗ . In what follows, we also prove some important Lemmas for our subsequent usage. Lemma 2.2.2 Let .ID & be defined in (2.2.27) and if .δ ∈ R+ with .δ ⪡ 1, then for & ϕ ∈ H −1/2 (∂D)3 , there holds

.

Lλ,μ ID ϕ ](& x) = −SD ϕ ](& x), & [& & [&

.

& & x ∈ D.

(2.2.33)

Proof Recall that .Gδ is the fundamental solution to .Lλ,μ + δ 2 , which shows that (Lλ,μ + δ 2 )Gδ (& x −& y) = 0,

.

& & & & x ∈ D, y ∈ ∂ D.

(2.2.34)

By substituting (2.2.24) into (2.2.34) and comparing the coefficients of .δ 2 one then has Lλ,μ Λ(& x −& y) = −G0 (& x −& y),

.

& & & & x ∈ D, y ∈ ∂ D,

(2.2.35)

where .Λ is given in (2.2.28). By using the definition of .ID & in (2.2.27), one finally obtains (2.2.33), which completes the proof. ⨆ ⨅ & where .C is a constant vector Lemma 2.2.3 Suppose that .SD ϕ ] = C holds on .∂ D, & [& and .& ϕ ∈ H −1/2 (∂D)3 , then  .

I −1 ∗ − + KD & SD & [C] = 0. 2

(2.2.36)

2.2 Elastic Problem

57

& one can easily obtain that .SD Proof When .SD ϕ ] = C on .∂ D, ϕ ] = C in the & [& & [& & Indeed, let us introduce the Dirichlet boundary value problem domain .D.  .

& x) = 0, & x ∈ D, Lλ,μ u(& & & u(& x) = C, x ∈ ∂ D,

which has a unique solution (cf. [5]), and .u(& x) = C is the solution. Then by the jump condition (2.2.14) one can obtain that  .

I x) = 0, − + K∗D & [ϕ](& 2

& & x ∈ ∂ D.

Thus the proof is completed since the operator .SD & is invertible (cf. [21]).

⨆ ⨅

We shall also need the following important lemma about the spectrum of the operator .KD & , which can be found in [21]. Lemma 2.2.4 Assume that .a ∈ R3 , then .a is an eigenfunction of the operator .KD & and the corresponding eigenvalue is . 12 , i.e. KD & [a] =

.

1 a. 2

2.2.3 Far-Field Expansion In this section, we derive the far-field expansion for the elastic wave field .u(x) to the Eq. (2.2.5). First of all, by Taylor series expansion, .F(y) given in (2.2.19) has the following expansion since D is a nanoparticle F(y) = F(δ& y + z) =

.

+∞

1 |β| β β δ & y ∂ F(z). β!

(2.2.37)

|β|=0

&β ) be the solution to the following equation Let .(& ϕβ , ψ  .

δω1 δω2 SD −SD & &   δω2 ∗ δω1 ∗ I − − (K ) c − 2I + (KD & ) & 2 D



& ϕβ & ψβ



 =

 & yβ ∂ β F(z) . ∂ β β y ∂ F(z) ∂& ν&

(2.2.38)

58

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

From the identities (2.2.29) and (2.2.32) for the single layer potential operator and the NP operator, the linearity of the Eq. (2.2.23) and together with the help of the following relationship

.

+∞ ∂ 1 |β|−1 β β ∂ & δ F(y) = y ∂ F(z), ∂& ν β! ∂ν |β|=0

& with the following expression one can conclude that .(& ϕ , ψ) +∞

& ϕ=

.

ϕβ , δ |β|−1&

&= ψ

|β|=0

+∞

&β , δ |β|−1 ψ

(2.2.39)

|β|=0

is the solution to (2.2.23). Therefore from (2.2.18), we have the following expansion for .u(x) in .R3 \ D u(x) = F(x) +

+∞ +∞



δ

.

1+|α|+|β| (−1)

|α|

α!β!

|α|=0 |β|=0

 α

∂ Gω (x − z)

& ∂D

&β (& & yα ψ y)ds(& y). (2.2.40)

For .α, β ∈ N3 , define  Mα,β :=

.

& ∂D

&β (& & yα ψ y)ds(& y).

(2.2.41)

Then the following lemma holds. Lemma 2.2.5 Let .u be the solution to the system (2.2.2)–(2.2.6). Then for .x ∈ R3 \D, one has u(x) = F(x) +

+∞ +∞



.

δ 1+|α|+|β|

|α|=0 |β|=0

(−1)|α| α ∂ Gω2 (x − z)Mα,β . α!β!

(2.2.42)

In order to derive the far-field expansion for the displacement field .u(x), we next analyze the term .Mα,β more precisely. From the expression for .Mα,β in (2.2.41), &β needs to be treated more carefully. .ψ &β still depend on .δ. Thus ϕ β and .ψ From (2.2.38) one can readily find out that .& & ϕ β and .ψ β could be further expanded by .& & ϕβ =

+∞

.

n=0

ϕ β,n , δ n&

&β = ψ

+∞

n=0

&β,n . δnψ

(2.2.43)

2.2 Elastic Problem

59

Then by using (2.2.38), Proposition 2.2.1 and Proposition 2.2.2, one can obtain that       & ϕ β,1 + δ 2& ϕ β,2 ϕ β,0 + δ& & yβ ∂ β F(z) 2 3 . A + δT + δ N β ∂ β F(z) + O(δ ), &β,1 + δ 2 ψ &β,0 + δ ψ &β,2 = ∂ & y ψ ∂ν (2.2.44) where .A , .T and .N are defined by  A =

.

 & &  −SD  SD , I ∗ c − 2I + K∗D & & − 2 − KD

 T =

 ω1 RD & −ω2 RD & , 0 0

(2.2.45)

and  2 ω12 ID & −ω2 ID & . .N = 2 cω12 PD & −ω2 PD & 

(2.2.46)

The operators .ID & and .PD & are defined in (2.2.27) and (2.2.31), respectively. Since &β that we need is of .δ 2 order, we neglect the details ϕ β and .ψ the highest order on .& of the higher-order terms in (2.2.44) and put them all in .O(δ 3 ). We mention that the invertibility of the operator .A −1 can be found in [5]. For the convergence of serials in (2.2.43) and (2.2.42), we refer to Remark 2.2.2 and Remark 2.2.4 for more details.

2.2.4 Asymptotics for the Potential From Lemma 2.2.5, the elastic wave field has the following asymptotic property when .δ ⪡ 1,   u(x) =F(x) + δ Gω2 (x − z)M0,0 ⎛ ⎞



+ δ2 ⎝ Gω2 (x − z)M0,β − ∂ α Gω2 (x − z)Mα,0 ⎠ |α|=1

|β|=1

⎛ .

+ δ3 ⎝

1



Gω2 (x − z)M0,β − ∂ α Gω2 (x − z)Mα,β β!

|β|=2

+

⎞

|α|=1 |β|=1

1 ∂ α Gω2 (x − z)Mα,0 ⎠ + O(δ 4 ). α!

|α|=2

(2.2.47)

60

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

We next analyze (2.2.47) term by term. For the sake of simplicity, we define the parameter .κc by κc :=

.

c+1 . 2(c − 1)

(2.2.48)

It is noted that key parts lying in (2.2.47) is .Mα,β with .|α| ≤ 2 and .|β| ≤ 2. From the expression for .Mα,β in (2.2.41), we should first derive the asymptotic expression &β , .|β| ≤ 2. We present the estimates in the following propositions. for .ψ &β,0 be defined in (2.2.43). Then one has for .β ∈ Proposition 2.2.3 Let .& ϕ β,0 and .ψ 3 N that    −1  ∂  1 I −1 ∗ β β ∗ β β & & KD y ∂ F(z)−c − +KD y ∂ F(z) , .ψ β,0 = & −κc I & SD & & (c − 1) ∂ν 2 (2.2.49) and   β β &β,0 + S−1 & & y ϕ β,0 = ψ ∂ F(z) . & D

.

(2.2.50)

&0,0 = 0. Furthermore, there holds that .ψ Proof With the help of (2.2.44), one can obtain that  .

& ϕ β,0 & ψ β,0

 =A

−1



 & y β ∂ β F(z) , ∂ β β y ∂ F(z) ∂ν &

namely &β,0 ] = & SD ϕ β,0 − ψ yβ ∂ β F(z), & [&

.

(2.2.51)

and   I I ∂ β β ∗ & & [& ϕ + K y ∂ F(z). c − + K∗D ] − β,0 & & [ψ β,0 ] = D 2 2 ∂ν

.

(2.2.52)

−1/2 (∂ D) & 3 to .H 1/2 (∂ D) & 3 , by solving Since the operator .SD & is invertible from .H (2.2.51) and (2.2.52), one thus has (2.2.49) and (2.2.50). Let .β = 0, then (2.2.49) turns to  −1  I c −1 ∗ ∗ & KD − + KD .ψ 0,0 = − & − κc I & SD & [F(z)] . (c − 1) 2

&0,0 = 0. Since .F(z) is a constant vector, by using (2.2.36) one thus has .ψ

2.2 Elastic Problem

61

⨆ ⨅

The proof is complete.

Remark 2.2.1 In Proposition 2.2.3, we have made use of the invertibility of the operator .K∗D & − κc I. In principle, we need to impose a certain condition on .κc , or equivalently on c, in order to have the invertibility. So far, we have only assumed that .c ∈ C and .ℑc ≥ 0. Further condition on c to guarantee the aforesaid invertibility is given in Theorem 2.2.84 in what follows. We are mainly interested in the case with .ℜc < 0. In such a case, one can conclude by Theorem 2.2.84 that if c is such chosen that ℑκc (κc2 − k02 ) /= 0,

.

where .k0 is given in (2.2.84), then the operator .K∗D & − κc I is invertible. The invertibility of .K∗D − κ I can hold in more general scenarios and we shall discuss c & this in more details in Section 3 (cf. Theorem 3.1.1 and Remark 3.1.2), and at this stage, we assume the invertibility of .K∗D & − κc I. It is also worth of pointing out that −κ I clearly also guarantees the well-posedness of the elastic the invertibility of .K∗D c & scattering system (2.2.1)–(2.2.6) for plasmonic nanoparticles. &β , we are in a position to derive After obtaining the asymptotic expression for .ψ the estimate for .Mα,β with .|α| ≤ 2 and .|β| ≤ 2. Proposition 2.2.4 There holds the following estimate for .Mα,β with .α ∈ N3 and .β = 0  Mα,0 = δ 2

.

& ∂D

&0,2 (& & yα ψ y)ds(& y) + O(δ 3 ),

(2.2.53)

where &0,2 = .ψ

   −1  −cω12  ∗ I −1 −1 ∗ , S P S + K KD I − κ I − − [F(z)] & & c & & D D & & D D D 2 (c − 1) (2.2.54)

with .ID & and .PD & defined in (2.2.27) and (2.2.31) respectively. Proof Firstly, from the proposition 2.2.3, one has that −1 &0,0 = 0 and & ϕ 0,0 = SD ψ & [F(z)] .

.

(2.2.55)

By comparing the coefficients of order .δ in (2.2.44), one has that  A

.

   & & ϕ 0,0 ϕ 0,1 = 0, &0,1 + T 0 ψ

(2.2.56)

62

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

where the operator .A is given in (2.2.45). By solving the Eq. (2.2.56) one derives that −1  I cω1  ∗ −1 ∗ & − + KD KD ϕ 0,0 ] . (2.2.57) .ψ 0,1 = & [& & − κc I & SD & RD (c − 1) 2 with .RD ϕ 0,0 ] is a constant, by Lemma 2.2.3 & given in (2.2.26). Since .RD & [& and (2.2.56) one can obtain that &0,1 = 0 ψ

.

−1 and & ϕ 0,1 = −ω1 SD ϕ 0,0 ]. & [& & RD

Furthermore, by comparing the coefficients of order .δ 2 in (2.2.44), one has that  A

.

     & & & ϕ 0,1 ϕ 0,0 ϕ 0,2 + T + N = 0, &0,2 0 0 ψ

(2.2.58)

where .T and .N are defined in (2.2.45) and (2.2.46), respectively. By solving the above equation one can find that &0,2 = ψ

.

 −1  −cω12  ∗ I −1 ∗ KD + K P S ϕ 0,0 ], − κ I − − I & & [& c & & D D & D D (c − 1) 2 (2.2.59)

with .ID & and .PD & defined in (2.2.27) and (2.2.31) respectively. Finally, by the definition of .Mα,0 in (2.2.41), one has that  Mα,0 = δ 2

.

& ∂D

&0,2 (& & yα ψ y)ds(& y) + O(δ 3 ),

(2.2.60) ⨆ ⨅

and the proof is completed. Proposition 2.2.5 One has the following estimate

.

M0,β = O(δ 2 ).

(2.2.61)

|β|=1

& Proof From the proof of Proposition 2.2.3, one has that for & .y ∈ ∂ D,  .

&β,0 ] = & ϕ β,0 ] − SD yβ ∂ β F(z), SD & [& & [ψ ∂ ∂ β β &β,0 ]|+ = ∂ & c ∂ν SD ϕ β,0 ]|− − ∂ν SD & [& & [ψ ∂ν y ∂ F(z).

(2.2.62)

2.2 Elastic Problem

63

For .|β| = 1, one has that  .

& ∂D

 ∂  β β & y ∂ F(z) ds(& y) = 0, ∂ν

which follows from Green’s formula and   yβ ∂ β F(z) = 0, .Lλ,μ &

& & y ∈ D.

Using integration by parts, one can similarly show that  .

& ∂D

∂ S & [& ϕ ]|− ds(& y) = 0 ∂ν D β,0

 and

& ∂D

∂ & ]|− ds(& S & [ψ y) = 0. ∂ν D β,0

Therefore from the second equation of (2.2.62), one can obtain that  .

∂ & ]|+ ds(& S & [ψ y) = 0. ∂ν D β,0

& ∂D

With the help of the jump relationship (2.2.14) and Lemma 2.2.4, one has that  .

& ∂D

&β,0 (& y)ds(& y) = 0, ψ

for |β| = 1.

(2.2.63)

By using (2.2.44) again, one can further obtain, similar to (2.2.57), that &β,1 = ψ

.

−1 cω1  ∗ KD − κ I c & (c − 1)



I −1 − + K∗D S R [& ϕ ] , & & D β,0 & D 2

(2.2.64)

and by using Lemma 2.2.3 and noting that .RD ϕ β,0 ] is a constant vector, one can & [& in turn derive that &β,1 = 0, ψ

.

for |β| = 1.

Thus from the definition (2.2.41), one can obtain that

.

M0,β = O(δ 2 ).

(2.2.65)

|β|=1

This proof is complete.

⨆ ⨅

64

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

From the last two Propositions 2.2.4 and 2.2.5, one can show that the first three terms in (2.2.47) satisfy ⎛ ⎞



  2 .δ Gω2 (x − z)M0,0 + δ ⎝ Gω2 (x − z)M0,β − ∂ α Gω2 (x − z)Mα,0 ⎠ |α|=1

|β|=1

= O(δ 4 ). Therefore from (2.2.47), the far-field expansion for the displacement .u(x) can be written as   &0,2 (& u(x) − F(x) =δ 3 Gω2 (x − z) y)ds(& y) ψ & ∂D



1 &β,0 (& + Gω (x − z) y)ds(& y) ψ 2 2 & ∂D

.

|β|=2



−

 α

∂ Gω2 (x − z)

|α|=1 |β|=1

& ∂D

 &β,0 (& & yα ψ y)ds(& y) + O(δ 4 ), (2.2.66)

&β,0 and .ψ &0,2 given in (2.2.49) and (2.2.54), respectively. The following with .ψ lemmas show the detailed calculation of the three terms appeared in the right hand side of (2.2.66). Lemma 2.2.6 There holds the following identity

 .

& |β|=2 ∂ D

&β,0 (& & y)ds(& y) = −2|D|(L ψ λ,μ F)(z).

(2.2.67)

&β,0 given in (2.2.49), one has that Proof Recalling the expression for .ψ (c − 1)

.



K∗D &

   ∂ β β I −1 ∗ β β & & y ∂ F(z) − c − + KD y − κc I [ψ β,0 ] = & ∂ F(z) . & SD & 2 ∂ν (2.2.68) 

& 3, By using Lemma 2.2.4, one can obtain that for .ϕ ∈ H −1/2 (∂ D)  & ∂D .

=

3

i=1

K∗D & [ϕ]ds =

3

 ei

i=1

 ei

& ∂D

ϕ · KD & [ei ] =

& ∂D

(K∗D & [ϕ]) · ei ds

 3

ei i=1

2

& ∂D

ϕ · ei ds =

1 2

 & ∂D

ϕds.

2.2 Elastic Problem

65

Therefore  .

& ∂D

   & (c − 1) K∗D − κ I [ ψ ]ds = − c β,0 &

& ∂D

&β,0 ds, ψ

(2.2.69)

and  .

& ∂D



  I −1 ∗ − + KD yβ ∂ β ∂ β F(z) ds = 0. & SD & & 2

(2.2.70)

With the help of Green’s formula, there holds the following

 .

& |β|=2 ∂ D

 ∂ β β & & y ∂ F(z)ds = yβ ∂ β F(z)dv = 2|D|(L Lλ,μ& λ,μ F)(z), ∂ν & D |β|=2

(2.2.71) & is the volume of the domain .D. & Finally, from (2.2.68), (2.2.69), (2.2.70) where .|D| and (2.2.71), one can thus obtain (2.2.67), which completes the proof. ⨆ ⨅ Lemma 2.2.7 There holds the following identity  .

& ∂D

&0,2 (& & y)ds(& y) = −ω2 |D|F(z). ψ

(2.2.72)

Proof From (2.2.54) one has (c −1)



.

K∗D &

    I −1 −1 2 ∗ & SD − κc I [ψ 0,2 ] = −cω1 PD & & − − + KD & SD & ID & [F(z)] . 2 (2.2.73) 

Similar to the proof in Lemma 2.2.6, one has  .

& ∂D

&0,2 (& y)ds(& y) = cω12 ψ

 & ∂D

  −1 (& y)ds(& y). PD [F(z)] & SD &

(2.2.74)

By using the definition of .PD & in (2.2.31), integration by parts and (2.2.33) one obtains       ∂ −1 −1 (& y)ds(& y) (& y)ds(& y) = I S PD [F(z)] [F(z)] & SD & D & & D & & ∂ν ∂D ∂D    −1 y)d& y = (Lλ,μ ID . & ) SD & [F(z)] (& & D



=−

& D

  −1 & SD (& y)d& y = −|D|F(z). [F(z)] & SD & (2.2.75)

66

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

By substituting (2.2.75) into (2.2.74) one thus has (2.2.72). The proof is complete.

⨆ ⨅

&β,0 be defined in (2.2.39). Then for .|β| = 1, one has Lemma 2.2.8 Let .ψ  −1  ∂  β β &β,0 = − K∗ − κc I & y ∂ F(z) . ψ & D ∂ν

(2.2.76)

.

&β,0 in (2.2.49). For .|β| = 1, it can be easily verified Proof Recall the formula for .ψ that   β β Lλ,μ & y ∂ F(z) = 0

.

in R3 .

(2.2.77)

Define   −1 β β & y ϕ := SD ∂ F(z) &

.

on

& ∂ D.

Then one has SD yβ ∂ β F(z) & [ϕ] = &

.

on

& ∂ D.

(2.2.78)

Together with (2.2.77) one can show that SD yβ ∂ β F(z) & [ϕ] = &

.

& in D.

(2.2.79)

Thus one has from (2.2.14) and (2.2.79) that     I I −1 ∗ ∗ β β − + KD y ∂ F(z) = − + KD & [ϕ] & SD & & 2 2

(2.2.80)

.

∂ ∂ β β y ∂ F(z). S & [ϕ]|− (x) = & = ∂ν ∂ν D Together with (2.2.49), one finally has (2.2.76). The proof is complete.

⨆ ⨅

Based on the above results, we can finally show the far-field expansion for the displacement .u(x) as follows Theorem 2.2.1 Let .u(x) be the solution to the system (2.2.2)–(2.2.6). Then for .x ∈ R3 \D, there holds u(x) = F(x) + δ

.

3

3

j =1 |α|=1 |β|=1

j

∂ α Gω (x − z)∂ β Fj (z)Mα,β + O(δ 4 ),

(2.2.81)

2.2 Elastic Problem

67 j

where the Elastic Moment Tensor (EMT) .(Mα,β ) is defined by j .M α,β

  −1  ∂ ∗ β & & y), y ej ds(& y KD = & − κc I ∂ν & ∂D 

α

(2.2.82)

and ∂ β F(z) = (∂ β F1 (z), ∂ β F2 (z), ∂ β F3 (z)).

.

Proof With the help of Lemmas 2.2.6 and 2.2.7, one can obtain that  .

& ∂D

&0,2 (& y)ds(& y) + ψ

1 &β,0 (& y)ds(& y) ψ 2 ∂D &

|β|=2

 & − ω2 F(z) − (Lλ,μ F)(z) = 0, =|D| 

where the second identity follows from (2.2.21) and the assumption that .f is compactly supported in .R3 \ D. Therefore one can derive (2.2.81) thanks to (2.2.66)  −1 and the linearity of the operator . K∗D − κ I . c & This proof is complete. ⨆ ⨅ We remark that in [15], the authors proved the asymptotic expansion for static elastic problem in the presence of small inclusions. Our asymptotic expansion result (2.2.81) is in accordance with their results but has more details in describing the EMT in (2.2.82). This helps us to analyze the EMT in a much more elaborate way and so the phenomenon of polariton resonance.

2.2.5 Resolvent Analysis From the analysis of the last section, we know that the polariton resonance only j occurs when the energy of the EMT, .(Mα,β ), blows-up (see (2.2.81) and [12]). From (2.2.82), it then remains to analyze the resolvent of the Neumann-Poincaré operator ∗ .K . We have the following auxiliary result & D Theorem 2.2.2 For the operator .K∗D & , we have the following resolvent estimate −1  ‖ kα I − K∗D ‖L (H −1/2 (∂ D) & 3 ,H −1/2 (∂ D) & 3) ≤ &

.

C , d(h(kα ), σ (GD & ))

(2.2.83)

where h(kα ) = kα (kα2 − k02 ),

.

and

∗ ∗ 2 2 GD & = KD & ((KD & ) − k0 I),

(2.2.84)

68

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

with k0 =

.

μ , 2(2μ + λ)

and the constant C depends on the .K∗D & ) is the spectrum & , .kα and .k0 . In (2.2.83), .σ (GD of the operator .GD & and .d(h(kα ), σ (GD & )) is the distance between .h(kα ) and .σ (GD & ). Proof First we introduce the Hilbert space .H with the following inner product (g, h)H := −〈g, SD & [h]〉,

.

& 3, g, h ∈ H −1/2 (∂ D)

(2.2.85)

& 3 and .H 1/2 (∂ D) & 3 , and the operator .S & where .〈., .〉 is the pair between .H −1/2 (∂ D) D ∗ & is defined on .∂ D. Then the operator .KD & is a self-adjoint operator in the space .H . Furthermore, since the norm .‖·‖H induced by the inner product .(., .)H is equivalent to .‖ · ‖H −1/2 (∂ D) & 3 , we thus denote the norm .‖ · ‖H by .‖ · ‖H −1/2 (∂ D) & 3 without any ambiguity. We refer to [21] for more details. One can find that the operator .GD & defined in (2.2.84) is a compact operator on .H (see [23]). By using the Calderón identity ∗ SD & KD & SD & & = KD

.

one can find that ∗ (GD & [g], h)H = −〈GD & [g], SD & [h]〉 = −〈g, GD & [h]〉 & SD

.

= −〈g, SD & GD & [h]〉 = (g, GD & [h])H ,

(2.2.86)

∗ 2 2 where .GD & ((KD & ) − k0 I). Thus it is a self-adjoint compact operator on .H & := KD and therefore from [69] one can derive the resolvent for .GD & by

−1  ‖ h(kα )I − GD ‖L (H −1/2 (∂ D) & & 3 ,H −1/2 (∂ D) & 3) ≤

.

C . d(h(kα ), σ (GD & ))

Direct calculation gives that .

   ∗ 2 ∗ 2 2 K∗D − k I (K ) + k K + (k − k )I = h(kα )I − GD &, α α α & & & 0 D D

and therefore .

 −1   −1  ∗ 2 ∗ 2 2 K∗D (KD = h(kα )I − GD & & − kα I & ) + kα KD & + (kα − k0 )I . (2.2.87)

2.2 Elastic Problem

69

Finally one can conclude that −1  ‖ kα I − K∗D ‖L (H −1/2 (∂ D) & 3 ,H −1/2 (∂ D) & 3) ≤ &

.

C , d(h(kα ), σ (GD & )) ⨆ ⨅

and the proof is completed.

Remark 2.2.2 With the help of Theorem 2.2.2, one can have the following estimate for the items .& ϕ β,n , n ≥ 0 shown in (2.2.43),  ‖& ϕ β,n ‖H −1/2 (∂ D) &3 ≤

.

C d(h(κc ), σ (GD & ))

n+1 (2.2.88)

,

& .β and .F. Indeed, when .n = 0, from the expression of .& ϕ β,0 where C depends on .D, shown in (2.2.49) and Theorem 2.2.2, the following estimate holds ‖& ϕ β,0 ‖H −1/2 (∂ D) &3 ≤

.

C . d(h(κc ), σ (GD & ))

ϕ β,1 , it is clear that it satisfies the following equation from (2.2.44) For the item .&  A

.

& ϕ β,1 &β,1 ψ



 =T

 & ϕ β,0 &β,0 . ψ

Following the similar calculation in Proposition 2.2.3 and with the help of Theorem 2.2.2, one can have that  ‖& ϕ β,1 ‖H −1/2 (∂ D) &3 ≤

.

C d(h(κc ), σ (GD & ))

2 .

By the analogous discussion, finally one can show (2.2.88). From Theorems 2.2.2 and 2.2.1, one can find that the spectra of the NeumannPoincaré operator .K∗D & are attributed to the polariton resonance of the elastic problem, which we shall discuss in more details in the next subsection.

2.2.6 Polariton Resonance for Elastic Nanoparticles Theorem 2.2.3 Let u(x) be the solution to the system (2.2.2)–(2.2.6). Suppose that c = c0 + iτ , where τ ∈ R+ is sufficiently small and c0 is chosen such that the quantity hc0 hc0 := kc0 (kc20 − k02 ),

.

70

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

with kc0 =

.

c0 + 1 2(c0 − 1)

and

k0 =

μ , 2(2μ + λ)

∗ is the eigenvalue of the operator GD & with respect to an eigenfunction φ . Suppose further that

(φ ∗ , φ F )H /= 0,

.

(2.2.89)

where    ∂   2 ∗ 2 2 φ F (& y) = (K∗D F (z) (& y). & ) + kc KD & + (kc − k0 )I ∂ν

.

Then one has |u(x)| ∼ δ 3 τ −1 ,

a.e.

.

x ∈ R3 \ D.

(2.2.90)

Proof Firstly, one can rewrite the asymptotic expansion (2.2.81) as follows u(x) = F(x) + δ 3 P(x) + O(δ 4 ),

.

(2.2.91)

where P is defined by  P(x) :=

.

& ∂D

 φ G (x,& y)

K∗D &

−1  ∂   F (z) ds(& y). − κc I ∂ν

(2.2.92)

with φ G (x,& y) =



∂ α Gω (x − z)& yα .

.

|α|=1

Recall that GD & defined in (2.2.84) is a self-adjoint compact operator on H , where & 3 and inner product defined by H is a Hilbert space with functions in H −1/2 (∂ D) ∞ (2.2.85). Thus the eigenfunctions {φ n }n=1 , corresponding the eigenvalues {λn }∞ n=1 of the operator GD & , form a norm basis on H and therefore the operator GD & admits &3 the follwoing eigenvalue decompositions in H , for φ ∈ H −1/2 (∂ D) GD & [φ] =

∞

.

κn φ n ,

n=1

where κn = λn (φ n , φ)H .

.

(2.2.93)

2.2 Elastic Problem

71

Since c = c0 + iτ , one has κc =

.

c+1 = kc + O(τ ). 2(c − 1)

By using the relations in (2.2.87), (2.2.89) and (2.2.93), one then has  −1  ∂    −1 F (z) = hc I − GD K∗D − κ I [φ F ] & + O(τ ) c & ∂ν −1 ∗  . [φ ] + O(1) = C hc I − GD & + O(τ ) = φ ∗ O(τ −1 ) + O(1). (2.2.94) Since there holds (Lλ,μ + ω2 )P(x) = 0,

.

x ∈ R3 \ D,

by using the unique continuation principle, one has that P(x) /= 0,

x ∈ R3 \ D.

a.e.

.

By substituting (2.2.94) into (2.2.92) one thus has |P(x)| ∼ τ −1 ,

.

a.e.

x ∈ R3 \ D,

which together with (2.2.91) readily implies (2.2.90). The proof is complete.

⨆ ⨅

Remark 2.2.3 We remark that if the polariton resonance occurs, the parameter c defined in (2.2.3) should have negative real part, namely ℜc = c0 < 0.

.

In fact, if ℜc > 0, one can show from the definition of κc in (2.2.48) that ℜκc >

.

1 . 2

Then in such a case, one cannot have the polarition resonance since −1  ‖ κc I − K∗D ‖L (H −1/2 (∂ D) & 3 ,H −1/2 (∂ D) & 3 ) ≤ C, &

.

−1/2 (∂ D) & 3 lies which is due to the fact that the spectrum of the operator K∗D & on H in (−1/2, 1/2] (cf. [21]).

72

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

Remark 2.2.4 At the polariton resonance, one has that d(h(κc ), σ (GD & )) ⪡ 1. Thus from Theorem 2.2.3, one has that the scattered elastic wave filed is enhanced. From Remark 2.2.2, it is noted that the items & ϕ β,n are controlled in norm by powers of 1/d(h(κc ), σ (GD & )). Thus the series shown in (2.2.43) are convergent when δ/d(h(κc ), σ (GD & )) ⪡ 1, which also ensures the convergence of the series (2.2.42). Therefore the asymptotic development given by Theorem 2.2.3 is valid.

Appendix We want to prove the jump formula (2.1.25) for .ν × ∇ × AD . The continuity of ADk [φ] is a consequence of the continuity of single layer potentials. Assume that .φ is a continuous tangential field. We first prove the jump relation for .k = 0. For 3 .z ∈ R \ ∂D, .

 ∇ × AD [φ](z) =

∇z × (φ(y)G(z − y)) ds(y).

.

∂D

So if .x ∈ ∂D and .z = x + hν(x), then by using vector calculus we have: ν(x) × ∇ × AD [φ](z) 

= (φ(y) · ν(x)) ∇z G(z − y) − (∇z G(z − y) · ν(x)) φ(y) ds(y).

.

∂D

Since .φ is tangential, we have .∀y ∈ ∂D, ν(y) · φ(y) = 0, so we can write ν(x) × ∇ × AD [φ](z) 

= (φ(y) · [ν(x) − ν(y)]) ∇z G(z − y) − (∇z G(z − y) · ν(x)) φ(y) ds(y).

.

∂D

Here, following the same idea as the one in the proof of the jump of the double layer potential in [44], we introduce  DD [1](z) =

.

∂D

∂G (z − y)ds(y), ∂ν(y)

z ∈ R3 \ ∂D,

which takes the following values ([44]): ⎧ ⎪ 0 if ⎪ ⎪ ⎪ ⎨ 1 .DD [1](z) = if − ⎪ 2 ⎪ ⎪ ⎪ ⎩ − 1 if

z ∈ R3 \ D, z ∈ ∂D, z ∈ D.

(2.2.95)

Appendix

73

We write ν(x) × ∇ × AD [φ](z) = φ(x)DD [1](z) + f(z)

.

with 



f(z) =

(φ(y) · [ν(x) − ν(y)]) ∇z G(z − y)

.

∂D

− (∇z G(z − y) · [ν(x) − ν(y)]) φ(y) − (∇z G(z − y) · ν(y)) φ(y)  ∂G (z − y)φ(x) ds(y). − ∂ν(y) Using the fact that .∇z G(z − y) = −∇y G(z − y) we get  f(z) =

 (φ(y) · [ν(x) − ν(y)]) ∇z G(z − y)

.

∂D

− (∇z G(z − y) · [ν(x) − ν(y)]) φ(y) +

 ∂G (z − y) (φ(y) − φ(x)) ds(y). ∂ν(y) (2.2.96)

Now, we have only to prove that .f is continuous across .∂D, i.e., when .t → 0, f(z) = f(x + tν(x)) −→ f(x). If we assume that it is true, then we can write for 3 .z ∈ R \ D, .



φ ν(x) × ∇ × AD [φ](z) = φ(x)DD [1](z) − φ(x)DD [1](x) + f(z) − (x), 2

.

since .DD [1](x) = −1/2. So, when .t → 0+ , we get 

φ ν(x) × ∇ × AD [φ](x)+ = − φ(x)DD [1](x) + f(x) − (x). 2

.

Now we see that since .φ(y) · ν(y) = 0,  .

− φ(x)DD [1](x) + f (x) = − ∂D



 +

∀y ∈ ∂D

∂G (x − y)φ(x)ds(y) ∂ν(y)

(φ(y) · ν(x)) ∇x G(x−y)−(∇x G(x − y) · ν(x))+ ∂D

 ∂G (x−y)φ(x) ds(y), ∂ν(y)

which is exactly  .

− φ(x)DD [1](x) + f(x) =

ν(x) × ∇x × [G(x − y)φ(y)] ds(y). ∂D

74

2 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime

So the limit can be expressed as  ν(x) × ∇ × AD [φ](x)+ =

 ν(x) × ∇x × [G(x − y)φ(y)] ds(y) −

.

∂D

φ (x). 2

The limit when .t → 0− is computed similarly and we find (2.1.25) for .k = 0. The extension to .k > 0 can be done because the difference between the double layer potential with kernel .Gk and .G is continuous. Now, we go back to the continuity of .f defined by (2.2.96). We apply several results from [44] to get the continuity. The following lemma, which we state here for the sake of completeness, can be found in [44]. Lemma 2.2.9 Assume that the kernel K is continuous for all .x in a neighborhood Dh of .∂D, .y ∈ ∂D and .x /= y. Assume that there exists .M > 0 such that

.

|K(x, y)| ≤ M|x − y|−2

.

and assume that there exists .m ∈ N such that |K(x1 , y) − K(x2 , y)| ≤ M

m

|x1 − y|−2−j |x1 − x2 |j

.

j =1

for all .x1 , x2 ∈ Dh , .y ∈ ∂D with .2|x1 − x2 | ≤ |x1 − y| and that       . K(z, y)ds(y) ≤ M  ∂D\Sx,r  for all .x ∈ ∂D and .z = x + hν(x) ∈ Dh and all .0 < r < R. Then,  u(z) =

K(z, y)[φ(y) − φ(x)]ds(y)

.

∂D

belongs to .C 0,α (Dh ) if .φ ∈ C 0,α (∂D). It can be shown that    ∂G(x − y) ∂G(z − y)  |x − z|   .  ∂ν(y) − ∂ν(y)  ≤ C |z − y|3 . Using the above lemma with .m = 1 and the kernel associated with the double layer potential gives  .

∂D

∂G (z − y) [φ(y) − φ(x)] ds(y) −→ ∂ν(y)

as .z → x ∈ ∂D.

 ∂D

∂G (x − y) [φ(y) − φ(x)] ds(y) ∂ν(y)

Appendix

75

We now make use of the following lemma from [44]. Lemma 2.2.10 Assume that the kernel .K(x, y) is continuous for all .x in a closed domain .Ω containing .∂D in its interior, .y ∈ ∂D and .x /= y. Assume that there exists .M > 0 and .α ∈]0, 2] such that |K(x, y)| ≤ M|x − y|α−2

.

and assume that there exists .m ∈ N such that |K(x1 , y) − K(x2 , y)| ≤ M

m

.

|x1 − y|α−2−j |x1 − x2 |j

j =1

for all .x1 , x2 ∈ Dh , .y ∈ ∂D with .2|x1 − x2 | ≤ |x1 − y|. Then  u(x) =

K(x, y)φ(y)ds(y),

.

x∈Ω

∂D

belongs to .C 0,β (Ω) if .φ ∈ C 0,α (∂D). .β ∈]0, α] if .α ∈]0, 1[, .β ∈]0, 1[ if .α = 1 and .β ∈]0, 1] if .α ∈]1, 2[. Using the fact that .∂D is of class .C 2 , we have |ν(x) − ν(y)| ≤ |x − y|,

.

∀x, y ∈ ∂D.

We can apply Lemma 2.2.10 with .α = 1 and .m = 1 to the second and third terms of .f and get the continuity of  .

∂D



 (φ(y) · [ν(x) − ν(y)]) ∇z G(z−y)−(∇z G(z − y) · [ν(x) − ν(y)]) φ(y) ds(y) (2.2.97)

when .z → x ∈ ∂D, which conclude the proof for a continuous tangential field .φ. The formula can be extended to .L2T by a density argument .

Chapter 3

Anomalous Localized Resonances and Their Cloaking Effect

In this chapter, we present the spectral analysis of anomalous localized resonances (ALRs) in elastostatics and electrostatics [10, 20, 53]. ALRs are induced by properly designed core-shell material structures with metamaterials located inside the shell. In a delicate manner, resonance can occur if the material structure is properly constructed. The resonant field exhibits highly oscillatory behaviour and the high oscillation is confined, namely localized, within a domain which is larger than the underlying material structure. Moreover, the resonance strongly depends on the location of the excitation source. In fact, there exists a critical radius such that if the source located within the critical radius, then resonance occurs, and otherwise resonance does not occur. As a significant application, ALR can induce cloaking effect. All these make the study of ALR highly intriguing and intricate.

3.1 Elastostatic Problem 3.1.1 Mathematical Setup of Elastostatics Problem Let us first briefly introduce the mathematical formulation of the elastostatic system and the cloaking due to anomalous localised resonance in elastostatics. We follow the treatment in [53]. We also refer to [20, 21, 83, 87, 95, 96] for more relevant discussions. Let .C(x) := (Cij kl (x))3i,j,k,l=1 , .x ∈ R3 , be a four-rank tensor such that Cij kl (x) := λ(x)δ ij δ kl + μ(x)(δ ik δ j l + δ il δ j k ), x ∈ R3 ,

.

(3.1.1)

where .λ, μ ∈ C are complex-valued functions, and .δ is the Kronecker delta. .C(x) describes an isotropic elastic material tensor distributed in the space .R3 , where .λ and .μ are referred to as the Lamé constants. For a regular elastic material, the Lamé © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Y. Deng, H. Liu, Spectral Theory of Localized Resonances and Applications, https://doi.org/10.1007/978-981-99-6244-0_3

77

78

3 Anomalous Localized Resonances and Their Cloaking Effect

constants are real-valued and satisfy the following strong convexity condition, μ>0

and

.

3λ + 2μ > 0.

(3.1.2)

For a polariton elastic material, the Lamé constants are allowed to be complexvalued with the real parts being negative and the imaginary parts signifying the loss parameters. In the sequel, we write .CR3 ,λ,μ to specify the dependence of the elastic tensor on the Lamé parameters .λ and .μ, and the domain of interest .R3 . We shall also simply write .Cλ,μ for .CR3 ,λ,μ if no confusion would arise in the context. Let .Σ and .Ω be bounded domains in .RN with connected Lipschitz boundaries such that .Σ  Ω. Let the elastic tensor .Cλ,μ be regular in the core .Σ and in the matrix .R3 \Ω, whereas in the shell .Ω\Σ, the elastic tensor is polariton. We also assume that in the shell, ℑλ(x) = ℑμ(x) = δ

.

for x ∈ Ω\Σ,

(3.1.3)

where .δ ∈ R+ is sufficiently small, signifying the lossy parameter. Let .f be an .R3 valued function that is compactly supported outside .Ω with a zero average,  .

R3

f(x) dx = 0.

(3.1.4)

f signifies an elastic source/forcing term. Let .uδ (x) ∈ C3 , .x ∈ R3 , denote the displacement field in the space that is occupied by the elastic configuration .(Cλ,μ , f) 1 (R3 )3 verifies the following described above. In the quasi-static regime, .uδ (x) ∈ Hloc Lamé system

.

⎧ ⎪ ⎪ ⎨ .

Lλ,μ uδ (x) = f(x),

x ∈ R3 ,

uδ |− = uδ |+ , ∂ν λ,μ uδ |− = ∂ν λ,μ uδ |+   uδ (x) = O |x|−1 as |x| → +∞,

⎪ ⎪ ⎩

on ∂Σ ∪ ∂Ω,

(3.1.5)

where the PDO .Lλ,μ is defined by Lλ,μ uδ := ∇ · Cλ,μ ∇ s uδ = μΔuδ + (λ + μ)∇∇ · uδ ,

.

with .∇ s signifying symmetric gradient ∇ s uδ :=

.

1 (∇uδ + ∇u⏉ δ ), 2

(3.1.6)

3.1 Elastostatic Problem

79

and the superscript .⏉ denoting the matrix transpose. In (3.1.5), the conormal derivative (or traction) is defined by ∂ν uδ =

.

∂uδ := λ(∇ · uδ )ν + μ(∇uδ + ∇u⏉ δ )ν ∂ν

on ∂Σ or ∂Ω,

(3.1.7)

where .ν denotes the exterior unit normal to .∂Σ/∂Ω, and the .± signify the traces taken from outside and inside of the domain .Σ/Ω, respectively. Next, for .u ∈ 1 (R3 )3 and .v ∈ H 1 (R3 )3 , we introduce Hloc loc  Pλ,μ (u, v) :=

.

R3



λ(∇ · u)(∇ · v)(x) + 2μ∇ s u : ∇ s v(x) dx,

(3.1.8)

3 where and also in what follows, .A : B = i,j =1 aij bij for two matrices .A = 3 3 (aij )i,j =1 and .B = (bij )i,j =1 . For the solution .uδ to (3.1.5), we define Eδ (Cλ,μ , f) :=

.

δ Pλ,μ (uδ , uδ ). 2

(3.1.9)

Then cloaking due to anomalous localised resonance occurs if the following two conditions are satisfied: .

lim sup E(uδ ) → ∞, δ→+0

|uδ (x)| < C,

|x| > r ' ,

(3.1.10)

for some constants C and .r ' independent of .δ.

3.1.2 Preliminaries on Layer Potentials We first introduce some function spaces that shall be needed in our subsequent study. Let D be a bounded Lipscthiz domain with a connected complement in .R3 . Let .∇∂D · 3 denote the surface divergence. Denote by .L2T (∂D) := {ϕ ∈ L2 (∂D) , ν · ϕ = 0}. Let .H s (∂D) be the usual Sobolev space of order .s ∈ R on .∂D. We also introduce the function spaces

 TH(div, ∂D) : = ϕ ∈ L2T (∂D) : ∇∂D · ϕ ∈ L2 (∂D) ,

 TH(curl, ∂D) : = ϕ ∈ L2T (∂D) : ∇∂D · (ϕ × ν) ∈ L2 (∂D) , .

80

3 Anomalous Localized Resonances and Their Cloaking Effect

equipped with the norms ‖ϕ‖TH(div,∂D) = ‖ϕ‖L2 (∂D) + ‖∇∂D · ϕ‖L2 (∂D) ,

.

‖ϕ‖TH(curl,∂D) = ‖ϕ‖L2 (∂D) + ‖∇∂D · (ϕ × ν)‖L2 (∂D) . In the following, we introduce some basic notions on layer potentials. For a density function .φ,denote by .SD : H −1/2 (∂D)m → H 1/2 (∂D)m , .m = 1, 3 the single layer potential operator, which is represented as follows  SD [ϕ](x) :=

G(x − y)ϕ(y)ds(y),

.

(3.1.11)

∂D

where .G(x) is the fundamental solution to Laplacian .Δ given by G(x) = −

.

1 . 4π |x|

(3.1.12)

We mention that the density function .φ can be either a scalar density function or a vector density function in .R3 . If .ϕ ∈ L2T (∂D) then .SD [ϕ] is continuous on .R3 and its curl satisfies the following jump formula:  ϕ ν × ∇ × SD [ϕ]± = ∓ + MD [ϕ] 2

on ∂D,

.

(3.1.13)

where .MD is the boundary operator defined by MD : L2T (∂D) −→ L2T (∂D) .



ϕ ‫׀‬−→ MD [ϕ](x) = ν x × ∇x ×

G(x, y)ν y × ϕ(y)ds(y). ∂D

(3.1.14) On the other hand, for a vector function .ϕ on .∂D, denote by .SD [ϕ](x) the single layer potential associated with the Lamé system (3.1.5),  SD [ϕ](x) :=

G(x − y)ϕ(y)ds(y),

.

x ∈ R3 \∂D,

(3.1.15)

∂D

where .G = (Gj,k )3j,k=1 is the Kelvin matrix of fundamental solutions to the Lamé operator .Lλ,μ and has the following representation Gj,k (x) = −

.

α2 xj xk α1 δj k − , 4π |x|3 4π |x|

(3.1.16)

3.1 Elastostatic Problem

81

with 1 .α1 := 2



1 1 + μ 2μ + λ

 and

1 α2 := 2



 1 1 − . μ 2μ + λ

(3.1.17)

From the definition of traction in (3.1.7), the vector valued single layer potential (3.1.15) enjoys the following jump relation .

  ∂ 1 SD [ϕ]|± (x) = ± I + K∗D [ϕ](x), ∂ν 2

a.e. x ∈ ∂D,

(3.1.18)

where .I denotes the identity matrix operator in .R3 and .K∗D is the NeumannPoincaré(N-P) operator defined by K∗D [ϕ](x) := p.v.



.

∂D

∂ G(x − y)ϕ(y)ds(y). ∂ν x

(3.1.19)

In (3.1.19), p.v. stands for the Cauchy principal value. Here and also what in follows, ∂ ∂ν x G(x − y)ϕ(y) is defined by

.

.

∂ ∂ G(x − y)ϕ(y) := (G(x − y)ϕ(y)). ∂ν x ∂ν x

3.1.3 Spectral Analysis of N-P Operator in Spherical Geometry We shall derive the spectral of the N-P operator, .K∗B associated with Lamé system on a ball. It has been pointed out that the .K∗D is not a compact operator even if the domain D has a smooth boundary [20], thus we cannot infer directly that the N-P operator has point spectrum on a general smooth domain. However when B is a ball, the properties of .K∗B is more elaborate. We shall derive the eigenvalues of the N-P operator .K∗B and its corresponding eigenfunctions when the domain B is a ball. Before this, we present several auxiliary lemmas. Lemma 3.1.1 Suppose .Br0 is a central ball in .R3 with radius .r0 . Then the N-P operator .K∗Br can be written in the following form 0

μ 1 3 μ ) SBr0 [ϕ](x) SBr0 [ϕ](x) + ( + 2 2(2μ + λ) r0 r0  μ  ∇ × SBr0 [ν × ϕ](x) − ∇SBr0 [ν · ϕ](x) . − 2μ + λ (3.1.20)

K∗Br [ϕ](x) = − 3 0

.

82

3 Anomalous Localized Resonances and Their Cloaking Effect

Proof Let .x and .y be vectors on .∂Br0 . By (3.1.16) and straightforward computations one can show that ∂ν x G(x − y) = −b1 K1 (x, y) + K2 (x, y),

.

(3.1.21)

where K1 (x, y) = .

ν x (x − y)⏉ − (x − y)ν ⏉ x , 4π |x − y|3

K2 (x, y) =b1

(x − y) · ν x (x − y) · ν x I + b2 (x − y)(x − y)⏉ , 4π |x − y|3 4π |x − y|5

(3.1.22)

with b1 =

.

μ 2μ + λ

and

b2 =

3(μ + λ) . 2μ + λ

(3.1.23)

Then by (3.1.19), we have

.

K∗Br [ϕ](x) 0



 = − b1

K1 (x, y)ϕ(y)ds(y) + ∂Br0

K2 (x, y)ϕ(y)ds(y) ∂Br0

:=L1 + L2 . (3.1.24) Since .Br0 is a central ball, for .x, y ∈ ∂Br0 , one has that (ν x − ν y )(x − y)⏉ = (x − y)(ν x − ν y )⏉

.

and thus K1 (x, y) = .

= =

ν x (x − y)⏉ − (x − y)ν ⏉ x , 4π |x − y|3 (ν x − ν y + ν y )(x − y)⏉ − (x − y)(ν x − ν y + ν y )⏉ , 4π |x − y|3 ν y (x − y)⏉ − (x − y)ν ⏉ y 4π |x − y|3

(3.1.25)

.

Next, it is also easy to verify that .

(x − y) · ν y 1 1 =− . 3 2r0 |x − y| |x − y|

(3.1.26)

3.1 Elastostatic Problem

83

By using vector calculus identity, (3.1.24) and (3.1.26), there holds  L1 = −b1

∇x G(x − y) × ν y × ϕ(y) ∂Br0

1 G(x − y)ϕ − ∇x G(x − y)(ν · ϕ)ds(y) 2r0   1 SBr0 [ϕ](x) − ∇SBr0 [ν · ϕ](x) = −b1 ∇ × SBr0 [ν × ϕ](x) + 2r0 (3.1.27) +

.

Then by direct calculation, one further has that K2 (x, y) = − .

b1 b2 (x − y)(x − y)⏉ G(x − y)I + 2r0 2r0 4π |x − y|3

b1 b2 b2 α1 − )G(x − y)I. G(x − y) + ( =− 2r0 α2 2r0 2r0 α2

(3.1.28)

Hence, there holds L2 = −

b2 2r0 α2

 G(x − y)ϕ(y)ds(y) + ( ∂Br0



b2 α1 b1 − ) 2r0 α2 2r0

G(x − y)ϕ(y)ds(y)

.

(3.1.29)

∂Br0

=−

b2 b1 b2 α1 − )SBr0 [ϕ](x). SB [ϕ](x) + ( 2r0 α2 2r0 2r0 α2 r0

Finally, by combining (3.1.27) and (3.1.29), we have   K∗Br [ϕ](x) = − b1 ∇ × SBr0 [ν × ϕ](x) − ∇SBr0 [ν · ϕ](x) 0

.

−

b1 b2 α1 b2 − )SBr0 [ϕ](x). SBr0 [ϕ](x) + ( 2r0 α2 r0 2r0 α2

(3.1.30)

By calculating the coefficients in the above equation we arrive at (3.1.20), which completes the proof. ⨆ ⨅ Remark 3.1.1 We mention that the last two terms in (3.1.20) are defined by Cauchy principal values, and it is clearly that the related two operators in the last two terms are not compact operators, which shows that .K∗Br is not a compact operator in L2 (∂Br0 )3 .

.

0

84

3 Anomalous Localized Resonances and Their Cloaking Effect

In the following, we shall define orthogonal vectorial polynomials which will be quite important in the analysis of the spectral of the N-P operator .K∗Br . Let .r = |x| 0 and .Ynm (ˆx), .−n ≤ m ≤ n be spherical harmonics on the unit sphere S. Define three vectorial polynomials Tnm (x) = ∇(r n Ynm (ˆx)) × x,

n ≥ 1,

.

−n ≤ m ≤ n,

(3.1.31)

and Mnm (x) = ∇(r n Ynm (ˆx)),

n ≥ 1,

.

−n ≤ m ≤ n,

(3.1.32)

anm m − r 2 )∇(r n−1 Yn−1 (ˆx)), 2n − 1

(3.1.33)

and m Nnm (x) = anm r n−1 Yn−1 (ˆx)x + (1 −

.

where anm =

.

2(n − 1)λ + 2(3n − 2)μ , (n + 2)λ + (n + 4)μ

n ≥ 1,

−(n − 1) ≤ m ≤ n − 1.

(3.1.34)

By directly using the trace theorem, the traces of .Tnm , .Mnm and .Nnm on the unit m m sphere .S, denoted by .Tm n , .Mn and .Nn , have the following form m Tm x) × ν x , n (x) =∇S Yn (ˆ .

m Mm x) + nYnm (ˆx)ν x , n (x) =∇S Yn (ˆ

(3.1.35)

anm m m Nm (−∇S Yn−1 (ˆx) + nYn−1 (ˆx)ν x ). n (x) = 2n − 1 We have the following fundamental result Lemma 3.1.2 The polynomials .Tnm , .Mnm and .Nnm are solutions to the elastic m m equation .Lλ,μ u(x) = 0. Moreover, (.Tm n , .Mn , .Nn ) defined in (3.1.35) forms an 2 3 orthogonal basis on .L (S) . Proof It is easy to find that .Tnm and .Mnm are spherical harmonic functions and divergence free (see Theorem 2.4.7 in [115]). Thus from (3.1.6) one can easily obtain that Lλ,μ Mnm = Lλ,μ Tnm = 0.

.

For .Nnm , note that by (3.1.33) there holds m ∇ · Nnm = (anm (n + 2) − 2(n − 1))r n−1 Yn−1 (ˆx),

.

3.1 Elastostatic Problem

85

and m ∇ × ∇ × Nnm = n(anm + 2)∇(r n−1 Yn−1 ).

.

Hence by using (3.1.6) again and (3.1.34), one has Lλ,μ Nnm = μΔNnm + (λ + μ)∇∇ · Nnm = 0.

.

(3.1.36)

m m The orthogonality of (.Tm n , .Mn , .Nn ) can be obtained by straightforward computations (cf. [115]). The proof is complete. ⨆ ⨅

Lemma 3.1.3 Suppose that the domain .Br0 is a central ball with radius .r0 . Let (.Tm n, m m .Mn , .Nn ) be defined in (3.1.35), then there holds the following on .∂Br0 r0 Tm , 2n + 1 n r0 m Mm , . SBr [Mn ] = − 0 (2n − 1) n r0 SBr0 [Nm Nm , n+1 ] = − 2n + 3 n+1 SBr0 [Tm n]=−

(3.1.37)

where .n ≥ 0 and .−n ≤ m ≤ n. Proof We shall only prove the second identity in (3.1.37) and the other two can be proved similarly. Without loss of generality we suppose that .Br0 is a unit sphere. By using the jump formula (3.1.18) we have .

 ∂SBr0 [Mm 1 n ] + KB∗r [Mm  = − Mm n ], 0 − ∂ν 2 n

on

∂Br0 .

(3.1.38)

Since .Br0 is a ball, there holds the following identity (cf. [15]) 1 m KB∗r [Mm n ] = − SBr0 [Mn ], 0 2

.

and thus .

 ∂SBr0 [Mm 1 1 n ] − SBr0 [Mm  = − Mm n ] on − 2 n 2 ∂ν

∂Br0 .

(3.1.39)

Suppose .SBr0 [Mm n ] has the following form in .Br0   n m n m 2 n m SBr0 [Mm n ] = c1 ∇(r Yn ) + c2 (2n + 1)r Yn x − r ∇(r Yn ) ,

.

(3.1.40)

86

3 Anomalous Localized Resonances and Their Cloaking Effect

where .c1 and .c2 are constants which depends on n. Then by substituting (3.1.40) into (3.1.39) and using the trace theorem there holds m c1 (n − 1)Mm x) + (n + 1)Ynm (ˆx)ν x ) n + c2 (n + 1)(−∇S Yn (ˆ .

1 1 m m = − Mm x) + (n + 1)Ynm (ˆx)ν x )), n − (c1 Mn + c2 (−∇S Yn (ˆ 2 2

(3.1.41)

and by using the orthogonality property one has c1 (n − 1) = −1/2 − c1 /2, .

(3.1.42)

c2 (n + 1) = −c2 /2.

By solving (3.1.42) we get that c1 = −

.

1 , 2n − 1

c2 = 0.

(3.1.43)

Finally by substituting (3.1.43) into (3.1.40) and the trace theorem we obtain the first equation in (3.1.37). The proof is complete. ⨅ ⨆ Theorem 3.1.1 Suppose that the domain .Br0 is a central ball of radius .r0 , then the the eigenvalues of the operator .K∗Br are given by 0

.

ξ1n =

3 , 4n + 2

ξ2n =

3λ − 2μ(2n2 − 2n − 3) , 2(λ + 2μ)(4n2 − 1)

ξ3n =

−3λ + 2μ(2n2 + 2n − 3) , 2(λ + 2μ)(4n2 − 1)

(3.1.44)

where .n ≥ 1 are nature numbers, and the corresponding eigenfunctions are m m respectively .Tm n , .Mn and .Nn . m Proof Without loss of generality we set .r0 = 1. First, letting .ϕ = Tm n = ∇S Yn × ν and using the results in Lemma 3.1.3, one can show that

SBr0 [∇S Ynm × ν] = −

.

1 ∇(r n Ynm ) × x, 2n + 1

in

Br0 .

(3.1.45)

Furthermore, there holds ∇ × SBr0 [ν × ∇S Ynm × ν] = ∇ × SBr0 [∇S Ynm ] =

.

n ∇(r n Ynm ) × x, 2n + 1

in Br0 . (3.1.46)

3.1 Elastostatic Problem

87

and by using the jump formula (3.1.13) there also holds 1 n ∇S Ynm × ν − ∇S Ynm × ν, 2 2n + 1

∇ × SBr0 [ν × ∇S Ynm × ν] =

.

on

∂Br0 . (3.1.47)

Hence, by using (3.1.20), (3.1.45) and (3.1.47) one obtains K∗Br [∇S Ynm × ν] = −3μSBr0 [∇S Ynm × ν] −

.

0

3 ∇S Ynm × ν. 2(2n + 1)

(3.1.48)

By combining the jump formula (3.1.18) one can suppose that SBr0 [∇S Ynm × ν] = c∇(r n Ynm ) × x in Br0 ,

.

(3.1.49)

and by using (3.1.7) one can calculate ∂ SBr0 [∇S Ynm × ν]|− = cμ(∇(∇(r n Ynm ) × x) + (∇(∇(r n Ynm ) × x))⏉ )ν ∂ν . = cμ(n − 1)∇S Ynm × ν. (3.1.50) By substituting (3.1.50) into (3.1.7) and using (3.1.48) and (3.1.49), one can show m .cμ(n−1)∇S Yn ×ν

  1 3 m ∇S Ynm ×ν. = − ∇S Yn ×ν− 3cμ + 2(2n + 1) 2

(3.1.51)

Therefore, we have c=−

.

1 . (2n + 1)μ

(3.1.52)

Finally by substituting (3.1.52) into (3.1.48), one can obtain that K∗Br [∇S Ynm × ν] =

.

0

3 ∇S Ynm × ν. 2(2n + 1)

(3.1.53)

Next, by letting .ϕ = Mm n , one can show that SBr0 [Mm n]=−

.

1 ∇(r n Ynm ) in 2n − 1

Br0 ,

and m ∇ × SBr0 [ν × Mm n ] = ∇ × SBr0 [ν × ∇S Yn ] =

.

n+1 ∇(r n Ynm ) 2n + 1

in Br0 .

88

3 Anomalous Localized Resonances and Their Cloaking Effect

Straightforward computations also yields that m ∇SBr0 [ν · Mm n ] = ∇SBr0 [nYn ] = −

.

n ∇(r n Ynm ) in 2n + 1

Br0 .

Then by using the jump formulas there holds m ∇ × SBr0 [ν × Mm n ] − ∇SBr0 [ν · Mn ] =

.

1 m M 2 n

on

∂Br0 .

n m Next, we assume that .SBr0 [Mm n ] = c∇(r Yn ) in .Br0 and then one can show that

.

∂ ∂ SBr0 [Mm ∇(r n Ynm ) = 2cμ(n − 1)Mm n ]|− = 2cμ n. ∂ν ∂ν

Hence, there holds 1 m m K∗Br [Mm n ] = − Mn − 3μSBr0 [Mn ] − 0 2



.

 μn 3 + Mm n. 2(2n − 1) (2μ + λ)(2n − 1)

By using the jump formula, one has 3 μn 1 + ), cμ(2n + 1) = −( + 2 2(2n − 1) (2μ + λ)(2n − 1)

.

and thus K∗Br [Mm n]=

.

0

3λ − 2μ(2n2 − 2n − 3) m Mn . 2(2μ + λ)(4n2 − 1)

Finally, by letting .ϕ = Nm n+1 , one can show that SBr0 [Nm n+1 ] = −

.

m   an+1 1 (2n + 1)r n Ynm x − r 2 ∇(r n Ynm ) 2n + 3 2n + 1

in Br0 ,

and ∇ × SBr0 [ν × Nm n+1 ] = −

.

=−

m an+1

2n + 1

∇ × SBr0 [ν × ∇S Ynm ]

m n + 1 an+1 ∇(r n Ynm ) 2n + 1 2n + 1

in Br0 .

3.1 Elastostatic Problem

89

Straightforward computation gives that ∇SBr0 [ν · Nm n+1 ] =

.

m an+1

2n + 1

=−

∇SBr0 [(n + 1)Ynm ]

m n + 1 an+1 ∇(r n Ynm ) 2n + 1 2n + 1

in Br0 .

Then by using the jump formulas, there holds m an+1

1 1 ( ∇S Ynm − (n + 1)Ynm ν) 2n + 1 2 2 1 = − Nn+1 on ∂Br0 . 2

m ∇ × SBr0 [ν × Nm n+1 ] − ∇SBr0 [ν · Nn+1 ] =

.

We next assume that   m an+1 2 n m m m n m − r )∇(r Y (ˆ x )) , SBr0 [Nm ] = cN (x) = c a r Y (ˆ x )x + (1 − n n n+1 n+1 n+1 2n + 1

.

in B and then one can show   ∂ 2n m . SB [N ]|− = c (λ + μ)(n + 2 − m ) + λ Nm n+1 ∂ν r0 n+1 an+1 = cμ(

2(2n + 1) − 3)Nm n+1 . m an+1

Hence, there holds 3 (n + 1)μ m 1 1 ( − )Nn+1 . − 3μSBr0 [Nm ]− K∗Br [Nm ] = − Nm n+1 n+1 n+1 0 2 2n + 3 2 2μ + λ

.

By using the jump formula, one has cμ = −

.

nλ + (3n + 1)μ , (2n + 3)(2n + 1)(2μ + λ)

and thus K∗Br [Nm n+1 ] =

.

0

−3λ + 2μ(2n2 + 6n + 1) m N . 2(2μ + λ)(2n + 1)(2n + 3) n+1

m Note that when .n = 0, both .Tm n and .Mn vanish. By arranging the values of n, we complete the proof. ⨆ ⨅

90

3 Anomalous Localized Resonances and Their Cloaking Effect

Remark 3.1.2 Note that only the first eigenvalues .ξ1n in Theorem 3.1.1 associated with the eigenfunctions .Tm n converge to zero as n goes to infinity. These are the only possible spectra of the N-P operator .K∗Br which can induce cloaking due to 0 anomalous localised resonance.

3.1.4 Anomalous Localized Resonances and Their Cloaking Effect In this section, we shall present the main mathematical theory on cloaking due to anomalous localised resonance for the elastostatic system in the three dimensional case. In what follows, we let .Br denote the central ball of radius .r ∈ R+ . Let .0 < ri < re < +∞. Set .( λ,  μ) to be the Lamé constants in .Bri of the following form ( λ,  μ) = cn (λ, μ),

.

(3.1.54)

where the coefficient .cn > 0 depending on n will be specified later, and .(λ, μ) satisfies the convexity condition (3.1.2). Set .(λ˘ , μ) ˘ to be the Lamé constants in .Bre \Bri of the following form (λ˘ , μ) ˘ = (ϵn + iδ)(λ, μ),

.

(3.1.55)

where .ϵn < 0 depends on n and .δ > 0. Define two elastic tensors . C(x) = ˘ ( Cij kl (x))3i,j,k,l=1 and .C(x) = (C˘ ij kl (x))3i,j,k,l=1 as follows:  λ(x)δij δkl +  μ(x)(δik δj l + δil δj k ), x ∈ Bri , Cij kl (x) := 

(3.1.56)

C˘ ij kl (x) := λ˘ (x)δij δkl + μ(x)(δ ˘ ik δj l + δil δj k ), x ∈ Bre \Bri .

(3.1.57)

.

and .

˘ defined above, and .C defined in (3.1.1), we next introduce the elastic C and .C With . tensor .CB to be ˘ Br \Br + CχR3 \B , CB :=  CχBri + Cχ e re i

.

(3.1.58)

where .χ denotes the characteristic function. Associated with the elastic material tensor in (3.1.58), we consider the following transmission problem in elastostatics  .

ˆ δ = f in R3 , ∇ · CB ∇u uδ (x) = O(|x|−1 ) as |x| → ∞,

(3.1.59)

3.1 Elastostatic Problem

91

where .f is a source function compactly supported in .R3 \Bre and satisfies the condition (3.1.4). To simplify the notation, we denote by .Υi the boundary of .Bri , i.e., .∂Bri and .Υe for .∂Bre . The operator .∂ν i means taking the traction on the boundary .Υi (see (3.1.7) for the definition) and it is same for .∂ν e . The transmission problem is equivalent to the following system: ⎧ Lλ,μ uδ (x) = 0, in Bri , ⎪ ⎪ ⎪ ⎪ ⎪ in Bre \Bri , Lλ,μ uδ (x) = 0, ⎪ ⎪ ⎪ ⎪ in R3 \Bre , ⎨ Lλ,μ uδ (x) = f, . on Υi , uδ |− = uδ |+ , ⎪ ⎪ ⎪ ∂ u | = (ϵ + iδ)∂ u | , c n νi δ − n ν i δ + on Υi , ⎪ ⎪ ⎪ ⎪ on Υe , uδ |− = uδ |+ , ⎪ ⎪ ⎩ (ϵn + iδ)∂ν e uδ |− = ∂ν e uδ |+ , on Υe .

(3.1.60)

For analysis of anomalous localized resonance, we need consider the energy .Eδ defined in (3.1.9), which is related to the solution in (3.1.60). To that end, we define  .F(x) := G(x − y)f(y)dy, (3.1.61) R3

then the solution to the system (3.1.60) can be represented as follows: uδ (x) = SBri [ϕ i ] + SBre [ϕ e ] + F,

.

(3.1.62)

where .ϕ i ∈ L2 (Υi )3 and .ϕ e ∈ L2 (Υe )3 . By using the transmission conditions, we can obtain the following equations ⎧     ⎪ (ϵ + iδ)∂ S [ϕ ] − c ∂ S [ϕ ] ⎪  n ν i Bri n ν i Bri i i  +(ϵn − cn + iδ)∂ν i SBre [ϕ e ] ⎪ ⎪ − + ⎪ ⎪ ⎪ ⎪ ⎨ =(cn − ϵn − iδ)∂ν i F, .   ⎪   ⎪ ⎪ −(ϵn + iδ)∂ν e SBre [ϕ e ] + ∂ν e SBre [ϕ e ] +(1 − ϵn − iδ)∂ν e SBri [ϕ i ] ⎪ ⎪ − + ⎪ ⎪ ⎪ ⎩ =(ϵn − 1 + iδ)∂ν e F, (3.1.63) that hold on .Υi and .Υe , respectively. By using the jump formula (3.1.18) on .Υi and Υe respectively, the above equations can be rewritten as:

.

 .

−a1,δ + K∗Υi ∂ν i SBre ∂ν e SBri −a2,δ + K∗Υe



ϕi ϕe





 ∂ν i F =− , ∂ν e F

(3.1.64)

92

3 Anomalous Localized Resonances and Their Cloaking Effect

where a1,δ =

.

cn + ϵn + iδ 2(cn − ϵn − iδ)

and

a2,δ =

1 + ϵn + iδ . 2(−1 + ϵn + iδ)

(3.1.65)

By using interior and exterior vector spherical harmonic functions and direct calculations, one has that m .SBr [Tn (x)] 0

=

d1 T m (x), r0n−1 n d1 r0n+2 ∇(r −(n+1) Ynm ) × x,

|x| ≤ r0 , |x| > r0 ,

(3.1.66)

where d1 =

.

−1 . μ(2n + 1)

(3.1.67)

By (3.1.50) it is easily found that the traction of .T1m (x) along the surface of any sphere vanishes, namely .

∂ (T m (x)) = 0, ∂ν 1

(3.1.68)

∂F and hence in the following, we only consider the situation from .n ≥ 2. If . ∂ν and i .

∂F ∂ν e

are given as follows  .

∂F ∂ν i ∂F ∂ν e

 =

n  n,m  +∞   gi m n,m Tn (x), g e m=−n

(3.1.69)

n=2

by substituting (3.1.66) into (3.1.64) and with the help of theorem 3.1.1, one has that the solution to the system (3.1.64) can be represented as follows: ϕi =

n +∞  

ϕin,m Tm n (x),

n=2 m=−n .

ϕe =

n +∞   n=2 m=−n

(3.1.70) ϕen,m Tm n (x),

3.1 Elastostatic Problem

93

where ϕin,m = −

gin,m (ξ1n − a2,δ ) − d1 gen,m μ(n − 1)(ri /re )n−1 (ξ1n − a1,δ )(ξ1n − a2,δ ) + d12 μ2 (n − 1)(n + 2)(ri /re )2n+1

.

ϕen,m = −

gen,m (ξ1n − a1,δ ) + d1 gin,m μ(n + 2)(ri /re )n+2 (ξ1n − a1,δ )(ξ1n − a2,δ ) + d12 μ2 (n − 1)(n + 2)(ri /re )2n+1

, (3.1.71) ,

with .ξ1n given in (3.1.44). Since the source .f is located outside .Bre , one has that .F defined in (3.1.61) satisfies Lλ,μ F(x) = 0,

.

x ∈ Bre ,

(3.1.72)

Tnm (x) + c,

(3.1.73)

and .F(x) can be represented as follows: F(x) =

.

n ∞  

gen,m

μ(n − 1)ren−1 n=2 m=−n

for .|x| ≤ re , where .gen,m is given in (3.1.69). Thus one can conclude that gin,m = (ri /re )n−1 gen,m .

(3.1.74)

.

Substituting (3.1.74) into (3.1.71), we then get ϕin,m =

gen,m (a2,δ − ξ1n + d1 μ(n − 1))(ri /re )n−1 (ξ1n − a1,δ )(ξ1n − a2,δ ) + d12 μ2 (n − 1)(n + 2)(ri /re )2n+1

.

ϕen,m = −

,

gen,m (ξ1n − a1,δ + d1 μ(n + 2)(ri /re )2n+1 ) (ξ1n − a1,δ )(ξ1n − a2,δ ) + d12 μ2 (n − 1)(n + 2)(ri /re )2n+1

(3.1.75) .

If we define cn0 = (n0 + 2)2 /(n0 − 1)2 , .

ϵn0 = −1 − 3/(n0 − 1),

(3.1.76)

with .n0 chosen properly later, then the denominator of .ϕin0 ,m and .ϕen0 ,m has the following relationship |(ξ1n0 − a1,δ )(ξ1n0 − a2,δ ) + d12 μ2 (n0 − 1)(n0 + 2)(ri /re )2n0 +1 | ≈ δ 2 + (ri /re )2n0 . (3.1.77)

.

94

3 Anomalous Localized Resonances and Their Cloaking Effect

With the help of (3.1.66), one has that SBri [ϕ i ] + SBre [ϕ e ] =

n ∞  

.

n=2 m=−n

d1

 ren+2  n,m ϕi + ϕen,m Tm n (x) n+1 r

|x| > re , (3.1.78)

and SBri [ϕ i ] =

n ∞  

d1

rin+2  n,m  m ϕ Tn (x), r n+1 i

d1

r n  n,m  m ϕe Tn (x), n−1

n=2 m=−n .

SBre [ϕ e ] =

n ∞   n=2 m=−n

re

ri < |x| ≤ re , (3.1.79) ri < |x| ≤ re .

Thus when .ri < |x| ≤ re , we denote SBri [ϕ i ] + SBre [ϕ e ] = Gn0 + Gn0 ,

.

(3.1.80)

where Gn0 .

Gn0

 rin+2  n,m  r n  n,m  = d1 n+1 ϕi + n−1 ϕe Tm n (x), r re n=2,n/=n0 m=−n   n +2 n0  ri 0  n0 ,m  r n0  n0 ,m  Tm + n −1 ϕe = d1 n +1 ϕi n0 (x). 0 0 r re m=−n0 ∞ 

n 



(3.1.81)

 Since the energy . Br \Br |∇F|2 dx < ∞, the phenomenon of the first equation e i in (3.1.10) occurs if and only if E(uδ − F) = E(SBri [ϕ i ] + SBre [ϕ e ]) → ∞,

.

as

δ → 0.

(3.1.82)

From (3.1.80), one has that

.

E(SBri [ϕ i ] + SBre [ϕ e ])   ˆ ˆ =δ ∇(Gn0 ) : C∇(Gn0 )dx + Bre \Bri

 Bre \Bri

ˆ n0 ) : C∇(G ˆ n0 )dx ∇(G (3.1.83)

3.1 Elastostatic Problem

95

By direct calculation, though a bit tedious, one can conclude that  Bre \Bri

ˆ n0 )dx ˆ n0 ) : C∇(G ∇(G

∞ 

.

≤

n=2,n/=n0

n  |gen,m |2 n m=−n



n4 n2 + (n − n0 )2 (n − n0 )4



ri re

2n 

(3.1.84) ≤ C,

where C is independent of .δ. With the help of (3.1.77), one can obtain  .

Bre \Bri

n0 

ˆ n0 )dx ≈ ˆ n0 ) : C∇(G ∇(G

m=−n0

|gen0 ,m |2 . n0 (δ 2 + (ri /re )2n0 )

(3.1.85)

Finally, combining (3.1.83), (3.1.84) and (3.1.85), we have E(uδ ) ≈

n0 

.

m=−n0

δ|gen0 ,m |2 . n0 (δ 2 + (ri /re )2n0 )

(3.1.86)

We are now in the position of presenting the main theorem on the cloaking due to anomalous localised resonance result on the three dimensional elastostatic system. We define the critical radius by r∗ =

.

 re3 /ri .

(3.1.87)

Theorem 3.1.2 Let the elasticity tensor .CB be given (3.1.58) with .cn0 and .ϵn0 given in (3.1.76). If the source .f is supported in .re < |x| < r∗ . Then cloaking due to anomalous localised resonance occurs, namely, the condition (3.1.10) is satisfied. Moreover, if the source .f is supported outside .Br∗ , then resonance does not occur, namely .E(uδ ) < ∞. Proof We first prove the second condition in (3.1.10). In other words, .uδ (x) is bounded outside some region. From the expression (3.1.75) and approximation (3.1.77), one has that (ri /re )n + δ (ri /re )2n + δ 2  (ri /re )n n0 ,m + = Cge (ri /re )2n + δ 2

|ϕin0 ,m + ϕen0 ,m | ≤ Cgen0 ,m

.



1 (ri /re )2n δ

+δ

(3.1.88)

96

3 Anomalous Localized Resonances and Their Cloaking Effect

 ≤

Cgen0 ,m

≤ Cgen0 ,m

(ri /re )n + (ri /re )2n



1 (ri /re )2n δ

+δ

1 , (ri /re )n

where the constant C may change from one inequality to another. If .n /= n0 , one has that   n2 n n |ϕin,m + ϕen,m | ≤ Cgen,m + (r /r ) i e n − n0 (n − n0 )2 . (3.1.89) ≤ Cgen,m . From the expression (3.1.78) outside .Bre , one has that if .|x| > re2 /ri , n ∞  

|uδ (x)| ≤ |F| + C

.

d1 gen,m

n=2 m=−n

ren+2 1 ≤ C, n+1 (ri /re )n r

(3.1.90)

where the constant C depends only on the source .f. Thus the second condition in (3.1.10) is satisfied. Next we consider the energy .E(uδ ). Let .n0 be chosen such that (ri /re )n0 < δ ≤ (ri /re )n0 −1 ,

(3.1.91)

.

and hence from the expression (3.1.86), one has that n0 

E(uδ ) ≈

m=−n0

.

δ|gen0 ,m |2 n0 (δ 2 + (ri /re )2n0 )

n0  C |g n0 ,m |2 n0 (ri /re )n0 m=−n e

≥

(3.1.92)

0

1 ren0 ≥ C n0 ri n0 (2n0 + 1)



n0 

2 |gen0 ,m |

m=−n0

Since .f is supported inside .r∗ , the potential .F given in (3.1.73) can not converge at |x| = r∗ . Then the following holds

.

.

lim sup n→∞

 m=n 1/n  |gen,m | m=−n

nren−1

 > 1/

re3 , ri

(3.1.93)

3.1 Elastostatic Problem

97

namely,

.

lim sup

2

 m=n 

n→∞

|gen,m |

> Cn2

m=−n

rin . ren

(3.1.94)

Finally, we have that .

sup E(uδ ) → ∞ as

δ → 0.

(3.1.95)

If the source .f is supported outside the ball .Br∗ , then the potential .F given in (3.1.73) converges at .|x| = r∗ + τ , for sufficiently small .τ ∈ R+ . With .n0 again chosen in (3.1.91) and from the expression (3.1.86), one has that E(uδ ) ≤ C

.

n0  C |g n0 ,m |2 ≤ C||f||2L2 (R3 )3 . n0 (ri /re )n0 m=−n e

(3.1.96)

0

⨆ ⨅

The proof is complete.

3.1.5 Cloaking by Anomalous Localized Resonance on a Coated Structure in Two Dimensional Case We shall review the results for CALR on a coated structure in elastic system in two dimensional case, one can find more details in [20]. First, it is proved in [21] that elastic NP eigenvalues, with Lamé parameters .(λ, μ), on smooth boundaries of two-dimensional domains shall accumulate at either .k0 or .−k0 where k0 =

.

μ . 2(λ + 2μ)

(3.1.97)

First, consider a homogeneous structure. Let the pair of Lamé constants .(λ, μ) be ( λ,  μ) = (c + iδ)(λ, μ)

.

(3.1.98)

where c is a negative constant and .δ is a small positive constant representing dissipation. The asymptotics of NP eigenvalues are derived and it is proved (see [21]) that CALR occurs on ellipses if the constant c in (3.1.98) satisfies z(c) :=

.

c+1 = ±k0 . 2(1 − c)

(3.1.99)

98

3 Anomalous Localized Resonances and Their Cloaking Effect

The condition (3.1.99) can be satisfied only when c is negative since .0 < k0 < 1/2. The elastic NP eigenvalues on two dimensional disks are .

1 , 2

−

λ , 2(2μ + λ)

±k0 .

(3.1.100)

It is also showed that the first two of the above eigenvalues have multiplicity one, while the last two are of infinite multiplicities. The CALR phenomena does not occur around .±k0 , since they are not the accumulating points of elastic NP eigenvalues. In what follows, consider a coated structure, in which the Lamé parameters are given by  δ δ = . λ ,μ

⎧ ⎪ ⎪ ⎨(λ, μ),

in R2 \Bre ,

(c + iδ)(λ, μ), ⎪ ⎪ ⎩(λ, μ),

in Bre \Bri ,

(3.1.101)

in Bri .

Here c is a negative constant and .δ is a parameter such that .δ ⪡ 1. To analyze the CALR for elastic system, one common way is to use the layer potential technique in representing the solution of the system. Let us recall the definitions of layer potential and NP operator related to the Lamé system (see, for example, [15] ). With the pair of Lamé constants .(λ, μ) satisfying .μ > 0 and 2  .λ + μ > 0, the isotropic elasticity tensor .C = Cij kl is defined by i,j,k,l=1   Cij kl := λδij δkl + μ δik δj l + δil δj k .

.

Then the corresponding Lamé system of elasticity equations is defined to be Lλ,μ := ∇ · C∇ s where .∇ s is the symmetric gradient, namely,

.

∇ s u :=

.

 1 ∇u + ∇u⏉ 2

(⏉ for transpose ).

Let .Ω be a connected bounded domain with the Lipschitz boundary in .R2 . The single layer potential of the density function .ϕ on .∂Ω associated with the Lamé operator .Lλ,μ is defined by  SΩ [ϕ](x) :=

G(x − y)ϕ(y)ds(y),

.

x ∈ R2

∂Ω

 2 where ds is the arc length of .∂Ω and .G(x) = Gij (x) i,j =1 is the Kelvin matrix of the fundamental solution to the Lamé system in .R2 , namely, Gij (x) =

.

α2 xi xj α1 δij ln |x| − , 2π |x|2 2π

(3.1.102)

3.1 Elastostatic Problem

99

with 1 .α1 = 2



1 1 + μ 2μ + λ

 and

1 α2 = 2



 1 1 − . μ 2μ + λ

Let .H 1/2 (∂Ω)2 be the usual .L2 -Sobolev space of order .1/2 and .H −1/2 (∂Ω)2 be its dual space with respect to .L2 -pairing .〈·, ·〉. Let .Ψ be the subspace of .H −1/2 (∂Ω)2 spanned by .

  1 , 0

  0 , 1



y −x

 (3.1.103)

−1/2

Let .HΨ (∂Ω) be the collection of all .ϕ ∈ H −1/2 (∂Ω)2 such that .〈ϕ, f〉 = 0 for all .f ∈ Ψ . The .NP operator .K∗Ω (see (1.3.10)) is bounded on .H −1/2 (∂Ω)2 , and maps −1/2 .H (∂Ω) into itself. If .λ ∈ / [−1/2, 1/2), then .λI − K∗Ω is invertible on Ψ −1/2 −1/2 (∂Ω)2 (and on .H .H (∂Ω) ). Even if .K∗Ω is not self-adjoint with respect Ψ 2 to the usual .L -inner product, it can be realized as a self-adjoint operator on −1/2 .H (∂Ω). In fact, it is proved in [21] (following the discovery of [79]) that the Ψ product .(·, ·)∗ , defined by (ϕ, ψ)∗ := − 〈ϕ, SΩ [ψ]〉 ,

.

(3.1.104)

−1/2

is actually an inner product on .HΨ (∂Ω) which yields a norm equivalent to the usual .H −1/2 -norm. Then .K∗Ω is self-adjoint with respect to this new inner product due to the so called Plemelj’s symmetrization principle: SΩ K∗Ω = KΩ SΩ ,

.

where .KΩ is the .L2 -adjoint operator of .K∗Ω . It is well-known (see, for example, [48]) that .K∗Ω is not a compact operator on .H −1/2 (∂Ω) even if .∂Ω is smooth. However, it is proved in [21] that the spectrum of .K∗Ω consists of pure point spectrum converging to .±k0 , which is defined in (3.1.97). Let .Bri := {|x| < ri } and .Bre := {|x| < re }, .0 < ri < re , and write .Sri = ∂Bri and .Sre = ∂Bre . The distribution of Lamé parameters are given by (3.1.101), and the elasticity tensor 2  δ δ .C = C is given by ij kl i,j,k,l=1

  Cijδ kl := λδ δij δkl + μδ δik δj l + δil δj k .

.

(3.1.105)

100

3 Anomalous Localized Resonances and Their Cloaking Effect

With the parameter setup, one can now consider the following elasticity equation: ∇ · Cδ ∇ s uδ = f

.

in R2

(3.1.106)

 ⏉ with the decaying condition .uδ (x) → 0 as .|x| → ∞, where .uδ = uδ1 , uδ2 and the source .f is supposed to be compactly supported in .R2 \Bre which satisfies  .

R2

f = 0.

(3.1.107)

Let .uδ be the solution to (3.1.106) and define    2 2  δ := .E u λ ∇ · uδ  + 2μ ∇ s uδ  . Bre \Bri

(3.1.108)

    Here and afterwards, .A : B for two matrices .A = aij and .B = bij denotes  2 . i,j aij bij , and .|A| = A : A. To realize the so called CALR, the following two conditions should be satisfied: • Energy blow up on the annulus:   E δ := δE uδ → ∞

.

as δ → 0,

(3.1.109)

  • Find a radius .a > re such that .uδ (x) < C for some .C > 0 on .|x| > a as .δ → 0. It is worth mentioning that .E δ is the imaginary part of the total energy, namely,  E =ℑ

.

δ

R2

Cδ ∇ s uδ : ∇ s uδ

 2  2 2 −1/2  Sri × Let .H ∗ = H −1/2 Sri × H −1/2 Sre and .HΨ∗ := HΨ 2 −1/2  Sre . Let .F be the Newtonian potential of .f, that is .H Ψ  F(x) :=

.

R2

G(x − y)f(y)dy,

x ∈ R2 .

(3.1.110)

Note that .F satisfies .Lλ,μ F = f in .R2 and .F(x) → 0 as .|x| → ∞ since .f satisfies (3.1.107). Let .SBri and .SBre be the single layer potentials on .Sri and .Sre , respectively, with respect to the Lamé parameters .(λ, μ). Then the solution .uδ to (3.1.106) can be represented by



uδ (x) = F(x) + SBri ϕ δi (x) + SBre ϕ δe (x)

.

(3.1.111)

3.1 Elastostatic Problem

101

  for some . ϕ δi , ϕ δe ∈ HΨ∗ . By using the transmission conditions on .Sri and .Sre , one can obtain   (c + iδ)∂ν uδ + = ∂ν uδ − on Sri .   ∂ν uδ  = (c + iδ)∂ν uδ  on Sr +

−

e

By substituting the  integral  formulation to the above equations, one thus find out that the densities . ϕ δi , ϕ δe should satisfy the following system of integral equations: ⎧  

⎪ (c + iδ)∂ν i SBri ϕ δi  − ∂ν i SBri ϕ δi  + (c − 1 + iδ)∂ν i SBre ϕ δc ⎪ ⎪ ⎪ + − ⎨ = (1 − c − iδ)∂ F on Sri ν i . δ

δ   ⎪  − (c + iδ)∂ν SB ϕ δ  ϕ + ∂ ϕ (1 − c − iδ)∂ S S ⎪ ν B ν B e ri ε re e re e + e − ⎪ i ⎪ ⎩ = (c − 1 + iδ)∂νe F on Sre where .∂ν i and .∂ν e are the conormal derivatives on .Sri and .Sre , respectively. Using the jump formula (1.3.9), one then has  .

−zδ I + K∗Br

i

∂ν e SBri



∂ν i SBre zδ I + K∗Br

e

ϕ δi ϕ δϵ





∂ F = − νi ∂ν e F

 (3.1.112)

where zδ =

.

1 + c + iδ . 2(1 − c) − 2iδ

(3.1.113)

Next define the matrix operator .K∗ : H ∗ → H ∗ as  K∗ :=

.

−K∗Br −∂ν i SBre i

∂ν e SBri

 (3.1.114)

K∗Br

e

Then the Eq. (3.1.112) can be rewritten as .

 

zδ I + K∗ Φ δ = P,

(3.1.115)

where  ϕ δi , .Φ := ϕ δe 

δ



∂ν i F P := −∂ν e F



The operator .K∗ is the NP operator associated with two circles.

102

3 Anomalous Localized Resonances and Their Cloaking Effect

Next, define .A on .H ∗ as  A :=

.

SBri SBre SBri SBre

 (3.1.116)

Define, for .Φ, Ψ ∈ H ∗ , (Φ, Ψ )∗ := −〈Φ, A[Ψ ]〉

(3.1.117)

.

 2  2 where .〈·, ·〉 denotes the pairing of .L2 Sri × L2 Sre . Following the same arguments in subsection .3.3 of [9], one can show that .(·, ·)∗ is actually an inner product on .HΨ∗ , and .K∗ is a self-adjoint operator with respect to this inner product. Let .‖ · ‖∗ be the norm induced by the new inner product. One can show in the same way as in [21] that this norm is equivalent to the usual .H −1/2 -norm. Let .uδ be the solution to (3.1.106) given in the form (3.1.111). By using Green’s formula one has    . u · ∂ν vds = u · Lλ,μ v + λ(∇ · u)(∇ · v) + 2μ∇ s u : ∇ s v ∂Ω

Ω

Ω

one can see that  .

Bre \Bri

 

2  λ ∇ · SBri ϕ δi + SBre ϕ δe 

 

2  + 2μ ∇ s SBri ϕ δi + SBre ϕ δe  = Φ δ

Thus there are constants .C1 and .C2 such that      2 Φ δ ∗ − 1 ≤ E uδ ≤ C2 Φ δ .C1

2 ∗

 +1 .

2 . ∗

(3.1.118)

Next, we shall show the asymptotic results of the NP eigenvalues. First, let .Sr0 be the circle of radius .r0 . For an integer m let  ϕm =

.

cos mω sin mω



 and

 ϕm =

− sin mω cos mω

 (3.1.119)

The following computations are from [21]. For .x = (r cos ω, r sin ω), if .m = 1, then α1 −α2

r < r0 , 2 rϕ 1 (ω), ϕ (x) = (3.1.120) . − SBr 2 1 0 α1 −α2 r0 r ϕ 1 (ω), r > r0 . 2

3.1 Elastostatic Problem

103

If .m ≥ 2, then

.



ϕ m (x) =

− SBr0

⎧ ⎪ ⎨ α1



rm

α2 2

m−1 ϕ m (ω) +

2m r 0 ⎪ ⎩ α1 r0m+1 ϕ (ω), m 2m r m

r02 r

 −r

r m−1 ϕ (ω), r0m−1 −m+2

r < r0 r > r0 (3.1.121)

If .m ≤ −1, then

.



ϕ m (x) =

− SBr0

⎧ α1 r |m| ⎪ ⎨ 2|m| |m|−1 ϕ m (ω), r α1 ⎪ ⎩ 2|m|

0 |m|+1 r0 r |m|

ϕ m (ω) +

α2 2

 r−

r02 r



r < r0 |m|+1

r0 r|m|+1 ϕ −m+2 ,

r > r0 (3.1.122)

It is also shown that if .m ≥ 2, then ⎧ ⎪ rm ⎨ α1 m−1  ϕ m (ω) −

2m r 0  ϕ m (x) = . − SBr 0 m+1 ⎪ ⎩ α1 r0  2m r m ϕ m (ω),

 α2 2

r02 r

 −r

r m−1  (ω), ϕ r0m−1 −m+2

r < r0 , r > r0 . (3.1.123)

If .m ≤ −1, then

.



 ϕ m (x) =

− SBr0

⎧ α1 r |m| ⎪ ϕ m (ω), ⎨ 2|m| |m|−1  r α1 ⎪ ⎩ 2|m|

0 |m|+1 r0 r |m|

α2 2

 ϕ m (ω) −

 r−

r02 r



r < r0 , |m|+1

r0  ϕ (ω), r |m|+1 −m+2

r > r0 . (3.1.124)

As consequences of these computations, eigenvalues of .K∗Br are obtained in [21]: 0



ϕ m = −k0 ϕ m (m < 0),

K∗Br

.

0



ϕ m = −k0 ϕ m (m < 0) K∗Br 

(3.1.125)



K∗Br  ϕ m = k0  ϕ m (m ≥ 2)

(3.1.126)

0

and K∗Br

.

0



ϕ1 = −

λ ϕ , 2(2μ + λ) 1

0

104

3 Anomalous Localized Resonances and Their Cloaking Effect

For .n = 0, 1, 2, . . ., define  ϕ n+1 , 0   0 := ϕ −n+1 

Φ n,1 :=

.

Φ n,4

 Φ n,2 :=

 ϕ −n+1 , 0

 Φ n,3 :=

0

 ,

ϕ n+1

(3.1.127)

Here .0 denotes two-dimensional zero vector, and so .Φ n,j are four-dimensional vector-valued functions. Define finite dimensional subspaces .Vn of .H ∗ by " ! Vn := span Φ n,1 , Φ n,2 , Φ n,3 , Φ n,4 ,

.

n≥0

(3.1.128)

It is worth mentioning that .V0 is of two dimensions and spanned by .Φ 0,1 and .Φ 0,3 . Similarly, define    ϕ n+1  , .Φ n,1 := 0   0  Φ n,4 :=  ϕ −n+1

   ϕ −n+1  Φ n,2 := , 0

 n,3 := Φ



0  ϕ n+1

 , (3.1.129)

and define ! "  n,1 , Φ  n,2 , Φ  n,3 , Φ  n,4 , Vn = span Φ

n≥0

.

(3.1.130)

Using (3.1.120)–(3.1.124) in the previous subsection, one can see that A (Vn ) ⊂ Vn

and

.

  A Vn ⊂ Vn

for each n

(3.1.131)

For example, according to the definition (2.18), .A Φ n,1 is given by

 ⎤ SBri ϕ n+1  ⎢ Sr ⎥ =⎣

 i ⎦ SBri ϕ n+1  ⎡



A Φ n,1

.



Sre

So, using (3.1.120)–(3.1.122), one can see that .A Φ n,1 ∈ Vn . Moreover, one can derive the block matrix representing .−A on each .Vn or .Vn . In the following, that  4 .−A is represented by a block matrix . aij on .Vn means i,j =1

.

 − A Φ n,i = aij Φ n,j , j

In fact, they are given as follows. Set .ρ := ri /re .

i = 1, . . . , 4.

3.1 Elastostatic Problem

105

(i) On .Vn , n ≥ 2, ⎡

α1 2(n+1) ri

⎤

0

⎢ ⎢ 0 ⎢ .−A = ⎢ ⎢ α1 ri2 n ⎢ 2(n+1) re ρ ⎣ 0

α2 2

α1 ri 2(n−1)  r2 re − rie ρ n

α1 r ρn 0 2(n+1)  2 i  ⎥ α1 re2 n ⎥ α2 re n 2 ri − ri ρ 2(n−1) ri ρ ⎥ α1 2(n+1) re

0

0

α1 2(n−1) re

α1 n 2(n−1) re ρ

⎥. ⎥ ⎥ ⎦

(3.1.132) )n , n ≥ 2, (ii) On .V ⎡

α1 2(n+1) ri

0

⎤

⎢ α1 ⎢ 0 ⎢ 2(n−1) ri   .−A = ⎢ r2 ⎢ α1 ri2 n α2 ⎢ 2(n+1) re ρ − 2 re − rie ρ n ⎣ α1 n 0 2(n−1) re ρ

α1 r ρn 0 2(n+1)  2 i  re2 n ⎥ α α2 re 1 ⎥ − 2 ri − ri ρ n 2(n−1) ri ρ ⎥ α1 2(n+1) re

0

0

α1 2(n−1) re

⎥. ⎥ ⎥ ⎦

(3.1.133) As a consequence one can obtain the following lemma. Lemma 3.1.4 (i) It holds that .Vm ⊥ Vn and .Vm ⊥ Vn if .m /= n, and .Vm ⊥ Vn for all m and n, with respect to the inner product .(, ·). (ii) For .j = 1, . . . , 4, .

  1 Φ n,j , Φ n,j ∗ ≈ , n

   n,j , Φ  n,j ≈ 1 Φ ∗ n

(3.1.134)

    Φ  n,i , Φ  n,j  ≲ ρ n . ∗

(3.1.135)

(iii) If .i /= j , then .

    Φ n,i , Φ n,j  ≲ ρ n , ∗

  In the above and afterwards . Φ n,j , Φ n,j ∗ ≈ .C1 and .C2 independent of n such that .

1 n

mean that there are constants

 C1  C2 ≤ Φ n,j , Φ n,j ∗ ≤ n n

   and . Φ n,i , Φ n,j ∗  ≲ ρ n means that there is a constant C independent of n such that .

    Φ n,i , Φ n,j  ≤ Cρ n . ∗

106

3 Anomalous Localized Resonances and Their Cloaking Effect

∗  Now, one should represent the

NP operator .K on .Vn and .Vn . According to the ∗ definition (3.1.114), .K Φ n,1 , for examples, is given by



 ⎤ ∗ ϕ − K n+1  Bri

⎢ Sr ⎥ ∗ Φ n,1 = ⎣ .K

 i ⎦ ∂ν 2 SBri ϕ n+1  ⎡

Sre



So, using (3.1.121) and (3.1.125), one can compute .K∗ Φ n,1 . In this way, one can show that   K∗ (γn ) ⊂ Vn and K∗ Vn ⊂ Vn for each n.

.

(3.1.136)

One can also get the following results by straight forward computations. Lemma 3.1.5 The NP-operator .K∗ admits the following matrix representation: (i) On .Vn , n ≥ 2, K ∗ = J + Mn ,

.

(3.1.137)

where ⎡

−1 ⎢ 0 .J = k0 ⎢ ⎣ 0 0

0 1 0 0

⎤ 0 0 0 0 ⎥ ⎥ 1 0 ⎦ 0 −1

and ⎡

⎤ 0 0 0 μα2   ⎢ 0 0 −(n − 1)μα2 1 − ρ −2 μα1 ρ −2 ⎥ n ⎥.   .Mn = ρ ⎢ ⎣ μα1 ρ 2 −(n + 1)μα2 ρ 2 − 1 0 0 ⎦ 0 0 0 μα2 )n , n ≥ 2 : (ii) On .V n , K∗ = J + M

.

where ⎡

⎤ 0 0 0 μα2   ⎢ 0 (n − 1)μα2 1 − ρ −2 μα1 ρ −2 ⎥ ⎥ n = ρ n ⎢ 0   .M ⎣ μα1 ρ 2 (n + 1)μα2 ρ 2 − 1 ⎦ 0 0 0 0 0 μα2

3.1 Elastostatic Problem

107

It turns out that the exact expression of the eigenvalues of .K∗ on .Vn , or of the matrix .J + Mn in (3.1.137), is extremely lengthy and complicated. However, sometimes it is enough to have their asymptotic behavior as .n → ∞, which we obtain in this part using the perturbation theory. To investigate the asymptotic behavior of the eigenvalues, it is more convenient to express .K∗ = J + Mn on .Vn as follows: K∗ = Pn + ρ n Q,

.

where ⎤ −k0 0 0 0   ⎢ 0 −nρ n μα2 1 − ρ −2 0 ⎥ k0 ⎥   .Pn = ⎢ ⎣ 0 −nρ n μα2 ρ 2 − 1 k0 0 ⎦ 0 0 0 −k0 ⎡

and ⎤ 0 0 0 μα  2 −2  ⎢ 0 μα1 ρ −2 ⎥ 0 μα2 1 − ρ ⎥   .Q = ⎢ ⎦ ⎣ μα1 ρ 2 −μα2 ρ 2 − 1 0 0 0 0 0 μα2 ⎡

It is worth emphasizing that Q is independent of n. It is easy to find exact eigenvalues and eigenfunctions of .Pn . In fact, they are given as follows: λ0n,1 = −k0 ,

.

λ0n,2 λ0n,3

E0n,1 = [1, 0, 0, 0]⏉ , .

  = +k0 − nρ n μα2 ρ − ρ −1 , E0n,2 = [0, 1, ρ, 0]⏉ , .   = +k0 + nρ n μα2 ρ − ρ −1 , E0n,3 = [0, −1, ρ, 0]⏉ , .

λ0n,4 = −k0 ,

E0n,4 = [0, 0, 0, 1]⏉ .

(3.1.138) (3.1.139) (3.1.140) (3.1.141)

Here and afterwards, .E0n,j written as a vector in .R4 actually represents a four dimensional vector-valued function. For example, .E0n,2 in (3.1.139) represents .Φ n,2 + ρΦ n,3 . Note that, if n is large, the matrix .ρ n Q becomes a small perturbation matrix. So one can derive asymptotic formula for eigenvalues using standard arguments of the eigenvalue perturbation theory (see, for example, section XII of [117]). For the cases of .λ01 = λ04 , one can apply the degenerate perturbation theory.

108

3 Anomalous Localized Resonances and Their Cloaking Effect

Let .λn,j and .En,j (n ≥ 1, j = 1, 2, 3, 4) be the eigenvalues and eigenvectors of K∗ , respectively. Then

.

λn,2

    μ2 α1 α2 ρ 4   + O ρ 3n , En,1 = E0n,1 + O nρ n , . k0 1 + ρ 2 (3.1.142)       = +k0 − nρ n μα2 ρ − ρ −1 + O ρ n , En,2 = E0n,2 + O ρ n , .

λn,3

(3.1.143)       = +k0 + nρ n μα2 ρ − ρ −1 + O ρ n , En,3 = E0n,3 + O ρ n , .

λn,1 = −k0 − ρ 2n

.

(3.1.144) 



  μ2 α1 α2 ρ −2   + O ρ 3n , En,4 = E0n,4 + O nρ n . 2 k0 1 + ρ (3.1.145)

λn,4 = −k0 − ρ 2n

Similarly, one can also obtain the asymptotic behavior of eigenvalues of .K∗ on n as follows .V   μ2 α1 α2 ρ 4   + O ρ 3n , . k0 1 + ρ 2     = +k0 + nρ n μα2 ρ − ρ −1 + O ρ n , .     = +k0 − nρ n μα2 ρ − ρ −1 + O ρ n , .

2n  .λn,1 = −k0 − ρ

(3.1.146)

 λn,2

(3.1.147)

 λn,3

 λn,4 = −k0 − ρ 2n

  μ2 α1 α2 ρ −2   + O ρ 3n , k0 1 + ρ 2

(3.1.148) (3.1.149)

and the corresponding eigenvectors are    En,1 = [1, 0, 0, 0]⏉ + O nρ n , .    En,2 = [0, 1, ρ, 0]⏉ + O ρ n , .    En,3 = [0, −1, ρ, 0]⏉ + O ρ n , .    En,4 = [0, 0, 0, 1]⏉ + O nρ n .

.

(3.1.150) (3.1.151) (3.1.152) (3.1.153)

Note that all eigenvalues in (3.1.142)–(3.1.149) converge to either .k0 or .−k0 .

3.1 Elastostatic Problem

109

In view of (3.1.134), (3.1.134), (3.1.134)–(3.1.134), and (3.1.134)–(3.1.134), one can obtain the following results.      En,j , En,j ∗ ≈ n−1 , En,j ,  En,j ∗ ≈ n−1 for 1 ≤ j ≤ 4, . (3.1.154)        n,j ∗ ≈ n−1 for j = 1, 4, . | En,j , Φ n,j ∗ ≈ n−1 ,   En,j , Φ (3.1.155)         En,j , Φ n,k · ≲ ρ n ,    n,k · ≲ ρ n for j = 1, 4, k /= j, . En,j , Φ (3.1.156)        n,k · ≈ n−1 for j = 2, 3, k = 2, 3, . En,j , Φ | En,j , Φ n,k ∗ ≈ n−1 ,   (3.1.157)       n n  En,j , Φ n,k · ≲ ρ ,    n,k · ≲ ρ En,j , Φ for j = 2, 3, k = 1, 4. (3.1.158) .

Now, consider the integral Eq. (3.1.115). Since .P ∈ HΨ∗ , P is orthogonal to .V0 in particular. So .P is uniquely represented as P=

∞ 

.

Pn +

n=0

∞ 

 Pn ,

n . Pn ∈ Vn ,  Pn ∈ V

n=1

It may be helpful to mention that .P1 has no component of .Φ 1,2 and .Φ 1,4 since .P is perpendicular to constant vectors. The solution .Φ δ to (3.1.115) is given by Φδ =

∞ 

.

n=0

Φ δn +

∞ 

 δn , Φ

(3.1.159)

n=1

nδ ∈ Vn are, respectively, the solutions to the finite dimensional where .Φ δn ∈ Vn and .Φ equations .

  zδ I + K∗ Φ δn = Pn on Vn ,

 δ  n =  zδ I + K ∗ Φ Pn on Vn .

(3.1.160)

Since .En,j , 1 ≤ j ≤ 4, is an orthogonal basis for .Vn , there holds δ .Φn

=

4  j =1



 Pn , En,j ∗     En,j , En,j , En,j ∗ zδ + λn,j

and hence

.

2 Φ δn ∗

=

4  j =1

    Pn , En,j 2 ∗ 2 . 2 En,j ∗ zδ + λn,j 

(3.1.161)

110

3 Anomalous Localized Resonances and Their Cloaking Effect

Note that .zδ → z(c) as .δ → 0 where .z(c) is defined in (3.1.99). On the other hand, .λn,j approaches to either .k0 or .−k0 as .n → ∞. So, if .z(c) = / ±k0 , then   .zδ + λn,j  ≥ C for some constant C for all sufficiently large n if .δ is small. So the norm given in (3.1.161) does not blow up. So, one can assume that .z(c) is either .k0 or .−k0 . Suppose that .z(c) = k0 . If .j = 2, 3, then we infer from (3.1.143) and (3.1.144) that .| zδ + λn,j |≥ C for some constant C independent of n. Thus, there holds  .

j =2,3

    Pn , En,j 2 2 ∗ 2 ≲ ‖Pn ‖∗∗ 2   En,j ∗ zδ + λn,j

(3.1.162)

If .j = 1, 4, then one can infer from (3.1.142) and (3.1.145) that .

  zδ + λn,j 2 ≈ δ 2 + ρ 4n ,

(3.1.163)

and hence        Pn , En,j 2  Pn , En,j 2  ∗ ∗ 2 ≈  2  2 2 4n  En,j ∗ zδ + λn,j j =1,4 En,j ∗ δ + ρ

 .

j =1,4

(3.1.164)

n , one then has Since .Pn ∈ Vn and  .Pn ∈ V Pn =

4 

.

k=1

pn,k Φ n,k ,

 Pn =

4 

 n,k , p n,k Φ

(3.1.165)

k=1

    4 for some constants .pn,k and .p n,k . Since . Pn , En,1 ∗ = k=1 pn,k Φ n,k , En,1 ∗ , one can see from (3.1.154)–(3.1.158) that   pn,1 2 .

n2

−ρ

2n

  pn,1 2   2    2 pn,k  ≲  Pn , En,1  ≲ pn,k 2 . + ρ 2n ∗ 2 n k/=1

k/=1

(3.1.166) Likewise, one has   pn,4 2 .

n2

−ρ

2n

  pn,4 2    2   2 pn,k 2 . pn,k  ≲  Pn , En,4  ≲ + ρ 2n ∗ 2 n k/=4

k/=4

(3.1.167)

3.1 Elastostatic Problem

111

It then follows from (3.1.154)–(3.1.158) and (3.1.164)–(3.1.167) that   2  2  2  2  n−1 pn,1  + pn,4  − nρ 2n pn,2  + pn,3  δ 2 + ρ 4n

.

 j =1,4

≲

  2  2  2  2     n−1 pn,1  + pn,4  + nρ 2n pn,2  + pn,3   Pn , En,j 2 ∗ . 2 ≲ 2  δ 2 + ρ 4n En,j ∗ zδ + λn,j  (3.1.168)

So, by (3.1.161), (3.1.162), and (3.1.168) there holds   2  2  2  2  n−1 pn,1  + pn,4  − nρ 2n pn,2  + pn,3 

− ‖Pn ‖2∗ ≲ δ 2 + ρ 4n   2  2  2  2  n−1 pn,1  + pn,4  + nρ 2n pn,2  + pn,3  ≲ + ‖Pn ‖2∗∗ δ 2 + ρ 4n (3.1.169)

.

Φ δn

2 ∗

 δn One can estimate . Φ

2 ∗

in a similar way to obtain

  2  2  2  2  gn,1  + p n,2  + p n,4  − nρ 2n p n,3  n−1  δ2

.

2  δn Φ ∗

≲

+ ρ 4n

Pn − 

2 ∗

≲

  2  2  2  2  n,1  + p n,2  + p n,4  + nρ 2n p n,3  n−1 p δ 2 + ρ 4n

Pn + 

2 ∗

(3.1.170) Let, for ease of notation,  2  2  2  2 n,1  + p n,4  In := pn,1  + pn,4  + p

(3.1.171)

 2  2  2  2 n,2  + p n,3  . I In := pn,2  + pn,3  + p

(3.1.172)

.

and .

Since ∞  .

n=0

‖Pn ‖2∗ +

∞  n=1

 Pn

2 ∗

= ‖P‖2∗ ,

112

3 Anomalous Localized Resonances and Their Cloaking Effect

by summing (3.1.169) and (3.1.170) over n that there are constants .C1 and .C2 independent of .δ such that  C1

.

∞  n−1 In − nρ 2n I In n=1

δ 2 + ρ 4n

 2 Φδ ∗

−1 ≤

≤ C2

∞  n−1 In + nρ 2n I In δ 2 + ρ 4n

n=1

 +1

(3.1.173) If .z(c) = −k0 , then in a similar way there holds C1

.



∞  n−1 I In − nρ 2n In n=1

−1 ≤

δ 2 + n2 ρ 2n

2 Φδ ∗

≤ C2

∞  n−1 I In + nρ 2n In δ 2 + n2 ρ 2n

n=1

 +1 .

(3.1.174)   It is worth emphasizing that the quantity .n δ 2 + n2 ρ 2n in the denominator is different from that in (3.1.173). This discrepancy is caused by the different asymptotic behaviors of eigenvalues near .k0 and .−k0 as shown in (3.1.142)– (3.1.149). To sum up, one can obtain the following proposition from (3.1.118), (3.1.173) and (3.1.174). Proposition 3.1.1 There are constants .C1 and .C2 independent of .δ such that (i) if .z(c) = k0 , then C1

.

∞  n−1 In − nρ 2n I In n=1

≤ C2

δ2

+ ρ 4n



  − 1 ≤ E uδ

∞  n−1 In + nρ 2n I In n=1

δ 2 + ρ 4n

 +1 ,

(3.1.175)

(ii) if .z(c) = −k0 , then C1

.

∞  n−1 I In − nρ 2n In n=1

≤ C2

δ2

+ n2 ρ 2n



  − 1 ≤ E uδ

∞  n−1 I In + nρ 2n In n=1

δ 2 + n2 ρ 2n

 +1

(3.1.176)

3.1 Elastostatic Problem

113

Now assume that the source function .f in (3.1.106) is given by the dipole-type function, namely, f(x) = b⏉ ∇x



.

  δz (x) 0 a 0 δz (x)

(3.1.177)

where .z = (z1 , z2 )⏉ is the location of the dipole outside .Bre , and .a = (a1 , a2 )⏉ , b = (b1 , b2 )⏉ are constant vectors. With this source function one obtain the following theorem using Proposition 3.1.1. Theorem 3.1.3 Let .f be given by (3.1.177), and let .ρz = re /|z|. It holds that (i) if .z(c) = k0 , then ln ρz   E uδ ∼ | ln δ|3 δ ln ρ −2

.

as δ → 0

(3.1.178)

(ii) if .z(c) = −k0 , then 2 ln ρz 2 ln ρz   E uδ ∼ | ln δ|−1− ln ρ δ ln ρ −2

.

as

δ→0

(3.1.179)

As an immediate consequence we obtain the following corollary. Corollary 3.1.1 Suppose that .f is given by (3.1.177). (i) If .z(c) = k0 , let .r∗ := re2 /ri . As .δ → 0,  δ ∞ .δE u → 0 (ii) If .z(c) = −k0 , let .r∗∗ :=

*

if re < |z| ≤ r∗ if |z| > r∗

(3.1.180)

rc3 /ri . As .δ → 0,

 δ ∞ → .δE u 0

if re < |z| < r∗∗ if |z| ≥ r∗∗

(3.1.181)

Recall that



uδ = F + SBri ϕ δi + SBre ϕ δe .

.

Next, one should show that .

 δ  u (x) ≤ C,

|x| = r > r0 ,

(3.1.182)

114

3 Anomalous Localized Resonances and Their Cloaking Effect

for some .r0 > re . Note that the above boundedness means that CALR happens, since it is already shown the energy .E δ blows up when a point source located inside the circle radius.



One shall only consider .SBre ϕ δe for simplicity. The potential .SBri ϕ δi can be estimated in a similar way. Suppose the solutions of the equations (3.1.160) are .Φ n,j  n,j . and .Φ Φ δn =

4 

.

j =0

pn,j Φ n,j , zδ + λn,j

 φn = Φ

4  j =1

p n,j  n,j , Φ zδ +  λn,j

where .pn,j and .p n,j are given by (3.1.165). Then, from (3.1.159), one has ϕ δe = .

∞   pn,3 ϕ n+1 n=0

zδ + λn,3

+

p n,3 ϕ n+1 zδ +  λn,3

+ +

∞   pn,4 ϕ −n+1 n=1

zδ + λn,4

+

p n,4 ϕ −n+1 zδ +  λn,4

+

ϕ δe,3 + ϕ δe,4 +  ϕ δe,4 . =: ϕ δe,3 +  Hence, there holds



δ



δ

ϕ e,3 + SBre ϕ δe,4 + SBre  ϕ e,4 . SBre ϕ δe = SBre ϕ δe,3 + SBre 

.

(3.1.183)

, Suppose .z(c) = k0 . First one should estimate .SBre ϕ δe,4 in (3.1.183). The other terms can be estimated in the same manner. From (3.1.122), one has   2 2n   ∞  ∞   δ   pn,4  ren 1 n−1 2 re   ϕ e,4 (x) ≲ ≲ n . . SBr e ri r|z| δ + ρ 2n r n−1 n=1

(3.1.184)

n=1

 , -    Therefore, for .|x| = r > re3 /ri2 , we have .SBre ϕ δe,4 (x) < C for some .C > 0. .z(c) = −k0 . For the same reason, one can only estimate Next, suppose , δ ϕ .SBr . From (3.1.121), there holds e,3 e   n   ∞  ∞   δ   pn,3  ren 1 n−1 1 re3 ≲ . SBr ≲ . ϕ (x) e,3 e δ + nρ n r n−1 n ri r|z| n=1

(3.1.185)

n=1

 , -    Therefore, for .|x| = r > re2 /ri , one has .SBre ϕ δe,3 (x) < C for some .C > 0. Since .F(x) → 0 as .|x| → ∞, (3.1.184) and (3.1.185) together with Theorem 3.1.3 yield the following theorem. Theorem 3.1.4 Suppose that .f be given by (3.1.177).

3.2 Electrostatic Problem

115

2 (i) If .z(c) = k0 , then CALR  δ  occurs with the critical radius .r∗ = re /ri ; more δ precisely, .E = δE u → ∞ as .δ → 0 if .re < |z| ≤ r∗ and there is a constant C such that

 δ  u (x) < C

.

for |x| = r > re3 /ri2

as δ → 0.

* (ii) If .z(c) = −k0 , then CALR occurs with the critical madius .r∗∗ = re3 /ri ; more precisely, .E δ → ∞ as .δ → 0 if .re < |z| < r∗∗ and there is a constant C such that .

 δ  u (x) < C

for |x| = r > re2 /ri

as δ → 0.

3.2 Electrostatic Problem This part is a collection of important results for analysis of cloaking due to anomalous localized resonance (CALR) in electrostatics problem [10]. For the sake of completeness, we would like to introduce the CALR techniques which started from electro static case.

3.2.1 Background We shall first introduce the mathematical formulation. Suppose that .Ω is a bounded domain in .Rd , d = 2, 3. Let D be a domain whose closure is contained in .Ω. Given a loss parameter .δ > 0, let the permittivity distribution in .Rd be given by ⎧ ⎪ ⎪ ⎨1 .ϵδ = ϵs + iδ ⎪ ⎪ ⎩ϵ c

¯ in Rd \Ω, ¯ in Ω\D, in D,

where .−ϵs and .ϵc are positive. The configuration of parameter is designed by a core with permittivity .ϵc coated by the shell .Ω\D¯ with permittivity .ϵs + iδ. Let the function f be compactly supported in .Rd \Ω which satisfies  .

R2

f dx = 0.

Now consider the following problem ∇ · ϵδ ∇Vδ = f

.

in Rd ,

(3.2.1)

116

3 Anomalous Localized Resonances and Their Cloaking Effect

together with the decay condition .Vδ (x) → 0 as .|x| → ∞. The problem of cloaking by anomalous localized resonance (CALR) can be formulated as the problem of identifying the sources f such that there holds  δ |∇Vδ |2 dx → ∞

Eδ :=

.

as δ → 0

(3.2.2)

Ω\D

√ and moreover, .Vδ / Eδ goes to zero outside some radius a, that is, .

 *   Vδ (x)/ Eδ  → 0

as δ → 0

when |x| > a.

(3.2.3)

Physically the quantity .Eδ is proportional to the electro-magnetic power dissipated into heat by the time harmonic electrical field averaged over time. By making use of the integration by parts, there holds  Eδ = ℑ

.

Rd

 (ϵδ ∇Vδ ) · ∇Vδ dx = −ℑ

Rd

f Vδ dx

which equates the power dissipated into heat with the electromagnetic power produced by the source. Here .Vδ denotes the complex conjugate of .Vδ . Hence .(3.2.2) implies an infinite amount of energy dissipated per unit time √ in the limit .δ → 0 which is unphysical. If we rescale the source f by a factor of .1/ Eδ then the source will√produce the same power independent of .δ and the new associated potential .Vδ / Eδ will, by .(3.2.3), approach zero outside the radius a, in which case the cloaking due to anomalous localized resonance (CALR) occurs. In [9], the authors develop a spectral approach to analyze the CALR phenomenon. In particular, they show that if D and .Ω are concentric disks in .R2 and .ϵc = −ϵs = 1, then there is a critical radius .r∗ such that for any source f supported outside .r∗ CALR does not occur, and for sources f satisfying a mild condition CALR takes place. The critical radius .r∗ is given by r∗ =

.

 re3 /ri ,

(3.2.4)

where .re and .ri are the radii of .Ω and D, respectively. It turns out that the occurrence of CALR depends on the distribution of eigenvalues of the Neumann-Poincaré (NP) operator associated with the structure. The NP operator is compact with its eigenvalues accumulating towards 0 which is different with the NP type operator in Elastic system. It is proved in [9] that in two dimensions the NP operator associated with the circular structure has the eigenvalues .±ρ n for .n = 1, 2, . . ., where .ρ = ri /re . The exponential convergence of the eigenvalues in two dimensions is responsible for the occurrence of CALR and the slow convergence (at the rate .1/n ) in three dimensions is responsible for the non-occurrence if the parameter choice does not depend on the convergence rate (the number n).

3.2 Electrostatic Problem

117

3.2.2 Layer Potential Formulation and Spectral Theory of a Neumann-Poincaré-Type Operator To better describe the CALR effect, one should first introduce the layer potential techniques in representing of the solution to the system (3.2.1). Let .F be the Newtonian potential of f , i.e.,  F (x) =

.

Rd

G(x − y)f (y)dy,

x ∈ Rd .

Then .F satisfies .ΔF = f in .Rd , and the solution .Vδ to .(3.2.1) may be represented as Vδ (x) = F (x) + SD [ϕi ] (x) + SΩ [ϕe ] (x)

.

 for some density functions .ϕi ∈ L20 (Γi ) and .ϕe ∈ L20 (Γe ) L20 is the collection of all square integrable functions with zero mean-value). By transmission conditions along the interfaces .Γe and .Γi one can derive that   ∂Vδ  ∂Vδ  = ϵc (ϵs + iδ) ∂ν + ∂ν −   ∂Vδ  ∂Vδ  . = (ϵs + iδ) ∂ν + ∂ν −

on Γi , on Γe .

Hence the pair of potentials .(ϕi , ϕe ) is the solution to the following system of integral equations:   ⎧ ∂SD [ϕi ]  ∂SD [ϕi ]  ∂SΩ [ϕe ] ⎪ + iδ) − ϵ (ϵ  ⎪ s c ∂ν i ∂ν i − + (ϵs − ϵc + iδ) ∂ν i ⎪ ⎪ + ⎪ ⎨ = (−ϵs + ϵc − iδ) ∂F on Γ , ∂ν i  i  . ⎪ (−1 + ϵs + iδ) ∂SD [ϕi ] − ∂SΩ [ϕe ]  + (ϵs + iδ) ∂SΩ [ϕe ]  ⎪ ⎪ ∂ν e ∂ν e ∂ν e ⎪ + − ⎪ ⎩ = (1 − ϵs − iδ) ∂F on Γ . e ∂ν e Note that the notation .ν i and .ν e are used to indicate the outward normal on .Γi and Γe , respectively. Using the jump formula for the normal derivative of the single layer potentials, the above equations can be rewritten as

.

 .

ziδ I − KD∗ − ∂ν∂ i SΩ ∂ ∗ δ ∂ν e SD ze I + KΩ



ϕi ϕe



 =

∂F ∂ν i − ∂F ∂ν e

 (3.2.5)

118

3 Anomalous Localized Resonances and Their Cloaking Effect

on .H0 = L20 (Γi ) × L20 (Γe ), where ziδ =

.

ϵc + ϵs + iδ , 2 (ϵc − ϵs − iδ)

zeδ =

1 + ϵs + iδ . 2 (1 − ϵs − iδ)

(3.2.6)

Let .H = L2 (Γi ) × L2 (Γe ) and let the Neumann-Poincar-type (NPt) operator ∗ .K : H → H be defined by  K∗ :=

.

−KD∗ − ∂ν∂ i SΩ ∂ KΩ∗ ∂ν e SD

 ,

and let 



 ϕi .Φ := , ϕe

g :=

∂F ∂ν i − ∂F ∂ν e

 .

(3.2.7)

Then, .(3.2.5) can be rewritten in the form .

δ  I + K∗ Φ = g,

(3.2.8)

where the matrix type operator .Iδ is given by  Iδ =

.

 ziδ I 0 . 0 zeδ I

The eigenvalues of the aforementioned NPt operator is essential in analysis of CALR. It is proved in [9] that for arbitrary-shaped domains .Ω and D the spectrum of the NPt operator .K∗ lies in .[−1/2, 1/2], and if .Ω and D are concentric disks, the eigenvalues of .K∗ on .H0 are .±ρ n /2, .n = 1, 2, . . .. We shall review the eigenvalues of .K∗ on .H when .Ω and D are concentric disks or balls. First, let .Sr0 = {|x| = r0 } in two dimensions. It is known that for each integer .n /= 0 SBr0

.

⎧   ⎨− r0 r |n| einθ , 2|n| r0 einθ (x) = ⎩− r0  r0 |n| einθ 2|n| r

if |x| = r < r0 , if |x| = r > r0 .

(3.2.9)

Moreover, KB∗r

,

.

0

einθ = 0 ∀n /= 0,

(3.2.10)

3.2 Electrostatic Problem

119

and KBr0 [1] =

.

1 . 2

(3.2.11)

In other words, .KBr0 is a rank 1 operator whose only non-zero eigenvalue is .1/2. Using .(3.2.10), it is proved that eigenvalues of .K∗ on .H0 are .±ρ 2 /2 (see [9]). It can be shown that .±1/2 are also eigenvalues of .K∗ on .H0 . These eigenvalues are of interest in relation to estimation of stress concentration [11]. In fact, by using .(3.2.11) one can obtain SBr0 [1](x) =

.

log r0 log |x|

if |x| = r < r0 , if |x| = r > r0 ,

and hence ∂ 0 . SBr0 [1](x) = 1 ∂r r

if |x| = r < r0 , if |x| = r > r0 .

It then follows that    1   −2 0 a a = , .K 1 1 b b re 2 ∗

where a and b are constants. So .±1/2 are eigenvalues of .K∗ . To sum up, one has the following results: Proposition 3.2.1 The eigenvalues of .K∗ defined on concentric circles in two dimensions and .

1 1 1 1 − , , − ρ n , ρ n , n = 1, 2, . . . , 2 2 2 2

and corresponding eigenfunctions are  .

1 − r1e

      ±inθ  e±inθ e 0 , , , n = 1, 2, . . . , ±inθ 1 ρe −ρe±inθ

120

3 Anomalous Localized Resonances and Their Cloaking Effect

Next, for three dimensional problem, let .Ynm (ˆx)(m = −n, −n + 1, . . . , 0, 1, . . . , n) x be the spherical harmonics of degree n and order m. Here .xˆ = |x| . Then .|x|n Ynm (ˆx) is harmonic in .R3 . Lemma 3.2.1 Let .Sr0 = {|x| = r0 } in three dimensions. We have for .n = 0, 1, . . . KB∗r

.

0

m

Yn (x) =

1 Y m (ˆx), 2(2n + 1) n

|x| = r0 ,

m = −n, . . . , n.

(3.2.12)

Lemma 3.2.1 says that the eigenvalues of .KB∗r on .L2 (Sr0 ) when .Sr0 is a sphere are .

1 2(2n+1) , n

0

= 0, 1, . . ., and their multiplicities are .2n + 1. By direct computations,

∂ n . SΩ Ynm (x) = − 2n + 1 ∂ν i



ri re

n−1 Ynm (ˆx),

|x| = ri .

Similarly, one also has

SD Ynm (x) = −

.

1 rin+2 m Y (ˆx), 2n + 1 r n+1 n

for |x| = r ≥ ri ,

and hence .



∂ n+1 SD Ynm (x) = 2n + 1 ∂ν e



ri re

n+2 Ynm (ˆx),

|x| = re .

(3.2.13)

To sum up, one has for constants a and b K∗

.



aYnm bYnm



⎡

⎤  a n − 2(2n+1) ρ n−1 Ynm + b 2n+1 ⎦  = ⎣  n+1 b a 2n+1 ρ n+2 + 2(2n+1) Ynm    n 1 n−1  − 2(2n+1) ρ aYnm 2n+1 = n+1 n+2 . 1 bYnm 2n+1 ρ 2(2n+1)

Thus one has following result regarding the eigenvalues of NPt operator. Proposition 3.2.2 The eigenvalues of .K∗ defined on two concentric spheres are .

±

 1 1 + 4n(n + 1)ρ 2n+1 , 2(2n + 1)

n = 0, 1, . . . ,

3.2 Electrostatic Problem

121

and corresponding eigenfunctions are

.

   * 1 + 4n(n + 1)ρ 2n+1 − 1 Ynm 2(n + 1)ρ n+2 Ynm

,

  *  − 1 + 4n(n + 1)ρ 2n+1 − 1 Ynm

, m = −n, . . . , n,

2(n + 1)ρ n+2 Ynm respectively.

Again, we mention that the spectral properties of the NPt operator arising in the related system are essential in the analysis of CALR.

3.2.3 Analysis of Cloaking Due to Anomalous Localized Resonance By using the spectral properties in the last section, it is easy to observe that if we let 1 2 = λ0 ≥ λ1 ≥ . . . be the eigenvalues of .KBr0 for a disk or a sphere enumerated according to their multiplicities, then the eigenvalues .μn of .K∗ satisfy

.

  μn = ±λn + O ρ n .

.

Let us first consider that the domains .Ω and D are concentric disks. Observe that .ziδ and .zeδ converges to non-zero numbers as .δ tends to 0 if .ϵc /= −ϵs /= 1. So, in this case CALR does not occur regardless of the location of the source. Furthermore, if .ϵc = ϵs = 1, a thorough study was done in [9]. It is proved in [9] that if the source * f is supported inside the critical radius .r∗ = re3 /ri , then the weak CALR occurs, namely, .

lim sup Eδ = ∞. δ→0

Moreover, if .F is the Newtonian potential of f and the Fourier coefficients .gen of ∂F .− ∂ν e satisfies the following gap property: [GP] There exists a sequence .{nk } with .|n1 | < |n2 | < · · · such that

.

lim ρ |nk+1 |−|nk |

k→∞

 nk 2 g e  |nk | ρ |nk |

= ∞,

122

3 Anomalous Localized Resonances and Their Cloaking Effect

then CALR occurs, namely .

lim Eδ = ∞,

δ→0

* √ and .Vδ / Eδ goes to zero outside the radius . re3 /ri . It holds the following elementary results (see e.g., [9]): , ∂ 1 SD einθ (x) = ρ |n|+1 einθ , ∂ν e 2 . , 1 ∂ SΩ einθ (x) = − ρ |n|−1 einθ . ∂ν i 2 Using these identities, one can see that if g defined by .(3.2.7) has the Fourier series expansion g=

.

  gn  i

gen

n/=0

einθ ,

then the integral Eqs. (3.2.8) are equivalent to .

ρ |n|−1 n 2 ϕe ρ |n|+1 n 2 ϕi

ziδ ϕin + zeδ ϕen +

= gin ,

(3.2.14)

= gen

for every .|n| ≥ 1. It is readily seen that the solution .Φ = (ϕi , ϕe ) to .(3.2.14) is given by ϕi = 2

 2zeδ g n − ρ |n|−1 gen i

n/=0 .

ϕe = 2

4ziδ zeδ − ρ 2|n|

 2zδ gen − ρ |n|+1 g n i

i

n/=0

4ziδ zeδ − ρ 2|n|

einθ ,

einθ .

If the source is located outside the structure, i.e., f is supported in .|x| > re , then the Newtonian potential of f , which is denoted by .F , is harmonic in .|x| ≤ re and F (x) = c −



.

n/=0

gen

||n|−1

|n|re

r |n| einθ ,

|x| ≤ re .

(3.2.15)

Thus one has gin = −gen ρ |n|−1 .

.

(3.2.16)

3.2 Electrostatic Problem

123

And so ⎧  δ  |n|−1 n  ρ ge inθ ⎪ ⎨ ϕi = −2 n/=0 2ze +1 e , 2|n| 4ziδ zeδ −ρ   . δ 2|n| n 2zi +ρ ge ⎪ϕ = 2 ⎩ e n/=0 4zδ zδ −ρ 2|n| einθ . i e

Therefore, from .(3.2.9) one can find out that   2|n| 2|n| 2 ri zeδ − re ziδ gen inθ .SD [ϕi ] (x) + SΩ [ϕe ] (x) = e ,   |n| |n|−1 4ziδ zeδ − ρ 2|n| r n/=0 |n|re 

re < r = |x|, (3.2.17)

and SD [ϕi ] (x) = −

.



2|n| 

ri

2zeδ + 1

|n|−1  2|n| ρ n/=0 |n|re



− 4ziδ zeδ



gen inθ e , r|n|

ri < r = |x| < re , (3.2.18)

 δ  2zi + ρ 2|n| n |n| inθ .SΩ [ϕe ] (x) = g r e , |n|−1  2|n| δ zδ e ρ |n|r − 4z e n/=0 i e 

ri < r = |x| < re . (3.2.19)

One then has the following result: Lemma 3.2.2 There exists .δ0 such that ⎧ 2  δ |gen | ⎪ ⎨ n/=0 |n|(δ 2 +ρ 4|n| ) , .Eδ ≈ 2|n| n 2  | ⎪ ⎩ n/=0 δρ 2 |ge4|n| , |n|(δ +ρ )

if ϵc /= ϵs = 1, if ϵc = ϵs /= 1,

(3.2.20)

uniformly in .δ ≤ δ0 . The remaining two cases are when .ϵc /= −ϵs = 1 and .ϵc = −ϵs /= 1. In these cases, one has the following theorem. Theorem 3.2.1 (i) If .ϵc = −ϵs /= 1, then CALR does not occur, i.e., Eδ ≤ C

.

for some .C > 0. (We note, however, that there will be CALR for appropriately placed sources inside the core, as can be seen from the fact that the equations are invariant under conformal transformations, and in particular under the

124

3 Anomalous Localized Resonances and Their Cloaking Effect

inverse transformation .1/z where .z = x1 + ix2 , which in effect interchanges the roles of the matrix and core.) (ii) If .ϵc /= −ϵs = 1, then weak CALR occurs and the critical radius is .r∗ = re2 ri−1 , i.e., if the source function is supported inside .r∗ (and its Newtonian potential does not extend harmonically to .R2 ), then .

lim sup Eδ = ∞ δ→0

and there exists a constant C such that |Vδ (x)| < C

.

for all .x with .|x| > re3 /ri2 . (iii) In addition to the assumptions of (ii), the Fourier coefficients .gen of .− ∂F ∂ν e satisfies the following gap property: [GP2] There exists a sequence .{nk } with .|n1 | < |n2 | < · · · such that

.

lim ρ

2(|nk+1 |−|nk |)

k→∞

 nk 2 g e  |nk | ρ |nk |

=∞

then the CALR occurs, i.e., .

lim Eδ = ∞

δ→0

√ and .Vδ / Eδ goes to zero outside the nadius .re3 /ri2 , as implied by (4.5). Proof First, if .ϵc = −ϵB /= 1, then Eδ ≈



.

n/=0

 2    gen 2 δρ 2|n| gen  1 ∂F  ≤ ≤ 2|n| 2 ∂ν e |n| δ 2 + ρ 4 |n| n/=0

L2 (Γe )

≤ C‖f ‖L2 (R2 ) .

Suppose that .ϵc /= −ϵs = 1, and let Nδ =

.

log δ 2 log ρ

If .|n| ≤ Nδ , then .δ ≤ ρ 2|n| , and hence  .

n/=0

 2  2     δ gen 2 δ gen  δ gen    ≥ ≥ . |n|ρ 4|n| |n| δ 2 + ρ 4|n| |n| δ 2 + ρ 4|n| 0/=|n|≤N 0/=|n|≤N δ

δ

(3.2.21)

3.2 Electrostatic Problem

125

If the following holds  n 2 g  e =∞ . lim sup 2 n→∞ |n|ρ |n|

(3.2.22)

then one can show as in [2] that there is a sequence .{|nk |} such that .

lim Eρ |nk | = ∞.

k→∞

Suppose that the source function f is supported inside the critical radius .r∗ = re2 ri−1 (and outside .re ). Then its Newtonian potential .F cannot be extended harmonically in .|x| < r∗ in general. So, if .F is given by F =c−



.

an r |n| einθ ,

r < re + ϵ

n/=0

for some .ϵ > 0, then the radius of convergence of the series is less than .r∗· . Thus we have .

2|n|

lim sup |an |2 r∗ |n|→∞

= ∞.

|n|−1

, .(3.2.22) holds. Since .gen = |n|an re By .(3.2.17), one has

.

|Vδ | ≤ |F | + C

 n   |n| g   re3|n| gen  re e ≤ |F | + C ≤ C' 2|n| r |n| δ + ρ 2|n| r |n| r

 n/=0

n/=0 i

if .r > re3 /ri2 . Thus (ii) is proved. It is now remaining to prove (iii). First [GP2] implies .(3.2.22). On the other hand [GP2] holds if

.

lim

n→∞

 n 2 g  e

|n|ρ 2|n|

= ∞.

(3.2.23)

Suppose that [GP2] holds. By taking .δ = ρ 2α and letting .k(α) be the number such that .

    nk(α)  ≤ α < nk(α)+1  ,

126

3 Anomalous Localized Resonances and Their Cloaking Effect

then  .

0/=|n|≤Nδ

 2  δ gen  2α = ρ |n|ρ 4|n|

0/=|n|≤α

 n 2 g  e |n|ρ 4|n|

   nk(α) 2 g e   ≥ ρ 2(|nk(α)+1 |−|nk(α) |)  → ∞, nk(α)  ρ 2|nk (α)|

as .α → ∞. Combined with Lemma 3.2.2 and (4.16), one finally has (iii).

⨆ ⨅

Next, one can see that that CALR does not occur in a radially symmetric three dimensional coated sphere structure when the core, matrix and shell are isotropic. There holds the following theorem. For other constructions which ensure the occurrence of CALR in 3D, we refer to [90]. Theorem 3.2.2 Suppose that .Γe and .Γi are concentric spheres. For any .ϵc and .ϵs , there is a constant C independent of .δ such that if .Vδ is the solution to .(3.2.1), then  .

Ω\D

δ |∇Vδ |2 ≤ C‖f ‖2L2 (R3 )

(3.2.24)

Proof Suppose that . ∂ν∂ e F has the Fourier series expansion .

∞  n  ∂ e F =− gmn Ynm . ∂ν e m=−n n=0

Then one can show as in .(3.2.16) that

.

n ∞   ∂ e F =− gmn ρ n−1 Ynm . ∂ν i m=−n n=0

By solving the integral equation .(3.2.5) using .(3.2.13) , we obtain ϕi = −

n ∞  



n−1 Δ−1 n ρ

zeδ

n=0 m=−n .

ϕe =

n ∞  

n−1 Δ−1 n ρ

 ziδ

n=0 m=−n

 1 + ge Y m, 2 mn n

 1 n+1 2n+1 e − Ynm , + ρ gmn 2(2n + 1) 2(2n + 1)

where  Δn := ziδ −

.

1 2(2n + 1)

 zeδ +

1 2(2n + 1)

 +

n(n + 1) 2n+1 ρ . (2n + 1)2

3.2 Electrostatic Problem

127

Suppose for simplicity that .ϵc = −ϵs = 1, so that .ziδ and .zeδ given by (3.2.6) simplify to ziδ = zeδ =

.

iδ . 2(2 − iδ)

Then one can see that if .δ is sufficiently small, then .

|Δn | ≈ δ 2 + n−2 .

So we have δ ‖ϕi ‖2L2 (Γ ) i

≤C ≤C

.

∞  n 

 e 2 δρ 2n  2 gmn  2 −2 n=0 m=−n δ + n

∞  n 

 e 2  n3 ρ 2n gmn

n=0 m=−n

≤C

∞   e 2 g  ≤ C‖f ‖2 2 3 . mn L (R ) n=0

and δ ‖ϕe ‖2L2 (Γ ) ≤ C

.

∞  n 

δ2

e

n=0 m=−n

Therefore we have   δ |∇Vδ |2 = Ω\D .



δρ 2n  e 2 g ≤ C‖f ‖2L2 (R3 ) . + n−2 mn

 δ |∇ (SD [ϕi ] + SΩ [ϕe ])|2

δ|∇F |2 + Ω\D

≤



Ω\D

δ |∇ (SD [ϕi ] + SΩ [ϕe ])|2

δ|∇F |2 + Ω\D

 ≤

Ω\D

Ω\D

  δ|∇F |2 + δ ‖ϕi ‖2L2 (Γ ) + ‖ϕe ‖2L2 (Γ ) ≤ C‖f ‖2L2 (R3 ) . i

e

If .ϵs /= −1 or /and .ϵc /= 1, then the same argument can be applied to arrive at (3.2.24). This completes the proof. ⨅ ⨆

.

Chapter 4

Localized Resonances for Anisotropic Geometry

In Chaps. 2 and 3, we consider localized resonances associated with nanoparticles. In this chapter, we consider plasmon resonances associated with nanorods. Nanorods possess high aspect ratios and present an anisotropic geometric setup. We shall follow the treatment in [59, 65]. The results in this chapter provides a quantitative understanding of the curvature effect of the material structure on the localized resonance.

4.1 Conductivity Problem Initially focusing on the mathematics, but not the physics, we consider the following elliptic PDE system in .R2 [65]: ⎧ ⎨ .

⎩

  ∇ · σ (x)∇u(x) = 0,

x = (x1 , x2 ) ∈ R2 ,

u(x) − H (x) = O(|x|−1 ), |x| → ∞,

(4.1.1)

1 (R2 ) is a potential field, and .σ (x) is of the following form: where .u(x) ∈ Hloc

σ := (σ0 − 1)χ (D) + 1, σ0 ∈ R+ and σ0 /= 1,

.

(4.1.2)

with D being a bounded domain with a connected complement .R2 \D. .H (x) in (4.1.1) is a (nontrivial) harmonic function in .R2 , which stands for a background potential. We are mainly concerned with the quantitative properties of the solution .u(x) in (4.1.1) and in particular, its dependence on the geometry of D. To that end, we next introduce the rod-geometry of D for our subsequent study. Let .Γ0 be a straight line of length .L ∈ R+ with .Γ0 = (x1 , 0), .x1 ∈ (−L/2, L/2). Let .n := (0, 1) and define the two points .P := (−L/2, 0) and .Q := (L/2, 0). Then the rod © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Y. Deng, H. Liu, Spectral Theory of Localized Resonances and Applications, https://doi.org/10.1007/978-981-99-6244-0_4

129

130

4 Localized Resonances for Anisotropic Geometry

D is introduced as .D = D a ∪ D f ∪ D b , where .D f is defined by D f := {x; x = Γ0 ± tn, t ∈ (−δ, δ)}, δ ∈ R+ .

.

(4.1.3)

The two end-caps .D a and .D b are two half disks with radius .δ and centering at P and Q, respectively. More precisely, D a = {x; |x − P | < δ, x1 < −L/2},

.

D b = {x; |x − Q| < δ, x1 > L/2}.

It can be verified that D is of class .C 1,α for a certain .α ∈ R+ . In what follows, we define .S c := ∂D c = ∂(D a ∪ D b ), and .S f := ∂D f . Specially, .S f = Γ1 ∪ Γ2 , where .Γj , .j = 1, 2 are defined by Γ1 = {x; x = Γ0 − δn},

.

Γ2 = {x; x = Γ0 + δn}.

(4.1.4)

Moreover, we shall always use .zx and .zy to signify the projections of .x ∈ S f and f .y ∈ S on .Γ0 , respectively. The elliptic PDE system (4.1.1) describes the perturbation of an electric field .H (x) due to the presence of a conductive body D. .u(x) signifies the electric potential and .σ (x) signifies the conductivity of the space. The homogeneous background space possesses a conductivity being normalized to be 1, whereas the conductive rod D possesses an inhomogeneous conductivity being .σ0 . The perturbed electric potential is .u − H and the gradient field .∇(u − H ) is the corresponding electric current.

4.1.1 Some Auxiliary Results In this section, we shall establish several auxiliary results for our subsequent use. We first present some preliminary knowledge on the layer potential operators for solving the conductivity problem (4.1.1), and we also refer to [15, 115] for more related results and discussions. By using the layer-potential techniques, one can readily find the integral solution to (4.1.1) by u = H + SD [ϕ],

.

(4.1.5)

where the density function .ϕ ∈ H −1/2 (∂D) is determined by −1  ∂H    . ϕ = λI − KD∗  ∂ν ∂D

.

(4.1.6)

4.1 Conductivity Problem

131

Here, the constant .λ is defined by λ :=

.

σ0 + 1 . 2(σ0 − 1)

In what follows, we always suppose that .δ ⪡ 1. We shall present some asymptotic expansions of the Neumann-Poincaré operator with respect to .δ. Recalling that .∂D = S a ∪ S f ∪ S b , we decompose the Neumann-Poincaré operator into several parts accordingly. To that end, we introduce the following boundary integral operator:

〈x − y, ν x 〉 ' 1 φ(y)dsy , f or S ∩ S ' = ∅. .KS ,S ' [φ](x) := χ (S ) 2π S |x − y|2 (4.1.7) It is obvious that .KS ,S ' is a bounded operator from .L2 (S ) to .L2 (S ' ). For the case .S = S c , we mean .S c = S a ∪ S b . In what follows, we define .S1a and .S1b by S1a = {x; |x − P | = 1, x1 < −L/2},

.

S1b = {x; |x − Q| = 1, x1 > L/2}. (4.1.8)

For the subsequent use, we also introduce the following regions: ιδ (P ) := {x; |P − zx | = O(δ), x ∈ S f }, .

(4.1.9)

ιδ (Q) := {x; |Q − zx | = O(δ), x ∈ S f }.

(4.1.10)

.

a b ( x) := φ(x), where .x ∈ S a , S b and  Define .φ .x ∈ S , S . 1 1 We can prove the following result on the asymptotic expansion of the NeumannPoincaré operator.

Lemma 4.1.1 The Neumann-Poincaré operator .KD∗ admits the following asymptotic expansion: KD∗ [φ](x) = K0 [φ](x) + δK1 [φ](x) + O(δ 2 ),

.

(4.1.11)

where .K0 is defined by

  1 (y) + χ (S b ) φ K0 [φ](x) = χ (S a ) KS f ,S a [φ](x) + 4π S1a

  1  φ × K f ,S b [φ](x) + S . 4π S1b + AΓ2 ,Γ1 [φ] + AΓ1 ,Γ2 [φ] + χ (ιδ (P ))KS a ,S f [φ](x) + χ (ιδ (Q))KS b ,S f [φ](x),

(4.1.12)

132

4 Localized Resonances for Anisotropic Geometry

and K1 [φ] = χ (S b ) .

〈x − P , ν x 〉 2π |x − P |2

S1a

 + χ (S a ) φ

〈x − Q, ν x 〉 2π |x − Q|2

S1b

 φ



  δ δ ( + χ (S \ ιδ (P )) (1 − 〈 y − P , ν x 〉)φ y)dsy + o |x − P |2 S1a |x − P |2

  δ δ ( (1 − 〈 y − Q, ν x 〉)φ y)dsy + o + χ (S f \ ιδ (Q)) . |x − Q|2 S1b |x − P |2 (4.1.13) f

Here, the operators .AΓ1 ,Γ2 and .AΓ2 ,Γ1 are defined by

δ 1 φ(y)dsy , AΓ1 ,Γ2 [φ](x) = χ (Γ2 ) 2 π Γ1 |x − y| .

δ 1 φ(y)dsy . AΓ2 ,Γ1 [φ](x) = χ (Γ1 ) π |x − y|2 Γ2

(4.1.14)

Proof First, one has the following separation:



〈x − y, ν x 〉 〈x − y, ν x 〉 1 1 KD∗ [φ](x) = φ(y)ds + φ(y)dsy y 2 f a b 2π 2π |x − y| |x − y|2 S S ∪S . =:A1 [φ](x) + A2 [φ](x). (4.1.15) Note that for .x, y ∈ Γj , .j = 2, 3, one can easily obtain that .〈x − y, ν x 〉 = 0. Thus one has

〈x − y, ν x 〉 1 A1 [φ](x) = φ(y)dsy 2π S f |x − y|2

〈x − y, ν x 〉 1 φ(y)dsy = χ (S a ∪ S b ) 2π S f |x − y|2

〈(x − 2δν x − y) + 2δν x , ν x 〉 1 + χ (Γ1 ) φ(y)dsy 2π Γ2 |x − y|2 (4.1.16) .

〈(x − 2δν x − y) + 2δν x , ν x 〉 1 φ(y)ds + χ (Γ2 ) y 2π Γ1 |x − y|2

1 1 φ(y)dsy = KS f ,S c [φ](x) + δχ (Γ1 ) π Γ2 |x − y|2

1 1 φ(y)dsy . + δχ (Γ2 ) π Γ1 |x − y|2

4.1 Conductivity Problem

133

On the other hand,

〈x − y, ν x 〉 1 A2 [φ](x) = φ(y)dsy 2π S a ∪S b |x − y|2



〈x − y, ν x 〉 〈x − y, ν x 〉 1 b 1 φ(y)ds + χ (S ) φ(y)dsy =χ (S a ) y 2π S a |x − y|2 2π S b |x − y|2



〈x − y, ν x 〉 〈x − y, ν x 〉 b 1 a 1 + χ (S ) φ(y)dsy + χ (S ) φ(y)dsy 2 b a 2π 2π |x − y| |x − y|2 S S .

〈x − y, ν x 〉 1 + χ (S f ) φ(y)dsy 2π S a ∪S b |x − y|2



b 1 a 1  ( φ ( y)dsy + χ (S ) φ y)dsy + KS b ,S a [φ] =χ (S ) 4π S1a 4π S1b + KS a ,S b [φ] + KS c ,S f [φ]. (4.1.17) For .y ∈ S a and .x ∈ S b , by using Taylor’s expansion one has |x − y| = |x − P − (y − P ))| = |x − P − δ( y − P )|

.

= |x − P | + δ〈x − P , y − P 〉 + O(δ 2 ).

(4.1.18)

Thus one has 〈x − P , ν x 〉 .KS a ,S b [φ](x) = δ 2π |x − P |2

S1a

 + O(δ 2 ). φ

(4.1.19)

 + O(δ 2 ). φ

(4.1.20)

Similarly, one can obtain 〈x − Q, ν x 〉 .KS b ,S a [φ](x) = δ 2π |x − Q|2

S1b

For .x ∈ S f .y ∈ S a , by direct computations, one can obtain

KS a ,S f [φ](x) = δ 2

.

S1a

|x − P |2

1 − 〈 y − P , νx〉 ( φ y)dsy . − 2δ〈x − P , y − P 〉 + δ2

We decompose .S f = (S f \ ιδ (P )) ∪ ιδ (P ), then one has KS a ,S f [φ](x) =

.

δ2 |x − P |2 +o



S1a

( (1 − 〈 y − P , ν x 〉)φ y) dsy

δ2  , |x − P |2

x ∈ S f \ ιδ (P ).

(4.1.21)

134

4 Localized Resonances for Anisotropic Geometry

Similarly, one can derive the asymptotic expansion for .KS b ,S f . By substituting (4.1.19)–(4.1.21) back into (4.1.17) and combining (4.1.16) one finally achieves (4.1.11), which completes the proof. The proof is complete. ⨆ ⨅ Lemma 4.1.2 The operators .AΓj ,Γk , .{j, k} = {1, 2}, {2, 1}, defined in (4.1.14) are bounded operators from .L2 (Γj ) to .L2 (Γk ). Furthermore, the operators 2 a .χ (ιδ (P ))KS a ,S f and .χ (ιδ (Q))KS b ,S f are bounded operators from .L (S ) to 2 f 2 b 2 f .L (S ), and from .L (S ) to .L (S ), respectively. Proof We only prove that .AΓ2 ,Γ1 is a bounded operator .L2 (Γ2 ) to .L2 (Γ1 ). First, for 2 2 .φ1 ∈ L (Γ1 ) and .φ2 ∈ L (Γ2 ), one has |〈AΓ2 ,Γ1 [φ2 ], φ1 〉L2 (Γ1 ) |



 δ 1   = φ (y)ds φ (x)ds   2 y 1 x 2π Γ1 Γ2 |x − y|2





1 δ δ 1 2 ≤ φ (y)ds ds + φ 2 (x)dsy dsx y x 4π Γ1 Γ2 |x − y|2 2 4π Γ1 Γ2 |x − y|2 1

L/2 L/2 δ 1 dx1 φ22 (y)dy1 = 4π −L/2 −L/2 |x1 − y1 |2 + 4δ 2

L/2 L/2 δ 1 dy1 φ12 (x)dx1 + 4π −L/2 −L/2 |x1 − y1 |2 + 4δ 2

L/2  −L/2 − y1  2 L/2 − y1 1 − arctan φ2 (y)dy1 arctan = 2δ 2δ 8π −L/2

L/2  −L/2 − x1  2 L/2 − x1 1 − arctan φ1 (x)dx1 arctan + 8π −L/2 2δ 2δ

.

≤ C(‖φ1 ‖2L2 (Γ ) + ‖φ2 ‖2L2 (Γ ) ), 1

2

where the constant C is independent of .δ. By following a similar arguments as in ⨆ [15] (pp. 18), one can show that .AΓ2 ,Γ1 is a bounded operator .L2 (Γ2 ) to .L2 (Γ1 ). ⨅ Lemma 4.1.3 Suppose .x ∈ S c , then for any function .φ ∈ L2 (S f ), which satisfies φ(y) = −φ(y + 2δn),

y ∈ Γ1 ,

(4.1.22)

KS f \(ιδ (P )∪ιδ (Q)),S c [φ](x) = o(1).

(4.1.23)

.

there holds .

4.1 Conductivity Problem

135

Proof Note that .S f = Γ1 ∪ Γ2 . Straightforward computations show that KS f \(ιδ (P )∪ιδ (Q)),S c [φ](x) = = .

= =

1 2π 1 2π 1 2π





1 2π

S f \(ιδ (P )∪ιδ (Q))

S f \(ιδ (P )∪ιδ (Q))

〈x − zy − δν y , ν x 〉 φ(y)dsy |x − zy − δν y |2

S f \(ιδ (P )∪ιδ (Q))

〈x − zy , ν x 〉 φ(y)dsy + o(1) |x − zy |2

Γ1 \(ιδ (P )∪ιδ (Q))

〈x − zy , ν x 〉 φ(y)dsy |x − zy |2

1 + 2π



Γ1 \(ιδ (P )∪ιδ (Q))

〈x − y, ν x 〉 φ(y)dsy |x − y|2

〈x − zy , ν x 〉 φ(y + 2δn)dsy + o(1) = o(1), |x − zy |2 ⨆ ⨅

which completes the proof.

4.1.2 Quantitative Analysis of the Electric Field In this section, we present the quantitative analysis of the solution to the conductivity equation (4.1.1) as well as its geometric relationship to the inclusion D. Recall that u is represented by (4.1.5). We first derive some asymptotic properties of the density function .ϕ in (4.1.6). Let .zx ∈ Γ0 be defined by  zx =

.

x + δn, x ∈ Γ1 , x − δn, x ∈ Γ2 .

(4.1.24)

One has the following asymptotic expansion for H around .Γ0 : H (x) = H (zx ) + ∇H (zx ) · (x − zx ) + O(|x − zx |2 )

.

= H (zx ) + δ∇H (zx ) · ( x − zx ) + O(δ 2 ),

(4.1.25)

f

for .x ∈ S f and  .x ∈ S . Similarly, one has 1 H (x) = H (P ) + ∇H (P ) · (x − P ) + O(|x − P |2 )

.

= H (P ) + δ∇H (P ) · ν x + O(δ 2 ),

(4.1.26)

136

4 Localized Resonances for Anisotropic Geometry

a for .x ∈ S a and  .x ∈ S . Moreover, 1

H (x) = H (Q) + ∇H (P ) · (x − Q) + O(|x − Q|2 )

.

= H (Q) + δ∇H (Q) · ν x + O(δ 2 ),

(4.1.27)

b for .x ∈ S b and  .x ∈ S . 1 We now can show the following asymptotic result.

Lemma 4.1.4 Suppose .ϕ is defined in (4.1.6), then one has ⎧ (λI + Aδ )−1 [(−1)j ∂x2 H (·, 0)](x1 ) + δ(λI − Aδ )−1 [∂x22 H (·, 0)](x1 ) ⎪ ⎪ ⎨ +χ (ιδ ϵ (P ) ∪ ιδ ϵ (Q))O(δ 2(1−ϵ) ) + O(δ 2 ), x ∈ Γj \ (ιδ (P ) ∪ ιδ (Q)), .ϕ(x) = ⎪(λI − K1∗ )−1 [∇H (P ) · ν] + o(1), x ∈ S a ∪ ιδ (P ), ⎪ ⎩ ∗ −1 x ∈ S b ∪ ιδ (Q), (λI − K2 ) [∇H (Q) · ν] + o(1), (4.1.28) where .0 < ϵ < 1 and the operator .Aδ is defined by

1 .Aδ [ψ](x1 ) := 2π

L/2

−L/2

δ ψ(y1 )dy1 , (x1 − y1 )2 + 4δ 2

ψ ∈ L2 (−L/2, L/2). (4.1.29)

The operators .K1∗ and .K2∗ are defined by K1∗ [ϕ1 ](x) := .

K2∗ [ϕ2 ](x) :=

S a ∪ιδ (P )



S b ∪ιδ (Q)

〈x − y, ν x 〉 ϕ1 (y)dsy + χ (ιδ (P ))AS f ∩ιδ (P ) [ϕ1 ](x) |x − y|2 〈x − y, ν x 〉 ϕ1 (y)dsy + χ (ιδ (Q))AS f ∩ιδ (Q) [ϕ2 ](x), |x − y|2 (4.1.30)

respectively. Proof Since    ∂H   λI − KD∗ [ϕ] = .  ∂ν ∂D

.

By combining (4.1.11) and (4.1.25) one can readily verify that   λI − K0 [ϕ](x) = ∇H (zx ) · ν x + o(1),

.

x ∈ Sf .

4.1 Conductivity Problem

137

By using (4.1.12), one thus has λϕ(x) −



1 π

Γ2

δ ϕ(y)dsy |x − y|2

= −∇H (zx ) · n + o(1), .

1 δ λϕ(x) − ϕ(y)dsy π Γ1 |x − y|2 = ∇H (zx ) · n + o(1),

  x ∈ Γ1 \ ιδ (P ) ∪ ιδ (Q) ,

(4.1.31)

  x ∈ Γ2 \ ιδ (P ) ∪ ιδ (Q) .

By direct computations, one can show

.

λϕ(x1 , δ) −



1 π

λϕ(x1 , −δ) −

1 π

L/2 −L/2



δ ϕ(y1 , δ)dy1 (x1 − y1 )2 + 4δ 2 = −∂x2 H (x1 , 0) + o(1),

L/2

−L/2

|x1 | ≤ L/2 − O(δ),

δ ϕ(y1 , −δ)dy1 (x1 − y1 )2 + 4δ 2 = ∂x2 H (x1 , 0) + o(1),

|x1 | ≤ L/2 − O(δ). (4.1.32)

Thus one can derive that .ϕ(x1 , −δ) = −ϕ(x1 , δ) + o(1), for .|x1 | ≤ L/2 − O(δ). Furthermore, for .x ∈ S1a , by making use of (4.1.12), (4.1.26) and Lemma 4.1.6, one has (λI − K1∗ )[ϕ](x) = ∇H (P ) · ν x + o(1),

.

in S a ∪ ιδ (P ).

(4.1.33)

in S b ∪ ιδ (Q).

(4.1.34)

In a similar manner, one can show that (λI − K2∗ )[ϕ](x) = ∇H (Q) · ν x + o(1),

.

and so the last equation in (4.1.28) follows. Next, by combining (4.1.11), (4.1.12), and (4.1.13) again for .x ∈ Γj \ (ιδ (P ) ∪ ιδ (Q)), .j = 1, 2, and using the second and third equations in (4.1.28), one has 1 .λϕ(x1 , (−1) δ) − 2π j



L/2 −L/2

δ ϕ(y1 , (−1)j +1 δ)dy1 (x1 − y1 )2 + 4δ 2

= (−1) ∂x2 H (x1 , 0) + δ∂x22 H (x1 , 0) j

+ χ (ιδ ϵ (P ) ∪ ιδ ϵ (Q))O(δ 2(1−ϵ) ) + O(δ 2 ),

0 < ϵ < 1,

(4.1.35)

which verifies the first equation in (4.1.28) and completes the proof.

⨆ ⨅

138

4 Localized Resonances for Anisotropic Geometry

Before presenting our main result, we need to further analyze the operator .Aδ defined in (4.1.29) Lemma 4.1.5 Suppose .Aδ is defined in (4.1.29), then it holds that Aδ [y1n ](x1 ) =

.

1 n x + o(1), 2 1

  x ∈ Γj \ ιδ (P ) ∪ ιδ (Q) ,

n ≥ 0.

(4.1.36)

  Proof We use deduction to prove the assertion. Since .x ∈ Γj \ ιδ (P ) ∪ ιδ (Q) , one has |L/2 − x1 | = O(δ ϵ ),

.

and

|L/2 + x1 | = O(δ ϵ ),

0 ≤ ϵ < 1.

Then for .n = 0, it is straightforward to verify that

δ 1 L/2 dy1 Aδ [1](x1 ) = π −L/2 (x1 − y1 )2 + 4δ 2 . L/2 − x1 −L/2 − x1  1 1  arctan − arctan = + o(1). = 2π 2δ 2δ 2 (4.1.37) Next, we suppose that (4.1.36) holds for .n ≤ N. Then by using change of variables, one can derive that

δ 1 L/2 N +1 y N y1 dy1 Aδ [y1 ](x1 ) = π −L/2 (x1 − y1 )2 + 4δ 2 1 1 = 2π .



L/2−x1 2δ −L/2−x1 2δ



1 y N (2δt + x1 )dt 1 + t2 1

L/2−x1 2δ

1 1 y1N tdt + x1 x1N + o(1) 2 2 1+t   1 1 1 = δO ln(1 + δ 2(ϵ−1) ) + x1N +1 + o(1) = x1N +1 + o(1), π 2 2 (4.1.38)

1 = δ π

−L/2−x1 2δ

which completes the proof.

⨆ ⨅

The following lemma is also of critical importance Lemma 4.1.6 There holds the following that

 1 −1 (λI − K1∗ )−1 [∇H (P ) · ν] = − 2δ λ − ∂x1 H (P ) + o(δ), 2 S a ∪ιδ (P ) .

 1 −1 (λI − K2∗ )−1 [∇H (Q) · ν] =2δ λ − ∂x1 H (Q) + o(δ). 2 S b ∪ιδ (Q) (4.1.39)

4.1 Conductivity Problem

139

Proof For any .f ∈ L2 (∂D), we consider the following boundary integral equation (λI − KD∗ )[φ] = f.

(4.1.40)

.

By using the decomposition (4.1.12) (see also (4.1.33) and (4.1.34)), one has χ (S a ∪ ιδ (P ))(λI − K1∗ + o(1))[φ] + χ (S b ∪ ιδ (Q))(λI − K2∗ + o(1))[φ] + AΓ2 ,Γ1 [φ] + AΓ1 ,Γ2 [φ]

.

+ χ (ιδ ϵ (P ) ∪ ιδ ϵ (Q))O(δ 2(1 − ϵ) ) + O(δ 2 ) = f,

0 < ϵ < 1. (4.1.41)

Note that .∂D is of .C 1,α . By taking the boundary integral of both sides of (4.1.40) on .∂D and making use of (4.1.41), one then has 

.

1 λ− 2



φ= ∂D

S a ∪ιδ (P )

(λI − K1∗ + o(1))[φ]



+

S b ∪ιδ (Q)

+ Γ1

(λI − K2∗ + o(1))[φ]

AΓ2 ,Γ1 [φ] +

(4.1.42)



Γ2

AΓ1 ,Γ2 [φ] + o(δ) =

f. ∂D

By assuming .f = χ (S a ∪ ιδ (P ))∇H (P ) · ν and plugging into (4.1.42), one thus has



 1 . λ− (λI − K1∗ + o(1))−1 [∇H (P ) · ν] = ∇H (P ) · ν 2 S a ∪ιδ (P ) S a ∪ιδ (P ) = −2δ∂x1 H (P ), (4.1.43) which verifies the first equation in (4.1.39). Similarly, by assuming .f = χ (S b ∪ ιδ (Q))∇H (q) · ν, one can prove the second equation in (4.1.39). The proof is ⨆ ⨅ complete. With Lemmas 4.1.4, 4.1.5, and 4.1.6, we can now establish one of the main results as follows.

140

4 Localized Resonances for Anisotropic Geometry

Theorem 4.1.1 Let u be the solution to (4.1.1) and (4.1.2), with D of the rod-shape described in Sect. 4.1. Then for .x ∈ R2 \ D, it holds that

(x1 − y1 )2 + x22 2 1 −1 L/2 1  λ− u(x) =H (x) + δ ln ∂y H (y1 , 0) dy1 2π 2 (x1 + L/2)2 + x22 2 −L/2

1 −1 L/2 x2 1 λ− +δ ∂ H (y1 , 0) dy1 . 2 y2 2 π 2 −L/2 (x1 − y1 ) + x2 +δ

1 −1 (x1 − L/2)2 + x22 1  λ− ln ∂x1 H (L/2, 0) + o(δ). 2 2π (x1 + L/2)2 + x22 (4.1.44)

Proof By using (4.1.5) and Taylor’s expansion along with .Γ0 , one has

u(x) =H (x) +

S f \(ιδ (P )∪ιδ (Q))

G(x − zy )ϕ(y) dsy



+δ

.

+

+

S f \(ιδ (P )∪ιδ (Q))

S a ∪ιδ (P )

S b ∪ιδ (Q)

∇y G(x − zy ) · ν y ϕ(y) dsy (4.1.45)

G(x − zy )ϕ(y) dsy G(x − zy ) ϕ ( y) dsy + O(δ 2 ).

First, by using (4.1.35), one can derive that

Sf

G(x − zy )ϕ(y) dsy



. =2δ

=δ

Γ1 \(ιδ (P )∪ιδ (Q))

G(x − zy )(λI − Aδ )−1 [∂x22 H (·, 0)](y1 )dy1 + o(δ)

1 −1 L/2 1  λ− ln((x1 − y1 )2 + x22 )∂y22 H (y1 , 0) dy1 + o(δ). 2 2π −L/2 (4.1.46)

Similarly, one has

Sf

.

=

∇y G(x − zy ) · ν y ϕ(y) dsy

1 −1 L/2 x2 1 λ− ∂ H (y1 , 0) dy1 + o(1). 2 y2 2 2 π (x − y −L/2 1 1 ) + x2

(4.1.47)

4.1 Conductivity Problem

141

By using Lemma 4.1.6, one then obtains that



S b ∪ιδ (Q)

.

G(x − zy )ϕ(y) dsy =

S b ∪ιδ (Q)

G(x − Q)ϕ(y) dsy + o(δ)

 1 1 −1 ∂x1 H (Q) + o(δ) ln |x − Q| λ − π 2 1 −1 1  λ− ln((x1 − L/2)2 + x22 )∂x1 H (L/2, 0) + o(δ). =δ 2 2π

=δ

(4.1.48) Noting that



ϕ(y)dsy =

.

∂D

 −1  ∂H (y)dsy = 0, λI − KD∗ ∂ν ∂D

and by combining (4.1.46), one can readily show that



S a ∪ιδ (P )

=− .

G(x − zy )ϕ(y) dsy =

 1 ln |x − P | 2π

+2

1  λ− =−δ 2π 1  −δ λ− 2π

S b ∪ιδ (Q)

2

G(x − P )ϕ(y) dsy + o(δ)

ϕ(y) dsy

Γ1 \(ιδ (P )∪ιδ (Q))

1 −1

S a ∪ιδ (P )

 (λI − Aδ )−1 [∂x22 H (·, 0)](y1 )dy1 + o(δ)

ln((x1 + L/2)2 + x22 )∂x1 H (L/2, 0)

1 −1 ln((x1 + L/2)2 + x22 ) 2



L/2 −L/2

∂y22 H (y1 , 0) dy1 + o(δ). (4.1.49)

Finally, by substituting (4.1.46)–(4.1.49) into (4.1.45) one has (4.1.44), which ⨆ ⨅ completes the proof. We finally can derive the sharp asymptotic expansion of the electric field as follows. Theorem 4.1.2 Suppose .H (x) = a · x, where .a = (a1 , a2 ) ∈ R2 . Then for .x ∈ R2 \ D, the electric field u satisfies   L/2 − x   L/2 + x  1 −1 1 1 1 λ− + arctan a2 arctan 2 x2 x2 π (x1 − L/2)2 + x22 1 −1 1  λ− a1 ln + o(δ). +δ 2 2π (x1 + L/2)2 + x22 (4.1.50)

u(x) =a · x + δ .

142

4 Localized Resonances for Anisotropic Geometry

Furthermore, the perturbed gradient field admits the following asymptotic expansion:   1 −1 f2 (x)a1 − f1 (x)a2 1 + o(δ), λ− (4.1.51) .∇u(x) = a + δ f1 (x)a1 + f2 (x)a2 π 2 where the functions .fj , .j = 1, 2 are defined by f1 (x) :=

x2 x2 − (x1 + L/2)2 + x22 (x1 − L/2)2 + x22

f2 (x) :=

x1 − L/2 x1 + L/2 − . (x1 − L/2)2 + x22 (x1 + L/2)2 + x22

.

(4.1.52)

Proof The proof is given by using (4.1.44) together with direct computations.

⨆ ⨅

Define the following vector field Es := δ

.

  1 −1 f2 (x)a1 − f1 (x)a2 1 . λ− f1 (x)a1 + f2 (x)a2 π 2

(4.1.53)

According to (4.1.51), .Es is the leading order term of the perturbed gradient field. It is noted that the distribution of .|Es | is independent of the uniform gradient potential .a. In fact, one has |Es |2 = δ 2

.

1  1 −2 2 λ − (a1 + a22 )(f1 (x)2 + f2 (x)2 ). 2 π2

(4.1.54)

Moreover, further computations show that 2 1 1 − .f1 (x) + f2 (x) = |x − Q| |x − P |   〈x − P , x − Q〉 2 . 1− + |x − P ||x − Q| |x − P ||x − Q| 

2

2

(4.1.55)

One can thus derive that .|Es | is maximized near the two caps (high curvature parts) of the inclusion D. In fact, near the caps one has |x − P | = δ + o(δ),

.

or

|x − Q| = δ + o(δ).

By (4.1.55) one then has f1 (x)2 + f2 (x)2 = δ −2 (1 + o(1)),

.

(4.1.56)

4.1 Conductivity Problem

143

while near the centering parts of the rod, f1 (x)2 + f2 (x)2 = O(1).

.

To better illustrate the result, we next present some numerical solutions with different background fields. The parameters of the rod-shape inclusion are selected as follows: σ0 = 2,

.

L = 10,

δ = 5 ∗ tan(π/36) ≈ 0.4374.

(4.1.57)

We choose three different uniform background fields, i.e., .a = (1, 0), (0, 1), (1, 1), respectively, and plot the absolute values of the perturbed fields as well as the corresponding gradient fields, which are scaled for better display. It is clearly shown from Figs. 4.1, 4.2, and 4.3 that the gradient fields behave much stronger near the high curvature parts of the inclusion.

Fig. 4.1 .a = (1, 0). Left: Perturbed field .|u−a·x| (scaled) Right: Perturbed gradient field .|∇u−a| (scaled)

Fig. 4.2 .a = (0, 1). Left: Perturbed field .|u−a·x| (scaled) Right: Perturbed gradient field .|∇u−a| (scaled)

144

4 Localized Resonances for Anisotropic Geometry

Fig. 4.3 .a = (1, 1). Left: Perturbed field .|u−a·x| (scaled) Right: Perturbed gradient field .|∇u−a| (scaled)

4.1.3 Application to Calderón’s Inverse Inclusion Problem In this section, we consider the application of the quantitative results derived in the previous section to the Calderón inverse inclusion problem. To that end, we let D denote a generic rod inclusion that is obtained through rigid motions performed on special case described in before. We write .D(L, δ, z0 , σ0 ) to signify its dependence on the length L, width .δ, position .z0 (which is the geometric centre of D) as well as the conductivity parameter .σ0 . Consider the conductivity system (4.1.1) associated with a generic inclusion described above. The inverse inclusion problem is concerned with recovering the shape of the inclusion, namely .∂D, independent of its content .σ0 , by measuring the perturbed electric field .(u − H ) away from the inclusion. This is one of the central problems in EIT, which forms the fundamental basis for the electric prospecting. The case with a single measurement, namely the use of a single probing field H , is a longstanding problem in the literature. The existing results for the single-measurement case are mainly concerned with specific shapes including discs/balls and polygons/polyhedrons [24, 26, 39, 62, 102, 122] as well as the other general shapes but with a-priori conditions; see [2–4, 35, 67, 74]. As discussed earlier, in [103], the local recovery of the highly-curved part of .∂D was also considered. Next, using the asymptotic result quantitative result in Theorem 4.1.2, we shall show that one can uniquely determine a conductive inclusion up to an error level .δ ⪡ 1. (j )

(j )

Theorem 4.1.3 Let .Dj = Dj (Lj , δj , z0 , σ0 ), .j = 1, 2, be two conductive (j ) rods such that .Lj ∼ 1, δj ∼ δ ⪡ 1 and .σ0 ∼ 1 for .j = 1, 2. Let .uj be the corresponding solution to (4.1.1) associated with .Dj and a given nontrivial .H (x) = a · x. Suppose that u1 = u2

.

on ∂Σ,

(4.1.58)

4.2 Helmholtz Problem

145

where .Σ is a bounded simply-connected Lipschitz domain enclosing .Dj . Then it cannot hold that dist(D1 , D2 ) ⪢ δ.

.

(4.1.59)

Proof First, by (4.1.58), we know that .u1 = u2 in .R2 \Σ and hence by unique continuation, we also know that .u1 = u2 in .R2 \(D1 ∪D2 ). Next, since the Laplacian is invariant under rigid motions, we note that the quantitative result in Theorem 4.1.2 still holds for .Dj . By contradiction, we assume that (4.1.59) holds. It is easily seen that there must be one cap point, say .Θ0 ∈ ∂D1 , which lies away from .D2 and .dist(Θ0 , D2 ) ⪢ δ. Hence, one has .u1 (Θ0 ) = u2 (Θ0 ). Now, we arrive at a contradiction by noting that using Theorem 4.1.2, one has .u1 (Θ0 ) ∼ 1 whereas .u2 (Θ0 ) ∼ δ ⪡ 1. The proof is complete. ⨆ ⨅

4.2 Helmholtz Problem We present the mathematical setup of our study by first introducing the anisotropic nanorod geometry. We shall adopt the notations in [56, 57, 97] where anisotropic geometries were introduced in a different context. Let .Γ0 be a smooth simple and non-closed curve in .R3 , and the two endpoints of .Γ0 are .P0 and .Q0 . Let .r ∈ R+ . Denote by .N (x) the normal plane of the curve .Γ0 at .x ∈ Γ0 . We note that .N(P0 ) and .N (Q0 ) are, respectively, defined by the left and right limits along .Γ0 . For any .x ∈ Γ0 , we let .Sr (x) denote the disk lying on .N(x), centered at .x and of radius r. It is assumed that there exists .r0 ∈ R+ such that when .r ≤ r0 , .Sr (x) intersects .Γ0 f only at .x. We start with a thin structure .Dr given by f

Dr := Sr (x) × Γ0 (x), x ∈ Γ 0 ,

.

(4.2.1)

where we identify .Γ0 with its parametric representation .Γ0 (x). Clearly, the facade f f of .Dr , denoted by .Sr and parallel to .Γ0 , is given by f

Sr := {x + r · ν(x); x ∈ Γ0 , ν(x) ∈ N(x) ∩ S2 },

.

(4.2.2)

f

and the two end-surfaces of .Dr are the two disks .Sr (P0 ) and .Sr (Q0 ). Let .Dra0 and .Drb0 be two simply connected sets with .∂Dra0 = Sra0 ∪ Sr0 (P0 ) and .∂Drb0 = f

Srb0 ∪ Sr0 (Q0 ). It is assumed that .Sr0 := Sr0 ∪ Srb0 ∪ Sra0 is a smooth boundary of the f

domain .Dr0 := Dra0 ∪ Dr0 ∪ Drb0 . For .0 < r < r0 , we set Dra :=

.

r (D a − P0 ) + P0 = r0 r0



 r · (x − P0 ) + P0 ; x ∈ Dra0 , r0

146

4 Localized Resonances for Anisotropic Geometry

and similarly, .Drb := r/r0 · (Drb0 − Q0 ) + Q0 . Let .Sra and .Srb , respectively, denote the f

boundaries of .Dra and .Drb excluding .Sra and .Srb . Now, we set .Dr := Dra ∪Dr ∪Drb , f and .Sr := Sr ∪ Srb ∪ Sra = ∂Dr . .Dr represents the geometry of a curved nanorod in f our study with .Dra,b signifying the two end-parts and .Dr signifying the facade-part. According to our earlier description, it is obvious that for .0 < r ≤ r0 , .Dr is a simply connected set with a smooth boundary .Sr , and .Dr1  Dr2 if .0 ≤ r1 < r2 ≤ r0 . Moreover, .Dr degenerates to .Γ0 if one takes .r = 0. Without loss of generality, we assume that .r0 ≡ 1. In what follows, we let .δ ∈ R+ be the size parameter and let f

Sδ := Sδ ∪ Sδb ∪ Sδa ,

.

(4.2.3)

denote the boundary surface of .Dδ . In order to ease the exposition, we drop the dependence on r if one takes .r = 1. For example, D and S denote, respectively, .Dr and .Sr with .r = 1. It is emphasized that in all of our subsequent argument, D can always be replaced by .Dτ0 with .0 < τ0 ≤ r0 being a fixed number. Hence, we indeed shall not lose any generality of our study by assuming that .r0 ≡ 1. Finally, we would like to note that a particular case is to take .Γ0 to be a straight line-segment and, .Srb and .Sra to be two semi-spheres of radius r and centered at .P0 and .Q0 respectively. Hence, though we term .Dδ as a curved nanorod, it actually also includes as a special case the “straight” nanorod as considered in the physics literature. .y ∈ D as Next we introduce a blowup transformation which maps .y ∈ Dδ to  follows A(y) =  y :=

.

1 (y − zy ) + zy , δ

f

y ∈ Dδ ,

(4.2.4)

whereas  A(y) =  y :=

.

y−P0 δ + P0 , y−Q0 + Q0 , δ

y ∈ Dδa , y ∈ Dδb ,

(4.2.5)

which will be used to analyze the asymptotic expansions of some layer potential operators. We consider a long and thin nanorod occupying a bounded and simply connected domain .Dδ ⊂ R3 as described above, whose boundary .∂Dδ (≡ Sδ ) is .C 1,γ for some .γ ∈ (0, 1). We present the mathematical description of the electro-magnetic (EM) scattering from the nanorod .Dδ . In principle, the propagation of light in the nanostructures is described by the Maxwell equations. Due to technical reasons, we shall mainly consider the Helmholtz equation, which in 2D describes the transverse EM propagation and in 3D the acoustic wave propagation (cf. [94]). In order to avoid repeating the discussions, we present the results mainly for the 3D case and the extension to the 2D case should be clear and can be straightforwardly done. As remarked earlier, the extension to the full Maxwell system will be presented in our forthcoming work. In order to ease the exposition, we stick to the physical

4.2 Helmholtz Problem

147

terminologies associated with the electro-magnetic scattering and the physical interpretation of our results for the acoustic scattering can be easily given. The material properties of the nanorod .Dδ are characterized by the electric permittivity .εc and the magnetic permeability .μc , while the homogeneous medium in .R3 \Dδ is characterized by electric permittivity .εm and the magnetic permeability .μm . We shall be mainly concerned with the time-harmonic scattering and let .ω ∈ R+ signify the angular frequency of the wave. Let .ℜεc < 0, .𝔍εc > 0, .ℜμc < 0, .𝔍μc > 0 be constants. Define the wavenumbers to be √ √ kc = ω εc μc , km = ω εm μm .

.

(4.2.6)

Set εDδ = εc χ (Dδ ) + εm χ (R3 \ Dδ ), μDδ = μc χ (Dδ ) + μm χ (R3 \ Dδ ),

.

(4.2.7)

where and also in what follows, .χ signifies the characteristic function. It is pointed out that nano-metal materials with the electric permittivity and magnetic permeability satisfying .ℜεc < 0, .ℜμc < 0 are called double negative materials, which show several unusual properties, such as the counter directance between the group velocity and the phase vector, negative index of refraction and the reverse Doppler and Cherenkov effects [119, 123, 130]. It is emphasized that the double negative materials inside the nanorod are the key ingredient accounting for the plasmon resonance in our study. The positive imaginary parts of electric permittivity and magnetic permeability signify the dissipation of the plasmonic nanostructures. For the background medium in .R3 \Dδ , we assume that .εm , .μm are real and strictly positive. √ Let .ui = eikm d·x , .i := −1, be the time-harmonic incident plane wave. Here .d is a unit vector which represents the incident direction. The wave scattering due to the impingement of the incident field .ui on the nanorod .Dδ is described by the following Helmholtz system, ⎧ ⎪ ∇ · εD1 ∇u + ω2 μDδ u = 0 ⎪ ⎪ ⎪ δ ⎪ ⎨ u|+ = u|− .   1 ∂u  ⎪  ⎪ ε1m ∂u ⎪ ∂ν + = εc ∂ν − ⎪ ⎪ ⎩us := u − ui

in R3 \ ∂Dδ , on ∂Dδ , on ∂Dδ satisfies the Sommerfeld radiation condition. (4.2.8)

By the Sommerfeld radiation condition, we mean that the scattered wave .us satisfies .

∂us − ikm us = O(|x|−2 ) ∂|x|

as |x| → +∞,

(4.2.9)

which holds uniformly in the angular variable .xˆ = x/|x|. The Sommerfeld radiation condition characterises the outgoing nature of the scattered field.

148

4 Localized Resonances for Anisotropic Geometry

Our study of the plasmon resonance associated with the Helmholtz system (4.2.8) heavily relies on the layer-potential techniques. To that end, we introduce the boundary layer potential operators for the analysis of the solution to (4.2.8), and we also refer to [44] for more relevant discussions on layer potential techniques. Let 2 .Gk (x − y) be the 3D Green’s function for the PDO .Δ + k , which is given by Gk (x − y) = −

.

eik|x−y| . 4π |x − y|

(4.2.10)

Let .Σ be a bounded domain with a .C 1,γ boundary .∂Σ, which could be .Dδ or D in our subsequent study. The single layer potential for the Helmholtz equation is defined by

k .SΣ [φ](x) = Gk (x − y)φ(y)ds(y), x ∈ R3 , (4.2.11) ∂Σ

1

where .φ ∈ H − 2 (∂Σ) signifies a boundary density function. The following jump formula holds    ∂(SΣk [φ])  1 k ∗ [φ](x), a.e. x ∈ ∂Σ, I + (K (x) = ± ) (4.2.12) . Σ  ∂ν 2 ± where (KΣk )∗ [φ](x) = p.v.



.

∂Σ

∂Gk (x − y) φ(y)ds(y), ∂ν(x)

(4.2.13)

is known as the Neumann-Poincaré operator. Here p.v. stands for the Cauchy principle value. In what follows, we let .SΣ and .KΣ∗ respectively denote the operators .SΣk and .(KΣk )∗ by formally taking .k = 0, which are known as the static single-layer and Neumann-Poincaré operators, respectively. By using the single layer potential (4.2.11) and the jump formula (4.2.12), one has the following integral representation for the solution to (4.2.8):  SDkcδ [φ](x), x ∈ Dδ , (4.2.14) .u(x) = km i u (x) + SDδ [ψ](x), x ∈ R3 \ Dδ , 1

1

where .(φ, ψ) ∈ H − 2 (∂Dδ ) × H − 2 (∂Dδ ) satisfy the following integral system ⎧ ⎨S kc [φ] − S km [ψ] = ui on ∂Dδ , D Dδ    δ . i ⎩ ε1 − 21 I + (KDkc )∗ [φ] − ε1 12 I + (KDkm )∗ [ψ] = ε1 ∂u on ∂Dδ . ∂ν c

δ

m

δ

m

(4.2.15) We focus on our analysis in the quasi-static regime, namely .ω · diam(Dδ ) ⪡ 1. By a standard scaling argument and without loss of generality, we could assume that .diam(Dδ ) ∼ 1 and .ω ⪡ 1. It is emphasised that .δ ⪡ 1 is also an asymptotic

4.2 Helmholtz Problem

149 f

parameter and hence .diam(Dδ ) ∼ diam(Dδ ). In fact, .δ is the key parameter that characterises the anisotropy of the geometry of the nanorod .Dδ and it shall be related to .ω in what follows. This should become more evident in our subsequent discussion. It is known that for .ω small enough, .SDk δ is invertible (cf. [56]). Therefore, by using the first equation in (4.2.15), one can directly obtain that  −1   φ = SDkcδ SDkmδ [ψ] + ui .

.

(4.2.16)

Then, from the second equation in (4.2.15), we have that ADδ (ω)[ψ] = f,

.

(4.2.17)

where    1 1 1 I + (KDkδm )∗ + I − (KDkδc )∗ (SDkcδ )−1 SDkmδ , . 2 εc 2 (4.2.18)   1 ∂ui 1 1 f =− (4.2.19) − I − (KDkδc )∗ (SDkcδ )−1 [ui ]. εm ∂ν εc 2

ADδ (ω) =

.

1 εm



Clearly,     1 1 1 1 I + KD∗δ + I − KD∗δ εm 2 εc 2     1 1 1 1 1 = + − I + KD∗δ . 2 εm εc εm εc

ADδ (0) = ADδ ,0 =

.

(4.2.20)

Similarly, for .ω ⪡ 1, we can also deduce the following operator equation of the density .φ: ADδ (ω)[φ] = f,

.

(4.2.21)

where .

   1 1 1 I − (KDkδc )∗ + I + (KDkδm )∗ (SDkmδ )−1 SDkcδ , . 2 εm 2 (4.2.22)   1 1 1 ∂ui (4.2.23) + I + (KDkδm )∗ (SDkmδ )−1 [ui ]. f = − εm ∂ν εm 2

1 ADδ (ω) = εc



Notice that .ADδ (0) = ADδ ,0 = ADδ (0).

150

4 Localized Resonances for Anisotropic Geometry

Finally, we introduce the formal definition of the plasmon resonance for our subsequent study. Definition 4.1 Consider the wave scattering system (4.2.8) associated with the nanorod .Dδ , where the material configuration is described in (4.2.6) and (4.2.7). (i) Plasmon resonance occurs if the following condition is fulfilled: .

  ∇us 

L2 (R3 \Dδ )

⪢ 1.

(4.2.24)

(ii) If the internal energy of wave field inside the nanorod significantly increases, i.e., .

‖∇u‖L2 (Dδ ) ⪢ 1,

(4.2.25)

then we also say that plasmon resonance occurs. Remark 4.1 According to (4.2.24) or (4.2.25), we see that when plasmon resonance occurs, the resonant wave field exhibits highly oscillatory behaviours that cause the blowup of the resonant energy in different senses. The high oscillation is a hallmark of the plasmon resonance and moreover the high oscillation is mainly confined within the vicinity of the nanostructure, which is the fundamental basis for many plasmonic technologies. Remark 4.2 We would like to emphasize that compared to the other definitions in the mathematical literature on the plasmon resonance, say e.g. Definition 1 in [17], our definition of the plasmon resonance is a more refined one. Indeed, in Definition 4.1, we characterize the resonance with respect to the wave fields both outside and inside the nanorod, as well as the electric energy. It is pointed out that in Definition 4.1, (i) and (ii) are essentially equivalent; see Theorems 4.4 and 4.6.

4.2.1 Asymptotic and Quantitative Analysis of the Scattering Field In this section, we conduct asymptotic analysis of the scattering system (4.2.8) associated with the nanorod .Dδ . We shall derive several asymptotic formulas of the wave field .us with respect to .ω ⪡ 1 and .δ ⪡ 1, which are crucial to our subsequent analysis of the resonant behaviours of the field, in particular the anisotropy of the resonant behaviours that is related to the anisotropic geometry of the nanorod. First, we derive several asymptotic expansion formulas of the layer potential operators with respect to the asymptotic wave number k as well as the anisotropic size parameter .δ of the nanorod. Those asymptotic results pave the way for the asymptotic analysis of the scattering system (4.2.8). It is pointed out that some of the asymptotic results can be found in [17, 56], which we still include for the self-

4.2 Helmholtz Problem

151

containedness, and some are new from the current setup of study whose proofs shall be given. First, it is recalled the layer potential operators .SΣk , (KΣk )∗ and .SΣ , KΣ∗ introduced earlier. For the subsequent use, we introduce the function space .H ∗ (∂Σ) which is a Hilbert space equipped with the following inner product 〈u, v〉H ∗ (∂Σ) = −(u, SΣ [v])− 1 , 1 ,

(4.2.26)

.

2 2

1

where .(·, ·)− 1 , 1 is the duality pairing between the Sobolev spaces .H − 2 (∂Σ) and 2 2

1

H 2 (∂Σ). It is noted that .‖ · ‖H ∗ (∂Σ) is equivalent to .‖ · ‖H −1/2 (∂Σ) (cf. [9]). The following lemma collects the necessary asymptotic results of the layer potential operators with respect to .k ⪡ 1, whose proof can be found in [17].

.

Lemma 4.1 For .k ⪡ 1, the following asymptotic results hold. (i) The single layer potential operator can be expanded as follows: SDk = SD +

∞ 

k j SD,j ,

(4.2.27)

|x − y|j −1 ψ(y)ds(y).

(4.2.28)

.

j =1

where ij .SD,j [ψ](x) = − 4πj !

∂D

Moreover, the norms .‖SD,j ‖L (H ∗ (∂D),H ∗ (∂D)) are uniformly bounded with respect to .j ≥ 1, and the series in (4.2.27) is convergent in ∗ ∗ .L (H (∂D), H (∂D)). (ii) It holds that (SDk )−1 = SD−1 + kBD,1 + k 2 BD,2 + · · · ,

.

(4.2.29)

where BD,1 = −SD−1 SD,1 SD−1 ,

.

BD,2 = −SD−1 SD,2 SD−1 + SD−1 SD,1 SD−1 SD,1 SD−1 .

(4.2.30)

Furthermore, the series in (4.2.29) is convergent in .L (H ∗ (∂D), H ∗ (∂D)). (iii) The Neumann-Poincaré operator .(KDk )∗ can be expanded as follows: (KDk )∗ = KD∗ +

∞ 

.

j =1

k j KD,j ,

(4.2.31)

152

4 Localized Resonances for Anisotropic Geometry

where KD,j [ψ](x) = −

.

ij (j − 1) 4πj !



|x−y|j −3 (x−y, ν(x))ψ(y)ds(y)

(4.2.32)

∂D

Furthermore, the norms .‖SD,j ‖L (H ∗ (∂D),H ∗ (∂D)) are uniformly bounded with respect to j , and the series in (4.2.31) is convergent in .L (H ∗ (∂D), H ∗ (∂D)). In particular, we notice the following result from Lemma 4.1: SD,1 [ψ](x) = −

.

i 4π

and

ψ(y)ds(y)

KD,1 = 0.

(4.2.33)

∂D

Let the boundary integral operators .KDδ ,j and .SDδ ,j be defined by (4.2.28) and (4.2.32) respectively. We next derive further asymptotic expansions of .KDδ ,j , .SDδ ,j , the single layer potential operator .SDδ and the static Neumann-Poincaré operator .KD∗δ with respect to the size scale .δ. ( Lemma 4.2 Let .ψ ∈ H ∗ (∂Dδ ) and .ψ .x ∈ ∂D, and x) = ψ(x) for .x ∈ ∂Dδ and  let .ι1,δ t ( x), .t = 12 or 1, be the region defined by ι1,δ t ( x) := { y | |zx − zy | < δ t , y ∈ ∂D}.

.

(4.2.34)

Then we have the following asymptotic results: (i) ∗  KD∗δ [ψ](x) = Kδ,c [ψ ]( x) + δKS∗f \ι

.

1,δ 1/2

]( [ψ x) + o(δ), c = {a, b}, (4.2.35)

where ∗  Kδ,c [ψ ]( x) :=

.

KS∗f \ι

1,δ 1/2

1 4π



]( [ψ x) :=

S c ∩ι1,δ ( x)

1 4π

( x − y + (δ −1 − 1)(zx − zy ), ν x ) ( ψ y)ds( y), . | x − y + (δ −1 − 1)(zx − zy )|3 (4.2.36)

S f \ι1,δ 1/2 ( x)

(zx − zy , ν x ) ( ψ y)ds( y). |zx − zy |3

(4.2.37)

(ii) (2)  (2) KDδ ,2 [ψ](x) = δ 2 Kδ,c [ψ ]( x) + δKS f \ι

.

1,δ 1/2

]( [ψ x) + o(δ), c = {a, b}, (4.2.38)

4.2 Helmholtz Problem

153

where ]( Kδ,c [ψ x) :=

.

(2)

(2)

KS f \ι

1,δ 1/2

1 8π



]( [ψ x) :=

( x − y + (δ −1 − 1)(zx − zy ), ν x ) ( ψ y)ds( y), . | x − y + (δ −1 − 1)(zx − zy )|

S c ∩ι1,δ ( x)

1 8π

(4.2.39)

S f \ι1,δ 1/2 ( x)

(zx − zy , ν x ) ( ψ y)ds( y). |zx − zy |

(4.2.40)

(iii) ]( ]( (KDδ ,j [ψ](x) = δ j KS c [ψ x) + δ j −1 KS f [ψ x), j ≥ 3, c = {a, b}, (4.2.41) where (j )

.

]( KS c [ψ x) := −

.

(j )

(j )  KS f [ψ ]( x)

(j )



ij (j − 1) 4πj !

Sc

( x − y + (δ −1 − 1)(zx − zy ), ν x ) ( ψ y)ds( y), . | x − y + (δ −1 − 1)(zx − zy )|3−j (4.2.42)



ij (j − 1) := − 4πj !

Sf

(zx − zy , ν x ) ( ψ y)ds( y). |zx − zy |3−j

(4.2.43)

Proof The proofs of (i) and (ii) can follow from a similar argument to that of Lemma 4.2 in [56], so we only give the proof of (iii) in what follows. From the definition of .KDδ ,j , we see that, for .j ≥ 3, the operator .KDδ ,j has no singularity. Then by the transformation formula (4.2.4), we have KDδ ,j [ψ](x) = −

.

=− =− −

ij (j − 1) 4πj ! − 1) 4πj !

ij (j

ij (j − 1) 4πj ! ij (j − 1) 4πj !



|x − y|j −3 (x − y, ν(x))ψ(y)ds(y)

∂Dδ



f

Sδc ∪Sδ



δj Sc



Sf

|x − y|j −3 (x − y, ν(x))ψ(y)ds(y)

( x − y + (δ −1 − 1)(zx − zy ), ν x ) ( ψ y)ds( y) | x − y + (δ −1 − 1)(zx − zy )|3−j

δ j −1

( x − y + (δ −1 − 1)(zx − zy ), ν x ) ( ψ y)ds( y) | x − y + (δ −1 − 1)(zx − zy )|3−j

]( ]( x) + δ j −1 · KS f [ψ x), = δ j · KS c [ψ (j )

which yields the assertion in (iii). The proof is complete.

(j )

⨆ ⨅

154

4 Localized Resonances for Anisotropic Geometry

Remark 4.3 Since the transformation A defined by (4.2.4) is a diffeomorphism from .∂Dδ onto .∂D, we can rewrite (4.2.35) as ∗ KD∗δ [ψ](x) = Kδ,c [A−1 ♢ ψ](A(x)) + δKS∗f \ι

.

1,δ 1/2

[A−1 ♢ ψ](A(x)) + o(δ),

c = {a, b},

(4.2.44)

( x) := ψ(A−1 ( x)) = ψ x), known as the pull-back transformation. where .(A−1 ♢ψ)( ∗ Thus, the integral operators .Kδ,c and .KS∗f \ι in the RHS of (4.2.35) should be 1,δ 1/2

seen as the operators from .H ∗ (∂Dδ ) to .H ∗ (∂Dδ ). In what follows, we always hold this view in our asymptotic analysis and spectral expansions.

We next present the expansion formulas for the operators .SDδ ,j and .SDδ with respect to the size parameter .δ ⪡ 1. ( .x ∈ ∂D. Then Lemma 4.3 Let .ψ ∈ H ∗ (∂Dδ ) and .ψ x) = ψ(x) for .x ∈ ∂Dδ and  the following asymptotic results hold: (i) ]( SDδ [ψ](x) = δSδ,c [ψ x) + δSS f \ι

.

1,δ 1/2

]( [ψ x) + o(δ), c = {a, b}, (4.2.45)

where ]( Sδ,c [ψ x) := −

.

SS f \ι

1,δ 1/2

]( [ψ x) := −

1 4π 1 4π

S c ∩ι1,δ ( x)

| x − y + (δ −1

1 ( ψ y)ds( y), . − 1)(zx − zy )| (4.2.46)

S f \ι1,δ 1/2 ( x)

1 ( ψ y)ds( y). |zx − zy |

(4.2.47)

(ii) ]( ]( SDδ ,j [ψ](x) = δ j +1 SS c [ψ x) + δ j SS f [ψ x), j ≥ 1, c = {a, b}, (4.2.48) (j )

.

(j )

where (j )  .S c [ψ ]( x) S

ij := − 4πj !

Sc

( | x − y + (δ −1 − 1)(zx − zy )|j −1 ψ y)ds( y), . (4.2.49)

(j )  SS f [ψ ]( x) := −

ij 4πj !

Sf

( | x − y + (δ −1 − 1)(zx − zy )|j −1 ψ y)ds( y). (4.2.50)

4.2 Helmholtz Problem

155

From (4.2.49) and (4.2.50), it is directly verified that (1)  .S c [ψ ]( x) S

ij := − 4πj !



( ψ y)ds( y),

(1)  SS f [ψ ]( x)

Sc

ij := − 4πj !



( ψ y)ds( y).

Sf

(4.2.51) Next, in order to analyze the plasmon resonance for the scattering system (4.2.8), we first establish the asymptotic expansion formula of the scattered field with respect to the angular frequency .ω ⪡ 1. To that end, we resent the following lemma. Lemma 4.4 ([9, 17]) Let .Dδ be defined in Section 4.1. Then (i) .KD∗δ is a compact self-adjoint operator in the Hilbert space .H ∗ (∂Dδ ).   (ii) Let . λj,δ ; ϕj,δ , j = 0, 1, 2, · · · , be the eigenvalue and eigenfunction pair of 1 1 1 ∗ .K Dδ , where .λ0 = 2 . Then, .λj,δ ∈ (− 2 , 2 ], and .λj,δ → 0 as .j → ∞. ∗ ∗ (iii) .H (∂Dδ ) = H0 (∂Dδ ) ⊕ {cϕ0 }, c ∈ C, where H0∗ (∂Dδ ) = {φ ∈ H ∗ (∂Dδ ) :

φds = 0}.

.

∂Dδ

1

(iv) For any .ψ ∈ H − 2 (∂Dδ ), it holds that KD∗δ [ψ] =

∞ 

.

λj,δ

j =0

〈ψ, ϕj,δ 〉H ∗ (∂Dδ ) ϕj,δ . 〈ϕj,δ , ϕj,δ 〉H ∗ (∂Dδ )

(4.2.52)

From now on, we use .(·, ·) as the standard inner product in .R3 . The inner product (4.2.26) and the corresponding norm on .∂Dδ are denoted by .〈·, ·〉 and .‖ · ‖ in short, respectively. .A ≲ B means .A ≤ CB for some generic positive constant C. .A ≈ B means that .A ≲ B and .B ≲ A. Owing to (4.2.52) and (4.2.20), we have ADδ ,0 [ψ] =

∞ 

.

τj,δ

j =0

〈ψ, ϕj,δ 〉 ϕj,δ , 〈ϕj,δ , ϕj,δ 〉

(4.2.53)

where τj,δ =

.

1 2



1 1 + εm εc



 +

1 1 − εm εc

 λj,δ .

(4.2.54)

In what follows, for notational convenience, we define aj,δ := 〈ϕj,δ , ϕj,δ 〉H ∗ (∂Dδ ) .

.

(4.2.55)

156

4 Localized Resonances for Anisotropic Geometry

We present the following lemma on the asymptotic expansion of the operator ADδ (ω) introduced in (4.2.18) with respect to .ω ⪡ 1, whose proof follows from straightforward computations and is omitted.

.

Lemma 4.5 The operator .ADδ (ω) : H ∗ (∂Dδ ) → H ∗ (∂Dδ ) has the following expansion: ADδ (ω) = ADδ ,0 + ω2 ADδ ,2 + O(ω3 )

.

(4.2.56)

with ADδ ,2 = (μm − μc )KDδ ,2 +

.

εm μm − εc μc εc



 1 I − KD∗δ SD−1 SDδ ,2 , δ 2 (4.2.57)

where .KDδ ,2 and .SDδ ,2 are defined in Sect. 2.1.3. For our subsequent use, we introduce the so-called index set of plasmon resonance. Definition 4.2 Let .τj,δ be introduced in (4.2.54). We say that .J ⊆ N is an index set of resonance if .τj,δ is close to zero when .j ∈ J and is bounded from below when c .j ∈ J := N\J . More precisely, we choose a threshold number .η0 > 0 independent of .ω and .δ such that .|τj,δ | ≥ η0 > 0, for .j ∈ J c . Next, we impose the following two mild conditions throughout our study: (C1) Each eigenvalue .λj,δ for .j ∈ J is a simple eigenvalue of the operator .KD∗δ . (C2) Suppose that .εc + εm /= 0. It is noted that in (4.2.6) and (4.2.7), we assume that .𝔍εc > 0, whereas .εm is a real constant. Hence, condition (C2) is easily fulfilled. We would like to point out that by condition (C2), one can deduce that the index set J is finite. Noting also that for 1 1 .j = 0, .λ0,δ = 2 , we see that .τ0,δ = μm ∼ 1. Thus we exclude 0 from the index set J . Lemma 4.6 In the quasi-static regime and under conditions (C1) and (C2), the scattering field .us to (4.2.8) has the following representation us = SDkmδ [ψ],

.

(4.2.58)

where ψ=

.

√ −1  iω μm εm (1/εc − 1/εm )aj,δ 〈d · ν, ϕj,δ 〉ϕj,δ + O(ω2 ) j ∈J

τj,δ + O(ω2 )

+ O(ω). (4.2.59)

4.2 Helmholtz Problem

157

Proof Note that the incident wave .ui = eikm d·x admits the following asymptotic expansion: ui = 1 + ikm d · x + O(ω2 ).

(4.2.60)

.

By combing (4.2.17), (4.2.18), (4.2.20), together with the spectral expansion (4.2.53), and the fact that  .

   I −1 i −1 I ∗ − KDδ SDδ [u ] = SDδ − KDδ [ui ] 2 2   −1 I 2 = ikm SD − K Dδ [x · d] + O(ω ) δ 2

(4.2.61) ⨆ ⨅

one can obtain (4.2.59).

Equations (4.2.58) and (4.2.59) give the asymptotic expansion of the scattered wave .us with respect to .ω ⪡ 1. It is noted that the anisotropic size parameter .δ is also asymptotically small. We shall derive further asymptotic expansions with respect to the size parameter .δ. We stress that this part is essential in our analysis, since the nanorod is anisotropic with respect to its dimensional sizes. First, by using the expansions of the layer potential operators .SDδ and .KD∗δ with respect to .δ, we have the following lemma. Lemma 4.7 The operator .ADδ (ω) : H ∗ (∂Dδ ) → H ∗ (∂Dδ ) has the expansion formula as follows ADδ (ω) = ADδ ,0 + ω2 δ ADδ ,2 + o(ω2 δ) + O(ω3 ),

(4.2.62)

.

where (2) ADδ ,2 = (μm − μc )KS f \ι

1,δ 1/2



× Sδ,c + SS f \ι

1,δ 1/2

∗ , .K with .Kδ,c S f \ι (2)

1,δ 1/2

, .Sδ,c , .SS f \ι

1,δ 1/2

εm μm − εc μc εc −1 + o(1) SS(2) f

+

.



1 ∗ I − Kδ,c 2



(2)

and .SS f defined in Section 2.1.3.

Proof Since .SDδ is invertible, by substituting the expansion formulas (4.2.35), (4.2.38), (4.2.45) and (4.2.48) into (4.2.56), one can derive (4.2.62) by direct calculations. ⨆ ⨅

158

4 Localized Resonances for Anisotropic Geometry

For the subsequent use, we define the following regions associated with .Dδ and D: ιδ (P0 ) := {y | |P0 − zy | < δ, y ∈ ∂Dδ }, .

(4.2.63)

ιδ (Q0 ) := {y | |Q0 − zy | < δ, y ∈ ∂Dδ }, .

(4.2.64)

ι1,δ (P0 ) := { y | |P0 − zy | < δ, y ∈ ∂D}, .

(4.2.65)

y | |Q0 − zy | < δ, y ∈ ∂D}. ι1,δ (Q0 ) := {

(4.2.66)

.

In what follows, we use .‖ · ‖Γ to represent the norm in .H ∗ (Γ ) (cf. (4.2.26)) for any surface .Γ . The following lemma is of critical importance for our subsequent analysis. ∗ and .K ∗ Lemma 4.8 Let .Kδ,c S f \ι

1,δ 1/2

be defined in Lemma 4.2. Then the following

results hold. a  ∈ H ∗ (∂D), we have (i) If .x ∈ Sδa or ( .x ∈ S ), for any .ψ ∗  .Kδ,a [ψ ]( x)

KS∗f \ι

1,δ 1/2

=

]( KS∗a [ψ x)

]( [ψ x) = KS∗f ,P

0



( x − y, ν x ) ( ψ y)ds( y), . | x − y|3  1  ]( ‖∂D , [ψ x) + o δ 2 ‖ψ

1 := 4π

(4.2.67)

Sa

(4.2.68)

where ]( KS∗f ,P [ψ x) =

.

0



1 4π

Sf

(P0 − zy , ν P0 ) ( ψ y)ds( y). |P0 − zy |3

(4.2.69)

b  ∈ H ∗ (∂D), we have (ii) If .x ∈ Sδb or ( .x ∈ S ), for any .ψ ∗  .Kδ,b [ψ ]( x)

KS∗f \ι

1,δ 1/2

=

]( KS∗b [ψ x)

]( [ψ x) = KS∗f ,Q

0



( x − y, ν x ) ( ψ y)ds( y), . | x − y|3  1  ]( ‖∂D , [ψ x) + o δ 2 ‖ψ

1 := 4π

(4.2.70)

Sb

(4.2.71)

where ]( KS∗f ,Q [ψ x) =

.

0

1 4π

Sf

(Q0 − zy , ν Q0 ) ( ψ y)ds( y). |Q0 − zy |3

(4.2.72)

4.2 Helmholtz Problem

159

f f  ∈ H ∗ (∂D), we have (iii) If .x ∈ Sδ ∩ ιδ (P0 ) or ( .x ∈ S ∩ ι1,δ (P0 )), for any .ψ

  ∗  ]( ‖∂D , . Kδ,a [ψ ]( x) = KS∗a ,P0 [ψ x) + O δ‖ψ  1  ]( ]( ‖∂D , KS∗f \ι [ψ x) = KS∗f [ψ x) + O δ 2 ‖ψ

.

1,δ 1/2

(4.2.73) (4.2.74)

where ∗ ]( .KS a ,P [ψ x) 0

]( KS∗f [ψ x)

1 = 4π

1 = 4π

Sa

Sf

( x1 −  y, νx1 ) ( ψ y)ds( y),  x1 ∈ S a ∩ S f , . | x1 −  y|3 (4.2.75)

(zx − zy , ν x ) ( ψ y)ds( y). |zx − zy |3

(4.2.76)

f  ∈ H ∗ (∂D), we have (iv) If .x ∈ Sδ ∩ ιδ (Q0 ) or ( .x ∈ S ∩ ι1,δ (Q0 )), for any .ψ f

  ∗  ]( ‖∂D , . Kδ,b [ψ ]( x) = KS∗b ,Q [ψ x) + O δ‖ψ 0  1  ]( ]( ‖∂D , KS∗f \ι [ψ x) = KS∗f [ψ x) + O δ 2 ‖ψ

.

1,δ 1/2

(4.2.77) (4.2.78)

where 1 = 4π

∗ ]( .K b [ψ x) S ,Q0

Sb

( x1 −  y, νx1 ) ( ψ y)ds( y),  x1 ∈ S b ∩ S f . | x1 −  y|3 (4.2.79)

f \ ι (P ) ∪ ι (Q )), for any .ψ  ∈ (v) If .x ∈ Sδ \ ιδ (P0 ) ∪ ιδ (Q0 ) or ( .x ∈ S 1,δ 0 1,δ 0 ∗ H (∂D), we have f

∗  Kδ,c [ψ ]( x) = 0, .

.

KS∗f \ι

1,δ 1/2





]( ]( ‖∂D . [ψ x) = KS∗f [ψ x) + O δ 2 ‖ψ 1

(4.2.80) (4.2.81)

Proof Since the proofs of (i)-(v) are similar, we only prove (i) in what follows. By ∗ and noting that .S a ∩ ι ( a .x ∈ S a , one has the definition of .Kδ,a 1,δ x) = S if  ∗  Kδ,a [ψ ]( x) =

.

=

1 4π 1 4π

S a ∩ι1,δ ( x)



Sa

( x − y + (δ −1 − 1)(zx − zy ), ν x ) ( ψ y)ds( y) | x − y + (δ −1 − 1)(zx − zy )|3

( x − y, ν x ) ( ψ y)ds( y). | x − y|3

160

4 Localized Resonances for Anisotropic Geometry

Moreover by direct asymptotic analysis, one has KS∗f \ι

.

1,δ 1/2

]( [ψ x) =

1 4π



S f \ι1,δ 1/2 ( x)

(zx − zy , ν x ) ( ψ y)ds( y) |zx − zy |3

(P0 − zy , ν P0 ) ( ψ y)ds( y) |P0 − zy |3

(zx − zy , ν x ) 1 ( ψ y)ds( y) − 4π ι 1/2 (P0 ) |zx − zy |3 1,δ

  1 (P0 − zy , ν P0 ) 1 ‖∂D . ( 2 ‖ψ = ψ y)ds( y) + o δ 4π S f |P0 − zy |3 =

1 4π

Sf

⨆ ⨅

The proof is complete.

In a similar manner, one can derive the expansion formulas of .Sδ,c and SS f \ι 1/2 as follows.

.

1,δ

Lemma 4.9 Let .Sδ,c , .SS f \ι 1/2 be defined in Lemma 4.3. Then the following 1,δ results hold. a  ∈ H ∗ (∂D), we have (i) If .x ∈ Sδa (or equivalently  .x ∈ S ), for any .ψ

SS f \ι

1,δ



1 ( ψ y)ds( y), . | x − y|  1  ](   2 ‖ψ [ ψ x) = S [ ψ ]( x) + o δ ‖ f ∂D , S ,P0 1/2

]( ]( Sδ,a [ψ x) = SS a [ψ x) := −

.

1 4π

(4.2.82)

Sa

(4.2.83)

where ]( SS f ,P0 [ψ x) = −

.

1 4π

Sf

1 ( ψ y)ds( y). |P0 − zy |

(4.2.84)

b  ∈ H ∗ (∂D), we have .x ∈ S ), for any .ψ (ii) If .x ∈ Sδb (or equivalently 

SS f \ι

1,δ



1 ( ψ y)ds( y), . x − y| S b |  1  ]( ]( ‖∂D , 2 ‖ψ [ ψ x) = S x) + o δ f ,Q [ψ S 1/2 0

]( ]( .Sδ,b [ψ x) = SS b [ψ x) := −

1 4π

(4.2.85) (4.2.86)

where ]( SS f ,Q0 [ψ x) = −

.

1 4π

Sf

1 ( ψ y)ds( y). |Q0 − zy |

(4.2.87)

4.2 Helmholtz Problem

161

f f  ∈ H ∗ (∂D), (iii) If .x ∈ Sδ ∩ ιδ (P0 ) (or equivalently  .x ∈ S ∩ ι1,δ (P0 )), for any .ψ we have

  ]( ]( ‖∂D , . Sδ,a [ψ x) = SS a ,P0 [ψ x) + O δ‖ψ  1  ]( ]( ‖∂D , SS f \ι 1/2 [ψ x) = SS f [ψ x) + O δ 2 ‖ψ

.

1,δ

(4.2.88) (4.2.89)

where ]( SS a ,P0 [ψ x) = −

.

]( SS f [ψ x) = −

1 4π

1 4π

Sa

Sf

1 ( ψ y)ds( y),  x1 ∈ S a ∩ S f , . | x1 −  y|

1 ( ψ y)ds( y). |zx − zy |

(4.2.90) (4.2.91)

Furthermore, the higher order terms in (4.2.88) and (4.2.89) are boundary integrals on subsets of .S f . f f  ∈ H ∗ (∂D), (iv) If .x ∈ Sδ ∩ ιδ (Q0 ) (or equivalently  .x ∈ S ∩ ι1,δ (Q0 )), for any .ψ we have   ]( ]( ‖∂D , . Sδ,b [ψ x) = SS b ,Q0 [ψ x) + O δ‖ψ  1  ]( ]( ‖∂D , x) = SS f [ψ x) + O δ 2 ‖ψ SS f \ι 1/2 [ψ

.

1,δ

(4.2.92) (4.2.93)

where ]( SS b ,Q0 [ψ x) = −

.

1 4π

Sb

1 ( ψ y)ds( y),  x1 ∈ S b ∩ S f . | x1 −  y|

(4.2.94)

Furthermore, the higher order terms in (4.2.92) and (4.2.93) are boundary integrals on subsets of .S f . f f (v) If .x ∈ Sδ \ ιδ (P0 ) ∪ ιδ (Q0 ) (or equivalently  .x ∈ S \ ι1,δ (P0 ) ∪ ι1,δ (Q0 )), for ∗  any .ψ ∈ H (∂D), we have ]( Sδ,c [ψ x) = 0, .

.

SS f \ι

1,δ 1/2





]( ]( ‖∂D . [ψ x) = SS f [ψ x) + O δ 2 ‖ψ 1

(4.2.95) (4.2.96)

Using the asymptotic results in Lemmas 4.8 and 4.9 and comparing with the asymptotic formula of Neumann-Poincaré operator .KD∗δ in (4.2.35), we can obtain the following more refined one, such that the asymptotic operators at the right hand side of (4.2.35) are independent of size scale .δ.

162

4 Localized Resonances for Anisotropic Geometry

( x) = ψ(x) for .x ∈ ∂Dδ and  Theorem 4.1 Let .ψ ∈ H ∗ (∂Dδ ) and .ψ .x ∈ ∂D. Then, we have that ]( ]( ‖∂D ), KD∗δ [ψ](x) = K0∗ [ψ x) + δK1∗ [ψ x) + o(δ‖ψ

.

(4.2.97)

where     ]( ]( ]( x) K0∗ [ψ x) :=χ S a KS∗a [ψ x) + χ S f ∩ ι1,δ (P0 ) KS∗a ,P0 [ψ .     ]( ]( + χ S f ∩ ι1,δ (Q0 ) KS∗b ,Q [ψ x) + χ S b KS∗b [ψ x), 0

(4.2.98) and       ]( ]( ]( ]( K1∗ [ψ x) := χ S a KS∗f ,P [ψ x)+χ S f KS∗f [ψ x)+χ S b KS∗f ,Q [ψ x).

.

0

0

Here, .χ denotes the characteristic function and the asymptotic operators .KS∗a , .KS∗b , ∗ ∗ ∗ ∗ ∗ .K f , .K a S ,P0 , .KS b ,Q , .KS f ,P and .KS f ,Q are defined in Lemma 4.8. S 0

0

0

Proof The proof follows from straightforward though a bit tedious calculations along with the use of Lemma 4.8 and (4.2.35). ⨆ ⨅ In what follows, we define . ϕj,δ ( x) := ϕj,δ (x) for .x ∈ ∂Dδ , and  .x ∈ ∂D. Let P denote the eigenprojection of .K0∗ on .H ∗ (∂Dδ ), i.e., .P : H ∗ (∂Dδ ) → Vλj , where ∗ .Vλj is the eigenspace associated with the eigenvalue .λj of .K . Furthermore, we 0 ∗ define the reduced resolvent for .λj of .K0 as (see [77]) Λ=

.

1 2π i

γj

(K0∗ − ξ )−1 dξ, ξ − λj

(4.2.99)

where .γj : |ξ − λj | = r is a circle in the complex plane enclosing the isolated eigenvalue .λj . By using the perturbation theory for the spectrum of a linear operator, one can obtain the following elementary result. Lemma 4.10 The operator .K0∗ defined in (4.2.98) admits the following spectral property, K0∗ [ ϕj ]( x) = λj  ϕj ( x)

.

(4.2.100)

with . ϕj ( x) := ϕj (x) and .λj and .ϕj satisfying λj,δ = λj + δλj,1 + O(δ 2 ),

.

ϕj,δ = ϕj − δΛK1∗ [ ϕj ] + O(δ 2 ),

(4.2.101)

4.2 Helmholtz Problem

163

where .λj,1 is the eigenvalue of the operator .P (K1∗ )P considered in the eigenspace .Vλj . Furthermore, one has  ϕj ( x) = 0,

.

 x ∈ Sf .

(4.2.102)

Proof Recall that .λj,δ and .ϕj,δ are the eigenvalues and eigenfunctions of .KD∗δ in ∗ .H (∂Dδ ), i.e., KD∗δ [ϕj,δ ] = λj,δ ϕj,δ .

.

By Theorem 4.1 and the perturbation theory for the spectrum of linear operators (see [77], pages 449, Theorem 2.9), one can find that the eigenvalues .λj,δ and eigenfunctions .ϕj,δ of .KD∗δ can be approximated in (4.2.101), where .λj and .ϕj satisfy (4.2.100). From the definition of .K0∗ in (4.2.98), one can easily see that  ϕj = 0 in

.

S f \ ι1,δ (P0 ) ∪ ι1,δ (Q0 ).

f Furthermore, if  .x ∈ S ∩ ι1,δ (P0 ), then one has

KS∗a ,P0 [ ϕj ]( x) = K0∗ [ ϕj ]( x) = λj  ϕj ( x),

.

and by the definition of .KS∗a ,P0 in (4.2.75), one thus has .

 ϕj ( x) = 0,

 x ∈ S f ∩ ι1,δ (P0 ).

 ϕj ( x) = 0,

 x ∈ S f ∩ ι1,δ (Q0 ).

Similarly, one has .

⨆ ⨅

One thus has (4.2.102) and completes the proof. ∗ .K 1

Remark 4.4 In fact, noting the definition of and using (4.2.102), we find K1∗ [ ϕj ] = 0, and then the expansion formula of the eigenfunction in (4.2.101) can be rewritten as

.

 ϕj,δ =  ϕj + O(δ 2 ).

(4.2.103)

.

By using Lemma 4.4, Lemma 4.10 and the anisotropic geometry of the nanorod Dδ , one can derive the following result.

.

Lemma 4.11 Let .ϕj be defined in Lemma 4.10, then it fulfils that



.

S a ∪S b

 ϕj,δ = O(δ 2 ),

and Sf

 ϕj,δ = o(δ 2 ),

j /= 0,

(4.2.104)

164

4 Localized Resonances for Anisotropic Geometry

∗ ∗ Proof Note that .{ϕj,δ }∞ j =0 are the eigenfunctions of .KDδ in .H (∂Dδ ). Direct computation shows

.

∂Dδ

ϕj,δ = −〈SD−1 [1], ϕj,δ 〉 = 0, δ

for

j /= 0.

(4.2.105)

By further using the asymptotic expansion



ϕj,δ =

.

∂Dδ

Sδa ∪Sδb





ϕj,δ +

f

Sδ

ϕj,δ = δ

2 S a ∪S b

 ϕj,δ + δ

Sf

 ϕj,δ ,

(4.2.106)

and the asymptotic expansion (4.2.103) one immediately achieves (4.2.104). The proof is complete.

⨆ ⨅

By using (4.2.97), together with Lemmas 4.11 and 4.10, the expression of the layer potential density associated with the nanorod can be deduced as follows, Theorem 4.2 Under the conditions (C1) and (C2), and assume that .ψ ∈ H ∗ (∂Dδ ) ( is the solution of the operator equation (4.2.17), where .ψ(x) = ψ x) for  .x ∈ ∂D. Then, for every .x ∈ ∂Dδ , the solution .ψ can be expressed as .ψ(x) = ψ1 (x) + O(ω), where .ψ1 is defined by ψ1 (x) =



.

j ∈J

√ −1 〈d · ν, ϕj,δ 〉 ϕj ( x) + O(ωδ 2 ) iω μm εm (1/εc − 1/εm )aj,δ     . 1 1 1 1 1 2 2 2 εm + εc + εm − εc λj + δλj,1 + o(δ) + O(ω δ) + O(ω ) (4.2.107)

In (4.2.107), .λj and .ϕj are respectively the eigenvalues and eigenfunctions of the operator .K0∗ . Moreover, for any surface .Γ , .Γ1 and .Γ2 , the operator .SˆΓ is defined by .SˆΓ [ ϕ ](x) =

Γ

G0 (x − zy ) ϕ ( y) d s( y),

(4.2.108)

and 〈·, ·〉H ∗ (Γ1 ,Γ2 ) := −(·, SˆΓ1 [·])L2 (Γ2 ) .

.

Setting 〈〈d · ν,  ϕj 〉〉 = 〈d · ν,  ϕj 〉H ∗ (S a ,S a ∪S f ) + 〈d · ν,  ϕj 〉H ∗ (S b ,S b ∪S f ) , .

.

aj,S c = 〈 ϕj ,  ϕj 〉H ∗ (S a ,S a ) + 〈 ϕj ,  ϕj 〉H ∗ (S b ,S b ) .

(4.2.109) (4.2.110)

4.2 Helmholtz Problem

165

Then, we have that .

〈〈d · ν,  ϕj 〉〉 + O(δ) 〈d · ν, ϕj,δ 〉 = . c aj,S + O(δ) aj,δ

(4.2.111)

Proof By Lemma 4.6 and using the eigenvalue and eigenfunction expansion formula in Lemma 4.10, one can deduce (4.2.107) by straightforward though tedious calculations. Next, we prove (4.2.111). By using (4.2.102) and (4.2.103), it deduces   〈d · ν, ϕj,δ 〉 = − d · ν, SDδ [ϕj,δ ] − 1 , 1 2 2



 G0 (x − zy ) + ∇G0 (x − zy )· =− d · ν(x)

.

∂Dδ

∂Dδ

 (y − zy ) + O(|y − zy |2 ) [ϕj,δ (y)]ds(y)ds(x) 



d · ν(x) G0 (x − zy ) ϕj ( y)δ 2 d s( y) + O(δ 3 ) d s( x) =− ∂Dδ

= −δ 3

Sa

d · ν( x)

Sb

d · ν( x)

−δ

Sf

Sf

Sb

d · ν( x)



− δ3

G0 ( x − zy ) ϕj ( y)d s( y)d s( x) G0 ( x − zy ) ϕj ( y)d s( y)d s( x)



3

Sa



− δ3

S a ∪S b





d · ν( x)

Sa

Sb

G0 (δ( x − zx ) + (zx − zy )) ϕj ( y)d s( y)d s( x) G0 (δ( x − zx ) + (zx − zy )) ϕj ( y)d s( y)

d s( x) + O(δ 4 )   = δ 3 〈d · ν,  ϕj 〉H ∗ (S a ,S a ∪S f ) + 〈d · ν,  ϕj 〉H ∗ (S b ,S b ∪S f ) + O(δ 4 ). ⨆ ⨅

Similarly, we can compute .aj,δ and then (4.2.111) holds.

Next, we present the asymptotic expansion of the scattering field in (4.2.8) associated with the nanorod .Dδ . Theorem 4.3 Let u be the solution to (4.2.8). Then, under conditions (C1) and (C2), the scattering field can be presented as u (x) =

.

s

 j ∈J

√ −1 iωδ 2 μm εm aj,δ 〈d · ν, ϕj,δ 〉SˆS c [ ϕj ](x) −1    λ εεmc − λj + δ ε1c − ε1m λj,1 + o(δ) + O(ω2 δ) + O(ω2 )

‖∂D ) + O(ωδ), + O(ωδ 3 ‖ψ

(4.2.112)

166

4 Localized Resonances for Anisotropic Geometry

where λ(t) =

.

t +1 . 2(t − 1)

(4.2.113)

Proof First, by the Taylor expansion around .y = zy for the Green function .Gkm (x − y), and using (4.2.102), (4.2.107), we have that SDkmδ [ψ](x)

u (x) =

.

s

− ∂Dδ

= ∂Dδ

Gkm (x − zy )ψ(y)ds(y)

‖∂D + ωδ). ∇Gkm (x − zy ) · (y − zy )ψ(y)ds(y) + O(δ 4 ‖ψ (4.2.114)

Furthermore, noticing that     ∂Dδ = S a ∪ S f ∩ ι1,δ (P0 ) ∪ S f \ ι1,δ (P0 ) ∪ ι1,δ (Q0 )   ∪ S f ∩ ι1,δ (Q0 ) ∪ S b ,

.

and then substituting (4.2.107) into (4.2.114) and using the results in Lemma 4.11, yield √ −1 〈d · ν, ϕj,δ 〉 iω μm εm (1/εc − 1/εm )aj,δ us (x) =

.

 1 j ∈J 2

+

−

−



 1 j ∈J 2

 1 j ∈J 2

 1 j ∈J 2

1 εm







+

1 εc



+ 



+

1 εm



ϕj ](x) + O(ωδ 2 ) + O(ω2 ) SˆS c [  δ2 1 2 2 − εc λj + δλj,1 + o(δ) + O(ω δ) + O(ω ) O(ωδ 2 )

+

1 εm

√ −1 iω μm εm (1/εc − 1/εm )aj,δ 〈d · ν, ϕj,δ 〉  ∂Gk ϕj ( y)d s( y) y ) S c ∂ν( y) (x − z    δ3 1 1 1 2 2 + εc + εm − εc λj + δλj,1 + o(δ) + O(ω δ) + O(ω )

1 εm

+

1 εc



+



1 εm

1 εm

−

1 εc

1 εc



λj + δλj,1 + o(δ) + O(ω2 δ) + O(ω2 )

δ

1 εm

1 εc

−



O(ωδ 2 ) λj + δλj,1 + o(δ) + O(ω2 δ) + O(ω2 )

δ2

‖∂D ), + O(ωδ) + O(δ 3 ‖ψ which proves the expansion formula (4.2.112). The proof is complete.

⨆ ⨅

4.2 Helmholtz Problem

167

Based on the asymptotic expansion formula of the scattering field .us in Theorem 4.3, we can perform some quantitative analysis of the scattering field associated with the nanorod .Dδ . We first consider the case that the nanorod is straight, namely .Γ0 is a straight line with length L. Moreover, we assume that .S a and .S b are two semi-spheres. Then using the definition in (4.2.108), the scattering field .us in (4.2.112) is given by us (x) = iωδ 2 p(x)



.

‖∂D ) + O(ωδ), κj,δ + O(δ 3 ‖ψ

(4.2.115)

j ∈J

where .κj,δ is defined by κj,δ :=

.

√

−1 μm εm aj,δ 〈d · ν, ϕj,δ 〉  ϕj , −1    εm 1 1 Sa λj,1 λ εc − λj + δ εc − εm

and the function .p(x) has the form p(x) :=

.

1 1 − . |x − P0 | |x − Q0 |

(4.2.116)

We mention that in order to get (4.2.115), we used the fact that



.

Sa

 ϕj = −

Sb

 ϕj ,

which can be verified by using (4.2.104). The amplitude of the scattering field us is mainly determined by the absolute value of function .p(x) in (4.2.116). By straightforward computations, one has

.

⎧   −1 −1  ⎪  ⎪ − l(x))2 + 1 − (L/2 + l(x))2 + 1 , x ∈ S f , ⎨ (L/2  −1 .|p(x)| = 1 − 1 + L2 + 2(x − P0 , P0 − Q0 ) x ∈ Sa , ⎪  ⎪ −1 ⎩ 1 − 1 + L2 + 2(x − Q0 , Q0 − P0 ) x ∈ Sb, (4.2.117) where  P0 + Q0   l(x) := zx − . 2

.

Finally, by some elementary analysis, one can find that .p(x) attains its maximum 1 − (1 + L)−1 at the two ending points of .Sδa and .Sδb , and attains its minimum 0 f on the centering parts (circular area) of .Sδ . The behaviour is more obvious when L is larger. Hence, we can conclude that the scattering field .us behaves stronger (in

.

168

4 Localized Resonances for Anisotropic Geometry

terms of the amplitude of the wave field) on the two end-parts of the nanorod .Sδa and f b .S than that on the facade-part of the nanorod .S . δ δ For the general case with a generically curved nanorod, we believe the same quantitative behaviours should hold in a certain sense. In what follows, we present two numerical examples to illustrate such quantitative behaviours for the scattering field of the Helmholtz system (4.2.8) associated with a curved nanorod .Dδ ; see Figs. 4.4 and 4.5. In the sequel, we adopt the coordinate notation .x = (xj )3j =1 ∈ R3 . In Fig. 4.4, .Dδ is generated by a straight .Γ0 (along the .x3 -axis) of length 4 and two end-caps being two semi-balls of radius .δ; whereas in Fig. 4.5, .Dδ is generated 1

1

5

-5

0

0

-5

0.8 0.6 0.4 0.2

5

2 0 -2

-20 2

-5

0.8 0.6

0

0.4 0.2

5 -2 0 2

-2 0 2

Fig. 4.4 Left: The geometry of a straight nanorod; Middle: Slice plot of the normalized scattering field .|ℜus | on the .(x2 , x3 )-plane, where the incident direction is .d = (1, 0, 0); Right: Slice plot of the normalized scattering field .|ℜus | on the .(x2 , x3 )-plane, where the incident direction is .d = (0, 0, 1). The incident wave is: .ui (x) = 103 eikm d·x 1

5

5

0

0

-5 -2 0 2

0.8 0.6 0.4 0.2

-5

-2 0 2

-2 0

2

1

5

0.8 0.6

0

0.4 0.2

-5 -2 0

2

Fig. 4.5 Left: The geometry of a curved nanorod; Middle: Slice plot of the normalized scattering field .|ℜus | on the .(x2 , x3 )-plane, where the incident direction is .d = (1, 0, 0); Right: Slice plot of the normalized scattering field .|ℜus | on the .(x2 , x3 )-plane, where the incident direction is .d = (0, 0, 1). The incident wave is: .ui (x) = 103 eikm d·x

4.2 Helmholtz Problem

169

by a curved .Γ0 and two end-caps being two semi-balls of radius .δ, where the parametrization of .Γ0 is .x(t) = (xj (t))3j =1 : ⎧ ⎪ ⎪ ⎨x1 (t) = 0, . x2 (t) = 12 (cos(t) − 1), ⎪ ⎪ ⎩x (t) = 2 sin(t), t ∈ [− π + 3 2

(4.2.118) 3 π 10 , 2

−

3 10 ].

The other key parameters are given as follows, 1 δ=√ , 2

.

ω = δ2,

εc = −1 + iδ 4 ,

εm = 1,

μDδ ≡ 1.

(4.2.119)

Figures 4.4 and 4.5 respectively plot the absolute value of the real parts of numerically computed wave fields, namely .|ℜus |, associated with the two nanorods described above. It is remarked that .ℜu and .ℜus are the physical fields. Here, in order to present a better display, we normalize the wave fields in the sense that the maximum value of .|ℜus | in both figures is 1. It is evident that the wave field attains its maximum amplitude at the two-end parts of the nanorod. It is emphasized that the choice of the material configuration in (4.2.119), in particular .ℜεc = −1 is mainly based on numerical simplicity and convenience. One can pick the other choices, say e.g. .ℜεc = −2, and would have similar numerical behaviours. The numerical simulations are not the main focus here, and we only present a typical example for illustration. On the other hand, we would like to point out that the material configuration (4.2.119) is nearly resonant according to Definition 4.2 with .η0 being set sufficiently small. We shall consider this example again in our subsequent resonance study and provide more relevant discussions in what follows.

4.2.2 Resonance Analysis of the Exterior Wave Field Using the asymptotic result in Theorem 4.3, we proceed to analyze the plasmon resonance of the scattering system (4.2.8). We first derive the gradient estimate of the scattering field .us outside the nanorod .Dδ . −1 Lemma 4.12 Let .ψ = ψc + a0,δ 〈ψ, ϕ0,δ 〉ϕ0,δ , where .ψc ∈ H0∗ (∂D). Then for the

scattering solution of (4.2.8), i.e., .us = SDkmδ [ψ], we have the following estimate .

  2    −1  〈ψ, ϕ0,δ 〉 . ‖∇us ‖2L2 (R3 \D ) − ‖ψc ‖2  ≲ ωa0,δ δ

(4.2.120)

170

4 Localized Resonances for Anisotropic Geometry

Proof Let .BR be a sufficiently large ball such that .Dδ ⊂ BR . By using the divergence theorem in .BR \ Dδ , the jump relation (4.2.12) and the Sommerfeld radiation condition, it follows that

.

BR \Dδ

∂us  ∂us ds us +ds + ∂ν ∂R ∂BR ∂Dδ BR \Dδ  



1 2 = k¯m |us |2 dx − SDkmδ [ψ] I + (KDkδm )∗ [ψ]ds 2 ∂Dδ BR \Dδ

+ us · ikm us + O(R −2 )ds.

2 |∇us |2 dx = k¯m





|us |2 dx −

us

∂BR

From the expansion formulas (4.2.27) and (4.2.31), we obtain  

    1   . SDkmδ [ψ] I + (KDkδm )∗ [ψ]ds    ∂Dδ 2 

     1   ∗ ≤ SDδ [ψ] I + KDδ [ψ]ds  + |E| ,  ∂Dδ  2

(4.2.121)

where

E=

.

∂Dδ

SDδ [ψ]

+

∞ 

j

km KDδ ,j [ψ]ds

j =1 ∞ 

∂Dδ j =1

 j km SDδ ,j [ψ]

 1 km ∗ I + (KDδ ) [ψ]ds. 2

Owing to .ω is sufficiently small, by Cauchy’s inequality, it is easy to see that .|E| ≲      2 2 km ‖ψ‖ ≲ ω‖ψ‖ . Next, we estimate . ∂Dδ SDδ [ψ] 12 I + KD∗δ [ψ]ds . Since −1 KD∗δ [ϕ0,δ ] = 12 ϕ0,δ and .SDδ [ϕ0,δ ] = 0, for .ψ = ψc + a0,δ 〈ψ, ϕ0,δ 〉ϕ0,δ , it implies that

.

 1 ∗ SDδ [ψ] I + KDδ [ψ]ds 2 ∂Dδ    

1 −1 I + KD∗δ [ψc ] ds = 〈ψ, ϕ0,δ 〉ϕ0,δ + SDδ [ψc ] a0,δ 2 ∂Dδ ⎡ ⎤

∞  SDδ ⎣ a −1 〈ψc , ϕj,δ 〉ϕj,δ ⎦ = 

.

∂Dδ

j,δ

j =1

4.2 Helmholtz Problem

171

−1 〈ψ, ϕ0,δ 〉ϕ0,δ a0,δ

×

∞ 

=

∂Dδ j =1

 +

 ! ∞

−1 〈ψ, ϕ0,δ 〉ϕ0,δ a0,δ

=

+

∞ 

−1 al,δ 〈ψc , ϕl,δ 〉

−1 −1 a0,δ 〈ψ, ϕ0,δ 〉aj,δ 〈ψc , ϕj,δ 〉

j =1

+

ds

l=1



l=1 ∞ 

" −1 al,δ 〈ψc , ϕl,δ 〉ϕl,δ

−1 aj,δ 〈ψc , ϕj,δ 〉SDδ [ϕj,δ ]

×

1 I + KD∗δ 2

 1 + λl,δ ϕl,δ ds 2

SDδ [ϕj,δ ]ϕ0,δ ds

∂Dδ



∞   1 −1 −1 〈ψc , ϕl,δ 〉aj,δ 〈ψc , ϕj,δ 〉 SDδ [ϕj,δ ]ϕl,δ ds. + λl,δ al,δ 2 ∂Dδ

j,l=1

Noting that

.

∂Dδ

SDδ [ϕj,δ ]ϕl,δ ds = −〈ϕl,δ , ϕj,δ 〉 =

 −aj,δ , l = j, 0, l /= j,

it deduces that

.



  ∞    2 1 1 −1  ∗ 〈ψc , ϕj,δ 〉 . SDδ [ψ] I + KDδ [ψ]ds = − + λj,δ aj,δ 2 2 ∂Dδ j =1

  Since .λj,δ ∈ − 12 , 12 .(j ≥ 1), we can obtain  

    1   ∗ . SDδ [ψ] I + KDδ [ψ]ds  ≈ ‖ψc ‖2 .   ∂Dδ 2

(4.2.122)

Moreover, by Cauchy’s inequality, it follows .



  

∂BR

  us · ikm us + O(R −2 )ds  ≲ω‖us ‖2L2 (∂B

R

+ )

≲ω‖ψ‖ + O(R 2

−1

|us · O(R −2 )|ds

∂BR

) · ‖ψ‖.

(4.2.123)

By combing (4.2.122) and (4.2.123), for .ω ⪡ 1, we obtain ‖∇us ‖2L2 (B

.

R \Dδ )

 2 −1  〈ψ, ϕ0,δ 〉 + O(R −1 ) · ‖ψ‖. ≲ ‖ψc ‖2 + ωa0,δ

172

4 Localized Resonances for Anisotropic Geometry

Similarly, by (4.2.122) and (4.2.123), we also deduce the inverse inequality as ‖∇us ‖2L2 (B

.

R \Dδ )

2  −1  〈ψ, ϕ0,δ 〉 − O(R −1 ) · ‖ψ‖. ≳ ‖ψc ‖2 − ωa0,δ

Hence, by letting .R → ∞, we see that the estimate (4.2.120) holds. The proof is complete.

⨆ ⨅

Before proceeding with the gradient analysis of the scattering field .us

outside the nanorod .Dδ , we consider the parameter choice of the permittivity with an imaginary part. In fact, in real applications, nano-metal materials always contain losses, which are reflected in the imaginary part of a complex electric permittivity .εc . As shall be shown, like the frequency .ω and the size .δ, the lossy parameter also plays a key role  in the plasmon resonance of the scattering field of the nanorod. Let .θ = ℜ ε1c ,   1 .ρ = 𝔍 εc < 0. Then .τj,δ given by (4.2.54) can be written as τj,δ =

.

1 1 −1 −1 ) − (θ − εm )λj,δ + ρ( − λj,δ )i. (θ + εm 2 2

(4.2.124)

Next, by considering the principal equation .ADδ ,0 [ψ0 ] = f0 , where .ADδ ,0 is defined by (4.2.20) and .ψ0 , f0 ∈ H ∗ (∂Dδ ), similar to (4.2.59), applying the eigenfunction expansion, it follows that ψ0 = AD−1 [f0 ] = δ ,0

.

∞ a −1 〈f , ϕ 〉  j,δ 0 j,δ

τj,δ

j =0

(4.2.125)

ϕj,δ .

Lemma 4.13 Under conditions (C1) and (C2), .ψ0 is given by (4.2.125) and has the decomposition .ψ0 = ψ0,c + cϕ0 , (.ψ0,c ∈ H0∗ (∂Dδ ), c is a constant). Then, for sufficiently small .|ρ|, it holds that ∗ ∗ (1) .‖AD−1 ‖ ≲ |ρ|−1 . δ ,0 L (H (∂Dδ ),H (∂Dδ ))

θ+ε−1

−1 m (2) If . 12 −1 /= λj,δ , (j ≥ 0), then .‖ADδ ,0 ‖L (H ∗ (∂Dδ ),H ∗ (∂Dδ )) ≲ C for some θ−εm positive constant C. 1   −1 θ+εm −1 a − 2 〈f , ϕ 〉. (3) If . 1 0 j,δ −1 = λj,δ for some .j ≥ 1, then .‖ψ0,c ‖ ≳ |ρ| j,δ

2 θ−εm

Proof

   −1  (1) For .j /= 0, since .τj,δ ≲ ‖ψ0 ‖2 ≲ |ρ|−2

1 |ρ|( 21 −λj,δ )

∞ 

.

≲ |ρ|−1 , it follows that

2  −1  〈f0 , ϕj,δ 〉 ≲ |ρ|−2 ‖f0 ‖2 . aj,δ

j =0 ∗ ∗ Hence, .‖AD−1 ‖ ≲ |ρ|−1 . δ ,0 L (H (∂Dδ ),H (∂Dδ ))

(4.2.126)

4.2 Helmholtz Problem

173

   θ+εm−1  /= λj,δ , then, for .j ≥ 0, we have . 12 − λ  ≥ c0 , where j,δ −1 θ−εm    −1  2 2 .c0 is a positive constant. Therefore, .τ j,δ  ≲ 1 and .‖ψ0 ‖ ≲ ‖f0 ‖ , i.e.,

(2) If

−1 1 θ+εm −1 2 θ−εm

.

∗ ∗ ‖AD−1 ‖ ≲ C. δ ,0 L (H (∂Dδ ),H (∂Dδ ))

.

θ+ε−1

1 m (3) When . 12 −1 = λj,δ for some .j ≥ 1, by (4.2.124), one has .τj,δ = ρ( 2 −λj,δ )i, θ−εm it then follows that

‖ψ0,c ‖ ≳

.

−1 aj,δ2

 −1    aj,δ2 〈f0 , ϕj,δ 〉  −1  〈ψ0,c , ϕj,δ 〉 ≳   ≳ |ρ|−1 aj,δ2 〈f0 , ϕj,δ 〉 ,  1  ρ( 2 − λj,δ ) (4.2.127)

which completes the proof. ⨆ ⨅ With Lemma 4.13, we can establish the following key result for estimating the gradient of the scattering field .us , which provide resonant and non-resonant conditions for the scattering system (4.2.8) associated with the nanorod .Dδ according to criterion (4.2.24) in Definition 4.1. Theorem 4.4 Let .us be the scattering solution of (4.2.8). Suppose that .|ρ|−1 ω2 δ ≤ c1 for a sufficiently small .c1 , then under conditions (C1) and (C2), we have the following results. θ+ε−1

m (1) If . 12 −1 /= λj,δ for any .j ≥ 0, there exists a constant C independent of .δ such θ−εm that

.

(2) If . 12 .

−1 θ+εm −1 θ−εm

  ∇us  2 3 ≤ C. L (R \D ) δ

(4.2.128)

−1

= λj,δ , and .aj,S2c 〈〈d · ν,  ϕj 〉〉 /= 0 for some .j ≥ 1, it holds

  ∇us 

3

L2 (R3 \Dδ )

1

≳ O(|ρ|−1 ωδ 2 ) + O(|ρ|−1 ωδ 2 ) + O(|ρ|−1 ω2 ) + O(ω 2 ). (4.2.129) 3

Furthermore, assuming that .|ρ| = o(ωδ 2 ) and .|ρ|−1 ω2 = O(1) (as .ω → 0, .δ → 0), then it follows that .

  ∇us 

L2 (R3 \Dδ )

→ ∞ as ω → 0.

(4.2.130)

174

4 Localized Resonances for Anisotropic Geometry

Proof (1) From (4.2.62), it deduces that    2  2 3 ω ADδ (ω) = ADδ ,0 I + AD−1 δ A + o(ω δ) + O(ω ) , D ,2 δ δ ,0

.

(4.2.131) and then   −1 ω2 δ ADδ ,2 + o(ω2 δ) + O(ω3 ) ψ = I + AD−1 AD−1 [f ]. δ ,0 δ ,0

.

(4.2.132) By Lemma 4.13 (1), one finds .

    −1  ADδ ,0 ω2 δ ADδ ,2 + o(ω2 δ) + O(ω3 ) 

L (H ∗ (∂Dδ ),H ∗ (∂Dδ ))

≲ |ρ|−1 ω2 δ. (4.2.133)

Hence, it follows that   −1  −1 2  2 3 ω . ‖ψ − ψ0 ‖ =  I + A δ A + o(ω δ) + O(ω ) Dδ ,2 Dδ ,0   

AD−1 [f ] − AD−1 [f ]  δ ,0 δ ,0     ≲ |ρ|−1 ω2 δ AD−1 [f ]  δ ,0 = |ρ|−1 ω2 δ ‖ψ0 ‖ . If . 12 .

−1 θ+εm −1 θ−εm

(4.2.134)

/= λj,δ , by Lemma 4.13 (2), we obtain

    ‖ψ‖ ≲ (1 + |ρ|−1 ω2 δ) ‖ψ0 ‖ = (1 + |ρ|−1 ω2 δ) AD−1 [f ]  ≲ (1 + c1 ) ‖f ‖ . ,0 δ

Then, from Lemma 4.12, it yields .

(2) If . 21

−1 θ+εm −1 θ−εm

  ∇us 2 2

L (R3 \Dδ )

≲ ‖ψ‖2 ≲ C.

= λj,δ , (.j ≥ 1), then, by using Lemma 4.13 (3), we have that  −1  ‖ψ0,c ‖ ≳ |ρ|−1 aj,δ2 〈f, ϕj,δ 〉 .

.

(4.2.135)

4.2 Helmholtz Problem

175

Moreover, from (4.2.134), it implies     |ρ|−1 ω2 δ ‖ψ0 ‖ ≳ ‖ψ − ψ0 ‖ ≳ ψc − ψ0,c  ≳ ψ0,c  − ‖ψc ‖ . (4.2.136)

.

Combining now (4.2.135) and (4.2.136), and noticing that .|ρ|−1 ω2 δ ≤ c1 for a sufficiently small .c1 , we have .

   −1  ‖ψc ‖ ≳ ψ0,c  − |ρ|−1 ω2 δ ‖ψ0 ‖ ≳ |ρ|−1 aj,δ2 〈f, ϕj,δ 〉 .

(4.2.137)

Therefore, we obtain from Lemma 4.12 that .

 s 2 ∇u  2

L (R3 \Dδ )

2  −1  〈ψ, ϕ0,δ 〉 ≳ ‖ψc ‖2 − ωa0,δ 2 2   −1  −1  〈f, ϕj,δ 〉 − ωa0,δ 〈ψ, ϕ0,δ 〉 . ≳ |ρ|−2 aj,δ

Furthermore, similar to the proof of Lemma 4.6, we find √ 〈f, ϕj,δ 〉 = iω μm εm (1/εc − 1/εm )〈d · ν, ϕj,δ 〉 + O(ω2 ).

.

On the other hand, from the proof of Theorem 4.2, it implies that 2  δ 3 〈〈d · ν,  ϕj 〉〉2 −1  〈d · ν, ϕj,δ 〉 = aj,δ + O(δ 4 ). aj,S c + O(δ)

.

Thus, by noting that .|〈ψ, ϕ0,δ 〉| is bounded, (4.2.129) holds and then by 3 choosing .|ρ| = o(ωδ 2 ) and such that .|ρ|−1 ω2 = O(1), one has (4.2.130). ⨆ ⨅

The proof is complete. Remark 4.5

It is noted that .|ρ|−1 ω2

=

O(1) implies that .|ρ|−1 ω2 δ

From (2) of Theorem 4.4, one readily sees that

−1 θ+εm when . 12 −1 θ−εm

→ 0 as .δ → 0. 3

= λj,δ , .|ρ| = o(ωδ 2 )

and .|ρ|−1 ω2 = O(1) (as .ω, δ → 0), the gradient of the scattering field blows up. According to (4.2.24) in Definition 4.1, we see that the plasmon resonance occurs. Notice that the last two conditions on the resonant material configuration are very flexible, and for example, one can take .|ρ| = ω2 , .δ = ωs , (.0 < s < 23 ). Remark 4.6 Note that case (2) in Theorem 4.2.122 is the resonance condition, i.e. −1 1 θ+εm . −1 = λj,δ , .(j ≥ 1). If the lossy parameter of the nanorod .𝔍(εc ) → 0, the 2 θ−εm     resonance condition is consistent with .τj,δ = 12 ε1m + ε1c + ε1m − ε1c λj,δ → 0, which appeared in (4.2.54) and (4.2.59).

176

4 Localized Resonances for Anisotropic Geometry

4.2.3 Resonance Analysis of the Interior Wave Field Owing to the relationship between the internal density .φ and the scattering density ψ, we can deduce the asymptotic formula of the interior field with respect to .ω as follows.

.

Lemma 4.14 In the quasi-static regime, under conditions (C1) and (C2), the interior field .u|Dδ has the following representation u(x) = SDkcδ [φ](x), x ∈ Dδ ,

(4.2.138)

.

where φ=

.

√ −1  iω μm εm (1/εc − 1/εm )aj,δ 〈d · ν, ϕj,δ 〉ϕj,δ + O(ω2 ) τj,δ + O(ω2 )

j ∈J

+ cϕ0,δ + O(ω), (4.2.139)

and c is a constant. Proof Substituting (4.2.27), (4.2.29) into (4.2.16), and using the Taylor expansion ui = 1 + ikm d · x + O(ω2 ), we conclude

.

φ = (SD−1 + kc BDδ ,1 + kc2 BDδ ,2 + · · · ) δ ⎡ ⎤ ∞  j × ⎣(SDδ + km SDδ ,j )[ψ] + 1 + ikm d · x + O(ω2 )⎦

.

j =1

= ψ + SD−1 [1] + O(ω). δ

(4.2.140)

From (4.2.59), and noticing .SD−1 [1] = cϕ0,δ (c is a constant), it follows that δ φ=

.

√ −1  iω μm εm (1/εc − 1/εm )aj,δ 〈d · ν, ϕj,δ 〉ϕj,δ + O(ω2 ) j ∈J

τj,δ + O(ω2 )

which readily completes the proof.

+ cϕ0,δ + O(ω), ⨆ ⨅

In what follows, we shall make use of Lemma 4.14 to derive the asymptotic form of the internal energy.

4.2 Helmholtz Problem

177

Theorem 4.5 Let u be the solution to (4.2.8). Then, under conditions (C1) and (C2), for every .x ∈ Dδ , the interior field can be presented as u(x) =



.

j ∈J

# −1 ωi μεmm aj,δ 〈d · ν, ϕj,δ 〉SˆS c [ ϕj ](x)δ 2 + o(ωδ 2 ) −1    λj,1 + o(δ) + O(ω) λ εεmc − λj + δ ε1c − ε1m

‖∂D ) + O(ωδ), + o(δ) + O(δ 3 ‖φ

(4.2.141)

where .aj,δ , .SˆS c , .λj,1 and function .λ(t) are defined in Theorem 4.3. Proof From Lemmas 4.14 and 4.10, it follows that φ=

.

√ −1  iω μm εm (1/εc − 1/εm )aj,δ 〈d · ν, ϕj,δ 〉 ϕj ( x) + o(ωδ)     1 1 1 1 1 j ∈J 2 εm + εc + εm − εc λj + δλj,1 + o(δ) + O(ω) x) + o(δ) + O(ω). + c ϕ0 (

Substituting the above identity into the interior field formula .u = SDkcδ [φ], and the by following a similar argument to that in the proof of Theorem 4.3, one can prove (4.2.141). ⨆ ⨅ We are now in a position to derive the energy estimate of the scattering system (4.2.8) within the curved nanorod .Dδ . Specifically, we shall show that under the same resonant condition in Theorem 4.4 (which characterises the resonance of the wave field outside the nanorod .Dδ ), the internal energy of the nanorod blows up as well. −1 Lemma 4.15 Let .φ = φc + a0,δ 〈φ, ϕ0,δ 〉ϕ0,δ , where .φc ∈ H0∗ (∂D). Then for the

interior solution of (4.2.8) in .Dδ , i.e., .u = SDkcδ [φ] in .Dδ , we have the following estimate:   2   −1  2 2 〈φ, ϕ0,δ 〉 . . ‖∇u‖ 2 − ‖φ ‖ (4.2.142)  ≲ ωa0,δ c L (Dδ ) Proof Using the divergence theorem in .Dδ , we have that

.

|∇u|2 dx = kc

2

= kc

2



Dδ



    ∂u    |u|2 dx +  u  ds    − ∂ν Dδ ∂Dδ



|u| dx + 2

Dδ

∂Dδ

 SDkcδ [φ]

 1 kc ∗ I + (KDδ ) [φ] ds. 2

Next, by using the same argument as that in the proof of Lemma 4.12, one can derive the estimate (4.2.142). The proof is complete. ⨆ ⨅

178

4 Localized Resonances for Anisotropic Geometry

Theorem 4.6 Let u be the solution of (4.2.8) inside .Dδ . Assume that .|ρ|−1 ω2 δ ≤ c1 for a sufficiently small .c1 , under conditions (C1) and (C2), we have the following results. θ+ε−1

m (1) If . 12 −1 /= λj,δ for any .j ≥ 0, then there exists a constant C independent of .δ θ−εm such that

‖∇u‖L2 (Dδ ) ≤ C.

.

(2) If . 21

−1 θ+εm −1 θ−εm

(4.2.143)

−1

ϕj 〉〉 /= 0 for some .j ≥ 1, it follows that = λj,δ , and .aj,S2c 〈〈d · ν,  1

3

.

‖∇u‖L2 (Dδ ) ≳ O(|ρ|−1 ωδ 2 ) + O(|ρ|−1 ωδ 2 ) + O(|ρ|−1 ω2 ) + O(ω 2 ). (4.2.144) 3

Furthermore, assuming that .|ρ| = o(ωδ 2 ) and .|ρ|−1 ω2 = O(1) (as .ω → 0, .δ → 0), then it holds that .

‖∇u‖L2 (Dδ ) → ∞ as ω → 0.

(4.2.145)

Proof (1) From the relation (4.2.140), one sees that .‖φ‖ ⩽ ‖ψ‖+‖ϕ0,δ ‖+O(ω). Noticing the estimate of .ψ, we have .

 2   ‖∇u‖2L2 (D ) = ∇SDkcδ [φ] 2 δ

L (Dδ )

≲ ‖φ‖2 ≲ ‖ψ‖2 + ‖ϕ0,δ ‖2 + O(ω2 ) ≲ C.

(2) Since −1 φc + a0,δ 〈φ, ϕ0,δ 〉ϕ0,δ = φ = ψ + cϕ0,δ + O(ω)

.

−1 〈ψ, ϕ0,δ 〉)ϕ0,δ + O(ω), = ψc + (c + a0,δ

one can deduce that 〈φ, ϕ0,δ 〉 = (c · a0,δ + 〈ψ, ϕ0,δ 〉) + O(ω),

.

〈φc , ϕj,δ 〉 = 〈ψc , ϕj,δ 〉 + O(ω). By Lemma 4.15, we have that .

 2 ‖∇u‖2L2 (D ) ≳ ‖φc ‖2 − ω 〈φ, ϕ0,δ 〉 δ  2 2    −1 −1  〈φ, ϕ0,δ 〉 ≳ ψc + (c + a0,δ 〈ψ − φ, ϕ0,δ 〉)ϕ0,δ + O(ω) − ωa0,δ

4.2 Helmholtz Problem

179

2    −1 ≳ ‖ψc ‖ − (c + a0,δ 〈ψ − φ, ϕ0,δ 〉) ϕ0,δ  − O(ω)  2 −1  c · a0,δ + 〈ψ, ϕ0,δ 〉 + O(ω) − ωa0,δ 2  −1  〈ψ, ϕ0,δ 〉 + O(ω) ≳ ‖ψc ‖2 − ωa0,δ 2 2   −1  −1  〈f, ϕj,δ 〉 − ωa0,δ 〈ψ, ϕ0,δ 〉 + O(ω). ≳ |ρ|−2 aj,δ Hence, following the proof of (2) of Theorem 4.4, one can prove the assertion. ⨆ ⨅ Up till now, we have analyzed the plasmon resonance of the Helmholtz system (4.2.8) associate with the nanorod .Dδ in terms of the blowup of the external energy (Theorem 4.4), the internal energy (Theorem 4.6) and the electric energy. Following similar analysis to as before, one can show more quantitative properties of the resonant fields, .∇us (x) for .x ∈ R3 \Dδ and .∇u(x) for .x ∈ Dδ . In particular, the anisotropy of the resonant behaviours due to the anisotropic geometry of .Dδ . In principle, one can expect that the resonance strength, namely .|∇us (x)| should be stronger in the vicinity of the two end-parts than that in the vicinity of the facade-part of the nanorod .Dδ . To achieve a concrete and vivid idea of such anisotropic resonant behaviours, under the same setup of the numerical examples in Figs. 4.4 and 4.5, we numerically compute the corresponding gradient fields and present them in Figs. 4.6 and 4.7. Before that, we notice from the material configuration in (4.2.119) one has .

−1 1 θ + εm 1 −1 + (1 + ω8 ) · → 0, as ω → 0. = −1 2 −1 − (1 + ω8 ) 2 θ − εm

Owing to 0 is the unique accumulation point of the eigenvalues of .KD∗δ , the θ+ε−1

m resonance condition . 12 −1 = λj,δ in general does not holds exactly, but holds θ−εm approximately for .ω ⪡ 1. In order to fulfil the resonance condition exactly, one

Fig. 4.6 Left: Slice plot of the normalized resonance strength for the scattering field in the middle figure of Fig. 4.4 on the .(x2 , x3 )-plane; 2 .Eo ≈ 8.31 × 10 and 3 .Ei ≈ 1.35 × 10 ; Right: Slice plot of the normalized resonance strength for the scattering field in the right figure of Fig. 4.4 on the .(x2 , x3 )-plane; 3 .Eo ≈ 6.72 × 10 and 2 .Ei ≈ 6.71 × 10

1

1

-5

0.8 0.6

0

0.4 0.2

5 -2 0 2

-5

0.8 0.6

0

0.4 0.2

5 -2 0 2

180

4 Localized Resonances for Anisotropic Geometry

1

1

5

0.8 0.6

0

0.4 0.2

-5 -2 0

2

5

0.8 0.6

0

0.4 0.2

-5 -2 0

2

Fig. 4.7 Left: Slice plot of the normalized resonance strength for the scattering field in middle figure of Fig. 4.5 on the .(x2 , x3 )-plane; .Eo ≈ 1.19 × 103 and .Ei ≈ 1.46 × 103 ; Right: Slice plot of the normalized resonance strength for the scattering field in middle figure of Fig. 4.5 on the 3 3 .(x2 , x3 )-plane; .Eo ≈ 8.33 × 10 and .Ei ≈ 8.48 × 10

needs to compute the eigenvalues of .KD∗δ , from which to determine an appropriate .ℜεc . This is not an easy task and is out of the scope here as we mentioned before that numerical study is not our focus. Nevertheless, according to our analysis made above, we know that there are many values of .ℜεc centering around .−1 such that the resonance condition is fulfilled exactly. We vary .ℜεc around .−1 and the obtained numerical behaviours are similar to the case with .ℜεc = −1. Hence it is unobjectionable that we stick to the material configuration in (4.2.119) for our numerical simulations of the resonant fields. With such a clarification, we can proceed with the numerical experiments. Set Θ(x) = |∇ℜus (x)|χR3 \Dδ (x) + |∇ℜ(u(x) − ui (x))|χDδ (x), x ∈ R3 ,

.

which is referred to as the resonance strength of the scattering field. In order to have a better display, similar to the plotting in Figs. 4.4 and 4.5, we shall normalize .Θ(x) such that its maximum value is 1. Let .Θnormal be the normalized scattering strength, which is a scalar function and can be used to depict the quantitative resonant behaviours for the scattering wave field. In addition, we define Eo = ‖∇ℜus (x)‖L2 (B\Dδ ) ,

.

Ei = ‖∇ℜu(x)‖L2 (Dδ ) ,

where B is the computing region in our numerical experiment. .Eo and .Ei are respectively the exterior and interior scattering energies of the resonant wave field. In Figs. 4.6 and 4.7, we respectively plot the normalized resonant strengths of the wave fields in Figs. 4.4 and 4.5. It can be easily seen that the resonant strength is generically stronger in the vicinity of the two end-parts than that in the vicinity of the facade-part of the nanorod.

4.2 Helmholtz Problem

181

4.2.4 Conclusion We have considered the plasmon resonance associated with the Hemholtz system in the quasi-static regime. The plasmon resonance is a delicate and subtle resonant phenomenon and its occurrence and quantitative behaviours critically depend on the metamaterial parameters (namely, the negative material parameters), the frequency of the impinging wave field and the geometry of the nanostructures that are coupled together in a highly intricate manner. Most of the existing theoretical studies are concerned with nanostructures of isotropic geometries that are uniformly small in all dimensions, which help to diminish the geometric influence in the resonance analysis. We investigated the plasmon resonance for curved nanorods, which present anisotropic geometries. The anisotropic geometries present significant challenges for the resonance analysis. By developing novel asymptotic analysis and spectral analysis techniques, we established comprehensive resonance and non-resonance results for this type of nanostructures through carefully analyzing the wave fields and their quantitative behaviours inside and outside the nanostructures. Moreover, through our delicate and subtle analysis, we could quantitatively characterize the anisotropy of the resonant behaviours that is caused by the anisotropic geometries of the curved nanorods. It is mentioned that the geometric properties of the plasmon resonance were also studied in two recent articles [6, 27]. In fact, it is shown that the Neumann-Poincaré (NP) eigenfunctions, namely the eigenfunctions for ∗ , localize at high-curvature places of .∂Σ in the Neumann-Poincaré operator .KΣ the high-mode-number limit. The localizing properties of the NP eigenfunctions indicate that the plasmon resonance is stronger at places on .∂Σ where the curvature is sufficiently high, provided the resonance field is caused by the high-mode-number NP eigenfunctions. It is emphasized that the geometric setup considered in [6, 27] cannot include the one considered here. Indeed, it is required in [6, 27] that the highcurvature place on .∂Σ is again of a locally isotropic nature where the curvature is uniformly high in all directions. Finally, we would like to point out that the mathematical methods developed in this work pave the way for many subsequent developments in studying the quantitative geometric properties of the plasmon resonances in different setups as well as considering the corresponding applications, in particular the one for the full Maxwell system that governs the general electromagnetic wave scattering.

Chapter 5

Localized Resonances Beyond the Quasi-Static Approximation

Based on the results in Chaps. 2–4, it is natural to consider the plasmon and polariton resonances beyond the quasi-static approximation. This can be obtained via considering the spectral properties of the layer potential operators introduced in the previous chapters with frequencies attached to integral kernels. This makes the spectral analysis much radically more challenging. In fact, the relevant layer potential operators are no longer self-adjoint in any function space and many classical spectral tools, say e.g. diagonalization, do not apply. Nevertheless, one still can manage to establish the resonance conditions which couple the geometric and medium parameters of the material structure as well as the frequency in a highly intricate and delicate manner. It turns out that the resonant fields possess distinct properties with some of them similar to the quasi-static resonances and some different. Moreover, it is inappropriate to discuss the polarisability of the material structure of a large size (compared to the operating wavelength). Hence, we refer to those resonances as atypical resonances. Interestingly, it is seemingly questionable to consider resonances from metamaterial structures beyond the quasistatic regime since in the current stage, metamaterials are mainly realized or observed in the nanoscale. However on the one hand, the corresponding resonance study is mathematically rigorous and sound, and hence it is of significant theoretical interest and on the other hand, it can serve as an inspiring source for technological advancement. In fact, it is not unusual in the scientific history that the development of theory is ahead of the corresponding technological application. The main results in this chapter follow from [55, 63, 91, 92, 94].

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Y. Deng, H. Liu, Spectral Theory of Localized Resonances and Applications, https://doi.org/10.1007/978-981-99-6244-0_5

183

184

5 Localized Resonances Beyond the Quasi-Static Approximation

5.1 Spectral System of Neumann-Poincaré Operators in Helmoholtz System and Its Asymptotic Behavior Let us consider the following Helmholtz system in .Rd , .d = 2, 3: ⎧ 2 ⎪ x ∈ Rd , ⎨ ∇ · (ε(x)∇u(x)) + k u(x) = 0,   . (d−1)/2 x ⎪ · ∇u(x) − iku(x) = 0, ⎩ lim |x| |x|→∞ |x|

(5.1.1)

where .ε(x) denotes for the material parameter. Suppose .ε(x) = (ε0 − 1)χ (D) + 1, where .χ (D) is the indicator function for inclusion D and .ε0 is the material parameter of D. The shape of the inclusion D is essentially connected with the spectral of the Neumann-Poincaré operator .(KDk )∗ . In static case, that is .k = 0 , the spectral of .(KD0 )∗ has been studied widely and the eigenvalues have been found elaborately for some special cases, i.e., D is a disk, ball or ellipse. In [94], the authors present one form of the eigenvalues of .(KDk )∗ when D is a ball for finite frequency k. However, the relation between asymptotic behavior of spectral of .(KDk )∗ and spectral of .(KD0 )∗ is still not known. We first derive some different forms of eigenvalues of .(KDk )∗ when D is a ball, and then show that the eigenvalues approach exactly to the eigenvalues of .(KD0 )∗ when k goes to zero. We also show that the eigenvalues converges to zero form the complex plane in general. Our main results for three dimensional case are listed in Theorems 5.1.1 and 5.1.2. The asymptotic behavior of spectral of .(KDk )∗ , where D is a disk, is listed in Theorem 5.1.4.

5.1.1 Layer Potential and Spectral Properties of Neumann-Poincaré Operator in R3 In this section, we present the spectral of Neumann-Poincaré operator. Before this, we recall some preliminary results for spherical Bessel and Neumann functions. It is known that the spherical Bessel and Neumann functions are solutions to the following spherical Bessel differential equation: t 2 f '' (t) + 2tf ' (t) + (t 2 − n(n + 1))f (t) = 0,

.

n = 0, 1, 2, . . . .

(5.1.2)

In the sequel, unless otherwise stated, n is always chosen for nonnegative nature numbers, i.e., .n = 0, 1, 2, . . .. The spherical Bessel and Neumann functions are then defined by jn (t) :=

∞ 

.

l=0

(−1)l 2l l!1 · 3 · · · (2n + 2l

+ 1)

t 2l+n ,

(5.1.3)

5.1 Spectral System of Neumann-Poincaré Operators in Helmoholtz System. . .

185

and ∞

yn (t) := −

.

(−1)l (2n)!  t 2l−n−1 , n l 2 n! 2 l!(−2n + 1)(−2n + 3) · · · (−2n + 2l − 1) l=0

(5.1.4) The linear combination h(1) n := jn (t) + iyn (t)

(5.1.5)

.

is called the spherical Hankel function of first kind of order n. For subsequent usage, we present the following famous Wronskian identity: '

(1) jn' (t)h(1) n (t) − jn (t)hn (t) = −

.

i . t2

(5.1.6)

For n sufficiently large enough, there holds the following asymptotic behavior : jn (t) =

.

 1  2n n!t n  1+O , (2n + 1)! n

(5.1.7)

uniformly on compact subsets of .R and yn (t) =

.

 1  (2n)!  , 1 + O n i2n n!t n+1

(5.1.8)

uniformly on compact subsets of .(0, ∞). In what follows we define by .Ynm the spherical harmonics of order n and degree m and .Pn (t) the Legendre polynomial of order n. For any .x ∈ R3 , let .xˆ := x/|x| be the unit vector. We present the following Funk-Hecke formula (cf. [114]): Lemma 5.1.1 Suppose that .f (t) is continuous for .t ∈ [−1, 1], then there holds that . f (ˆx · yˆ )Ynm (ˆy)ds = λYnm (ˆx), (5.1.9) S2

with λ := 2π

1

.

−1

f (t)Pn (t)dt.

(5.1.10)

We shall present the spectral of Neumann-Poincaré operator .(KBkR )∗ , where .BR is a ball with radius R. First, we have the following scaling result (see, e.g., [12]): Lemma 5.1.2 The spectral of .(KBkR )∗ is the same with the spectral of .(KBkR )∗ . 1

186

5 Localized Resonances Beyond the Quasi-Static Approximation

Without loss of generality, in the sequel, we only consider the spectral of .(KBk )∗ , where B is a unit ball. Lemma 5.1.3 There holds the following: (KBk )∗ [Ynm ] =

.



−

 1  1 ' m 2 ' (1) − ik 2 jn (k)h(1) − ik (k) Y = j (k)h (k) Ynm . n n n n 2 2 (5.1.11)

We mention that (5.1.11) is proved in [94], with some additional assumption. Here, we shall present a different proof without any additional assumption. Proof It is shown in [44] that m SBk [Ynm (ˆz)](ˆx) = −ikjn (k)h(1) x), n (k|x|)Yn (ˆ

.

|x| > 1.

(5.1.12)

By using the jump formula (1.2.5) from the outside of B one then has I .

2

 ' m + (KBk )∗ [Ynm (ˆz)](ˆx) = −ik 2 jn (k)h(1) x), n (k)Yn (ˆ

(5.1.13)

which proves the first equality in (5.1.11). By using (5.1.6) one thus has the second equality in (5.1.11). The proof is complete. ⨆ ⨅ We have another form of the spectral of Neumann-Poincaré operator. Before this, we present the following useful result (cf. [94]): Lemma 5.1.4 For any .φ ∈ H −1/2 (∂B), there holds the following identity: 1 ik (KBk )∗ [φ] = − SBk [φ] + EBk [φ], 2 2

.

(5.1.14)

where the operator .EBk : H −1/2 (∂B) → H 1/2 (∂B) is defined by EBk [φ](x) := −

.

1 4π

eik|x−y| φ(y)dsy .

(5.1.15)

∂B

Lemma 5.1.5 There holds the following: (KBk )∗ [Ynm ] =

.

ik m (jn (k)h(1) n (k) + cn,k )Yn , 2

(5.1.16)

where .cn,k is defined by cn,k := −

.

1 2



1

−1

eik

√

2(1−t)

Pn (t)dt.

(5.1.17)

5.1 Spectral System of Neumann-Poincaré Operators in Helmoholtz System. . .

187

Proof By using (5.1.12) and the continuous of .SBk across .∂B one has m SBk [Ynm ] = −ikjn (k)h(1) n (k)Yn .

.

(5.1.18)

Note that |x − y| =

.

2 − 2x · y,

x, y ∈ ∂B,

together with the definition (5.1.15) and Funk-Hecke formula (5.1.9), one thus has EBk [Ynm ] = cn,k Ynm .

.

(5.1.19)

By substituting (5.1.18) and (5.1.19) back into (5.1.14) one thus has (5.1.16), which completes the proof. ⨆ ⨅ By combining the two forms of eigenvalues (5.1.11) and (5.1.16), one can find the following result: Lemma 5.1.6 There holds the following: h(1) n (k) =

.

1 − ikcn,k , ik(jn (k) + 2kjn' (k))

(5.1.20)

where .cn,k is defined in (5.1.17). If .k = k0 , where .jn' (k0 ) = −1/(2k0 )jn (k0 ) then the above equality is realized as an limit for .k → k0 .

5.1.2 Asymptotic Behavior of Spectral System of Neumann-Poincaré Operator We shall derive the asymptotic behavior of the Neumann-Poincaré operator .(KBk )∗ when the frequency k is sufficiently small or n is sufficiently large enough. We first present some auxiliary results. Lemma 5.1.7 Let .cn,k be defined in (5.1.17). Suppose k is sufficiently small, then there holds the following: 4 c0,k = −1 − ik + O(k 2 ), 3

.

(5.1.21)

and cn,k

.

√ 1 √ 2 ik 1 − tPn (t)dt + O(k 2 ). =− 2 −1

(5.1.22)

188

5 Localized Resonances Beyond the Quasi-Static Approximation

Proof The proof is straight forward by using Taloy expansion and noticing the orthogonality of the Legendre polynomial .Pn (t). ⨆ ⨅ Lemma 5.1.8 Suppose k is sufficiently small, then there holds kjn' (k) = njn (k) + O(k n+2 ).

.

(5.1.23)

Proof We begin with .n = 0, then by .j0 (k) = sin(k)/k, one immediately has (5.1.23) by Taylor expansion. Now suppose .n /= 0, then by using (5.1.3) one has kjn' (k) = n

.

2n n! k n + O(k n+2 ), (2n + 1)!

jn (k) =

2n n! k n + O(k n+2 ), (2n + 1)! ⨆ ⨅

which verifies (5.1.23) and the proof is complete. We are now in the position of presenting our main results. Suppose that (KBk )∗ [Ynm ] = τn,k Ynm .

.

(5.1.24)

One can find the explicit form of .τn,k from (5.1.11) and (5.1.16). Theorem 5.1.1 Let .τn,k be defined in (5.1.24). Then there holds the following .

lim τn,k =

k→0

1 . 2(2n + 1)

(5.1.25)

Before the proof of Theorem 5.1.1, we want to make a remark. It is shown in [16] that (KB0 )∗ [Ynm ] =

.

1 Y m. 2(2n + 1) n

(5.1.26)

Thus Theorem 5.1.1 indicates that the spectral of Neumann-Poincaré operator is continuous near the origin. Proof By using (5.1.16) and (5.1.20) one has τn,k =

.

 ik  1 − ikcn,k + c jn (k) n,k ik(jn (k) + 2kjn' (k)) 2

(5.1.27)

If .n = 0, then by .j0 (k) = sin(k)/k, one can directly calculate (5.1.27) with .n = 0 to get .

lim τ0,k = lim

k→0

k→0

sin(k) i 1 = . 2 i sin(k) + 2i(k cos(k) − sin(k)) 2

(5.1.28)

5.1 Spectral System of Neumann-Poincaré Operators in Helmoholtz System. . .

189

If .n /= 0, by using Lemmas 5.1.7 and 5.1.8, one then has .

jn (k) 1 1 1 1 = lim = . ' ' k→0 2 1 + 2kjn (k)/jn (k) k→0 2 jn (k) + 2kjn (k) 2(2n + 1) (5.1.29)

lim τn,k = lim

k→0

⨆ ⨅

The proof is complete.

Theorem 5.1.2 Let .τn,k be defined in (5.1.24). Then for n sufficiently large enough, there holds the following:  k  i 1 + kcn,k + O 3/2 , 2(2n + 1) 2 n

τn,k =

.

(5.1.30)

where  1  cn,k = O √ . n

.

(5.1.31)

Proof Suppose that n is sufficiently large enough, then by using (5.1.7), one obtains that kjn' (k) =

.

 1  n2n n!k n  1+O . (2n + 1)! n

(5.1.32)

By using the orthogonality property of the Legendre polynomial .Pn (t), that is

1

.

−1

Pm (t)Pn (t) =

2 δmn , 2n + 1

(5.1.33)

where .δmn denotes for the Kronecker delta, which equals to one if .m = n and zero otherwise, and Cauchy-Schwarz inequality one derives that  1 1 √  1 1  √ 2 1/2 2   2(1−t) ik 2(1−t) ik  dt e Pn (t)dt  ≤ .|cn,k | =  − e 2n + 1 2 −1 2 −1 =√

1 2n + 1

.

Then by using (5.1.7), (5.1.27) and (5.1.32) one has τn,k =

.

 1  ik  k  i 1 − ikcn,k  1 1+O + cn,k = + kcn,k + O 3/2 , 2(2n + 1) n 2 2(2n + 1) 2 n (5.1.34)

which completes the proof.

⨆ ⨅

190

5 Localized Resonances Beyond the Quasi-Static Approximation

(0)

Fig. 5.1 Real part, imaginary part and modulus of .τn,k with respect to .0 ≤ k ≤ 20. The curves in the figure represent for .n = 0, 5, 10, 15, 20, respectively

From (5.1.35) one can find that the convergence rate of the spectral is no faster than .1/(4n). The coefficient .cn,k also plays an important role for the convergence of the spectral and it is a complex number in general, thus the spectral of .(KBk )∗ (0) should converge to zero from the complex plane in general. Let .τn,k := 1/(2(2n + (0)

1)) + 2i kcn,k be the leading order in (5.1.35), the numerical illustrations of .τn,k , .0 ≤ k ≤ 20 with different choice of n are presented in Figs. 5.1 and 5.2. In Fig. 5.3 (0) we illustrate the convergence of .|τn,k | for different choice of k. We mention that the estimate of .cn,k in (5.1.31) is not optimal. Actually we have a more elaborate estimate for .τn,k other than (5.1.35), when n is sufficiently large enough. We present the results as follows: Theorem 5.1.3 Let .τn,k be defined in (5.1.24). Then for n sufficiently large enough, there holds the following: τn,k =

.

where .O

  1 n

  1    1 1+O + iO jn (k)2 , 2(2n + 1) n

  and .O jn (k)2 are taking real values.

(5.1.35)

5.1 Spectral System of Neumann-Poincaré Operators in Helmoholtz System. . .

191

(0)

Fig. 5.2 Real part, imaginary part and modulus of .τn,k with respect to .0 ≤ k ≤ 20. The curves in the figure represent for .n = 6, 18, 30, 42, respectively

5.1.3 Two Dimensional Case We present the spectral of Neumann-Poincareé operator in .R2 for Helmholtz system (5.1.1). We suppose that Q is a disk of radius one and .x := (|x| cos θx , |x| sin θx ). Recall the Graf’s formula ([1, 19]) H0(1) (k|x − y|) =



.

Hn(1) (k|x|)einθx Jn (k|y|)e−inθy ,

|x| > |y|,

(5.1.36)

n∈Z (1)

where .Hn is the Hankel function of the first kind of order n, that is Hn(1) (t) = Jn (t) + iNn (t),

.

(5.1.37)

where .Jn and .Nn are Bessel functions of order n of the first and the second kind, respectively. Similar to the three dimensional case, we have the following result:

192

5 Localized Resonances Beyond the Quasi-Static Approximation

(0)

Fig. 5.3 Modulus of .τn,k with respect to .0 ≤ n ≤ 30. The curves in the figure represent for .k = 0, 7.5, 15, 22.5, 30, respectively

Lemma 5.1.9 There holds the following for .|x| > 1 SQk [einθ ](x) = −

.

iπ Jn (k)Hn(1) (k|x|)einθx . 2

(5.1.38)

Proof By using (5.1.36) and the orthogonality of .einθ on a unit circle, .n ∈ Z, one has  i Hm(1) (k|x|)eimθx Jm (k|y|)e−imθy einθy dsy SQk [einθ ](x) = − 4 ∂Q .

=− =−



i 4

m∈Z

2π

0



Hm(1) (k|x|)eimθx Jm (k)e−i(m−n)θy dθy

(5.1.39)

m∈Z

iπ Jn (k)Hn(1) (k|x|)einθx , 2 ⨆ ⨅

which completes the proof. In what follows, we present the Wronskian identity for .Jn and .Hn , which is '

Jn' (t)Hn(1) (t) − Jn (t)Hn(1) (t) = −

.

2i . πt

(5.1.40)

5.1 Spectral System of Neumann-Poincaré Operators in Helmoholtz System. . .

193

By using (5.1.38) and (5.1.40) one thus has the eigenvalues of .(KQk )∗ in the following: Lemma 5.1.10 There holds the following (KQk )∗ [einθ ] =

.



−

  1 iπ  1 iπ ' − kJn (k)Hn(1) (k) einθ = − kJn' (k)Hn(1) (k) einθ . 2 2 2 2 (5.1.41)

Proof The proof follows similar to the proof of Lemma 5.1.3.

⨆ ⨅

of .(KQk )∗

We have presented the spectral when Q is a unit disk in (5.1.41). We shall show the asymptotic behavior of the spectral when k is sufficiently small, or n (1) is sufficiently large. Recall that .Jn (k) and .Hn (k) admits the following asymptotic behavior: ⎧ k2 ⎪ ⎪ ⎪ 1 − + O(k 4 ), n = 0 and k re . (1) Let .jn (t) and .hn (t) be, respectively, the spherical Bessel and Hankel functions of order .n ∈ N, and .Yn (ˆx) be the spherical harmonics. Let .Gk (x) be the fundamental solution of the operator .Δ + k 2 , namely Gk (x) = −

.

eik|x| . 4π |x|

(5.2.13)

The Newtonian potential of .f (x) is defined as F (x) =

.

R3

Gk (x − y)f (y)dy,

x ∈ R3 ,

(5.2.14)

which verifies .(Δ + k 2 )F (x) = 0, x ∈ BR1 . For .x ∈ BR1 , the potential .F (x) can be written as F (x) =

∞ 

.

n=0

βn jn (kr)Yn (ˆx).

(5.2.15)

5.2 Helmholtz System

201

With the above preparations, the solution to (5.2.11) can be written for .x ∈ BR1 as  u(x) =

.

∞

an jn (k1,δ r)Yn (ˆx), x ∈ Bre , (1) x) + cn hn (kr)Yn (ˆx), x ∈ BR1 \Bre . n=0 bn jn (kr)Yn (ˆ

n=0 ∞

(5.2.16)

Applying the third condition in (5.2.11) on .∂Bre to u represented in (5.2.16), we have  (1) an jn (k1,δ re ) = bn jn (kre ) + cn hn (kre ), . (5.2.17) √ (1)' ϵs + iδan jn' (k1,δ re ) = bn jn' (kre ) + cn hn (kre ). Solving the equations in (5.2.17), we further have ⎧ (1)' jn' (kre )h(1) ⎪ n (kre ) − hn (kre )jn (kre ) ⎪ ⎪ , ⎨ an = bn √ ϵs + iδjn' (k1,δ re )h(1) (kre ) − hn(1)' (kre )jn (k1,δ re ) n√ . ' ' ⎪ ⎪ c = b jn (kre )jn (k1,δ re ) − ϵs + iδjn (k1,δ re )jn (kre ) . ⎪ n√ ⎩ n (1) (1)' ϵs + iδjn' (k1,δ re )hn (kre ) − hn (kre )jn (k1,δ re )

(5.2.18)

Since when .|x| > re , .u(x) − F (x) satisfies (Δ+k 2 )(u(x)−F (x)) = 0 and

(u(x)−F (x)) → 0 as |x| → ∞,

(5.2.19)

bn = βn .

(5.2.20)

.

one can show that there holds .

Therefore, the solution to (5.2.11) is given by (5.2.16) with the coefficients defined in (5.2.18) and (5.2.20). We are in a position to give the representation of the energy dissipation .Eδ [u]. Direct calculations together with the help of Green’s formula yield that  Eδ [u] = δ (k1,δ )



|u| dx +

2

2

Bre .

∂Bre

 ∞  2 = δ|an | (k1,δ )2

re 0

n=0

∂u uds(x) ∂ν

|jn (k1,δ r)r|

2



dx + k1,δ re2 jn' (k1,δ re )jn (k1,δ re )

 .

(5.2.21) Clearly, (5.2.21) indicates that if there exists .n0 ∈ N such that δ|an0 |2 → ∞ as δ → δ0 ,

.

(5.2.22)

202

5 Localized Resonances Beyond the Quasi-Static Approximation

then plasmon resonance occurs. To that end, we next analyze .an . Using the fact jn (t)hn(1)' (t) − jn' (t)h(1) n (t) =

.

i , t2

one has from (5.2.18) and (5.2.20) that an =

.

−iβn 1 . √ 2 (1) (1)' ' (kre ) ϵs + iδjn (k1,δ re )hn (kre ) − hn (kre )jn (k1,δ re )

(5.2.23)

In order to make (5.2.22) happen, one should have by using (5.2.23) that

(1)' ϵs + iδjn' (k1,δ re )h(1) n (kre ) − hn (kre )jn (k1,δ re ) → 0 as δ → δ0 .

.

(5.2.24)

That is, we need to determine an appropriate plasmon material distribution .ϵs + iδ such that (5.2.24) can occur. It is noted that (5.2.24) is a nonlinear equation in terms of .ϵs + iδ and n. The rest of this subsection is devoted to analyzing this nonlinear equation. Based on our earlier calculations, we first numerically solve the nonlinear Eq. (5.2.24) to obtain some plasmonic structures that can induce the resonance. For simplicity, we set .k = 1 and .re = 1, and choose the source f such that the Newtonian potential .F (x) in (5.2.15) satisfy .βn = 1 when .n = n0 and .βn = 0 when .n /= n0 . Let the plasmon parameters be chosen as follows, ϵs = ϵn 0

.

and

δ = δn0 ,

(5.2.25)

which depends on .n0 . We numerically find the following parameters that can induce resonance, ⎧ ϵn ⎪ ⎪ ⎨ 0 ϵn 0 . ⎪ ϵ ⎪ ⎩ n0 ϵn 0

= −1.303728 = −1.237160 = −1.224395 = −1.190550

δn0 δn0 δn0 δn0

= 0.498620 = 0.038434 = 0.001203 = 0.000019

for for for for

n0 n0 n0 n0

= 1, = 2, = 3, = 4.

(5.2.26)

In Fig. 5.7, we plot the energy .Eδ against the change of the loss parameter .δ when ϵn0 is fixed for .n0 = 1, 2, 3, 4, which clearly demonstrate the resonance results at those critical values in (5.2.26). The numerical results also motivate us that as .n0 increases, the plasmon parameter .ϵs approaches .−1 and the loss parameter .δ approaches 0. We next present the construction of a general plasmonic structure that can induce resonance. To that end, we first recall the following asymptotic properties of the (1) spherical Bessel and Hankel functions, .jn (t) and .hn (t) for sufficiently large n , .

  jn (t) = jˆn (t) 1 + jˇn (t) ,

.

  ˆ (1) ˇ (1) h(1) n (t) = hn (t) 1 + hn (t) ,

(5.2.27)

5.2 Helmholtz System n=1

6

3.5

203

x 10

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0 0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

n=3

7

x 10

n=2

7

3.5

x 10

0 0.01

0.02

0.03

0.05

0.06

0.07

n=4

7

14

12

0.04

x 10

12 10 10 8

8

6

6

4

4

2 0

2 0.8

1

1.2

1.4

1.6

−3

0 0

0.5

x 10

1

1.5

2

2.5

3

3.5

4

−5

x 10

Fig. 5.7 The change of the dissipation energy .Eδ with respect to the change of .δ when .ϵn0 is fixed, = 1, 2, 3, 4, in .R3

.n0

where .fˆn (t), .fˇn (t), .hˆ n (t) and .hˇ n (t) are defined by (1)

.jˆn (t) =

tn , jˇn (t) = O (2n + 1)!!

(1)

    (2n − 1)!! ˇ (1) 1 1 (1) ˆ , hn (t) = . , hn (t) = O n+1 n n it (5.2.28)

We have the following theorem regarding the plasmon resonance in three dimensions. Theorem 5.2.1 Consider the Helmholtz system (5.2.11), where the Newtonian potential of the source f is given in (5.2.15). Let .n0 ∈ N fulfil the following two conditions: (1)

1. .n0 is sufficiently large such that the asymptotic properties of .jn (t) and .hn (t) in (5.3.39) hold for .n ≥ n0 ; 2. .βn0 /= 0.

204

5 Localized Resonances Beyond the Quasi-Static Approximation

Let the plasmon parameters be chosen of the following form ϵs = −1 −

.

1 , n0

δ ∈ R+

and

δ ⪡ 1.

(5.2.29)

Then if the plasmon configuration fulfils the following condition, 

  ϵs + iδ jˇn0 (t)' (k1,δ re ) 1 + hˇ (1) jˆn0 (k1,δ re )hˆ (1) n0 (kre ) n0 (kre )   ˇn0 (k1,δ re ) = 0, (kr ) 1 + j −hˇ n(1)' e 0

.

(5.2.30)

one has that .Eδ [u] ∼ δ −1 . Proof Suppose that .ϵs = −1 − 1/n0 with .n0 satisfying the two conditions stated in the theorem. By straightforward calculations, we first have that      1 

 ' (1) (1)' . .  ϵs + iδjn (k1,δ re )hn (kre ) − hn (kre )jn0 (k1,δ re ) ≈ δ 1 + O 0 0 0 n0 (5.2.31) From the solution given in (5.2.18) and with the help of Green’s formula, one has that ∂u 2 uds(x) Eδ [u] = δ |∇u|2 dx = δk1,δ |u|2 dx + δ Bre Bre ∂Bre ∂ν 2 |an0 jn0 (k1,δ r)Yn0 (ˆx)|2 dx ≥ δk1,δ Bre



.

+δ ∂Bre

≈

 ˆ 2 ds(x) an0 k1,δ jn' 0 (k1,δ re ) an0 jn0 (k1,δ re ) |Yn0 (x)|

|βn0 |2 n0 δ



 1+O

1 n0

 ,

which readily completes the by noting that .βn0 /= 0.

⨆ ⨅

Remark 5.2.1 Since .δ ⪡ 1, we see that the plasmon configuration in Theorem 5.2.1 induces resonance. The next thing one needs to verify is that the Eq. (5.2.30) yields a nonempty set of parameters. This is indeed the case and we next present a numerical example for illustration. We set .re = 1, n0 = 500, ϵs = 1−1/n0 and .δ = 0.5n0 , and let k be a free parameter. Figure 5.8 plots the quantity in the LHS of (5.2.30) against k around .k = 8. One readily sees that there do exist k’s such that (5.2.30) holds. Hence, resonance occurs with the aforesaid parameters at those k’s. One can also fix k and determine the other parameters by solving (5.2.30).

5.2 Helmholtz System

205

Fig. 5.8 The real and imaginary parts of the LHS quantity in (5.2.30) with respect the change of the wavenumber k

Next, we consider the Helmholtz system (5.2.5)–(5.2.6) in .R3 with .ri /= 0, and show that ALR can be induced. Let the Newtonian potential of the source f is again given in (5.2.15). The solution to (5.2.5)–(5.2.6) in .BR1 can be expressed as ⎧ ∞ kr x ∈ Bri , ⎪ ⎨ n=0 an jn ( √ϵc )Yn (ˆx), ∞ (1) .u(x) = bn jn (k1,δ r)Yn (ˆx) + cn hn (k1,δ r)Yn (ˆx), x ∈ Bre \Bri ⎪ ⎩ n=0 (1) ∞ x) + dn hn (kr)Yn (ˆx), x ∈ BR1 \Bre . n=0 en jn (kr)Yn (ˆ (5.2.32) By applying the transmission conditions across .∂Bri and .∂Bre , we can have from (5.2.32) that ⎧ (1) ⎪ an jn ( √krϵic ) = bn jn (k1,δ ri ) + cn hn (k1,δ ri ), ⎪ ⎪   ⎪ ⎪ ⎨ √ϵ a j ' ( √kri ) = √ϵ + iδ b j ' (k r ) + c h(1)' (k r ) , c n n s n 1,δ i n 1,δ i n n ϵc . (1) ⎪ bn jn (k1,δ re ) + cn h(1) (k r ) = en jn (kre) + dn hn (kre ), ⎪ 1,δ e n ⎪  ⎪ √ ⎪ (1)' ' ⎩ ϵs + iδ bn j ' (k1,δ re ) + cn h(1)' n (k1,δ re ) = en jn (kre ) + dn hn (kre ). n (5.2.33) Solving the equations in (5.2.33), one has that .an = a˜ n /gn , bn = b˜n /gn , cn = c˜n /gn , dn = d˜n /gn , where  

(1)' a˜ n =en ϵs + iδ × ζn × jn' (k1,δ ri )h(1) (k r ) − h (k r )j (k r ) ,. 1,δ i 1,δ i n 1,δ i n n

.

b˜n =en × ζn ×

 √

(5.2.34) 

kri kri (1)' ϵc jn' ( √ )h(1) (k r ) − + iδh (k r )j ( ) ,. ϵ √ 1,δ i s 1,δ i n n ϵc n ϵc (5.2.35)

206

5 Localized Resonances Beyond the Quasi-Static Approximation

c˜n =en × ζn ×



kri kri √ ϵs + iδjn' (k1,δ ri )jn ( √ ) − ϵc jn' ( √ )jn (k1,δ ri ) ϵc ϵc

 ,. (5.2.36)

(1)' ζn :=jn' (kre )h(1) n (kre ) − hn (kre )jn (kre ),

(5.2.37)

and .

  √  (1) ' kri ' d˜n =en ϵc hn (k1,δ ri )jn (k1,δ re ) − h(1) n (k1,δ re )jn (k1,δ ri ) jn ( √ )jn (kre )+ ϵc   kri ' (1)' ' (k r )j (k r ) − h (k r )j (k r ) jn ( √ )jn (kre )+ (ϵs + iδ) h(1)' 1,δ i 1,δ e 1,δ e 1,δ i n n n n ϵc  

kri ' (1)' ϵs + iδ jn' (k1,δ ri )h(1) n (k1,δ re ) − hn (k1,δ ri )jn (k1,δ re ) jn ( √ )jn (kre )+ ϵc 

√  (1)' ϵs + iδ ϵc hn (k1,δ re )jn (k1,δ ri ) − jn' (k1,δ re )h(1) n (k1,δ ri )  kri (5.2.38) jn' ( √ )jn (kre ) , ϵc

  kri ' (1)' ' (1) gn =(ϵs + iδ) h(1)' n (k1,δ re )jn (k1,δ ri ) − hn (k1,δ ri )jn (k1,δ re ) hn (kre )jn ( √ )+ ϵc  

kri ' (1) (1)' ϵs + iδ h(1)' n (k1,δ ri )jn (k1,δ re ) − jn (k1,δ ri )hn (k1,δ re ) hn (kre )jn ( √ )+ ϵc   √ (1) ' kri (1)' ϵc h(1) n (k1,δ re )jn (k1,δ ri ) − hn (k1,δ ri )jn (k1,δ re ) jn ( √ )hn (kre )+ ϵc   √

(1)' ϵc ϵs + iδ jn' (k1,δ re )h(1) (k r ) − h (k r )j (k r ) 1,δ i 1,δ e n 1,δ i n n

.

kri (kre ), jn' ( √ )h(1) ϵc n

(5.2.39)

with .k1,δ given in (5.2.12). By a similar reasoning to that in (5.2.19) and (5.2.20), one can show that .en = βn . With the above series representation of the solution, we next consider the occurrence of ALR, namely configurations to make both (5.2.10) and (5.2.9) fulfilled. To that end, we need to impose a certain constraint on the source .f (x) such that the potential .F (x) is of the following form F (x) =

∞ 

.

n=N

βn jn (kr)Yn (ˆx),

(5.2.40)

5.2 Helmholtz System

207

for some sufficiently large N so that when .n ≥ N, the asymptotic properties in (5.3.39) hold. In the following, in order to simplify the statement, we also need to introduce the following two functions:    ˇ (1) ˇ' ϕ1 (n, b1 , b2 , r1 , r2 ) = jˆn (r1 )hˆ (1) n (r2 ) b1 jn (r1 ) 1 + hn (r2 )   . −b2 hˇ n(1)' (r2 ) 1 + jˇn (r1 ) , (1)' ϕ2 (n, b1 , b2 , r1 , r2 ) = b1 jn' (r1 )h(1) n (r2 ) − b2 hn (r2 )jn (r1 ),

ˇ (1) where .fˆn (t), .fˇn (t), .hˆ (1) n (t) and .h n (t) are defined in (5.3.39)–(5.2.28). Theorem 5.2.2 Consider the Helmholtz system (5.2.5)–(5.2.6) in .R3 with the Newtonian potential F of the source f satisfying (5.2.40). Let the plasmon configuration be chosen of the following form, ϵc = (1 + 1/n0 )2 ,

.

ϵs = −1 − 1/n0

and

δ = ρ n0 ,

(5.2.41)

where .ρ := ri /re < 1 and .n0 ∈ N with .n0 ⪢ 1. If the plasmon configuration fulfils the following condition, ϕ1 (n0 , .

√

kri ϵc , τs,δ , √ , k1,δ ri )ϕ2 (n0 , τs,δ , 1, k1,δ re , kre ) ϵc

+ ϕ1 (n0 , τs,δ , 1, k1,δ re , kre )   kri kri √ √ × ϕ2 (n0 , ϵc , τs,δ , √ , k1,δ ri ) − ϕ1 (n0 , ϵc , τs,δ , √ , k1,δ ri ) = 0, ϵc ϵc (5.2.42)

√ where .τs,δ := ϵs + iδ, then there is a critical radius .r∗ := re3 /ri such that if n0 f lies within this radius, .Eδ [u] ≥ μ0 n0 with .μ0 > 1, and .u(x) remains bounded for .|x| > re2 /ri ; and if f lies outside this radius, .Eδ [u] is bounded by a constant depending only on .f, k and .re . βn Proof For notational convenience of the proof, we set .β˜n := (2n+1)!! , .n ≥ N. By (5.2.41) and (5.2.42), together with the use of (5.3.39) one can derive the following estimates when .n = n0 ,

gn0 ≈ δ 2 + ρ 2n0 ,

.

c˜n0 ≈

.

n0 (kri )2n0 βn , (1 · 3 · · · (2n0 + 1))2 0

d˜n0 ≈

b˜n0 ≈ iδβn0 ,

(5.2.43)

δn0 (kre )2n0 βn , (1 · 3 · · · (2n0 + 1))2 0

(5.2.44)

208

5 Localized Resonances Beyond the Quasi-Static Approximation

and when .n /= n0 , gn ≈

.

n(kri )2n βn , (1 · 3 · · · (2n + 1))2

cn ≈

.

n − n0 βn , nn0

(5.2.45)

n − n0 (kre )2n βn , (1 · 3 · · · (2n + 1))2 n0

(5.2.46)

(n − n0 )2 , n2 n20 d˜n ≈

bn ≈

Noting .δ = ρ n0 , from (5.2.43) and (5.2.44) and by direct calculations, one can show that n0 β˜n20 (kre )2n0 δ

Eδ [u] ≥

.

δ 2 + ρ 2n0

≥ n0 β˜n20



k 2 re3 ri

n0 (5.2.47)

.

Consider first that the source .f (x) is supported inside the critical radius .r∗ =

re3 /ri . By (5.2.40) and the asymptotic property of .jn (t) in (5.3.39), one can verify that there exists .τ ∈ R+ such that .

lim sup(β˜n )1/n =



n→∞

ri 2 k re3

+ τ.

(5.2.48)

Combining (5.4.96) and (5.2.47), one can obtain that  Eδ [u] ≥ n0

.

ri +τ 2 k re3

 n0 

k 2 re3 ri

n0 .

Next, we suppose that the source is supported outside the critical radius .r∗ . Then there exists .η > 0 such that .

lim sup(β˜n )1/n ≤ n→∞

1 , kr∗ + η

and the dissipation energy .Eδ can be estimated as follows Eδ [u] ≈



 n3

n≥N,n/=n0 .

≤

 n≥N,n/=n0

 3

n

n0 n − n0 n0 n − n0

2

2

(kre )2n β˜n2 + ρ + n0 β˜n20

which means that resonance does not occur.

n

n0 β˜n20 (kre )2n0 δ



δ 2 + ρ 2n0 k 2 re3 ri

n0 ≤ C,

,

5.2 Helmholtz System

209

Next we prove the boundedness of the solution .u(x) when .|x| > re2 /ri . From (5.2.43) and (5.2.44), when .n = n0 one has that δ 1 · 3 · · · (2n0 − 1) |β˜n0 |(kre )2n0 n0 2 2n 0 1 · 3 · · · (2n0 + 1) δ + ρ (kr)n0 +1  2  n0 r 1 , ≤ |β˜n0 |(kre )n0 e r n0 ri

|dn0 h(1) n0 (kr)| ≤ .

(5.2.49)

and when .n /= n0 , one can obtain from (5.2.45) and (5.2.46), |dn h(1) n (kr)| ≤ .

n2 n0 1 · 3 · · · (2n − 1) |β˜n |(kre )2n , 1 · 3 · · · (2n + 1) n − n0 (kr)n+1

≤ |β˜n |(kre )n

nn0 ren . n − n0 r n

(5.2.50)

Hence from (5.2.49) and (5.2.50), one has that |u(x) − F F (x)| ≤



.

|β˜n |(kre )n ≤ C,

when

|x| ≥ re2 /ri .

(5.2.51)

n≥N

The proof is complete.

⨆ ⨅

Remark 5.2.2 Similar to Remark 5.2.1, we can numerically verify the equation in (5.2.42) yields a nonempty set of parameters. For illustration, we set .n0 = 300, ri = 0.5, re = 1, ϵc = (1 + 1/n0 )2 , ϵs = −1 − 1/n0 and .δ = (ri /re )n0 , and let k be a free parameter. Figure 5.9 plots the quantity in the LHS of (5.2.42) against k over an interval. It can be seen that there do exist k’s such that (5.2.42) holds.

Fig. 5.9 The real and imaginary parts of the LHS quantity in (5.2.42) with respect the change of the wavenumber k

210

5 Localized Resonances Beyond the Quasi-Static Approximation

5.2.2 Spectral System of the N-P Operator and Its Application to Atypical Resonance in R3 With the help of the results obtained in the previous section, we are able to derive the complete spectral system of the N-P operator associated with the Helmholtz equation at finite frequencies in the radial geometry. The spectral result is of independent mathematical interest and it can also help us to derive the resonance and ALR results. First we introduce the single layer potential that is defined as k .SB [ϕ](x) re

=

Gk (x − y)ϕ(y)ds(y),

x ∈ R3 ,

Bre

where .ϕ(x) ∈ L2 (∂Bre ) and .Gk (x) is given in (5.2.13). The conormal derivative of the single layer potential enjoys the following jump formula .

  ∗  ∂ 1 [ϕ](x), SBkre [ϕ]|± (x) = ± I + KBkre ∂ν 2

x ∈ ∂Bre ,

(5.2.52)

where  .

KBkre

∗

[ϕ](x) = ∂Bre

∂ Gk (x − y)ϕ(y)ds(y), ∂ν x

x ∈ ∂Bre ,

which is referred to as the N-P operator. In (5.2.52), the .± indicate the limit (to ∂Bre ) from outside  and inside of .Bre , respectively. Before giving the eigenvalues of

.

the operator . KBkr

e

∗

, we first introduce the following identity.

Lemma 5.2.1  .

KBkre

∗

1 ik [ϕ](x) = − S k [ϕ](x) − 8π re 2re Bre

eik|x−y| ϕ(y)ds(y), ∂Bre

x ∈ ∂Bre . (5.2.53)

Proof For .x, y ∈ ∂Bre , direct calculations yield that   〈x − y, ν x 〉 〈x − y, ν x 〉 − + ik |x − y|3 |x − y|2   ik 1 −eik|x−y| + − , = 2re |x − y| 2re 4π

∂ −eik|x−y| Gk (x − y) = 4π ∂ν x .

5.2 Helmholtz System

211

where the following fact is used .

re (1 − 〈ν y , ν x 〉) 1 〈x − y, ν x 〉 = = 2 . 2 2re |x − y| 2re (1 − 〈ν y , ν x 〉) ⨆ ⨅

The proof is complete.

We would like to point out that the identity (5.2.53) is an extension of (2.22) in [15], which plays  ∗ a very important role in calculating the spectrum of the N-P k operator . KBr . It is also remarked that .(KBkr )∗ is compact but non self-adjoint. e

e

In [13], it is shown for the quasistatic limit with .k = 0, .(KB0r )∗ is symmetrizable, e  ∗ but it is not the case with .k /= 0. The spectral system of the operator . KBkr is e contained in the following theorem.

Theorem 5.2.3 Let k and .re satisfy the following condition jn (kre ) /= jn+2 (kre )

for n ≥ 0.

(5.2.54)

 ∗ KBkre [Ym ](x) = λm,k,re Ym (ˆx),

(5.2.55)

1 1 2 2 (1)' − ik 2 re 2 jn' (kre )h(1) n (kre ) = − − ik re jn (kre )hn (kre ), 2 2

(5.2.56)

.

Then .

where λm,k,re =

.

and on the boundary .∂Bre , SBkre [Ym ](x) = χm,k,re Ym (ˆx),

.

x ∈ ∂Bre ,

(1)

where .χm,k,re = −ikre 2 hm (kre )jm (kre ). Remark 5.2.3 We believe the condition (5.3.20) should always hold. However, the verification of it is fraught with difficulties, and we include it as a condition. It is also remarked that from (5.3.22), one can verify that the eigenvalues are complex numbers for .k /= 0. After achieving the spectral system of the N-P operator, we can apply it to derive the resonance result for the Helmholtz system (5.2.11). Consider (5.2.11) and the solution could be represented as the following integral ansatz,  u(x) =

.

k

SBr1,δ [ϕ](x), x ∈ Bre , e k SBr [ψ](x) + F (x), x ∈ R3 \Bre , e

(5.2.57)

212

5 Localized Resonances Beyond the Quasi-Static Approximation

with .(ϕ, ψ) ∈ L2 (∂Bre ) × L2 (∂Bre ) and, .k1,δ and .F (x) given in (5.2.14) and (5.2.12) respectively. By using the transmission conditions across .∂Bre , one then has ⎧ ⎨ S k1,δ [ϕ](x) = S k [ψ](x) + F (x), Bre Bre     . (5.2.58) ⎩ (ϵs + iδ) ∂ SBk1,δ [ϕ](x) = ∂ SBk [ψ](x) + F (x) . ∂ν ∂ν r r e

e

Substituting the jump relation (5.2.52) into the last Eq. (5.2.58) yields that ⎡

⎤     −SBkr e  ∗     e ∗  ⎦ ϕ = F . .⎣ k ∂ ψ − 12 I + KBkr (ϵs + iδ) − 12 I + KBr1,δ ∂ν F k

SBr1,δ

e

e

(5.2.59) Since the potential .F (x) for .x ∈ Bre can be represented as in (5.2.15), then for ∂ x ∈ ∂Bre , .F (x) and . ∂ν F (x) can be written as

.

F (x) =

∞ 

.

∞

βn jn (kre )Yn (ˆx),

n=0

 ∂ kβn jn' (kre )Yn (ˆx). F (x) = ∂ν

(5.2.60)

n=0

Substituting (5.2.60) into the Eq. (5.2.59), one can obtain ∞ 

ϕ(x) =

.

ϕˆn Yn (ˆx),

(5.2.61)

n=0

where ϕˆn =

.

kχn,k,re jn' (kre ) − (1/2 + λn,k,re )jn (kre ) βn . (ϵs + iδ)(−1/2 + λn,k1 ,re )χn,k,re − (1/2 + λn,k,re )χn,k1 ,re

Therefore from Green’s formula, one has that 2 .Eδ [u] = δ(k1,δ ) |u|2 dx + δ Bre

= δ(k1,δ )2 

Bre

∂u uds(x) ∂Bre ∂ν k |SBr1,δ [ϕ]|2 dx + δ e

1  k × − + KBr1,δ e 2

∗ 

∂Bre

k

[ϕ]SBr1,δ [ϕ]ds(x) e

(5.2.62)

5.2 Helmholtz System

=

213 ∞ 

 δ|ϕˆn |2 (k1,δ )2 |γn,k1,δ ,re |2

re

|jn (k1,δ r)r|2 dr

0

n=0



1 +(− + λn,k1,δ ,re )χn,k1,δ ,re re2 . 2 Thus if we can determine .ϵs + iδ such that for some .n0 ∈ N, δ|ϕˆn0 |2 → ∞

.

as δ → δ0 ,

(5.2.63)

then the resonance occurs. Substituting the expressions of .λn,k,re , .χn,k1 ,re , .λn,k1 ,re and .χn,k,re given in Theorem 5.3.1 into (5.2.62), the condition (5.2.63) can be shown to be reduced to (5.2.24). Therefore one can derive the same results in Theorem 5.2.1. By using the spectral properties of the Nemann-Poincáre operator, one can also prove Theorem 5.4.4 in a similar manner.

5.2.3 Atypical Resonance and ALR Results in Two Dimensions In this section, we extend our 3D resonance and ALR results in Sect. 5.2 to the 2D case. The study is pretty much parallel to the 3D case and in what follows, we shall present the main results and then sketch their proofs. We first consider the Helmholtz system (5.2.11) in the two-dimensional case. Let .Jn (t) and .Hn(1) (t) be, respectively, the Bessel and Hankel functions of order .n ∈ N. The fundamental solution in (5.2.13) in the 2D case is replaced by .Gk (x) = (1) − 4i H0 (k|x|). The Newtonian potential the potential .F (x) defined in (5.2.14) has the following series representation for .x ∈ BR1 , ∞ 

F (x) =

.

βn Jn (kr)einθ ,

(5.2.64)

n=−∞

where and also in what follows, we assume that .J−n (t) = Jn (t). Following a similar argument to that in (5.2.16)–(5.2.21), one can show that  u(x) =

.

∞

an Jn (k1,δ r)einθ , x ∈ Bre , (1) inθ inθ + cn Hn (kr)e , x ∈ BR1 \Bre . n=−∞ bn Jn (kr)e

n=−∞ ∞

(5.2.65)

with ⎧ ⎪ ⎨ an = bn √ .

Jn' (kre )Hn (kre )−Hn (kre )Jn (kre ) (1)

(1)'

(1) (1)' ϵs +iδJn' (k1,δ re )Hn √(kre )−Hn (kre )Jn (k1,δ re ) Jn' (kre )Jn (k1,δ re )− ϵs +iδJn' (k1,δ re )Jn (kre ) n√ (1) (1)' ϵs +iδJn' (k1,δ re )Hn (kre )−Hn (kre )Jn (k1,δ re )

⎪ ⎩ cn = b

, (5.2.66) ,

214

5 Localized Resonances Beyond the Quasi-Static Approximation

and .bn = βn . Moreover, we have that 





Eδ [u] = δ (k1,δ )

.

|u| dx +

2

2

Bre ∞ 

=

∂Bre

 2π δ|an |2 (k1,δ )2

re

∂u uds(x) ∂ν



|Jn (k1,δ r)|2 rdx

0

n=−∞



+ k1,δ re Jn' (k1,δ re )Jn (k1,δ re )

,

(5.2.67)

δ|an0 |2 → ∞ as δ → δ0 ,

(5.2.68)

which shows that if there exists .n0 ∈ N such that .

then the atypical resonance occurs. Next, by using the fact that Jn (t)Hn(1)' (t) − Jn' (t)Hn(1) (t) =

.

2i , πt

one has that an =

.

−2iβn 1 . √ ∂ikre ϵs + iδJn' (k1,δ re )Hn(1) (kre ) − Hn(1)' (kre )Jn (k1,δ re )

Hence, if there exists atypical parameter .ϵs + iδ such that

.

ϵs + iδJn' (k1,δ re )Hn(1) (kre ) − Hn(1)' (kre )Jn (k1,δ re ) → 0,

(5.2.69)

as .δ → δ0 , then the condition (5.2.68) is fulfilled. Similar to our study in Section 5.2.1, one can numerically solve the nonlinear Eq. (5.2.69) to derive various atypical configurations which can induce resonance, and we refer to the arXiv preprint version [94] for some illustrative examples. Next, we present the construction of a general 2D plasmonic structure that can induce resonance. To that end, we recall the following asymptotic properties of the (1) functions .Jn (t) and .Hn (t) for sufficiently large .n ∈ N,   Jn (t) = Jˆn (t) 1 + Jˇn (t) ,

.

  Hn(1) (t) = Hˆ n(1) (t) 1 + Hˇ n(1) (t) ,

(5.2.70)

5.2 Helmholtz System

215

(1) (1) where .Jˆn (t), .Jˇn (t), .Hˆ n (t) and .Hˇ n (t) have the following properties,

tn Jˆn (t) = n , 2 n!

Jˇn (t) = O

.

  1 2n (n − 1)! , Hˆ n(1) (t) = , n π it n

Hˇ n(1) (t) = O

  1 . n (5.2.71)

Theorem 5.2.4 Consider the Helmholtz system (5.2.11) in .R2 , where the Newtonian potential F of the source f is given in (5.2.64). Suppose that for F in (5.2.64), there exists an index .n0 ∈ N such that .βn0 /= 0 and .n0 is sufficiently large such that when .n ≥ n0 , .Jn (t) and .Hn(1) (t) enjoy the asymptotic expansions in (5.2.70). Let the plasmon parameters be chosen of the following form ϵs = −1,

.

δ ∈ R+

and

δ ⪡ 1.

(5.2.72)

Then if the plasmon configuration fulfils the following condition, (kre ) Jˆn0 (k1,δ re )Hˆ n(1) 0 .



  ϵs + iδ Jˇn0 (t)' (k1,δ re ) 1 + Hˇ n(1) (kr ) e 0   ˇn0 (k1,δ re ) = 0, (kr ) 1 + J −Hˇ n(1)' e 0

(5.2.73)

one has .Eδ [u] ∼ δ −1 . Proof By using (5.2.72) and (5.2.73), together with straightforward calculations, one can show that    

 1   ' (1) (1)' . (kre )Jn0 (k1,δ re ) ≈ δ 1 + O .  ϵs + iδJn (k1,δ re )Hn (kre ) − Hn 0 0 0 n0 (5.2.74) From the solution given in (5.2.66) and with the help of Green’s formula, one has that ∂u 2 2 2 uds(x) Eδ = δ |∇u| dx = δk1,δ |u| dx + δ Bre Bre ∂Bre ∂ν 2 |an0 Jn0 (k1,δ r)ein0 θ |2 dx ≥ δk1,δ

.

Bre

+δ ∂Bre

|βn0 |2 ≈ δ

an0 k1,δ Jn' 0 (k1,δ re )

 an0 Jn0 (k1,δ re ) |ein0 θ |2 ds(x)

(5.2.75)

   1 , 1+O n0

which completes the proof by noting that .βn0 /= 0.

⨆ ⨅

216

5 Localized Resonances Beyond the Quasi-Static Approximation

Next, we consider the Helmholtz system (5.2.5)–(5.2.6) in .R2 with .ri /= 0 , and show that ALR can be induced. Let the Newtonian potential of the source f is again given in (5.2.64). By following a similar argument to that in deriving (5.2.32)– (5.2.39), one can show that the solution to (5.2.5)–(5.2.6) in .BR1 can be expressed as ⎧ ∞ kr inθ x ∈ Bri , ⎪ ⎨ n=−∞ an Jn ( √ϵc )e , ∞ (1) inθ inθ .u(x) = bn Jn (k1,δ r)e + cn Hn (k1,δ r)e , x ∈ Bre \Bri ⎪ ⎩ n=−∞ ∞ inθ + d H (1) (kr)einθ , x ∈ BR1 \Bre , n n n=−∞ en Jn (kr)e (5.2.76) with .an = a˜ n /gn , bn = b˜n /gn , cn = c˜n /gn , dn = d˜n /gn , where .

 

a˜ n =en ϵs + iδ × ζn × Jn' (k1,δ ri )Hn(1) (k1,δ ri ) − Hn(1)' (k1,δ ri )Jn (k1,δ ri ) , .

(5.2.77) 

kri ' kri (1) (1)' ˜bn =en × ζn × ϵc Jn ( √ )Hn (k1,δ ri ) − ϵs + iδHn (k1,δ ri )Jn ( √ ) , . ϵc ϵc (5.2.78) 

 kri kri √ c˜n =en × ζn × ϵs + iδJn' (k1,δ ri )Jn ( √ ) − ϵc Jn' ( √ )Jn (k1,δ ri ) , . ϵc ϵc (5.2.79)  √

ζn :=Jn' (kre )Hn(1) (kre ) − Hn(1)' (kre )Jn (kre ),

(5.2.80)

and   kri √  (1) ˜ ϵc Hn (k1,δ ri )Jn (k1,δ re ) − Hn(1) (k1,δ re )Jn (k1,δ ri ) jn' ( √ )jn' (kre )+ dn = en ϵc   kri τs,δ Hn(1)' (k1,δ ri )Jn' (k1,δ re ) − Hn(1)' (k1,δ re )Jn' (k1,δ ri ) Jn ( √ )Jn (kre )+ ϵc .   kri √ τs,δ Jn' (k1,δ ri )Hn(1) (k1,δ re ) − Hn(1)' (k1,δ ri )Jn (k1,δ re ) Jn ( √ )Jn' (kre )+ ϵc    √ √ (1)' ' (1) ' kri τs,δ ϵc Hn (k1,δ re )Jn (k1,δ ri ) − Jn (k1,δ re )Hn (k1,δ ri ) Jn ( √ )Jn (kre ) , ϵc (5.2.81)

5.2 Helmholtz System

217

and   kri gn = τs,δ Hn(1)' (k1,δ re )Jn' (k1,δ ri ) − Hn(1)' (k1,δ ri )Jn' (k1,δ re ) Hn(1) (kre )Jn ( √ )+ ϵc   kri √ τs,δ Hn(1)' (k1,δ ri )Jn (k1,δ re ) − Jn' (k1,δ ri )Hn(1) (k1,δ re ) Hn(1)' (kre )Jn ( √ )+ ϵc .   kri √ ϵc Hn(1) (k1,δ re )Jn (k1,δ ri ) − Hn(1) (k1,δ ri )Jn (k1,δ re ) Jn' ( √ )Hn(1)' (kre )+ ϵc   kri √ √ ϵc τs,δ Jn' (k1,δ re )Hn(1) (k1,δ ri ) − Hn(1)' (k1,δ re )Jn (k1,δ ri ) Jn' ( √ )Hn(1) (kre ), ϵc (5.2.82) with .τs,δ := ϵs + iδ and .k1,δ given in (5.2.12). One also has that .en = βn . Similar to (5.2.40), we impose the constraint on f such that the corresponding Newtonian potential .F (x) is of the following form ∞ 

F (x) =

.

βn Jn (kr)einθ ,

(5.2.83)

|n|=N (1)

for some sufficiently large .N ∈ N such that when .n ≥ N, .Jn (r) and .Hn (r) enjoy the asymptotic properties given in (5.2.70). For the subsequent use, we also define the following functions:     ϕ1 (n, b1 , b2 , r1 , r2 ) := Jˆn (r1 )Hˆ n(1) (r2 ) b1 Jˇn' (r1 ) 1 + Hˇ n(1) (r2 )   . −b2 Hˇ n(1)' (r2 ) 1 + Jˇn (r1 ) ,  ϕ2 (n, b1 , b2 , r1 , r2 ) := b1 Jn' (r1 )Hn(1) (r2 ) − b2 Hn(1)' (r2 )Jn (r1 ), where .Jˆn (t), .Jˇn (t), .Hˆ n (t) and .Hˇ n (t) are given (5.2.70) and (5.2.71). (1)

(1)

Theorem 5.2.5 Consider the Helmholtz system (5.2.5)–(5.2.6) with .ri /= 0 in .R2 with the Newtonian potential F of the source f satisfying (5.2.83). Let the plasmon configuration be chosen of the following form ϵs = −1,

.

ϵc = 1

and

δ = ρ n0 ,

(5.2.84)

where .ρ := ri /re < 1 and .n0 ∈ N with .n0 ⪢ 1. If the plasmon configuration fulfils the following condition, √ ϕ2 (n0 , τs,δ , 1, k1,δ re , kre ) +  ϕ1 (n0 , τs,δ , 1, k1,δ re , kre )  ϕ1 (n0 , ϵc , τs,δ , kri , k1,δ ri ) .

 √ √ ×  ϕ2 (n0 , ϵc , τs,δ , kri , k1,δ ri ) −  ϕ1 (n0 , ϵc , τs,δ , kri , k1,δ ri ) = 0, (5.2.85)

218

5 Localized Resonances Beyond the Quasi-Static Approximation

√ where .τs,δ = ϵs + iδ, then there is a critical radius .r∗ := re3 /ri such that if f lies within this radius, .Eδ [u] ≥ μn0 0 n0 with .μ0 > 1, and .u(x) remains bounded for .|x| > re2 /ri ; and if f lies outside this radius, .Eδ [u] is bounded by a constant depending only on .f, k and .re . Proof The proof is similar to that of Theorem 5.4.4 in the three dimensional case and in what follows, we mainly point out several major ingredients. Set .β˜n := 2βnnn! . One can derive the following estimates when .n ≥ N, 2n n(kri )2n ˜n ≈ − δn(kre ) βn . β , d gn ≈ δ 2 + ρ 2n , b˜n ≈ iδβn , c˜n ≈ n n 4 (n!)2 4n (n!)2

(5.2.86)

.

For .n0 sufficiently large, one can directly estimate with the help of (5.2.86) that  2 3 n0  nβ 2 (kre )2n k re δ n 2 ˜ ≥ n0 βn0 . .Eδ ≈ 2 2n n 2 ri (2 n!) δ + ρ

(5.2.87)

|n|≥N

If the source f is supported inside the critical radius .r∗ , by a similar reasoning to (5.4.96), one can show that there exists .τ ∈ R+ such that 

˜n . lim sup β

1/n

n→∞

=

ri 2 k re3

+ τ.

(5.2.88)

Combining (5.2.87) and (5.2.88), one can then show that .E [u] ≥ μn0 0 n0 for some 2 .μ0 > 1 as stated in the theorem. The boundedness of .u(x) for .|x| > re /ri can be shown as follows. By virtue of (5.2.86), one has that (1) .|dn Hn (kr)|

δ n(kre )2n |βn ||Hn(1) (kr)| ≤ |β˜n |(kre )n ≤ n 2 2 4 (n!) δ + ρ 2n



re2 ri

n

1 , rn (5.2.89)

which in turn implies that when .|x| > re2 /ri , |u(x) − F (x)| ≤



.

|β˜n |(kre )n ≤ C.

|n|≥N

Finally, the case when f is supported outside the critical radius can be treated by following a similar manner to that in the proof of Theorem 5.4.4. The proof is complete. ⨅ ⨆ Remark 5.2.4 Similar to Remark 5.2.2, we can numerically verify that the equation in (5.2.85) yields a nonempty set of parameters. For illustration, we set .n0 = 300, ri = 0.5, re = 1, ϵc = 1, ϵs = −1 and .δ = (ri /re )n0 , and let k be a free

5.3 Maxwell’s Problem

219

Fig. 5.10 The real part and imaginary part of left side of the Eq. (5.2.85) with respect the change of the frequency k

parameter. Figure 5.10 plots the quantity in the LHS of (5.2.84) against k over an interval. It can be seen that there do exist k’s such that (5.2.84) holds.

5.3 Maxwell’s Problem Next, we present the mathematical setup of our study by describing the electromagnetic scattering problem and the time-harmonic Maxwell system. Let .ϵm and .μm be two positive constants which, respectively, characterize the electric permittivity and magnetic permeability of the uniformly homogeneous space .R3 . Let D be a bounded domain in .R3 with a smooth boundary .∂D and a connected complement .R3 \D, which signifies the plamsonic inclusion embedded in the homogeneous space. The permittivity and permeability of D are specified by .ϵc and .μc . At this point, we only assume that .ϵc and .μc are complex numbers with nonnegative imaginary parts, that is, .ϵc , μc ∈ C and .ℑϵc , ℑμc ≥ 0. Eventually, we shall construct that .ϵc and .μc might be certain complex numbers with negative real parts. Let .ω ∈ R+ denote the frequency of the electro-magnetic waves. Set √ km = ω ϵm μm ,

.

√ kc = ω ϵc μc

with ℜkc > 0, ℑkc < 0,

(5.3.1)

and ϵD = ϵm χ (R3 \D) + ϵc χ (D),

.

μD = μm χ (R3 \D) + μc χ (D),

(5.3.2)

220

5 Localized Resonances Beyond the Quasi-Static Approximation

where and also in what follows .χ denotes the characteristic function. Let .(Ei , Hi ) be the incident electro-magnetic field, and it satisfies the following Maxwell system in the homogeneous space,

.

∇ × Ei − iωμm Hi =0

in R3 ,

∇ × Hi + iωϵm Ei =0

in R3 ,

(5.3.3)

√ where .i := −1 is the imaginary unit. The electro-magnetic scattering associated with the medium configuration described in (5.4.76) due to the incident field i i .(E , H ) in (5.3.3) is governed by the following transmission problem of the Maxwell system,

.

∇ × E − iωμD H =0

in R3 \∂D,

∇ × H + iωϵD E =0

in R3 \∂D,

(5.3.4)

ν × E|+ − ν × E|− =ν × H|+ − ν × H|− = 0 on ∂D, subject to the Silver-Müler radiation condition, .

lim |x|

|x|→∞

  x √ √ − ϵm (E − Ei )(x) μm (H − Hi )(x) × |x|

(5.3.5)

which holds uniformly in .x/|x| ∈ S2 . The radiation condition (5.3.5) characterizes the outgoing nature of the scattered wave fields .Es := E − Ei and .Hs := H − Hi . We are concerned with studying the quantitative behaviours of the scattered fields s s .E , H and their relationships with the proper choices of the plasmonic parameters i i .ϵc , μc and the incident fields .E , H . Specifically, as discussed earlier, we are mainly interested in the two phenomena of surface atypical resonance and invisibility cloaking. It is emphasized that we do not impose the quasi-static assumption in our study, namely, there is no such assumption that .ω · diam(D) ⪡ 1, and as also discussed earlier, this is one aspect that distinguishes the current study from the existing ones in the literature.

5.3.1 Integral Formulation of the Maxwell System In this section, using the layer potential techniques, we present the integral formulation of the electromagnetic scattering problem (5.4.75)–(5.3.5). We begin with the introduction of some Sobolev spaces and potential theory for the Maxwell system for the subsequent use. Let D be a bounded domain in .R3 with a smooth boundary .∂D and a connected complement .R3 \D. We use .∇∂D , .∇∂D · and .Δ∂D to respectively signify the surface

5.3 Maxwell’s Problem

221

gradient, surface divergence and Laplace-Beltrami operator on the surface .∂D. Denote by L2T (∂D) := {ϕ ∈ L2 (∂D)3 , ν · ϕ = 0},

.

where and also in what follows .ν signifies the exterior unit normal vector to the boundary of a domain concerned. Let .H s (∂D) be the usual Sobolev space of order .s ∈ R on .∂D. Introduce the following function spaces T H (div, ∂D) := {ϕ ∈ L2T (∂D) : ∇∂D · ϕ ∈ L2 (∂D)},

.

and T H (curl, ∂D) := {ϕ ∈ L2T (∂D) : ∇∂D · (ϕ × ν) ∈ L2 (∂D)}.

.

Define the vectorial surface curl by curl∂D ϕ = ∇∂D ϕ × ν,

.

for .ϕ ∈ L2 (∂D) and one can verify that ∇∂D · ∇∂D = Δ∂D

.

and

∇∂D · curl∂D = 0.

(5.3.6)

Next, we recall that the fundamental outgoing solution .Gk (x − y) to the Helmholtz operator .Δ + k 2 in .R3 is given by Gk (x − y) = −

.

eik|x−y| , 4π |x − y|

x, y ∈ R3 , x /= y.

(5.3.7)

For a density .ϕ ∈ T H (div, ∂D), we introduce the following vectorial single layer potential operator, k .AD [ϕ](x)

:=

Gk (x − y)ϕ(y)ds(y),

x ∈ R3 .

(5.3.8)

∂D

For a scalar density .ϕ ∈ L2 (∂D), the single layer potential operator is defined similarly by k .SD [ϕ](x)

:=

Gk (x − y)ϕ(y)ds(y), ∂D

x ∈ R3 .

(5.3.9)

222

5 Localized Resonances Beyond the Quasi-Static Approximation

We shall also need the following boundary integral operators, SDk [ϕ] : L2 (∂D) → H 1 (∂D) ϕ ‫ →׀‬SDk [ϕ](x) =

Gk (x − y)ϕ(y)ds(y), x ∈ ∂D; ∂D

 ∗ KDk [ϕ] : L2 (∂D) → L2 (∂D)

 ∗ k ϕ ‫ →׀‬KD [ϕ](x) =

.



KkD

∗

∂D

∂Gk (x − y) ϕ(y)ds(y), x ∈ ∂D; ∂ν x

∂D

∂Gk (x − y) ϕ(y)ds(y), x ∈ ∂D; ∂ν x

[ϕ] : L2 (∂D)3 → L2 (∂D)3  ∗ k ϕ ‫ →׀‬KD [ϕ](x) =

ADk [ϕ] : L2 (∂D)3 → H 1 (∂D)3 ϕ ‫ →׀‬ADk [ϕ](x) =

Gk (x − y)ϕ(y)ds(y),

x ∈ ∂D;

∂D

(5.3.10)

and MDk [ϕ] :T H (div, ∂D) → T H (div, ∂D) . k ϕ ‫ →׀‬MD [ϕ](x) = ν x × ∇x × Gk (x − y)ϕ(y)ds(y), ∂D

x ∈ ∂D; (5.3.11)

LDk [ϕ] :T H (div, ∂D) → T H (div, ∂D)   . ϕ ‫ →׀‬LDk [ϕ](x) = ν x × k 2 ADk [ϕ](x) + ∇ SDk [∇∂D · ϕ] ,

x ∈ ∂D. (5.3.12)

There holds the following jump formula for .ϕ ∈ L2T (∂D) (cf. [115]),  .

ν × ∇ × ADk

±

=∓

ϕ + MDk [ϕ] 2

on

∂D,

(5.3.13)

where  .

ν x × ∇ × ADk

±

(x) = lim ν x × ∇ × ADk [ϕ](x ± tν x ), t→0+

∀x ∈ ∂D.

In what follows, similar to (5.4.17), the subscripts .± signify the limits from the outside and inside of D, respectively. With the above preparations, we are in a position to derive the integral formulation of the electromagnetic scattering problem (5.4.75)–(5.3.5). Using the vectorial

5.3 Maxwell’s Problem

223

single layer potential operator (5.3.8), the solution to (5.4.75)–(5.3.5) can be given by the following integral ansatz (cf. [12]),  E(x) =

.

Ei (x) + μm ∇ × ADkm [ψ] + ∇ × ∇ × ADkm [ϕ](x), x ∈ R3 \D, x ∈ D, μc ∇ × ADkc [ψ] + ∇ × ∇ × ADkc [ϕ](x), (5.3.14)

and H(x) =

.

1 (∇ × E)(x) iωμD

x ∈ R3 \∂D.

(5.3.15)

By matching the boundary condition (the third condition in (5.3.4)) and using the jump formula (5.4.17), the pair .(ψ, ϕ) ∈ T H (div, ∂D) × T H (div, ∂D) is the solution to the following system of integral equations, (I + K)Φ = F,

(5.3.16)

.

where  μc +μm 2

I=

0 

.

K=

I 

0 kc2 2μc

+

2 km 2μm

μc MDkc − μm MDkm LDkc − LDkm 

ψ .Φ = ϕ





 I

,

LDkc − LDkm

kc2 kc μc MD

−

2 km km μm MD

 ,

 ν × Ei . F= iων × Hi 

and

Here and also in what follows, I signifies the identity operator. Based on the integral formulation (5.3.16), we next give the formal definitions on the surface plasmon resonance and invisibility cloaking effect associated with the scattering problem (5.4.75)–(5.3.5). At this point, we suppose that the operator +∞ ∞ .(I + K) possesses an eigen-system .{Ξ j } j =1 and .{τj }j =1 such that (I + K)[Ξ j ] = τj Ξ j .

.

Moreover, it is assumed that the set of functions .{Ξ j }+∞ j =1 is complete in the space 2 2 .L (∂D) . Definitely, we shall construct such an eigen-system in what follows under T a certain circumstance and this is one of the main technical contributions of the current study. Using such an eigen-system, the integral system (5.3.16) can be

224

5 Localized Resonances Beyond the Quasi-Static Approximation

solved via  the spectral resolution by firstly representing the RHS term .F in (5.3.16) as .F = +∞ j =1 fj Ξ j , and then by straightforward calculations that Φ=

.

+∞  fj j =1

τj

Ξj.

(5.3.17)

Thus from (5.3.14), the scattered wave field in .R3 \D to the problem (5.4.75)–(5.3.5) can be given as follows Es (x) =

.

+∞   fj  μm ∇ × ADkm [Ξ j,1 ] + ∇ × ∇ × ADkm [Ξ j,2 ](x) , τj

x ∈ R3 \D,

j =1

(5.3.18) where  Ξj =

.

 Ξ j,1 . Ξ j,2

Based on the representation of the scattered wave field .Es in (5.3.18), we introduce the following definitions for the phenomena of the surface plasmon resonance and invisibility cloaking. Definition 5.3.1 We say that the surface plasmon resonance occurs if there exists an eigenvalue .τj of the operator .(I + K) such that .|τj | ⪡ 1 with .fj /= 0. Definition 5.3.2 We say that the object D is invisible to the incident wave .(Ei , Hi ), if all the eigenvalues .τj of the operator .(I + K) with .fj /= 0 satisfy |τj | ⪢ 1.

.

In the situation as described in Definition 5.3.1, by (5.3.18), one readily sees that the corresponding scattered filed encounters a certain blowup. Indeed, in such a case, the significant field enhancement is mainly confined to the surface of the electro-magnetic inclusion D and it is referred to as the surface plasmon resonance; see our numerical illustrations in Figs. 5.1 and 5.2 in what follows. On the other hand, in the situation as described in Definition 5.3.2, one can easily verify that the corresponding scattered field should be nearly vanishing and hence invisibility cloaking effect can be observed. Indeed, in our subsequent numerical study, it is observed that not only the plasmonic structure but also certain inhomogeneous inclusions embedded in the structure are invisible under the wave impinging. Clearly, the occurrence of the surface plasmon resonance and the invisibility cloaking critically depends on the appropriate choice of the material constitution of the plasmonic inclusion, namely .(D; ϵc , μc ) and the source .F, namely .(Ei , Hi ). In what follows, we shall first construct the eigen-system for the operator .(I + K)

5.3 Maxwell’s Problem

225

with the spherical geometry. It is emphasized that the spectral structure of the operator .(I + K) with .ω = 0, namely in the quasi-static regime, is known [12]. In the finite-frequency regime with .ω ∼ 1, the operator .(I + K) is no longer selfadjoint and the construction is radically more challenging. Moreover, for the surface plasmon resonance and the invisibility cloaking, one shall require the exact spectral information of the operator .(I + K), and hence we mainly restrict to the spherical geometry. Nevertheless, as can be seen that even in such a case, the corresponding derivation is highly nontrivial. After the derivation of the exact spectral properties of the operator .(I + K), we can then use the corresponding result to construct the suitable plasmonic structures that can induce surface plasmon resonance or invisibility cloaking.

5.3.2 Spectral Analysis of the Integral Operators In this section, we derive the spectral properties of the integral operators introduced earlier within the spherical geometry. To that end, we assume that .BR is a central ball of radius .R ∈ R+ in what follows. Let .S be the unit sphere and we introduce the following vectorial spherical harmonics of order .n ∈ N0 := N ∪ {0},

.

In = ∇S Yn+1 + (n + 1)Yn+1 ν,

n ≥ 0,

Tn = ∇S Yn × ν,

n ≥ 1,

Nn = − ∇S Yn−1 + nYn−1 ν,

n ≥ 1,

(5.3.19)

where .∇S is the surface gradient on the unit sphere .S and .Yn is the spherical harmonics. It is noted that the set .(∇S Yn , ∇S Yn × ν)n∈N forms an orthogonal basis (1) of .L2T (S) (cf. [115]). Let .jn (t) and .hn (t) be, respectively, the spherical Bessel .n ∈ N0 . The following lemma shows the spectral and Hankel functions of order  ∗ k properties of the operator . K and .Sk (cf.[94]). BR

BR

Lemma 5.3.1 Suppose that k and R satisfy the following condition jm (kR) /= jm+2 (kR) for

.

Then it holds that  ∗ k . KB [Yn ](x) = λn,k,R Yn (ˆx) with R

m ≥ 0.

n ≥ 0,

(5.3.20)

(5.3.21)

where λn,k,R =

.

1 1 2 2 (1)' − ik 2 R 2 jn' (kR)h(1) n (kR) = − − ik R jn (kR)hn (kR). 2 2

(5.3.22)

226

5 Localized Resonances Beyond the Quasi-Static Approximation

On the boundary .∂BR , we have SBkR [Yn ](x) = χn,k,R Yn (ˆx),

.

x ∈ ∂BR ,

where χn,k,R = −ikR 2 h(1) n (kR)jn (kR).

.

Using the vectorial spherical harmonics in (5.3.19) and Lemma 5.3.1, one has by direct calculations that ⎧ ∗ ⎨ Kk [∇S Yn + nYn ν] = λn−1,k,R (∇S Yn + nYn ν), n ≥ 1,  BR ∗ . ⎩ Kk [−∇S Yn + (n + 1)Yn ν] = λn+1,k,R (−∇S Yn + (n + 1)Yn ν), n ≥ 0, BR and  .

n ≥ 1, ABkR [∇S Yn + nYn ν] = χn−1,k,R (∇S Yn + nYn ν), k  ABR [−∇S Yn + (n + 1)Yn ν] = χn+1,k,R (−∇S Yn + (n + 1)Yn ν), n ≥ 0.

By solving the above two systems, one can further show that ⎧ ∗ ⎨ Kk [∇S Yn ] = π1,n ∇S Yn + π2,n Yn ν, BR  ∗ . k ⎩ K [Yn ν] = π3,n ∇S Yn + π4,n Yn ν, BR

(5.3.23)

with (n + 1)λn−1,k,R + nλn+1,k,R , 2n + 1 λn−1,k,R − λn+1,k,R , = 2n + 1

π1,n = .

π3,n

π2,n = π4,n

n(n + 1) (λn−1,k,R − λn+1,k,R ), 2n + 1 (n + 1)λn+1,k,R + nλn−1,k,R ; = 2n + 1 (5.3.24)

and  .

ABkR [∇S Yn ] = σ1,n ∇S Yn + σ2,n Yn ν, Ak [Yn ν] = σ3,n ∇S Yn + σ4,n Yn ν, BR

(5.3.25)

5.3 Maxwell’s Problem

227

with (n + 1)χn−1,k,R + nχn+1,k,R , 2n + 1 χn−1,k,R − χn+1,k,R , = 2n + 1

σ1,n = .

σ3,n

σ2,n = σ4,n

n(n + 1) (χn−1,k,R − χn+1,k,R ), 2n + 1 (n + 1)χn+1,k,R + nχn−1,k,R . = 2n + 1

With these preparations, we next derive the spectral properties of the operators .MBkR and .LBkR . It is noted that .MBkR and .LBkR are entries of the matrix operator .K, and their spectral properties form a critical ingredient for our subsequent derivation of the spectral system for the operator .I + K. Proposition 5.3.1 The orthogonal base functions of .L2T (S), .∇S Yn and .∇S Yn × ν with .n ≥ 1, are eigenfunctions for the operator .MBkR defined in (5.3.11), and one has MBkR [∇S Yn × ν] = mk1,n ∇S Yn × ν and MBkR [∇S Yn ] = mk2,n ∇S Yn ,

.

(5.3.26)

with   1 mk1,n = π1,n + (σ1,n − n(n + 1)σ3,n ) R

.

and

 χn,k,R  mk2,n = λn,k,R + R (5.3.27)

where .π1,n , .σ1,n and .σ3,n , and, .λn,k,R and .χn,k,R are given in (5.3.23), (5.3.25) and Lemma 5.3.1, respectively. Proof With the help of vector calculus, the identity .∂x1 Gk (x − y) = −∂y1 Gk (x − y) and the definition of .MBkR given in (5.3.11), one can find that MBkR [ϕ] =

ν x × ∇x × Gk (x − y)ϕ(y)ds(y) ∂BR



= −ν x ×

∇y Gk (x − y) × ϕ(y)ds(y) ∂BR



.

= −ν x × ∂BR

((∇y Gk (x − y) · ν y )ν y + ∇∂BR y Gk (x − y)) × ϕ(y)ds(y)

= −ν x ×

∂BR

((∇x Gk (x − y) · ν x )ν y + ∇∂BR y Gk (x − y)) × ϕ(y)ds(y), (5.3.28)

where the last identity follows from ∇y Gk (x − y) · ν y = ∇x Gk (x − y) · ν x

.

for x, y ∈ ∂BR .

228

5 Localized Resonances Beyond the Quasi-Static Approximation

For .ϕ = ∇S Yn × ν, one has − νx ×

(∇x Gk (x − y) · ν x )ν y × ϕds(y) ∂BR

.

= − νx ×



KkBR

∗

(5.3.29) [∇S Yn ] = π1,n ∇S Yn × ν,

which follows from (5.3.23), and the following fact ∂BR

∇∂BR y Gk (x − y) × ϕ(y)ds(y)



=− ∂BR

=−

3 

(∇∂BR y Gk (x − y) · ∇S Yn )ν y ds(y)

ej ∂BR

j =1 .

=

3 

ej ∂BR

j =1

=

3 

Gk (x − y)∇∂BR · (∇S Yn (ν y · ej ))ds(y)

ej ∂BR

j =1

=

(∇∂BR y Gk (x − y) · ∇S Yn )(ν y · ej )ds(y)

Gk (x − y)(∇∂BR · ∇S Yn (ν y · ej ) + ∇S Yn · ∇∂BR (ν y · ej ))ds(y)

 1  −n(n + 1)ABkR [Yn ν] + ABkR [∇S Yn ] , R

where .ej , .j = 1, 2, 3, are the unit coordinate vectors in .R3 and the last identity follows from 3  .

j =1

 1 ej ∇S Yn · ∇∂BR (ν y · ej ) = ∇S Yn . R

From (5.3.25), one has that − νx × ∇∂BR y Gk (x − y) × (∇S Yn × ν)ds(y) ∂BR

.

(5.3.30)

1 = (σ1,n − n(n + 1)σ3,n )∇S Yn × ν. R Then by combining (5.3.28), (5.3.29) and (5.3.30), one can directly verify that k .MB [∇S Yn R

  1 × ν] = π1,n + (σ1,n − n(n + 1)σ3,n ) ∇S Yn × ν. R

5.3 Maxwell’s Problem

229

For .ϕ = ∇S Yn = ν × (∇S Yn × ν), by (5.3.19) and Theorem 5.3.1, one has that − νx ×

(∇x Gk (x − y) · ν x )ν y × ϕds(y) ∂BR

(5.3.31)

∗  =ν x × KBkR [∇S Yn × ν] = λn,k,R ∇S Yn .

.

Furthermore, one can deduce that ∇∂BR y Gk (x − y) × (ν y × (∇S Yn × ν y ))ds(y) ∂BR

.

= ∂BR

=

3 

(∇∂BR y Gk (x − y) · (∇S Yn × ν y ))ν y ds(y)

ej ∂BR

j =1

=−

3 

ej ∂BR

j =1

=−

3 

Gk (x − y)∇∂BR · ((∇S Yn × ν y )(ν y · ej ))ds(y)

ej ∂BR

j =1 .

(∇∂BR y Gk (x − y) · (∇S Yn × ν y ))(ν y · ej )ds(y)

Gk (x − y)((∇∂BR · (∇S Yn × ν y ))(ν y · ej ) (5.3.32)

+ (∇S Yn × ν y ) · ∇∂BR (ν y · ej ))ds(y) =−

3 

ej

j =1

=−

∂BR

Gk (x − y)((∇S Yn × ν y ) · ∇∂BR (ν y · ej ))ds(y)

 1  k ABR [∇S Yn × ν] . R

In the derivation of (5.3.32), we have used the fact that ∇∂BR · (∇S Yn × ν y ) = 0,

.

which follows from the identity (5.3.6). Thus from Lemma 5.3.1, one has that .

− νx × ∂BR

∇∂BR y Gk (x − y) × (∇S Yn × ν)ds(y) =

χn,k,R ∇S Y n . R

(5.3.33)

230

5 Localized Resonances Beyond the Quasi-Static Approximation

Finally, with the help of (5.3.28), (5.3.31) and (5.3.33), one can verify that  χn,k,R  ∇S Yn . MBkR [∇S Yn ] = λn,k,R + R

.

⨆ ⨅

The proof is complete.

Proposition 5.3.2 For the operator .LBkR defined in (5.3.12), and .∇S Yn and .∇S Yn × ν with .n ≥ 1, there hold the following identities, k k LBkR [∇S Yn × ν] = l1,n ∇S Yn and LBkR [∇S Yn ] = l2,n ∇S Yn × ν,

.

with  k l1,n = k 2 χn,k,R

.

and

k l2,n =

 n(n + 1) 2 χ − k σ n,k,R 1,n , R2

where .σ1,n and .χn,k,R are given in (5.3.25) and Theorem 5.3.1, respectively. Proof Recall that   LBkR [ϕ] = ν x × k 2 ABkR [ϕ](x) + ∇ SBkR [∇∂BR · ϕ] .

.

For .ϕ = ∇S Yn × ν, one can show that ν x × k 2 ABkR [∇S Yn × ν] = k 2 χn,k,R ∇S Yn .

.

The identity in (5.3.6) gives ∇∂BR · (∇S Yn × ν) = 0,

.

and thus from the last two identities, one has LBkR [∇S Yn × ν] = k 2 χn,k,R ∇S Yn .

.

For .ϕ = ∇S Yn , by (5.3.25), one can show ν x × k 2 ABkR [∇S Yn ] = −k 2 σ1,n ∇S Yn × ν.

.

As for another term in (5.3.34), one has that ν x × ∇SBkR [∇∂BR · ∇S Yn ] = ν x × ∇SBkR [ΔS Yn /R] .

=

−n(n + 1) n(n + 1) ν x × ∇SBkR [Yn ] = χn,k,R ∇S Yn × ν, R R2

(5.3.34)

5.3 Maxwell’s Problem

231

which follows from the identity (5.3.6), Lemma 5.3.1, and the fact ΔS Yn = −n(n + 1)Yn .

.

Therefore, there holds  LBkR [∇S Yn ] =

.

 n(n + 1) 2 ∇S Yn × ν. χ − k σ n,k,R 1,n R2 ⨆ ⨅

The proof is complete.

After achieving the spectral systems for the operators .MBkR and .LBkR , respectively, in Propositions 5.3.1 and 5.3.2, we proceed to derive the spectral system of the operator .I + K defined in (5.3.16). We have Theorem 5.3.1 The eigenvalues and their corresponding eigenfunctions for the operator .I + K given in (5.3.16) are given as follows with .n ∈ N, .

(I + K)[Ξ 1,n ] = τ1,n Ξ 1,n ,

(I + K)[Ξ 2,n ] = τ2,n Ξ 2,n ,

(I + K)[Ξ 3,n ] = τ3,n Ξ 3,n ,

(I + K)[Ξ 4,n ] = τ4,n Ξ 4,n ,

(5.3.35)

where kc km τ1,n = α1 (l1,n − l1,n )+

k2 k2 k2 kc2 kc m m2,n − m mk2,n + c + m , μc μm 2μc 2μm

kc km − l1,n )+ τ2,n = α2 (l1,n

k2 k2 k2 kc2 kc m + c + m , m2,n − m mk2,n μm 2μc 2μm μc

.

km − l2,n )+

kc2 kc k2 k2 k2 m m1,n − m mk1,n + c + m , μc μm 2μc 2μm

kc km − l2,n )+ τ4,n = α4 (l2,n

kc2 kc k2 k2 k2 m + c + m , m1,n − m mk1,n μm 2μc 2μm μc

τ3,n =

kc α3 (l2,n

and  α1 ∇S Yn × ν , = ∇S Yn   α3 ∇S Yn , = ∇S Y n × ν 

Ξ 1,n .

Ξ 3,n

 α2 ∇S Yn × ν Ξ 2,n = , ∇S Yn   α4 ∇S Yn Ξ 4,n = , ∇S Y n × ν 

(5.3.36)

232

5 Localized Resonances Beyond the Quasi-Static Approximation

with

α1 =

α2 = .

α3 =

α4 =

2 (2mkm − 1)μ + μ (μ (μ + 2mkc μ + μ − 2mkm μ ) km c m c c m 2,n 1,n c 1,m m √ kc 2 − kc (2m2,n + 1)) + β1 kc km 4(l1,n − l1,n )μc μm 2 (2mkm − 1)μ + μ (μ (μ + 2mkc μ + μ − 2mkm μ ) km c m c c m 2,n 1,n c 1,m m √ kc 2 − kc (2m2,n + 1)) − β1 kc km 4(l1,n − l1,n )μc μm 2 (2mkm − 1)μ + μ (μ (μ + 2mkc μ + μ − 2mkm μ ) km c m c c m 1,n 2,n c 2,m m √ kc 2 − kc (2m1,n + 1)) + β2 kc km 4(l2,n − l2,n )μc μm 2 (2mkm − 1)μ + μ (μ (μ + 2mkc μ + μ − 2mkm μ ) km c m c c m 1,n 2,n c 2,m m √ kc 2 − kc (2m1,n + 1)) − β2 kc km 4(l2,n − l2,n )μc μm

,

,

,

,

and kc km kc km β1 = 16(l1,n − l1,n )(l2,n − l2,n )μ2c μ2m +  2 2 m c m c km (2mk2,n − 1)μc +μm (μc (μc + 2mk1,n μc − 2mk1,n μm ) − kc2 (2mk2,n + 1)) , .

kc km kc km − l1,n )(l2,n − l2,n )μ2c μ2m + β2 = 16(l1,n  2 2 m c m c km (2mk1,n − 1)μc +μm (μc (μc + 2mk2,n μc − 2mk2,n μm ) − kc2 (2mk1,n + 1)) .

k , l k are, respectively, given in Propositions 5.3.1 and 5.3.2. Here .mk1,n , mk2,n and .l1,n 2,n   Moreover, the eigenfunctions . {Ξ i,n }4i=1 n∈N form a complete basis of .L2T (∂BR )2 .

Proof Equation (5.3.35) can be shown by direct calculations with the help of 5.3.2. The completeness of the set of eigenfunctions Propositions  5.3.1 and 4 2 2 . {Ξ i,n } on .L (∂BR ) follows from the fact that the family .(∇S Yn , ∇S Yn × T i=1 n∈N ⨆ ⨅ ν) with .n ≥ 1 forms an orthogonal basis of .L2T (S).

5.3 Maxwell’s Problem

233

5.3.3 Atypical Resonance and Its Cloaking Effect As applications of the spectral results in the previous section, in particular Theorem 5.3.1, we next construct the structures of the form (5.4.75)–(5.4.76) that can induce the atypical resonance and cloaking effect. First, let us consider the atypical resonance. By Definition 5.3.1 and the corresponding discussion at the end of Sect. 5.3.1, it is sufficient for us to design the electric permittivity .ϵc and the magnetic permeability .μc in the domain .B such that one of the eigenvalues of the associated operator .I + K is small enough. However, from the expressions of .{τi,n }4i=1 , .n ∈ N, in (5.3.36), these eigenvalues are too complicated to allow for the explicit design. With the aid of computer searching, we first show the existence of such plasmonic structures. Suppose .B is the unit ball and set the parameters in (5.4.75) and (5.4.76) to be μm = ϵm = 1, ω = 5,

.

μc = 1,

and

ϵc = −1.04018 + 0.00004i. (5.3.37)

One can verify from the expression of .τ1,n in (5.3.36) that |τ1,40 | ⪡ 1.

.

Let the incident wave be chosen to be a plane wave of the form   √ √ √ √ Ei = 4 eiω(x/ 2+y/ 2) , −eiω(x/ 2+y/ 2) , 0 ,

.

(5.3.38)

which guarantees that the Fourier coefficient in (5.3.17) corresponding to .τ1,40 is not vanishing. Hence, the conditions in Definition 5.3.1 are fulfilled and surface resonance occurs. Indeed, we plot the resonant electric field in Fig. 5.11 corresponding to the electro-magnetic configuration in (5.3.37) and (5.3.38). It can be readily seen that there is significant field enhancement near the boundary of the structure, namely the atypical resonance occurs. Next, by imposing a certain restriction on the incident fields, we can construct a general class of structures that can induce the atypical resonance. To that end, we (1) first note that for .n ⪢ 1, the spherical Bessel and Hankel functions .jn (t) and .hn (t) have the following asymptotic properties    1 tn , 1+O .jn (t) = n 1 · 3 · · · (2n + 1)

(5.3.39)

and (1) .hn (t)

   1 1 · 3 · · · (2n − 1) 1+O . = n+1 n it

(5.3.40)

Fig. 5.11 Plotting of the first, second and third components of the resonant electric field corresponding to the electro-magnetic configuration described in (5.3.37) and (5.3.38)

234 5 Localized Resonances Beyond the Quasi-Static Approximation

5.3 Maxwell’s Problem

235 (1)

With the help of the asymptotic properties for .jn (t) and .hn (t) in (5.3.39) and (5.3.40), one can show the following estimates for the parameters, .λn,k,R , k , .mk , .l k and .l k , given in Proposition 5.3.1, Lemmas 5.3.1 and 5.3.2 .χn,k,R , .m 1,n 2,n 1,n 2,n with .n ⪢ 1,       1 1 −R 1 1+O , χn,k,R = 1+O , λn,k,R = 2(2n + 1) n 2n + 1 n .       1 1 1 1 k k 1+O , m2,n = − 1+O , m1,n = 4n + 2 n 4n + 2 n and k l1,n =

.

   1 −Rk 2 1+O , 2n + 1 n

k =− l2,n

   Rk 2 (4n2 + 4n + 3) 1 n(n + 1) . + 3 1 + O 2 n (2n + 1)R 8n + 12n − 2n − 3

Using the above estimates, one can further derive the following asymptotic expressions for the eigenvalues .τi,n in (5.3.36) for .n ⪢ 1, τi,n

.

   1 , = τ˜i,n 1 + O n

i = 1, 2, 3, 4,

where τ˜1,n

.

 1 (n + 1)μc + nμm + ((n + 1)ϵm + nϵc )ω2 − = 2(2n + 1) !

((n + 1)μc + nμm − ((n + 1)ϵm + nϵc )ω2 )2

 4(ϵc μc − ϵm μm )2 (4n2 + 4n + 3)ω4 R 2 , 4n2 + 4n − 3  1 (n + 1)μc + nμm + ((n + 1)ϵm + nϵc )ω2 + = 2(2n + 1) −

τ˜2,n

!

((n + 1)μc + nμm − ((n + 1)ϵm + nϵc )ω2 )2  4(ϵc μc + ϵm μm )2 (4n2 + 4n + 3)ω4 R 2 , − 4n2 + 4n − 3

(5.3.41)

236

5 Localized Resonances Beyond the Quasi-Static Approximation

τ˜3,n

 1 = (n + 1)μm + nμc + ((n + 1)ϵc + nϵm )ω2 − 2(2n + 1) !

((n + 1)μm + nμc − ((n + 1)ϵc + nϵm )ω2 )2 −

 4(ϵc μc − ϵm μm )2 (4n2 + 4n + 3)ω4 R 2 , 4n2 + 4n − 3

and τ˜4,n = .

 1 (n + 1)μm + nμc + ((n + 1)ϵc + nϵm )ω2 + 2(2n + 1)

((n + 1)μm + nμc − ((n + 1)ϵc + nϵm )ω2 )2 !  4(ϵc μc − ϵm μm )2 (4n2 + 4n + 3)ω4 R 2 . − 4n2 + 4n − 3

Since the eigenfunctions .{Ξ i,n }4i=1 with .n ∈ N0 given in the Theorem 5.3.1 are complete on .L2T (S)2 , we can write the source term .F in (5.3.16) in the following form F=

4  +∞ 

.

fi,n Ξ i,n .

(5.3.42)

i=1 n=1

We can show the following resonance result. Theorem 5.3.2 Suppose the Newtonial potential .F in (5.3.16) has the representation in (5.3.42). Let .n0 ∈ N be sufficiently large such that the spherical Bessel and (1) Hankel functions .jn0 (t) and .hn0 (t) enjoy the asymptotic properties given in (5.3.39) and (5.3.40). Let the parameter .ϵc and .μc inside the domain .B be chosen such that the following conditions are fulfilled: ϵc = −ϵm

.

and

ℜ (μc + μm ) ≥ 0,

(5.3.43)

or μc = −μm

.

and

  ℜ (ϵc + ϵm )ω2 ≥ 0.

(5.3.44)

If the Fourier coefficients .f1,n0 /= 0 or .f3,n0 /= 0, then the atypical resonance occurs for both the medium configurations (5.3.43) and (5.3.44). Similarly, let the parameter .ϵc and .μc inside the domain .B be chosen such that the following conditions are fulfilled: ϵm = −ϵc ,

.

and

ℜ (μc + μm ) ≤ 0,

(5.3.45)

5.3 Maxwell’s Problem

237

or μm = −μc

.

  ℜ (ϵc + ϵm )ω2 ≤ 0.

and

(5.3.46)

If the Fourier coefficients .f2,n0 /= 0 or .f4,n0 /= 0, then atypical resonance occurs for both the medium configurations (5.3.45) and (5.3.46). Proof From the expression of .τ1,n0 in (5.3.41), one can show by direct calculations that  1 (n0 + 1)μc + n0 μm + ((n0 + 1)ϵm + n0 ϵc )ω2 − τ˜1,n0 = 2(2n0 + 1) ! .

((n0 + 1)μc + n0 μm − ((n0 + 1)ϵm + n0 ϵc )ω2 )2 −

4(ϵc μc − ϵm μm )2 (4n20 + 4n0 + 3)ω4 R 2



4n20 + 4n0 − 3

1 (μc + μm ) + (ϵc + ϵm )ω2 − 4   1 . +O n0 =

((μc + μm ) − (ϵc + ϵm )ω2 )2



If ϵm = −ϵc

.

and

ℜ (μc + μm ) ≥ 0,

one can find that  τ˜1,n0 = O

.

1 n0

 .

Clearly, one has resonance for this case since .n0 ⪢ 1. For the other case with μm = −μc

and

.

  ℜ (ϵc + ϵm )ω2 ≥ 0,

one can show by following a similar argument that  τ˜1,n0 = O

.

1 n0

 ,

and hence resonance occurs. The occurrence of resonance for the medium configurations (5.3.45) and (5.3.46) can be shown in a similar manner with the help of the asymptotic expressions in (5.3.41). The proof is complete. ⨆ ⨅

238

5 Localized Resonances Beyond the Quasi-Static Approximation

Corollary 5.3.1 By Theorem 5.3.2, it is readily seen that if the parameters .ϵc and μc inside the domain .B are chosen as

.

ϵc = −ϵm

.

and

μc = −μm ,

and ne of the Fourier coefficients .fi,n for .F in (5.3.42) is non-vanishing for some i ∈ {1, 2, 3, 4} and a certain .n = n0 ⪢ 1, then atypical resonance occurs.

.

Since we are considering the electro-magnetic scattering in the finite-frequency regime, it is more practical for the plasmon parameters to be constructed according to the Drude model (cf. [12]), which states that  ϵc = ϵ0 1 − 

.

μc = μ0



ωp2

,

ω(ω + iτ )

 ω2 1−F , ω2 − ω02 + iτ ω

(5.3.47)

where .ωp is the plasmon frequency of the bulk material, .τ > 0 is the object’s damping coefficient, .F is a filling factor and .ω0 is a localized resonant frequency. In the rest of the section, we show that by following a similar strategy as before via the use of the spectral result in Theorem 5.3.1, one can construct the desired plasmonic structures according to the Drude model. Indeed, if we take ϵ0 = μ0 = 1,

ω = 5,

.

τ = 0.0001,

ω0 = 2,

ωp2 = 51.0045 and

F = 0,

then by straightforward calculations one can obtain that ϵc = −1.04018 + 0.00004i

.

and

μc = 1,

which is exactly the one constructed earlier in (5.3.37). Let us consider another case with the filling factor .F /= 0, and set ϵ0 = μ0 = 1,

.

ω0 = 2,

ωp2

ω = 5, = 51.518

τ = 0.00001, and

F = 0.1.

(5.3.48)

Associated with (5.3.48), one has by direct calculations that ϵc = −1.06072 + 4.1 × 10−6 i and

.

μc = 0.880952 + 2.8 × 10−7 i.

(5.3.49)

If the incident electric field is taken to be (5.3.38) again, one can show that atypical resonance occurs. We plot the corresponding resonant electric field in Fig. 5.12 and one can readily see the surface resonance phenomenon.

Fig. 5.12 Plotting of the first, second and third components of the resonant electric field corresponding to the electro-magnetic configuration described in (5.3.49) and (5.3.38)

5.3 Maxwell’s Problem 239

240

5 Localized Resonances Beyond the Quasi-Static Approximation

5.3.4 Invisibility Cloaking Effect In this section, we construct structures that can induce the cloaking effect associated to certain incident waves. By Definition 5.3.2 as well as the discussion made at the end of Sect. 5.3.1, one needs to design the electric permittivity .ϵc and the magnetic permeability .μc in the domain D such that all those eigenvalues of the associated operator .I+K that correspond to the non-vanishing Fourier coefficients of the source .F, through the relation (5.3.17), should be large enough. That means the structures and the incident waves are related to each other and this makes the corresponding construction rather delicate and tricky. Similar to our study for the typical resonance in the previous section, we first present some specific constructions as well as the corresponding numerical simulations for illustrating the cloaking effects. Let R = 1,

.

ω = 5,

μm = ϵm = 1,

μc = 1 and

ϵc = −6.55806 + 0.000001i, (5.3.50)

and associated with such a configuration, one has from (5.3.36) that |τ3,1 | ⪢ 1.

(5.3.51)

.

We take the incident electric field to be  ⎡ √ 100y ω x2 +y2 +z2

 √  sin ω x2 +y2 +z2

− ⎤ ⎢ ω2 (x2 +y2 +z2 ) ⎢ Eix  ⎢   √ i i .E = ⎣ Ey ⎦ = ⎢ sin ω x2 +y2 +z2 ⎢ √−100x − ⎢ i ω2 (x2 +y2 +z2 ) Ez ⎣ ω x2 +y2 +z2 ⎡

 √ ⎤ cos ω x2 +y2 +z2

√

⎥

ω x2 +y2 +z2 ⎥ ⎥  √ ⎥, 2 2 2 cos ω x +y +z ⎥ ω

√

x2 +y2 +z2

⎥ ⎦

0 (5.3.52) whose trace on .∂D corresponds to an eigenfunction of the eigenvalue .τ3,1 in (5.3.51). Hence, the conditions in Definition 5.3.2 are fulfilled, and one should have cloaking effect associated with the configuration described in (5.3.50)– (5.3.52). In Fig. 5.13, we present the slice plotting of the incident, total and scattered electric fields associated with the configuration described in (5.3.50)–(5.3.52) for illustration of the cloaking effect. It is noted that the .z-component of the wave field is zero and hence it is not plotted. One can readily see that the scattered field outside the inclusion is nearly vanishing and hence the inclusion is nearly invisible under such a wave impinging. Furthermore, we next consider the cloaking effect if there is an inhomogeneous inclusion located inside the structure. Indeed, we let .B1/2 be a central ball of radius .1/2 which embraces an inhomogeneous medium with .ϵ = 5 and .μ = 1. In Fig. 5.14, we present the slice plotting of the incident, total and scattered electric fields associated with the aforesaid configuration. Since the

Fig. 5.13 Slice plotting of the x- and y-components of the incident, total and scattered electric fields associated to the electromagnetic configuration in (5.3.50)– (5.3.52)

5.3 Maxwell’s Problem 241

Fig. 5.14 Slice plotting of the .x- and .y-components of the incident, total and scattered electric fields associated with the electro-magnetic configuration in (5.3.50)–(5.3.52) that embraces an inclusion .(B1/2 ; ϵ = 5, μ = 1) inside the plasmonic structure

242 5 Localized Resonances Beyond the Quasi-Static Approximation

5.3 Maxwell’s Problem

243

corresponding .z-components are zero and we only plot the .x- and .y-components of the wave fields. It is interesting to note that not only the structure but also the embedded inclusion are nearly invisible under such a wave impinging. The latter cloaking effect could also be rigorously verified, but that would involve much lengthy and complicated analysis. Next, we show that if one imposes a certain restrictive condition on the impinging wave fields, one can construct a large class of structures that can induce cloaking effect. Theorem 5.3.3 Suppose that .F given in (5.3.16) due to the incident wave has the following representation F=

+∞  

.

fi,n Ξ i,n ,

(5.3.53)

i=1,3 n=N

where .N ∈ N is large enough such that when .n ≥ N the spherical Bessel and (1) Hankel functions .jn (t) and .hn (t) enjoy the asymptotic properties given in (5.3.39) and (5.3.40). Let the parameters .ϵc and .μc inside the domain D and the wave frequency .ω ∈ R+ be chosen such that   ℜ (μc + μm ) − (ϵc + ϵm )ω2 ≥ 0

.

and

|(ϵc + ϵm )ω2 | ⪢ 1,

(5.3.54)

|μc + μm | ⪢ 1,

(5.3.55)

or   ℜ (μc + μm ) − (ϵc + ϵm )ω2 < 0

.

and

then the corresponding scattered wave field is nearly vanishing outside D. That is, the inclusion D is nearly invisible. Proof By following a similar argument to the proof of Theorem 5.3.2, one can show that when .n ⪢ 1 and   2 ≥ 0, .ℜ (μc + μm ) − (ϵc + ϵm )ω there holds τ˜1,n

.

1 = (ϵc + ϵm )ω2 + O 2

  1 . n

(5.3.56)

Then by using (5.3.56) and the second condition in (5.3.54), one has |τ˜1,n | ⪢ 1,

.

(5.3.57)

244

5 Localized Resonances Beyond the Quasi-Static Approximation

which holds for .n ∈ N sufficiently large. Clearly, the conditions in Definition 5.3.2 are fulfilled and hence one has the cloaking effect. The other case in (5.3.55) can be proved in a similar manner. The proof is complete. ⨆ ⨅ Theorem 5.3.4 Suppose that .F given in (5.3.16) due to the incident wave has the following representation F=

+∞  

.

fi,n Ξ i,n ,

(5.3.58)

i=2,4 n=N

where .N ∈ N is large enough such that when .n ≥ N the spherical Bessel and (1) Hankel functions .jn (t) and .hn (t) enjoy the asymptotic properties given in (5.3.39) and (5.3.40). Let the parameters .ϵc and .μc inside the domain D and the wave frequency .ω ∈ R+ be chosen such that   ℜ (μc + μm ) − (ϵc + ϵm )ω2 ≥ 0,

.

and

|μc + μm | ⪢ 1,

(5.3.59)

|(ϵc + ϵm )ω2 | ⪢ 1,

(5.3.60)

or   ℜ (μc + μm ) − (ϵc + ϵm )ω2 < 0,

.

and

then the corresponding scattered wave field is nearly vanishing outside D. That is, the inclusion D is nearly invisible. Proof The proof is analogous to that of the Theorem 5.3.3.

⨆ ⨅

Remark 5.3.1 It is remarked that the conditions (5.3.54) or (5.3.55) in Theorem 5.3.3 can be easily fulfilled, say e.g., one can simply choose the parameter .ϵc such that ϵc < −ϵm ,

.

and the source .F is of the form (5.3.53) with .ω ⪢ 1, then (5.3.54) is satisfied. The same remark holds for the other conditions in Theorems 5.3.3 and 5.3.4. Remark 5.3.2 By Theorem 5.3.3 and 5.3.4, one can conclude that if the parameters ϵc and .μc inside the domain D, and the frequency .ω satisfy the following condition

.

|μc + μm | ⪢ 1 and

.

|(ϵc + ϵm )ω2 | ⪢ 1,

(5.3.61)

5.3 Maxwell’s Problem

245

then the inclusion .(D; ϵc , μc ) is nearly invisible to a general source term .F of the form F=

4  +∞ 

.

fi,n Ξ i,n ,

i=1 n=N

where .N ∈ N is large enough such that the spherical Bessel and Hankel functions jn (t) and .h(1) n (t) enjoy the asymptotic properties given in (5.3.39) and (5.3.40). However, it is also noted that the first condition in (5.3.61) may not be practical and hence this remark may be mainly of theoretical interest.

.

By Remarks 5.3.1 and 5.3.2, one can readily see that the high-frequency of the electro-magnetic wave plays a critical role for such a cloaking phenomenon and it may not be so realistic to consider such a phenomenon in the quasi-static regime. Finally, we present constructions following the Drude model (5.3.47) that can induce the cloaking effect. If we take ϵ0 = μ0 = 1,

.

ω0 = 2,

τ = 6.615 × 10−7 ,

ω = 5,

ωp2 = 188.952

and

F = 0,

then by (5.3.47) one can obtain that ϵc = −6.55806 + 0.000001i

.

and

μc = 1,

which is exactly the one constructed before in (5.3.50) and its cloaking effect has been shown in Figs. 5.3 and 5.4. Next, we show a case with a nonzero filling factor by taking ϵ0 = μ0 = 1,

.

ω0 = 2,

ω = 5,

τ = 0.00001,

ωp2 = 186.769

and

F = 0.02,

(5.3.62)

then by (5.3.47) one can obtain that ϵc = −6.47076 + 0.00001494i

.

and

μc = 0.97619 + 5.66893 × 10−8 i. (5.3.63)

Let the incident wave .Ei be given in (5.3.52). The slice plottings of the .x- and .ycomponents of the incident, total and scattered electric fields associated with the afore-described configuration are, respectively, presented in Figs. 5.15 and 5.16. It is noted that the corresponding .z-components are all zero. Apparently, there are cloaking effects observed.

Fig. 5.15 Slice plotting of the .x-components of the incident, total and scattered electric fields associated to the electro-magnetic configuration described in (5.3.62), (5.3.63) and (5.3.52)

246 5 Localized Resonances Beyond the Quasi-Static Approximation

Fig. 5.16 Slice plotting of the .y-components of the incident, total and scattered electric fields associated to the electro-magnetic configuration described in (5.3.62), (5.3.63) and (5.3.52)

5.3 Maxwell’s Problem 247

248

5 Localized Resonances Beyond the Quasi-Static Approximation

5.4 Elastic Problem To state the problem and results in a precise way, we first present the mathematical setup for our study. Set .C(x) := (Cij kl (x))3i,j,k,l=1 , .x ∈ R3 to be a four-rank tensor defined by Cij kl (x) := λ(x)δij δkl + μ(x)(δik δj l + δil δj k ), x ∈ RN ,

.

(5.4.1)

where .λ, μ ∈ C are complex-valued functions and referred as Lamé constants, and .δ is the Kronecker delta. In (5.4.1), .C(x) denotes an isotropic elasticity tensor distributed in the space. For a regular elastic material, the Lamé constants should satisfy the following two strong convexity conditions, (i). μ > 0

.

and

(ii). 3λ + 2μ > 0.

(5.4.2)

Let .D ⊂ Ω be two bounded domains in .R3 with connected Lipschitz boundaries. Assume that the domain .R3 \Ω is occupied by the regular elastic material parameterized by the Lamé constant .(λ, μ) satisfying the strong convexity conditions in (5.4.2). In the shell .Ω\D, the Lamé parameter is .(λˆ , μ) ˆ where λˆ = λˆ 1 + iδ

.

and

μˆ = μˆ 1 + iδ,

(5.4.3)

√ where and also in what follows, .i = −1. In (5.4.3), .λˆ 1 , μˆ 1 ∈ C with .ℑλˆ 1 ≥ 0, .ℑμ ˆ 1 ≥ 0 and .δ > 0. Moreover, the domain D is occupied by the regular elastic ˘ satisfying the strong convex conditions (5.4.2). Denote by .CR3 \Ω,λ,μ material .(λ˘ , μ) to specify the dependence on the domain .R3 \Ω and the Lamé parameters .(λ, μ). The same notations also hold for the tensors .CΩ\D,λˆ ,μˆ and .CD,λ˘ ,μ˘ . Furthermore, we define the following tensor C0 = CR3 \Ω,λ,μ + CΩ\D,λˆ ,μˆ + CD,λ˘ ,μ˘ .

.

(5.4.4)

Let .f signify an incident elastic source supported in .R3 \Ω. Thus the elastic displacement field .uδ corresponding to the configurations described above is governed by the following PEDs system  .

∇ · C0 ∇ s uδ (x) + ω2 uδ (x) = f

in R3 ,

uδ (x) satisfies the radiation condition,

(5.4.5)

where .ω ∈ R+ is the angular frequency, and the operator .∇ s is the symmetric gradient ∇ s uδ :=

.

 1 ∇uδ + ∇u⏉ δ , 2

(5.4.6)

5.4 Elastic Problem

249

with .∇uδ denoting the matrix .(∂j ui )3i,j =1 and the superscript T signifying the matrix transpose. In the Eq. (5.4.5), the radiation condition designates the following condition as .|x| → +∞ (cf. [86]), (∇ × ∇ × uδ )(x) × .

x − iks ∇ × uδ (x) =O(|x|−2 ), |x|

x · [∇(∇ · uδ )](x) − ikp ∇uδ (x) =O(|x|−2 ), |x|

(5.4.7)

where √ ks = ω/ μ and

.

kp = ω/ λ + 2μ,

(5.4.8)

with .λ and .μ defined in (5.4.4).

3 Next we introduce the following functional for .u, v ∈ H 1 (Ω\D) Pλˆ ,μˆ (u, v) =

∇ s u : C0 ∇ s u(x)dx Ω\D



.

=

  λˆ (∇ · u)(∇ · v)(x) + 2μ∇ ˆ s u : ∇ s v(x) dx,

(5.4.9)

Ω\D

where .C0 and .∇ s are defined (5.4.4) and (5.4.6), respectively. In (5.4.9) and also 3 3 in what follows, .A : B = i,j =1 aij bij for two matrices .A = (aij )i,j =1 and 3 ˆ ˆ defined in (5.4.3) contains .B = (bij ) i,j =1 . Since the Lamé parameter .(λ, μ) the imaginary part, thus there exists energy dissipation .E(uδ ) to the equation system (5.4.5) defined by E(uδ ) = ℑPλˆ ,μˆ (uδ , uδ ).

.

(5.4.10)

We are in a position to present the definitions of cloaking due to anomalous localized resonance. We say that the polariton resonance occurs if the following condition is satisfied, for any .M ∈ R+ , E(uδ ) ≥ M.

.

(5.4.11)

In addition to the condition (5.4.11), if the displacement field .uδ furthermore satisfies the following boundedness condition, namely |uδ | ≤ C,

.

when

˜ |x| > R,

(5.4.12)

with a fixed .R˜ ∈ R+ , then we say that the phenomenon of the CALR occurs. We refer to [110] for more relevant discussions. In this chapter, we consider the polariton resonance for the elastic system within the finite frequency regime beyond the quasi-static approximation. We first

250

5 Localized Resonances Beyond the Quasi-Static Approximation

derive the whole spectral system of the N-P operator corresponding to the elastic system within the finite frequency regime for the spherical geometry. Then through constructing two novel polariton configurations via choosing appropriate polariton parameters, we show that the polariton resonance and the CALR could occur for general source terms. Specifically, the spectral system of the N-P operator within the finite frequency regime is derived in Theorem 5.4.2. Indeed, the spectral system of the N-P operator in the static case has been derived in [53], which is in accordance with our results by taking the limit of the frequency to be zero in Theorem 5.4.2. We refer to Remark 5.4.1 for more discussions. If we set .D = ∅ and .Ω = BR with .BR denoting a central ball with radius R, then the polariton resonance occurs for a broad class of sources provided the Lamé parameter .μˆ inside the domain .Ω satisfies the condition (5.4.62) (see Theorem 5.4.3). Moreover, if we let .D = Bri and .Ω = Bre , the Newtonian potential .F of the source term .f be shown in (5.4.81), and the Lamé parameters .μ˘ as well as .μˆ satisfy the condition (5.4.81), then the phenomenon of the CALR occurs provided the source is supported inside the critical radius .r∗ = re3 /ri (see Theorem 5.4.4). The first major contribution of this study is the mathematical construction of core-shell-matrix polariton structures that can induce CALR within the finite frequency regime beyond the quasi-static approximation in elasticity. In existing studies concerning the polariton resonance for the elastic system [20, 21, 53, 91, 95, 96] as mentioned before, one always employs the quasi-static approximation in the analysis of the polariton resonance. Indeed, the quasi-static approximation plays a critical role. In [20, 21, 53, 95, 96], they directly consider the polariton resonance in the static case, and the paper [91] strictly verifies the quasi-static approximation. It is worth pointing out that this study is the first to consider the polariton resonance for the elastic system within the finite frequency regime beyond the quasi-static approximation. The second major contribution is that the polariton configurations considered are novel. In the study [20, 21, 53, 95, 96], both two strong convexity conditions shown in (5.4.2) are broken. However, in the current study, only the first strong convexity condition is broken to induce the polariton resonance for the polariton configurations. Although, the study [91] breaks off only one of the strong convexity conditions as well, however, they only consider the static case for the phenomenon of CALR. Nevertheless, the results on the CALR in the current study include the conclusions shown in [53, 91] as a special case. Several remarks are as follows. First, it is noted that in our polariton configurations, the domains .Ω and D are required to be balls, since in our analysis for the polariton resonance, the specific form of the eigensystem of the N-P operator is needed. Indeed, it is impossible to consider the polariton resonance for a general geometry from the mathematically theoretical perspective. Because one needs detailed information on the spectral properties of certain boundary integral operators. Even for the simplest model, i.e., the electrostatic case in two dimensions, only the radial geometry can be considered; see [22] for the ellipse geometry. Concerning the polariton resonance for other general geometry, one needs to resort to the assistance of the numerical simulations; see [34] for the electrostatic case. Second, when deriving the polariton resonance and the CALR, we only have the

5.4 Elastic Problem

251

constraint on the Lamé parameter .μˆ and require no restriction on another parameter λˆ , which makes our configuration easier for applications. Third, when showing the phenomenon of the CALR in Theorem 5.4.4, the Newtonian potential .F of the source term .f is assumed to have the expression in (5.4.81). Actually, the constraint on the source .f is only a technical issue. Indeed, the ALR is a spectral phenomenon at the limit point of eigenvalues of the N-P operator, which naturally requires that the order .n0 should be large as required in Theorem 5.4.4. Therefore, the phenomenon of the CALR could occur for a broad class of sources .f. We refer to Remark 5.4.6 for more discussions.

.

5.4.1 Preliminaries In this section, we show some preliminaries for the elastic system. We first introduce the elastostatic operator .Lλ,μ corresponding to the Lamé constants .(λ, μ) defined as Lλ,μ w := μΔw + (λ + μ)∇∇ · w,

.

(5.4.13)

for .w ∈ C3 . The traction (the conormal derivative) of .w on the .∂Ω is defined as follows ∂ν w = λ(∇ · w)ν + 2μ(∇ s w)ν,

.

(5.4.14)

where .∇ s is defined in (5.4.6) and .ν is the outward unit normal to the boundary .∂Ω. From [86], the fundamental solution .Gω = (Gωi,j )3i,j =1 for the operator .Lλ,μ + ω2 in three dimensions is given by (Gωi,j )3i,j =1 (x) = −

.

δij eikp |x| − eiks |x| 1 , eiks |x| + ∂i ∂j 2 |x| 4π μ|x| 4π ω

(5.4.15)

where .ks and .kp are defined in (5.4.8). Then the single layer potential associated with the fundamental solution .Gω is defined as ω .SΩ [ϕ](x) = Gω (x − y)ϕ(y)ds(y), x ∈ R3 , (5.4.16) ∂Ω

for .ϕ ∈ L2 (∂Ω)3 . On the boundary .∂Ω, the conormal derivative of the single layer potential satisfies the following jump formula   ∗

1 [ϕ](x) ∂ν SωΩ [ϕ]|± (x) = ± I + KωΩ 2

.

x ∈ ∂Ω,

(5.4.17)

252

5 Localized Resonances Beyond the Quasi-Static Approximation

where (KωΩ )∗ [ϕ](x) = p.v.



.

∂Ω

∂ν x Gω (x − y)ϕ(y)ds(y),

with .p.v. standing for the Cauchy principal value and the subscript .± indicating the limits from outside and inside .Ω, repectively. The operator .(KωΩ )∗ is called Neumann-Poincaré (N-P) operator. Let .Φ(x) be the fundamental solution for the operator .Δ+ω2 in three dimensions given as follows Φ(x) = −

.

eiωx . 4π |x|

Then for .ϕ ∈ L2 (∂Ω), define ω .SΩ [ϕ](x) = Φ(x − y)ϕ(y)ds(y),

(5.4.18)

x ∈ R3 .

(5.4.19)

∂Ω

Next, to facilitate the exposition, we present some notations and useful formulas. Let .N be the set of the positive integers and .N0 = N∪{0}. Set .Ynm with .n ∈ N0 , −n ≤ m ≤ n to be the spherical harmonic functions. Let .SR be the surface of the ball .BR and denote by .S for .R = 1 for simplicity. Furthermore, the operators .∇S , .∇S · and .ΔS designate the surface gradient, the surface divergence and the Laplace-Beltrami operator on the unit sphere .S. We also denote by .jn (t) and .hn (t), .n ∈ N0 , the spherical Bessel function and spherical Hankel function of the first kind of order n, respectively. In the analysis of the polariton resonance, sometimes we need do some asymptotic analysis for the functions .jn (t) and .hn (t) in terms of large n. Thus we introduce the following asymptotic expressions for the spherical Bessel and Hankel functions, .jn (t) and .hn (t), in terms of larger n ,   tn 1 + j`(t) , (2n + 1)!! .  (2n − 1)!!  ` hn (t) = 1 + h(t) , it n+1 jn (t) =

where .j`(t) = O

  1 n

and

` =O h(t)

  1 . n

(5.4.20)

5.4 Elastic Problem

253

Indeed, when the variable .t ⪡ 1, the spherical Bessel and Hankel functions, .jn (t) and .hn (t), satisfy the following asymptotic expressions with respect to t

 tn 1 + O(t) , (2n + 1)!! .  (2n − 1)!! hn (t) = 1 + O(t) . it n+1 jn (t) =

(5.4.21)

With the above preparation, we next introduce the following three important lemmas for our future analysis, which can be found in [115]. Lemma 5.4.1 For a vector field .w ∈ H 1 (S)3 and scalar .v ∈ H 1 (S), they satisfy the following relationships ∇S · (∇S v ∧ ν) = 0, .

ΔS v = ∇S · ∇S v, ∇S v · wds = − v∇S · wds, S

(5.4.22)

S

and ∇S · (wv) = ∇S · wv + w · ∇S v.

.

Lemma 5.4.2 The spherical harmonic functions .Ynm with .n ∈ N0 , −n ≤ m ≤ n, are the eigenfunctions of the Laplace-Beltrami operator .ΔS associated with the eigenvalue .−n(n + 1), namely ΔS Ynm + n(n + 1)Ynm = 0.

.

Lemma 5.4.3 The family .(Inm , Tnm , Nnm ), the vectorial spherical harmonics of order n, m m Inm =∇S Yn+1 + (n + 1)Yn+1 ν, m . Tn Nnm

=∇S Ynm

∧ ν,

n ≥ 0, n + 1 ≥ m ≥ −(n + 1), n ≥ 1, n ≥ m ≥ −n,

m m = − ∇S Yn−1 + nYn−1 ν,

n ≥ 1, n + 1 ≥ m ≥ −(n + 1),

forms an orthogonal basis of .(L2 (S))3 . From the Lemma 5.4.3, one has that m In−1 = ∇S Ynm + nYnm ν,

.

254

5 Localized Resonances Beyond the Quasi-Static Approximation

m can be expressed which is a vectorial spherical harmonics of order .n − 1, thus .In−1 by m In−1 = An−1,m Yn−1 ,

(5.4.23)

.

where −(n−1)

Yn−1 = [Yn−1

.

n−1 ⏉ , · · · , Yn−1 ] ,

and .An−1,m is a .3 × (2n − 1) matrix given as follows −(n−1)

An−1,m = [an−1,m , · · · , an−1 n−1,m ].

.

m of order .n + 1, it can be Similarly, for the vectorial spherical harmonics .Nn+1 expressed by m Nn+1 = Cn+1,m Yn+1 ,

(5.4.24)

.

where .Cn+1,m is a .3 × (2n + 3) matrix given as follows −(n+1) Cn+1,m = [cn+1,m , · · · , cn+1 n+1,m ].

.

With the help of these preliminaries, we prove the following three propositions. Proposition 5.4.1 The following identities hold .

S



q

S

and .

S

q



q

q

Y n−1 Ynm νds = Y n+1 Ynm νds =

Y p Ynm νds = 0,



an−1,m 2n + 1 q cn+1,m

2n + 1

, ,

q

S

q

S

Y n−1 ∇S Ynm ds = q

S

Y n+1 ∇S Ynm ds =

Y p ∇S Ynm ds = 0,

for

n+1 q a , 2n + 1 n−1,m −n q c , 2n + 1 n+1,m

p ≥ 0, p /= n − 1, n + 1,

where and also in what follows, the ovelrine denotes the conjugate. Moreover, the q coefficient vectors .an,m and .cm n+1,q , defined in (5.4.23) and (5.4.24), satisfy the following identity q

an,m =

.

2n + 3 m c . 2n + 1 n+1,q

(5.4.25)

5.4 Elastic Problem

255

Proof From Lemma 5.4.3 and the identities (5.4.23) as well as (5.4.24), one has that ∇S Ynm + nYnm ν = An−1,m Yn−1 , .

−∇S Ynm + (n + 1)Ynm ν = Cn+1,m Yn+1 .

(5.4.26)

q

Multiplying .Y n−1 on both sides of the identities (5.4.26) and integrating on the unit sphere .S yield that



q

.

S

Y n−1 ∇S Ynm ds + n

q

S

q

Y n−1 Ynm νds = an−1,m ,

(5.4.27)

and .

−



q

S

Y n−1 ∇S Ynm ds + (n + 1)

q

S

Y n−1 Ynm νds = 0.

(5.4.28)

Solving the Eqs. (5.4.27) and (5.4.28), one can obtain that

q

q

.

S

Y n−1 Ynm νds =

an−1,m 2n + 1

,

(5.4.29)

and .

S

q

Y n−1 ∇S Ynm ds =

n+1 q a , 2n + 1 n−1,m

which are the first two identities in the proposition. By the similar argument, the other four integral identities can be proved. The rest of the proof is to show the coefficient identity (5.4.25). Taking the conjugate on both sides of the Eq. (5.4.29) and replacing n with .n + 1 yield that

m

.

S

q

q

Y n+1 Yn νds =

an,m . 2n + 3

(5.4.30)

Comparing the Eq. (5.4.30) with the third integral identity of this proposition shows that q

an,m =

.

and this competes the proof.

2n + 3 m c , 2n + 1 n+1,q ⨆ ⨅

256

5 Localized Resonances Beyond the Quasi-Static Approximation

Proposition 5.4.2 The following identities hold .

(n + 1)(n − 1) q an−1,m , 2n + 1

q

S

(∇S Y n−1 · ∇S Ynm )νds =



n(n + 2) q , c 2n + 1 n+1,m

q

S

(∇S Y n+1 · ∇S Ynm )νds =

and

q

.

S

(∇S Y p · ∇S Ynm )νds = 0, q

for

p ≥ 0, p /= n − 1, n + 1,

q

where the coefficient vectors .an,m and .cn,m are defined in (5.4.23) and (5.4.24), respectively. Proof From Lemmas 5.4.1 and 5.4.2, direct calculation shows that S

=−

q (∇S Y p

· ∇S Ynm )νds

i=1 3 

ei

S

i=1

.

=−

=

3 

3 

ei

i=1

=n(n + 1)



S

ei

q

S

∇S Y p · ∇S Ynm (ν · ei )ds

 q Y p ∇S · ∇S Ynm (ν · ei ) ds  q Y p ΔS Ynm (ν · ei ) + ∇S Ynm · ∇S (ν · ei ) ds q

S



Y p Ynm ν −

S

q

Y p ∇S Ynm ds,

where and also in what follows, .ei , .i = 1, 2, 3 are the unit coordinate vectors. With the help of Proposition 5.4.1, one can obtain the integral identities in this proposition. The proof is completed. ⨆ ⨅ Proposition 5.4.3 The following identities hold .



q

S

∇S (∇S Y n−1 ) · ∇S Ynm ds = q

S

∇S (∇S Y n+1 ) · ∇S Ynm ds =

−n(n + 1)(n − 1) q an−1,m , 2n + 1 n(n + 1)(n + 2) q cn+1,m , 2n + 1

5.4 Elastic Problem

257

and

q

.

S

∇S (∇S Y p ) · ∇S Ynm ds = 0, q

p ≥ 0, p /= n − 1, n + 1,

for q

where the coefficient vectors .an,m and .cn,m are defined in (5.4.23) and (5.4.24), respectively. Proof From Lemmas 5.4.1 and 5.4.2, one has that

q

S

.

∇S (∇S Y p ) · ∇S Ynm ds =

ei

i=1



=n(n + 1)

3 

q

S

∇S (∇S Y p · ei ) · ∇S Ynm ds

q

S

∇S Y p Ynm ds.

Thus the integral identities in this proposition follow from the Proposition 5.4.1 and this completes the proof. ⨆ ⨅

5.4.2 Spectrum System of the Neumann-Poincaré Operator In this section, we derive the spectral system of the N-P operator for the elastic system within the finite frequency. To that end, we first deduce the spectrum of the associated single potential and then utilize the jump formulation (5.4.17) to obtain the spectral system of the N-P operator. The fundamental solution .Gω shown in (5.4.15) has the following decomposition Gω = Gω1 + Gω2 ,

.

(5.4.31)

where Gω1 = −

.

δij eiks |x| 4π μ|x|

and

Gω2 =

eikp |x| − eiks |x| 1 . ∂ ∂ i j |x| 4π ω2

Indeed, the first part, .Gω1 = Φ(x)δij /μ, where .Φ(x) is the fundamental solution of the operator .Δ + ω2 defined in (5.4.18). Moreover, the spectral system of the operator .SkSR defined in (5.4.19), associated with the kernel function .Φ(x), has been derived in [94]. For the convenience of readers, we include that in the following lemma.

258

5 Localized Resonances Beyond the Quasi-Static Approximation

Lemma 5.4.4 The eigen-system of the single layer potential operator .SkSR defined in (5.4.19) is given as follows SkBR [Ynm ](x) = −ikR2 jn (kR)hn (kR)Ym n,

.

x ∈ SR .

(5.4.32)

Moreover, the following two identities hold SkBR [Ynm ](x) = −ikR2 jn (k|x|)hn (kR)Ym n

x ∈ BR ,

.

and SkBR [Ynm ](x) = −ikR2 jn (kR)hn (k|x|)Ym n

.

x ∈ R3 \BR .

Thus, we mainly focus ourself on handling the second term .Gω2 given in (5.4.31). It is noted that the fundamental solution .Φ(x − y) defined in (5.4.18) has the following expansion Φ(x − y) = −ik

n ∞  

.

m

hn (k|x|)Ym x)jn (k|y|)Yn (ˆy) n (ˆ

for |y| < |x|.

n=0 m=−n

By direct calculations, there holds that ∇y Φ(x − y) = −ik

n ∞  



m hn (k|x|)Ym x)∇y jn (k|y|)Yn (ˆy) n (ˆ

n=0 m=−n .

= − ik

∞ 

n 

  m m ' (ˆ y )ˆ y + j (k|y|)∇ hn (k|x|)Ym (ˆ x ) j (k|y|)kY Y (ˆ y )/|y| , n S n n n n

n=0 m=−n

(5.4.33) and

.

n ∞    ∂ m ∇y Φ(x − y) = −ik hn (k|x|)Ym (ˆ x ) j''n (k|y|)k2 Yn (ˆy)ˆy n νy m=−n n=0

m m +jn' (k|y|)k∇S Y n (ˆy)/|y| − jn (k|y|)∇S Y n (ˆy)ˆy/|y|2

where .

∂ ∇y Φ(x − y) = ν y · ∇y2 Φ(x − y). νy

(5.4.34)

 ,

5.4 Elastic Problem

259

With the help the Propositions 5.4.1 and 5.4.2, one can derive the following important results: Proposition 5.4.4 There holds the following identities SR

.

∇x2 Φ(x − y) · ∇S Ynm (ˆy)ds =

  n(n + 1)(n − 1) n(n + 1) m − ikhn−1 (k|x|) j'n−1 (kR)kR In−1 − jn−1 (kR) 2n + 1 2n + 1   n(n + 1)(n + 2) n(n + 1) m + jn+1 (kR) Nn+1 − ikhn+1 (k|x|) j'n+1 (kR)kR . 2n + 1 2n + 1 (5.4.35)

and SR

∇x2 Φ(x − y) · (Ynm (ˆy)ν y )ds =



 n−1 1 m + j''n−1 (kR)k2 R2 In−1 jn−1 (kR) − j'n−1 (kR)kR 2n + 1 2n + 1 

 n+2 − ikhn+1 (k|x|) j'n+1 (kR)kR − jn+1 (kR) 2n + 1  1 m Nn+1 +j''n+1 (kR)k2 R2 . 2n + 1 (5.4.36) 

− ikhn−1 (k|x|) .

Proof Note that .∇x2 Φ(x − y) = ∇y2 Φ(x − y), and by using integration by parts as well as Lemma 5.4.2, there holds 2 m ∇x Φ(x − y) · ∇S Yn (ˆy)ds = ∇y2 Φ(x − y) · ∇S Ynm (ˆy)ds SR

.

SR

=−

1 R

SR

=n(n + 1)

∇y Φ(x − y)ΔS Ynm (ˆy)ds

1 R

SR

∇y Φ(x − y)Ynm (ˆy)ds,

260

5 Localized Resonances Beyond the Quasi-Static Approximation

Therefore, the integral identity (5.4.35) follows from the Proposition 5.4.1 and the identity (5.4.33). Next, we show another integral identity. Direct calculations yield that ∇x2 Φ(x − y) · (Ynm (ˆy)ν y )ds SR



.

  ∂ ∇SR (∇y Φ(x − y)) + (∇y Φ(x − y))ν y · (Ynm (ˆy)ν y )ds νy SR ∂ (∇y Φ(x − y))Ynm (ˆy)ds. = SR ν y

=

Thus, one can derive (5.4.36) from Proposition 5.4.1 and the identity (5.4.34). This completes the proof. ⨆ ⨅ Proposition 5.4.5 The following identity holds .

SR

∇x2 Φ(x − y) · (∇S Ynm (ˆy) ∧ ν y )ds = 0.

(5.4.37)

Proof By using integration by parts, there holds

∇x2 Φ(x − y) · (∇S Ynm (ˆy) ∧ ν y )ds

SR .=

1 R

=−

SR

=

SR

∇y2 Φ(x − y) · (∇S Ynm (ˆy) ∧ ν y )ds

∇S (∇y Φ(x − y)) · (∇S Ynm (ˆy) ∧ ν y )ds



SR

∇y Φ(x − y)∇S · (∇S Ynm (ˆy) ∧ ν y )ds = 0,

where the last identity follows from Lemma 5.4.1 and this completes the proof.

⨆ ⨅

With the above preparation, we are in a position to derive the spectral system of the single layer potential operator .SωBR . To that end, we first show the following m ] and .Sω [N m ] in the following single layer potentials .SωBR [Tnm ], .SωBR [In−1 BR n+1 theorem, which is fundamental for our further analysis. Theorem 5.4.1 The single layer potentials associated with the density functions m and .N m are given as follows when .x ∈ R3 \B Tnm , .In−1 R n+1

.

m SωBR [In−1 ](x)   (n + 1)ks jn−1,s hn−1 (ks |x|) nkp jn−1,p hn−1 (kp |x|) m In−1 + = −R 2 i . (λ + 2μ)(2n + 1) μ(2n + 1)   ks jn−1,s hn+1 (ks |x|) kp jn−1,p hn+1 (kp |x|) m 2 Nn+1 , − nR i − (λ + 2μ)(2n + 1) μ(2n + 1)

5.4 Elastic Problem

261

m SωBR [Nn+1 ](x)

 ks jn+1,s hn−1 (ks |x|) kp jn+1,p hn−1 (kp |x|) m − In−1 = −(n + 1)R i μ(2n + 1) (λ + 2μ)(2n + 1)   nks jn+1,s hn+1 (ks |x|) (n + 1)kp jn+1,p hn+1 (kp |x|) m + Nn+1 , − R2i μ(2n + 1) (λ + 2μ)(2n + 1) 

2

.

and SωBR [Tnm ](x) = −

.

iks R2 jn,s hn (ks |x|) m Tn , μ

here and in what follows, we denote .jn (ks R), .jn (kp R), .hn (ks R) and .hn (kp R) by jn,s , .jn,p , .hn,s and .hn,p for simplicity.

.

Proof The proof follows from the expression of the fundamental solution .Gω defined in (5.4.15), the Lemma 5.4.4, and Propositions 5.4.4 as well as 5.4.5. ⨆ ⨅ By the similar discussion as Theorem 5.4.1, when .x ∈ BR , the single layer m ] and .Sω [N m ] have the following expressions. potentials .SωBR [Tnm ], .SωBR [In−1 BR n+1 m ] Proposition 5.4.6 When .x ∈ BR , the single layer potentials .SωBR [Tnm ], .SωBR [In−1 m ω and .SBR [Nn+1 ] are expressed as follows m SωBR [In−1 ](x)   (n + 1)ks hn−1,s jn−1 (ks |x|) nkp hn−1,p jn−1 (kp |x|) m + In−1 = −R 2 i . μ(2n + 1) (λ + 2μ)(2n + 1)   ks hn−1,s jn+1 (ks |x|) kp hn−1,p jn+1 (kp |x|) m 2 Nn+1 − − nR i , (λ + 2μ)(2n + 1) μ(2n + 1) m SωBR [Nn+1 ](x)

.

 ks hn+1,s jn−1 (ks |x|) kp hn+1,p jn−1 (kp |x|) m In−1 − μ(2n + 1) (λ + 2μ)(2n + 1)   nks hn+1,s jn+1 (ks |x|) (n + 1)kp hn+1,p jn+1 (kp |x|) 2 m + Nn+1 −R i , μ(2n + 1) (λ + 2μ)(2n + 1) 

= −(n + 1)R 2 i

and SωBR [Tnm ](x) = −

.

iks R2 hn,s jn (ks |x|) m Tn . μ

262

5 Localized Resonances Beyond the Quasi-Static Approximation

From Theorem 5.4.1 and the continuity of the single layer potential operator .SωBR from .x ∈ R3 \BR to .x ∈ SR , one can conclude that for .x ∈ SR SωBR [Tnm ](x) = bn Tnm ,

(5.4.38)

.

m m m SωBR [In−1 ](x) = c1n In−1 + d1n Nn+1 ,

(5.4.39)

m m m SωBR [Nn+1 ](x) = c2n In−1 + d2n Nn+1 .

(5.4.40)

.

and .

where iks R2 jn,s hn,s , μ   jn−1,p hn−1,p kp n jn−1,s hn−1,s ks (n + 1) , = − R2i + μ(2n + 1) (λ + 2μ)(2n + 1)   jn−1,p hn+1,p kp jn−1,s hn+1,s ks 2 − , = − nR i μ(2n + 1) (λ + 2μ)(2n + 1)   jn+1,p hn−1,p kp jn+1,s hn−1,s ks 2 , − = − (n + 1)R i (λ + 2μ)(2n + 1) μ(2n + 1)   jn+1,s hn+1,s ks n jn+1,p hn+1,p kp (n + 1) + . = − R2i μ(2n + 1) (λ + 2μ)(2n + 1)

bn = − c1n .

d1n c2n d2n

The rest of the section is devoted to the derivation of the spectrum of the N-P operator through deriving the traction of the single layer potential on the .SR and the jump formula (5.4.17). First of all, we deduce the following two propositions. Proposition 5.4.7 The following identities hold for .n, p ∈ N0 :   ∇ · hn (k|x|)∇S Ypm = −p(p + 1)hn (k|x|)Ypm /|x|,

.

  ∇ · hn (k|x|)Ypm ν = (kh'n (k|x|) + 2hn (k|x|)/|x|)Ypm ,

.

and   ∇ · hn (k|x|)∇S Ypm ∧ ν = 0.

.

5.4 Elastic Problem

263

Proof By the vector calculus identity, one has that   ∇ · hn (k|x|)∇S Ypm = ∇hn (k|x|) · ∇S Ypm + hn (k|x|)∇ · ∇S Ypm .

= hn (k|x|)ΔS Ypm /|x| = −p(p + 1)hn (k|x|)Ypm /|x|,

where the last two identities follow from the Lemmas 5.4.1 and 5.4.2. Therefore, one can obtain the first identity in this proposition and the last two identities follow ⨆ ⨅ from the similar argument. This completes the proof. Proposition 5.4.8 The following identities hold for .n, p ∈ N0 :   ∇ hn (k|x|)∇S Ypm ν =kh'n (k|x|)∇S Ypm ,  ⏉ ∇ hn (k|x|)∇S Ypm ν = − hn (k|x|)∇S Ypm /|x|,   ∇ hn (k|x|)Ypm ν ν =kh'n (k|x|)Ypm ν, .

 ⏉ ∇ hn (k|x|)Ypm ν ν =kh'n (k|x|)Ypm ν + hn (k|x|)/|x|∇S Ypm ,   ∇ hn (k|x|)∇S Ypm ∧ ν ν =kh'n (k|x|)∇S Ypm ∧ ν,  ⏉ ∇ hn (k|x|)∇S Ypm ∧ ν ν = − hn (k|x|)/|x|∇S Ypm ∧ ν.

Proof We here only prove the first two identities and the others can be proved by similar arguments. With the help of vector calculus, one has that     ∇ hn (k|x|)∇S Ypm ν = ∇S Ypm ∇hn (k|x|)⏉ + hn (k|x|)∇∇S Ypm ν .

= kh'n (k|x|)∇S Ypm ,

where the last identity follows from  m .(∇∇S Yp )ν

=

 1 m ∇S ∇S Yp ν = 0. |x|

(5.4.41)

Thus the first identity is proved and it remains to show the second identity. Taking into account the symmetry of .∇∇Ypm and rewriting the Eq. (5.4.41) in the following      (∇∇S Ypm )ν = ∇ |x|∇Ypm ν = |x|∇∇Ypm + ∇S Ypm ν ⏉ ν = 0,

.

264

5 Localized Resonances Beyond the Quasi-Static Approximation

one can obtain that ⏉  ∇∇Ypm ν = ∇∇Ypm ν = −∇S Ypm /|x|.

.

(5.4.42)

Thus one can have that ⏉    ∇ hn (k|x|)∇S Ypm ν = ∇hn (k|x|)(∇S Ypm )⏉ + hn (k|x|)(∇∇S Ypm )⏉ ν  ⏉ ⏉   m m . ν = hn (k|x|) ∇∇Yp + ν ∇S Yp = −hn (k|x|)∇S Ypm /|x|, ⨆ ⨅

where the last identity follows from (5.4.42) and this completes the proof. Next, we derive the tractions of the single layer m ] on .S . and .SωBR [Nn+1 R

m ] potentials .SωBR [Tnm ], .SωBR [In−1

m ] Proposition 5.4.9 The traction of the single layer potentials .SωBR [Tnm ], .SωBR [In−1 m ω and .SBR [Nn+1 ] on .SR satisfy

∂ν SωBR [Tnm ](x) = bn Tnm ,

.

(5.4.43)

.

m m m ∂ν SωBR [In−1 ]|+ (x) = c1n In−1 + d1n Nn+1 ,

(5.4.44)

m m m ∂ν SωBR [Nn+1 ]|+ (x) = c2n In−1 + d2n Nn+1 ,

(5.4.45)

and .

where bn = −iks Rjn,s (ks Rh'n,s − hn,s ),  jn−1 (ks R)hn−1 (ks R)ks (n + 1) c1n = −2(n − 1)Ri 2n + 1  jn−1 (kp R)hn−1 (kp R)kp μn + (λ + 2μ)(2n + 1)   jn−1 (ks R)hn (ks R)ks2 (n + 1) + jn−1 (kp R)hn (kp R)kp2 n 2 +R i , 2n + 1   jn−1 (kp R)hn+1 (kp R)kp μ jn−1 (ks R)hn+1 (ks R)ks − d1n = 2n(n + 2)Ri 2n + 1 (λ + 2μ)(2n + 1)   2 −jn−1 (ks R)hn (ks R)ks + jn−1 (kp R)hn (kp R)kp2 2 + nR i , 2n + 1 .

5.4 Elastic Problem

265



 jn+1 (kp R)hn−1 (kp R)kp μ jn+1 (ks R)hn−1 (ks R)ks c2n = −2(n − 1)Ri − 2n + 1 (λ + 2μ)(2n + 1)   −jn−1 (ks R)hn (ks R)ks2 + jn−1 (kp R)hn (kp R)kp2 2 − (n + 1)R i , 2n + 1   jn+1 (ks R)hn+1 (ks R)ks n jn+1 (kp R)hn+1 (kp R)kp μ(n + 1) d2n = 2(n + 2)Ri + (2n + 1) (λ + 2μ)(2n + 1)   jn+1 (ks R)hn (ks R)ks2 n + jn+1 (kp R)hn (kp R)kp2 (n + 1) 2 −R i . 2n + 1 (5.4.46) 2

Proof The proof is directly from the definition of the traction in (5.4.14) and Propositions 5.4.7 as well as 5.4.8. ⨆ ⨅  ∗ We are in a position to show the spectrum of the N-P operator . KωBR .  ∗ Theorem 5.4.2 The spectral system of the NP operator . KωBR is as follows .

.

ω ∗ KBR [Tnm ] = λ1,n Tnm ,

(5.4.47)

ω ∗ KBR [Unm ] = λ2,n Unm ,

(5.4.48)

and

.

KωBR

∗

[Vnm ] = λ3,n Vnm ,

(5.4.49)

Where λ1,n = bn − 1/2,

.

if .d1n /= 0

(d2n − c1n )2 + 4d1n c2n .λ2,n = , 2

c1n + d2n + 1 − (d2n − c1n )2 + 4d1n c2n .λ3,n = , 2  

m m = c1n − d2n + (d2n − c1n )2 + 4d1n c2n In−1 + 2d1n Nn+1 , c1n + d2n + 1 +

Unm

.

 

m m Vnm = c1n − d2n − (d2n − c1n )2 + 4d1n c2n In−1 + 2d1n Nn+1 ;

.

266

5 Localized Resonances Beyond the Quasi-Static Approximation

if .d1n = 0 λ2,n = c1n − 1/2,

.

m Unm = In−1 ,

.

λ3,n = d2n − 1/2,

m m Vnm = c2n In−1 + (d2n − c1n )Nn+1 ,

with .Tnm , .Inm and .Nnm given in Lemma 5.4.3, and the parameters .bn , .c1n , .d1n , .c2n and .d2n defined in (5.4.46). Proof From the jump formula (5.4.17) and the identity (5.4.43), one can directly have that .

ω ∗ ∂ ω 1 SBR [Tnm ] − Tnm = (bn − 1/2)Tnm . KBR [Tnm ] = 2 ∂ν

Therefore the first identity (5.4.47) is proved. For the last two ones, i.e. (5.4.48) and (5.4.49), from the Eqs. (5.4.44) and (5.4.45), the eigenfunctions should be the m and .N m . Therefore we assume that the eigenfunclinear combinations of .In−1 n+1 m + N m , namely, tions have the following form .aIn−1 n+1 .

ω ∗ m m m m + Nn+1 ] = λ(aIn−1 + Nn+1 ). KBR [aIn−1

(5.4.50)

Again from the jump formula (5.4.17) and the identities (5.4.44) as well as (5.4.45), one has that

.

KωBR

∗

m m m [In−1 ] = (c1n − 1/2)In−1 + d1n Nn+1 ,

(5.4.51)

and .

ω ∗ m m m KBR [Nn+1 ] = c2n In−1 + (d2n − 1/2)Nn+1 .

Substituting the last two equations into (5.4.50) and comparing the coefficient on both sides yield that a 2 d1n + a(d2n − c1n ) − c2n = 0.

.

(5.4.52)

If .d1n /= 0, solving the Eq. (5.4.52) gives that a=

.

c1n − d2n ±

(d2n − c1n )2 + 4d1n c2n . 2d1n

(5.4.53)

Therefore, the last two identities, i.e. (5.4.48) and (5.4.49), follow from substituting (5.4.53) into (5.4.50).

5.4 Elastic Problem

267

If .d1n = 0, from the Eq. (5.4.51), one can directly have that

ω ∗ m m KBR [In−1 ] = (c1n − 1/2)In−1 ,

.

m is one of the eigenfunctions of the N-P operator .Kω which signifies that .In−1 SR corresponding to the eigenvalue .c1n − 1/2. For another eigenfunction containing m .N n+1 , solving the Eq. (5.4.52) yields that

a=

.

c2n . d2n − c1n

Substituting the last equation into (5.4.50) yields that .

ω ∗ m KBR [Vn ] = (d2n − 1/2)Vnm ,

where m m Vnm = c2n In−1 + (d2n − c1n )Nn+1 .

.

⨆ ⨅

The proof is complete.

Remark 5.4.1 In the paper [53], the full spectral system of the N-P operator .K0SR in the static case have been derived. Indeed, the results in [53] coincide with our conclusion on the spectral system of the N-P operator .KωSR in Theorem 5.4.2. This can be proved directly by taking .ω → 0 and applying the asymptotic properties of the spherical Bessel and Hankel functions, .jn (t) and .hn (t), for .t ⪡ 1 shown in (5.4.21).

5.4.3 Atypical Resonance Beyond the Quasi-Static Approximation In this section, we consider the atypical resonance for the structure containing no core. Thus, we assume that the domain .D = ∅ in the configuration .C0 defined in (5.4.4). Suppose that a source term .f is supported outside .Ω, then the elastic system (5.4.5) can be simplified as the following transmission problem ⎧ ⎪ L ˆ uδ (x) + ω2 uδ (x) = 0, ⎪ ⎪ ⎨ λ,μˆ Lλ,μ uδ (x) + ω2 uδ (x) = f, . ⎪ uδ (x)|− = uδ (x)|+ , ⎪ ⎪ ⎩ ∂νˆ uδ (x)|− = ∂ν uδ (x)|+ ,

x∈Ω x ∈ R3 \Ω x ∈ ∂Ω x ∈ ∂Ω,

(5.4.54)

268

5 Localized Resonances Beyond the Quasi-Static Approximation

where .∂ν is given in (5.4.14), .Lλ,μ is defined in (5.4.13) and .uδ satisfies the radiation condition (5.4.7). In (5.4.54) and also in what follows, .Lλˆ ,μˆ and .∂νˆ denote the Lamé ˆ and the operator and the traction operator associated with Lamé parameter .λˆ and .μ, same notations hold for the single layer potential operator .Sˆ ωΩ and N-P operator ˆ ω )∗ . .(K Ω Using the single layer potential defined in (5.4.16), the solution to the system (5.4.54) can be written as $

x ∈ Ω, Sˆ ωΩ [ψ 1 ](x), ω SΩ [ψ 2 ](x) + F, x ∈ R3 \Ω,

uδ =

.

(5.4.55)

where F(x) :=

.

R3

Gω (x − y)f(y)dy,

x ∈ R3 ,

(5.4.56)

is called the Newtonian potential of the source .f and .ψ 1 , ψ 2 ∈ L2 (∂Ω)3 . One can readily verify that the solution defined in (5.4.55) satisfy the first two conditions in (5.4.54). For the third and forth condition in (5.4.54) on .∂Ω, namely the transmission condition, one can obtain that  Sˆ ωΩ [ψ 1 ] − SωΩ [ψ 2 ] = F, . (5.4.57) x ∈ ∂Ω. ∂νˆ Sˆ ωΩ [ψ 1 ]|− − ∂ν SωΩ [ψ 2 ]|+ = ∂ν F, With the help of the jump formula (5.4.17), the Eq. (5.4.57) can be rewritten as  Aωδ

.

ψ1 ψ2



 =

 F , ∂ν F

(5.4.58)

where  ω .Aδ

=

 Sˆ ωΩ −SωΩ ˆ ω )∗ −1/2I − (Kω )∗ . −1/2I + (K Ω Ω

(5.4.59)

In the following analysis, we assume that the domain .Ω is a ball .BR . Since the source term .f is supported outside .BR , therefore there exists .ϵ > 0 such that when .x ∈ BR+ϵ , the Newtonian potential .F defined in (5.4.56) satisfies Lλ,μ F + ω2 F = 0.

.

5.4 Elastic Problem

269

Thus .F can be written as F=

.

n ∞  

 m m f1,n,m jn (ks |x|)Tnm + f2,n,m SωBR [In−1 ] + f3,n,m SωBR [Nn+1 ] , n=0 m=−n

(5.4.60) which follows from Lemma 5.4.3 and Proposition 5.4.6. Our main result in this section is stated in the following theorem. It characterizes the atypical resonance for the configuration without a core. Theorem 5.4.3 Consider the configuration .C0 with .D = ∅ defined in (5.4.4) and a source term .f supported outside the domain .Ω. If the Lamé parameter .μˆ inside the domain .Ω is chosen such that for any .M ∈ R+ .

ℑ(μ) ˆ > M,  |ψ1,n0 ,m |2

(5.4.61)

1,n0 ,m is defined in (5.4.67) and .f1,n0 ,m for some .n0 ∈ N with .f1,n0 ,m /= 0, where .ψ is given in (5.4.60), then the atypical resonance could occur. Furthermore, if .n0 ⪢ 1 is large enough such that the spherical Bessel (1) and Hankel functions, .jn (t) and .hn (t), enjoy the asymptotic expression shown in (5.4.20), then one can choose the Lamé parameter .μˆ inside the domain .Ω as follows μˆ = −μ + iδ + p1,n0 ,

(5.4.62)

p1,n0 + q1,n0 = O(δ),

(5.4.63)

.

where .p1,n0 should satisfy .

with .q1,n0 defined in (5.4.71), to ensure the occurrence of the atypical resonance. Proof From Propositions 5.4.6 and 5.4.9, one can conclude that both the displacement and traction of the term .jn (ks |x|)Tnm on the boundary .∂BR are orthogonal to m ] and .Sω [N m ]. Therefore, to show that of the other two terms, namely .SωBR [In−1 BR n−1 the atypical resonance, one could just consider the source only containing the terms m .jn (ks |x|)Tn , namely F=

.

n ∞  

 f1,n,m jn (ks |x|)Tnm . n=0 m=−n

(5.4.64)

270

5 Localized Resonances Beyond the Quasi-Static Approximation

Thanks to the orthogonality of the functions .Tnm , .Inm and .Nnm , the density functions expressed in (5.4.55) have the following expressions ψ1 =

+∞  n 

ψ1,n,m Tnm ,

n=0 m=−n .

ψ2 =

n +∞  

(5.4.65) ψ2,n,m Tnm .

n=0 m=−n

From the jump formula (5.4.17), and Propositions 5.4.6 and 5.4.9, the Eq. (5.4.58) can be written as follows      a11 a12 ψ1,n,m f1,n,m jn (ks R) = , . (5.4.66) a21 a22 ψ2,n,m g1,n,m where ikˆs R 2 jn (kˆs R)hn (kˆs R) , μ

a11 = −

.

a12 =

iks R2 jn (ks R)hn (ks R) , μ

  a21 = −ikˆs R 2 hn (kˆs R) kˆs Rjn' (kˆs R) − jn (kˆs R) ,

.

 a22 = −1 + iks R 2 hn (ks R) ks Rjn' (ks R) − jn (ks R) ,

.

and

 g1,n,m = f1,n,m μ ks Rjn' (ks R) − jn (ks R) /R.

.

With the help of the Wronskian identity jn (t)hn(1)' (t) − jn' (t)h(1) n (t) =

.

i , t2

for

t > 0,

solving the Eq. (5.4.66) yields that ψ1,n,m =

.

f1,n,m jn (ks R) , 1,n,m ψ

where  ' ˆ 1,n,m = ((μ − μ1 )jn (kˆs R) + kˆs muRj ψ ˆ n (ks R))hn (ks R) .  −ks μRjn (kˆs R)h'n (ks R) ks kˆs R 3 jn (ks R)hn (kˆs R).

(5.4.67)

5.4 Elastic Problem

271

Next we calculate the dissipation energy .E(uδ ). From the definition of the functional Pλ,μ (u, u) given in (5.4.9) and the following identity

.

∇ · uδ = ∇ · Sˆ ωΩ [Tnm ] = 0,

.

there holds that   E(uδ ) = ℑPλˆ ,μˆ (uδ , uδ ) = ℑ μP ˆ λˆ /μ,1 ˆ (uδ , uδ ) .

= ℑ(μ) ˆ

n  +∞     ˆ ω [Tnm ], Sˆ ω [Tnm ] . |ψ1,n,m |2 Pλˆ /μ,1 S Ω Ω ˆ

(5.4.68)

n=0 m=−n

Thus if there exists .n0 such that for any .M ∈ R+ 2 ℑ(μ)|ψ ˆ 1,n0 ,m | > M,

.

(5.4.69)

then the resonance occurs. From the expression of .ψ1,n,m in (5.4.67), the resonance condition (5.4.69) is equivalent to the following one .

ℑ(μ) ˆ > M,  |ψ1,n0 ,m |2

(5.4.70)

with .f1,n0 ,m /= 0. Next we do some asymptotic analysis for the left side of the condition (5.4.70) for .n0 ⪢ 1. From the asymptotic expression of the spherical Bessel and Hankel (1) functions, .jn (t) and .hn (t) shown in (5.4.20), one can obtain that 

1,n0 ,m = C μˆ + μ + q1,n0 ψ

.

(5.4.71)

where  q1,n0 = O

.

1 n0

 .

Thus if the parameter .μˆ inside the domain .Ω is chosen as stated in the theorem that .

μˆ = −μ + iδ + p1,n0 ,

(5.4.72)

p1,n0 + q1,n0 = O(δ),

(5.4.73)

where .

272

5 Localized Resonances Beyond the Quasi-Static Approximation

with .q1,n0 defined in (5.4.71), then the left side of condition (5.4.70) can be simplified as follows .

ℑ(μ) ˆ 1 ≥ . 2 1,n0 ,m | δ |ψ

(5.4.74)

Therefore if the parameter .δ ⪡ 1, then atypical resonance happens. The proof is complete. ⨆ ⨅ Remark 5.4.2 In Theorem 5.4.3, we only require the constrain on the Lamé parameter .μˆ and there is no restrict on the Lamé parameter .λˆ , which indicates that only the first strong convexity condition in (5.4.2) is broken. Remark 5.4.3 We do the numerical simulation to demonstrate that the condition (5.4.61) can be achieved. The parameters are chosen as follows n0 = 5, ω = 5, R = 1, μ = 1, and μˆ = −1.87988 + iδ,

.

which is the case beyond the quasi-static approximation from the values of .ω and R. The norm of the LHS quantity in (5.4.61) in terms of the parameter .δ is plotted in Fig. 5.17, which evidently demonstrates that the condition (5.4.61) is fulfilled. Remark 5.4.4 Indeed, the condition (5.4.63) is easy to achieve. Since the parameter .q1,n0 defined in (5.4.71) is of .O(1/n0 ), therefore one could choose .p1 = O(1/n0 ) to fulfill the condition (5.4.63). Moreover, we do the numerical simulation 4 x 108

3 x 108

2 x 108

1 x 108

5. x 10-7

1. x 10-6

1.5 x 10-6

Fig. 5.17 The norm of the LHS quantity in (5.4.61) in terms of the parameter .δ

5.4 Elastic Problem

273

5. x 10-10 4. x 10-10 3. x 10-10 2. x 10-10 1. x 10-10 - 0.0277901 -0.0277900 - 0.0277899 - 0.0277898

Fig. 5.18 The norm of the LHS quantity in (5.4.63) in terms of the parameter .p1

to demonstrate that the condition (5.4.63) can be fulfilled. The parameters are chosen as follows n0 = 100, ω = 5, R = 1, μ = 1, δ = 10−10 and μˆ = −μ + iδ + p1,n0 .

.

One can easily verify that this is the case beyond quasi-static approximation. The norm of the LHS quantity in (5.4.63) in terms of the parameter .p1,n0 is plotted in Fig. 5.18, which apparently demonstrates that the condition (5.4.63) is satisfied with .p1,n0 ≈ 0.02779005 = O(1/n0 ).

5.4.4 CALR Beyond the Quasi-Static Approximation In this section, we consider the cloaking effect induced by anomalous localized resonance. In the following, let .D = Bri and .Ω = Bre . To save the notations, we first define the following two functions j´n (t) = tjn' (t) − jn (t), .

h´ n (t) = th'n (t) − hn (t),

(5.4.75)

where .jn' (t) and .h'n (t) are the derivatives of the functions .jn (t) and .hn (t), respectively. We also introduce the following notations,

.

jn0i = jn (ks ri )

jn1i = jn (k˘s ri ),

jn2i = jn (kˆs ri ),

jn0e = jn (ks re )

jn1e = jn (k˘s re ),

jn2e = jn (kˆs re ),

(5.4.76)

274

5 Localized Resonances Beyond the Quasi-Static Approximation

and, the same notations hold for the spherical Hankel function .hn (t), the derivative of the Bessel and Hankel functions, .jn' (t) and .h'n (t), the functions .j`n (t) and .h` n (t) defined in (5.4.20), and the functions .j´n (t) and .h´ n (t) defined in (5.4.75). Moreover, ˘ ω )∗ , respectively, denote the Lamé operator, the we let .Lλ˘ ,μ˘ , .∂ν˘ , .S˘ ∂D and .(K ∂D associated conormal derivative, the single layer potential operator and the N-P operator associated with the Lamé parameters .(λ˘ , μ). ˘ Assume that the source .f is supported outside .Ω, then the elastic system (5.4.5) can be expressed as the following equation system ⎧ Lλ˘ ,μ˘ uδ (x) + ω2 uδ (x) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Lλˆ ,μˆ uδ (x) + ω2 uδ (x) = 0, . Lλ,μ uδ (x) + ω2 uδ (x) = f, ⎪ ⎪ ⎪ uδ |− = uδ |+ , ∂ν˘ uδ |− = ∂νˆ uδ |+ ⎪ ⎪ ⎩ uδ |− = uδ |+ , ∂νˆ uδ |− = ∂ν uδ |+

in D, in Ω\D, in R3 \Ω, on ∂D, on ∂Ω.

(5.4.77)

With the help of the potential theory, the solution to the equation system (5.4.77) can be represented by ⎧ ω x ∈ D, ⎨ S˘ D [ϕ 1 ](x), .uδ (x) = Sˆ ωD [ϕ 2 ](x) + Sˆ ωΩ [ϕ 3 ](x), x ∈ Ω\D, ⎩ ω x ∈ R3 \Ω, SΩ [ϕ 4 ](x) + F(x),

(5.4.78)

where .ϕ 1 , ϕ 2 ∈ L2 (∂D)3 , .ϕ 3 , ϕ 4 ∈ L2 (∂Ω)3 and .F is the Newtonian potential of the source .f defined in (5.4.56). One can easily see that the solution given (5.4.78) satisfies the first three condition in (5.4.77) and the last two conditions on the boundary yield that ⎧ ω ω ω ⎪ ⎪ S˘ D [ϕ 1 ] = Sˆ D [ϕ 2 ] + Sˆ Ω [ϕ 3 ], ⎪ ⎨ ˘ω ω ∂ν˘ SD [ϕ 1 |− = ∂νˆ (Sˆ D [ϕ 2 ] + Sˆ ωΩ [ϕ 3 ])|+ , . ⎪ Sˆ ωD [ϕ 2 ] + Sˆ ωΩ [ϕ 3 ] = SωΩ [ϕ 4 ] + F, ⎪ ⎪ ⎩ ∂νˆ (Sˆ ωD [ϕ 2 ] + Sˆ ωΩ [ϕ 3 ])|− = ∂ν (SωΩ [ϕ 4 ] + F)|+ ,

on on on on

∂D, ∂D, ∂Ω, ∂Ω.

(5.4.79)

With the help of the jump formula in (5.4.17), the system (5.4.79) further yields the following integral system, ⎡

⎤⎡ ⎤ ⎡ ⎤ −Sˆ ωD −Sˆ ωΩ 0 S˘ ωD ϕ1 0 ⎢ 1 ⎥ ⎢ ⎥ ˆ ω )∗ ˘ ω )∗ − 1 − (K ∂νˆ i Sˆ ωΩ 0 ϕ2 ⎥ ⎢ − 2 + (K ⎥⎢ Ω D 2 ⎥ = ⎢ 0 ⎥, .⎢ ⎥⎢ ω ω ω ⎣ ⎦ ⎣ ˆ ˆ F ⎦ ⎦ ϕ3 ⎣ −SΩ 0 SD SΩ 1 1 ω ω ω ∗ ∗ ˆ ) − − (K ) ϕ4 ∂ν F 0 ∂νˆ e Sˆ D − 2 + (K Ω Ω 2 (5.4.80) where .∂νˆ i and .∂νˆ e signify the conormal derivatives on the boundaries of D and .Ω, respectively.

5.4 Elastic Problem

275

In the following, we assume that the Newtonian potential .F of the source .f has the following expression F=

.

∞  n 

 f1,n,m jn (ks |x|)Tnm

for

x ∈ Ω,

(5.4.81)

n=N m=−n

where N is large enough such the spherical Bessel and Hankel functions, .jn (t) and hn (t), fulfill the asymptotic expansions shown in (5.4.20). From Theorem 5.4.1 and m and .N m , one can deduce that the the orthogonality of the functions .Tnm , .In−1 n+1 density functions .ϕ i , .i = 1, 2, 3, 4 can be written as follows

.

ϕ1 = .

ϕ3 =

+∞  n 

ϕ1,n,m Tnm ,

ϕ2 =

+∞  n 

ϕ2,n,m Tnm ,

n=N m=−n

n=N m=−n

+∞ 

+∞  n 

n 

ϕ3,n,m Tnm ,

n=N m=−n

ϕ4 =

(5.4.82) ϕ4,n,m Tnm .

n=N m=−n

With the help of the Eq. (5.4.38) as well as Theorem 5.4.2 and by substituting the expressions in (5.4.81) and (5.4.82) into the equation system (5.4.80), the integral system can be reduced to the following equation system ⎡

a11 ⎢ a21 .⎢ ⎣ 0 0

a12 a22 a32 a42

a13 a23 a33 a43

⎤⎡ ⎤ ⎡ ⎤ 0 ϕ1,n,m 0 ⎢ ⎥ ⎢ ⎥ 0 ⎥ 0 ⎥ ⎢ ϕ2,n,m ⎥ = ⎢ ⎥, ⎦ ⎣ ⎦ ⎣ a34 ϕ3,n,m f1,n,m jn0e ⎦ a44 ϕ4,n,m g1,n,m

(5.4.83)

where a11 =

.

−ik˘s ri2 jn1i hn1i , μ˘

a21 = −ik˘s ri j´n1i hn1i ,

.

a32 =

.

−ikˆs ri2 jn2i hn2e , μˆ

a42 = −ikˆs ri jn2i h´ n2e ,

.

a12 =

−ikˆs ri2 jn1i hn2i , μˆ

a22 = −ikˆs ri jn2i h´ n2i , a33 =

−ikˆs re2 jn2e hn2e , μˆ

a43 = −ikˆs re j´n2e hn2e ,

a13 =

a23 = −ikˆs re j´n2i hn2e , a34 =

iks re2 jn0e hn0e , μ

a44 = iks re jn0e h´ n0e ,

and 

' g1,n,m = f1,n,m μ ks re jn0e − jn0e /re .

.

−ikˆs re2 jn1i hn2e , μˆ

276

5 Localized Resonances Beyond the Quasi-Static Approximation

Solving the equation system (5.4.83) gives that ϕ1,nm = .

ϕ3,nm

 ϕ1,n,m , dn,m

 ϕ3,n,m , = dn,m

ϕ2,nm = ϕ4,nm

 ϕ2,n,m , dn,m

(5.4.84)

 ϕ4,n,m = , dn,m

where

 ϕ1,n,m =

.

 ϕ2,n,m =

.

 ϕ3,n,m =

.

−ik kˆ 2 ri re2 f1,n,m hn23 jn0e jn2i (h´ n0e jn0e − j´n0e hn0e ) (h´ n2i jn2i re − j´n2i hn2i ri ) μˆ ˆ i re2 f1,n,m hn1i hn2e jn0e (h´ n0e jn0e − j´n0e hn0e ) ik k˘ kr (j´n2i jn1i μr ˆ i − j´n1i jn2i μr ˘ e) μ˘ μˆ ˆ 3 re f1,n,m hn1i jn2i jn0e (h´ n0e jn0e − j´n0e hn0e ) ik k˘ kr i (j´n1i hn2i μ˘ − h´ n2i jn1i μ) ˆ μ˘ μˆ

  ik˘ kˆ 2 ri2 f1,n,m j´n0e μri jn1i (h´ n2i jn2e re − j´n2i hn2e ri )μˆ  ϕ4,n,m = .

+j´n1i (hn2e jn2i − hn2i jn2e )μr ˘ e

,

,

,

 ×

μ˘ μˆ 2

  ˆ e j´n1i (j´n2e hn2i ri − h´ n2e jn2i re )μ˘ hn1i hn2e jn2i jn0e μr +jn1i (h´ n2e j´n2i − h´ n2i j´n2e )μr ˆ i

 ,

μ˘ μˆ 2 and   k k˘ kˆ 2 ri2 re2 h´ n0e μri jn1i (h´ n2i jn2e re + j´n2i hn2e ri ) dn,m = .

μˆ − j´n1i (hn2e jn2i − hn2i jn2e )μr ˘ e



× μμ˘ μˆ 2   hn1i hn2e jn0e jn2i hn0e μr ˆ e j´n1i (h´ n2e jn2i re − j´n2e hn2i ri )μ˘ +jn1i (h´ n2i j´n2e − h´ n2e j´n2i )μr ˆ i μμ˘ μˆ 2

 .

5.4 Elastic Problem

277

To simplify the exposition, we introduce the following two notations ' ηn2e = n − 1 + nj`n2e − j`n2e ,

(5.4.85)

γn2e = n + 2 + (n + 1)h` 'n2e + h` n2e ,

(5.4.86)

.

and .

' , .j` , .h ' ` n2e are defined in (5.4.76). The same notations also hod where .j`n2e n2e ` n2e and .h for .ηn1i , .ηn21 , .γn0e and .γn2i . We also define the following function

q2,n (μ, ˘ μ, ˆ ri , r e ) = (μ˘ + μ)(μ ˆ + μ)n ˆ 2 re2 + (μr ˆ i − μr ˘ e )(μri − μr ˆ e )n2 ρ 2n −    ˘ e ηn11 γn2e ρ 2n (1 + j`n2i ) + (1 + h` n2i )ηn2e − μr ˆ e (1 + h` n0e ) μr .

  2n ` μ(1 ˆ + jn1i ) ri γn2e ηn2i ρ − re γn2i ηn2e −

(5.4.87)

   μγn0e μr ˘ e ηn1i re (1 + h` n2i )(1 + j`n2e ) − ri ρ 2n (1 + h` n2e )(1 + j`n2i )   μ(1 ˆ + j`n1i ) ri2 ρ 2n (1 + h` n2e )ηn2i + re2 (1 + j`n2e )γn2i , here and also in what follows, .ρ = ri /re . With the above preparation, we are in a position to show the CALR result, which is concluded in the following theorem. Theorem 5.4.4 Consider the configuration .(C0 , f) where .C0 is given in (5.4.4) with D = Bri and .Ω = Bre , and the Newtonian potential .F of the source term .f has the expression in (5.4.81). Moreover, Let the parameters in .C0 be chosen as follows

.

μ˘ = μ,

.

μˆ = −μ + iδ + p2,n0 ,

and

δ = ρ n0 ,

(5.4.88)

such that   2 2n0 , p2,n + q = O ρ 2,n0 0

.

(5.4.89)

where .n0 > N , with N defined in (5.4.81), and .q2,n0 is given in (5.4.87). Then the phenomenon of the CALR

could occur provided the source is supported inside the critical radius .r∗ = re3 /ri . If the source is supported outside .Br∗ , then the resonance does not occur.

278

5 Localized Resonances Beyond the Quasi-Static Approximation

Proof We first show the atypical resonance, namely the condition (5.4.11). For notational convenience of the proof, we set f˜1,n,m :=

.

f1,n,m , (2n + 1)!!

n ≥ N,

where N is defined in (5.4.81). From the asymptotic expressions of the spherical (1) Bessel and Hankel functions, .jn (t) and .hn (t), given in (5.4.20), one can show that the coefficients satisfy the following estimates | ϕ2,n,m | ≈

.

f1,n,m (kˆs ri )n , (2n + 1)!!

| ϕ3,n,m | ≈

| ϕ4,n,m | ≤

.

f1,n,m δ(kˆs re )n , (2n + 1)!!

(5.4.90)

f1,n,m (kre )n . (2n + 1)!!

(5.4.91)

Moreover, the condition (5.4.89) yields that when .n = n0 , |dn0 ,m | ≈ δ 2 + ρ 2n0 ,

(5.4.92)

|dn,m | ≥ δ 2 + ρ 2n .

(5.4.93)

.

and when .n /= n0 , .

Thus from (5.4.78), the displacement field .uδ to the system (5.4.77) in the shell Ω\D can be represented as

.

uδ = Sˆ ω∂D [ϕ 2 ](x) + Sˆ ω∂Ω [ϕ 3 ](x) .

=

∞  n  n=N m=−n

−

 i kˆs  ϕ2,n,m ri2 jn2i hn (kˆs |x|) + ϕ3,n,m re2 hn2e jn (kˆs |x|) Tnm , μˆ (5.4.94)

where .ϕ2,n,m and .ϕ3,n,m are defined in (5.4.84). Next we give the estimate of the dissipation energy .E(uδ ). From the definition of the dissipation energy .E(uδ ) in (5.4.10) and with the help of Green’s formula, one can have the following estimate  E(uδ ) = ℑPλˆ ,μˆ (uδ , uδ ) = ℑ .

≥

2 (kre )2n0 δ f˜1,n 0 ,m

δ 2 + ρ 2n0

≥



∂Ω

∂νˆ uδ uδ ds −

2 f˜1,n 0 ,m



n k 2 re3 0 ri

∂D

.

∂νˆ uδ uδ ds (5.4.95)

5.4 Elastic Problem

279

If the source .f is supported inside the critical radius .r∗ = re3 /ri , by (5.4.81) and the asymptotic property of .jn (t) in (5.4.20), one can verify that there exists .τ1 ∈ R+ such that ri 1/n + τ1 . (5.4.96) . lim sup(f˜1,n,m ) = k 2 re3 n→∞ Combining (5.4.95) and (5.4.96), one can obtain that  E(uδ ) ≥ n0

.

ri + τ1 k 2 re3

n0 

k 2 re3 ri

 n0 ,

which exactly shows that the atypical resonance occurs, namely the condition (5.4.11) is fulfilled. Then we consider the case where the source is supported outside the critical radius .r∗ . Thus there exists .τ2 > 0 such that .

lim sup(f˜1,n,m )1/n ≤ n→∞

1 , kr∗ + τ2

and the dissipation energy .E(uδ ) can be estimated as follows . E(uδ ) ≤

2  f˜1,n,m (kre )2n δ

δ 2 + ρ 2n

n≥N

≤



2 f˜1,n,m

n≥N



k 2 re3 ri

n0 ≤ C,

which means that resonance does not occur. Next we prove the boundedness of the solution .uδ when .|x| > re3 /ri2 . From (5.4.78), (5.4.82) and (5.4.84), the displacement field .uδ in .R3 \Ω can be represented as . uδ =

∞  n 

−

n=N m=−n

 iks  ϕ4,n,m re2 jn0e hn (kˆs |x|) Tnm + F(x), μ

(5.4.97)

Moreover, from (5.4.91), (5.4.92) and (5.4.93), one can obtain that

.

|uδ | ≤

∞  n 

 |f˜1,n,m |(kre )n0

n=N m=−n

when .|x| > re3 /ri2 . This completes the proof.

re3 ri2

n

1 + |F| ≤ C, rn

(5.4.98)

⨆ ⨅

280

5 Localized Resonances Beyond the Quasi-Static Approximation

Remark 5.4.5 Similar to Remark 5.4.2, in Theorem 5.4.4, we only require the constrain on the Lamé parameter .μˆ and there is no restrict on the Lamé parameter ˆ , which indicates that only the first strong convexity condition in (5.4.2) is broken. .λ Remark 5.4.6 In Theorem 5.4.4, the constraint on the source .f, whose Newtonian potential .F should have the expression in (5.4.81), is just a technical issue. Indeed, the phenomenon of the CALR could occur for a general source term .f. The reason we require N in (5.4.81) should be large is that we need to apply the asymptotic properties of the spherical Bessel and Hankel functions, .jn (t) and .hn (t) to prove the atypical resonance condition (5.4.11) and the boundedness condition (5.4.12). However, for the condition (5.4.11), the ALR is a spectral phenomenon at the limit point of eigenvalues of the N-P operator, which naturally requires that the order n should be large. While for the condition (5.4.12), if the item possessing the atypical resonance is bounded, then the other items are spontaneously bounded. Therefore the CALR could occur for a general source term .f. Remark 5.4.7 We do the numerical simulation to show that the condition (5.4.89) can be fulfilled. The parameters are chosen as follows n0 = 50, ω = 5, ri = 0.8, re = 1, μ˘ = μ = 1 and δ = (r1 /re )2n0 ≈ 2×10−10 ,

.

From the values of the parameters .ω and .re , one can readily verify that this is the case beyond quasi-static approximation. The norm of the LHS quantity in (5.4.89) in terms of the parameter .p2,n0 is plotted in Fig. 5.19, which apparently demonstrates that the condition (5.4.89) is satisfied. 6. x 10-10 5. x 10-10 4. x 10-10 3. x 10-10 2. x 10-10 1. x 10-10 -0.0513

-0.0512

-0.0511

-0.0510

- 0.0509

Fig. 5.19 The norm of the LHS quantity in (5.4.89) with respect the change of the parameter .p2

Chapter 6

Interior Transmission Resonance

In this chapter, we consider the interior transmission eigenvalue problem. Our focus is on justifying that the transmission eigenfunctions form a certain interior resonant modes. In fact, the results in this chapter show that the transmission eigenfunctions generically oscillate at frequencies much higher than the operating frequency, which is a typical resonance behaviour. Moreover, the high oscillation tend to localize on the boundary of the underlying the domain. This spectral phenomenon was first discovered in [41] for the acoustic transmission eigenfunctions and has triggered off a series of subsequent studies with intriguing discoveries and important applications in scattering theory. Since the transmission eigenvalue problem is non-elliptic and non-self-adjoint and couples two transmission eigenfunctions, the resonance possesses rich structures, say e.g. it is also discovered that the transmission eigenfunctions tend to vanish locally at a place with a high curvature or a singular tangential [29, 31]. However, we shall only discuss the high oscillation and boundary localization of the transmission eigenfunctions which justify that they form certain localized resonant modes as also mentioned above. We also refer to [7, 28, 30, 32, 33, 38, 46, 47, 49–51, 58, 60, 76, 89, 93, 99, 102–104] for more relevant discussion on the local behaviours of transmission eigenfunctions as well as the corresponding applications to invisibility cloaking and inverse scattering theory. The main results of this chapter follow from [41, 52, 58, 98].

6.1 Introduction The transmission eigenvalue problem was early studied in [80] and [45]. The exclusion of the transmission eigenvalues can guarantee the injectivity and dense range of the far-field operator and hence the validity of a certain reconstruction scheme for the inverse scattering problems. The spectral properties of the transmission eigenvalues have been extensively and intensively studied in the literature, and © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Y. Deng, H. Liu, Spectral Theory of Localized Resonances and Applications, https://doi.org/10.1007/978-981-99-6244-0_6

281

282

6 Interior Transmission Resonance

we refer to [37] and [99] for reviews and surveys on the existing developments on this aspect. However, the geometric properties of transmission eigenfunctions were not unveiled till the recent work [41] as well as the subsequent ones. As mentioned at the beginning of this chapter, we are mainly concerned with the boundary localization/concentration of transmission eigenfunctions; see Fig. 6.1 for two typical numerical illustrations, where the transmission eigenfunctions are plotted associated with different .(D, V )’s, where D is the underlying domain and 2 .V = n − 1 in (6.2.1) in what follows. It is highly intriguing to have the following observations: 1. It is clear that the transmission eigenfunctions are interior resonant modes which exhibit highly oscillatory patterns. Interestingly, the high oscillations of these resonant modes are localized on .∂D. The study on the eigenfunction concentration is a central topic in mathematical physics and spectral theory; see e.g. [129] and the references cited therein. However, the concentration phenomenon presented here is peculiar and different from the existing ones in the literature for the classical eigenvalue problems. Hence, it represents a new spectral phenomenon. 2. The physical intuition to explain the boundary-localizing behaviour can be described as follows. According to Theorem 6.1.1 in what follows, the transmission eigenfunctions are (at least approximately) restrictions of the incident and total wave fields when invisibility/transparency occurs. Hence, in order to reach the invisibility/transparency, a ‘smart’ way for the propagating wave is to ‘slide’ over the surface of the scattering object, namely .(D, V ), and return to its original path after passing through the object. This clearly gives rise to the regular pattern depicted in Fig. 6.1, where the wave fields inside the object clearly propagates along the surface .∂D. 3. The boundary-localization indicates that the transmission eigenfunctions carry the geometric information of the underlying scattering medium and hence is a global geometric rigidity property. This spectral property has been proposed for super-resolution wave imaging [41], generation of the so-called pseudo plasmon modes with a potential bio-sensing [41] application and the artificial electromagnetic mirage effect [41, 58]. In Sects. 6.2 and 6.3, we consider the acoustic and electro-magnetic transmission eigenfunctions, respectively. We show the boundary localization behaviours in different settings. Those eigen-modes are referred to as the boundary-localized eigenstates (SLEs) or boundary-localized eigenstates (BLEs). In the radial symmetric case with constant coefficients, we can have a much rigorous and comprehensive understanding of those SLEs, whereas in the general case we study those SLEs via extensive numerical experiments. Several interesting applications including superresolution imaging, generation of pseudo surface plasmon modes and artificial mirage are also presented and discussed. In Sect. 6.4, we present rigorous analysis on the boundary localization behaviour of generalized acoustic transmission eigenfunctions within general convex domains.

6.1 Introduction

283

(a) u-part,

= 5.5496

(b) v-part,

= 5.5496

(c) v-part,

= 3.1023

(d) v-part,

= 3.1023

Fig. 6.1 Transmission eigenfunctions to (6.2.1) associated with .n = 8 (or equivalently, .V = 63) for different D’s and transmission eigenvalue .κ’s. In Figures (a) and (b), the domain D is a central disk of radius 1. In Figure (c), D is an ellipsoid. Figure (d) is the slice plotting of Figure (c) at 3 3 .x3 = 0 for .x = (xj )j =1 ∈ R

We first introduce the time-harmonic acoustic wave scattering, which is the physical origin of the transmission eigenvalue problem in our study and moreover shall be used to motivate our mathematical analysis. Let D be an open connected and bounded domain in .Rd , .d ≥ 2, with a .C ∞ smooth boundary .∂D and a connected complement .Rd \D. In the physical setting, D signifies the support of an inhomogeneous medium scatterer located in an otherwise uniformly homogeneous space. The medium parameter is characterised by the refractive index which is normalised to be 1 in .Rd \D and is assumed to be .n ∈ R+ and .n /= 1 in D. Set .V = (n2 −1)χD +0χRd \D , which is referred to as the scattering potential. Let .ψ0 ∈ C ∞ (Rd ) be an impinging wave field which is an entire solution to .(Δ + κ 2 )ψ0 = 0 in .Rd , where .κ ∈ R+ signifies the angular frequency of the wave. The impingement of .ψ0 on the scattering potential .(D, V ), or equivalently on

284

6 Interior Transmission Resonance

the scattering medium .(D, n), leads to the following Helmholtz system for the total 1 (Rd ): wave field .ψ ∈ Hloc  .

Δψ + κ 2 (1 + V )ψ = 0

in Rd

(∂r − iκ)(ψ − ψ0 ) = O(r

− d+1 2

)

as r → ∞,

(6.1.1)

√ where .i := −1 and .r := |x| for .x ∈ Rd . The last limit in (6.1.1) is known as the Sommerfeld radiation condition which holds uniformly in the angular variable .xˆ := x/|x| ∈ Sd−1 and characterises the outgoing nature of the scattered .ψ s := ψ − ψ0 . The well-posedness of the scattering system (6.1.1) is known (cf. [44, 101]) and in particular it holds that ψ(x) = ψ0 (x) +

.

eiκr r (d−1)/2

 ψ∞ (x) ˆ +O



1 r (d+1)/2

as r → +∞.

(6.1.2)

In (6.1.2), .ψ∞ is referred to as the far-field pattern which encodes the scattering information of the underlying scatterer under the probing of the incident wave .ψ0 . An inverse problem of industrial importance is to recover .(D, V ) by knowledge of .ψ∞ . It is clear that the recovery fails if .ψ∞ ≡ 0, namely invisibility/transparency occurs. In such a case, one has by the Rellich theorem [44] that .ψ = ψ0 in .Rd \D. Hence, if setting .u = ψ|D and .v = ψ0 |D , it holds that ⎧ ⎪ ⎪ ⎨ .

⎪ ⎪ ⎩

Δu + κ 2 (1 + V )u = 0 Δv

+ κ 2v

u = v,

=0

∂ν u = ∂ν v

in D, in D,

(6.1.3)

on ∂D,

where and also in what follows .ν ∈ Sd−1 stands for the exterior unit normal to .∂D. That is, if invisibility/transparency occurs, the total and incident wave fields fulfil the spectral system (6.2.1), which is referred to as the transmission eigenvalue problem in the literature. Let us consider the spectral study of the transmission eigenvalue problem (6.2.1). It is clear that .u = v ≡ 0 are trivial solutions. If there exist nontrivial .u ∈ L2 (D) and .v ∈ L2 (D) such that .u − v ∈ H02 (D) and the first two equations in (6.2.1) are fulfilled, then .κ is referred to as a transmission eigenvalue and .u, v are the corresponding transmission eigenfunctions. It is emphasised that we are mainly concerned with real transmission eigenvalues, namely .κ ∈ R+ , though there exist complex transmission eigenvalues. The transmission eigenvalue problem is nonelliptic and non-selfadjoint, and this is partly evidenced by setting .w = u − v and verifying that

(Δ + κ 2 ) Δ + κ 2 (1 + V ) w = 0

.

in H02 (D),

(6.1.4)

6.2 Scalar Case (Helmholtz Equations)

285

which is a fourth-order PDE eigenvalue problem and quadratic in .λ = κ 2 . The following connection of the transmission eigenfunctions with the scattering problem (6.1.1)–(6.1.2) shall be a useful observation for our subsequent study. Theorem 6.1.1 ([29]) Suppose that .κ ∈ R+ is a transmission eigenvalue and u, v ∈ L2 (D) are the associated transmission eigenfunctions to (6.2.1). Then for any sufficiently small .ε > 0, there exists .gε ∈ L2 (Sd−1 ) such that

.

 ‖vgε − v‖L2 (D) < ε,

.

vgε (x)) :=

Sd−1

exp(iκx · θ )gε (θ ) ds(θ ).

(6.1.5)

Moreover, if taking .ψ0 = vgε in (6.1.1), one has .‖ψ∞ ‖L2 (Sd−1 ) ≤ CV ,κ ε and .‖u − ψ‖L2 (D) ≤ CV ,κ ε, where .CV ,κ is a positive constant depending only on V and .κ. In the physical setting, .vgε is referred to as a Herglotz wave, and Theorem 6.1.4 states that if .u, v are transmission eigenfunctions, they respectively correspond to the total and incident wave fields (restricted in D) from a nearly invisible/transparent scattering scenario.

6.2 Scalar Case (Helmholtz Equations) Let D be a bounded domain in .RN , .N = 2, 3, with a connected complement RN \D and .n ∈ L∞ (D) be a positive function. Consider the following transmission eigenvalue problem for .w ∈ H 1 (D) and .v ∈ H 1 (D):

.

⎧ ⎪ Δw + k 2 n2 w = 0 in D, ⎪ ⎨ in D, Δv + k 2 v = 0 . ⎪ ∂v ∂w ⎪ ⎩ w = v, = on ∂D, ∂ν ∂ν

(6.2.1)

where .ν is the exterior unit normal vector to .∂D. Clearly, .w ≡ v ≡ 0 are a pair of trivial solutions to (6.2.1). If there exists a non-trivial pair of solutions .(w, v) to (6.2.1), .k ∈ R+ is called a (real) transmission eigenvalue, and .w, v are the associated transmission eigenfunctions.

6.2.1 Boundary-Localized Transmission Eigenstates In this section, we first give the definite description of the SLEs. Next, we illustrate the existence of the SLEs for a special case theoretically. Then, for general situations, we provide extensive numerical examples to verify the existence of the SLEs and show their intriguing quantitative behaviours.

286

6 Interior Transmission Resonance

Definition 6.2.1 Consider a function .w ∈ L2 (Ω). It is said to be boundarylocalized if there exists a sufficiently small .ϵ0 ∈ R+ such that ‖w‖L2 (Nϵ .

0 (∂Ω))

‖w‖L2 (Ω)

= 1 − O(ϵ0 ),

where Nϵ0 (∂Ω) := {x ∈ Ω; dist(x, ∂Ω) < ϵ0 }.

.

First, we let .Ω := {x ∈ Rd : |x| < r0 ∈ R+ }, d = 2, 3, and n be a positive constant. For this case, we provide a comprehensive and accurate characterisation of the SLEs. To that end, we let .m ∈ N be a positive integer, .Jm (|x|) be the first kind Bessel function of order m, and .Jm' (|x|) be the derivative of .Jm (|x|). We also ' let .jm,s denote the s-th positive zero of .Jm (|x|), and .jm,s be the s-th positive zero of ' .Jm (|x|). Recalling from [1, Section 9.5, p. 370], one has ' ' ' m ≤ jm,1 < jm,1 < jm,2 < jm,2 < jm,3 < · · · ,.

(|x|/2)m ∞ |x|2 Jm (|x|) = . Π 1− 2 Γ (m + 1) s=1 jm,s

.

(6.2.2) (6.2.3)

To prove the localizing phenomena, we suppose that the order m of the Bessel function .Jm (x) is sufficiently large, and choose two sequences of integers s and .s ' as follows: s(m) := [mγ1 ],

.

s ' (m) = [mγ2 ],

(6.2.4)

where .[t] signifies the integer part of a real number t, namely .t = [t] + ϵt with 0 < ϵt < 1. The quantities .γ1 and .γ2 are two different constants that satisfy .0 < γ1 < γ2 < 1.

.

Lemma 6.2.1 Let .Ω = {x ∈ Rd : |x| < r0 ∈ R+ }, d = 2, 3, and .n > 1 be a constant. Let .{km,𝓁 } denote the subset of transmission eigenvalues k of (6.2.1), where m is the positive integer order of the Bessel function and .𝓁 denotes the .𝓁-th eigenvalue for a fixed m. Then there exists a subsequence of .{km,𝓁 }, denoted as .{km,s(m) }, such that for m sufficiently large, it holds that km,s ∈ (jm,s(m) , jm,s ' (m) ),

.

(6.2.5)

where .s(m) and .s ' (m) are defined in (6.3.20). Moreover, one has .

km,s → 1, m

as

m → ∞.

(6.2.6)

6.2 Scalar Case (Helmholtz Equations)

287

More specifically, there exists .ς ∈ (−2/3, 0) such that km,s = m(1 + mς + o(mς )).

(6.2.7)

.

Proof Without loss of generality, we assume that the radius of .Ω is .r0 = 1. Since n is a positive constant, we can expand the solutions w and v of the system (6.2.1) into Fourier series in terms of the Bessel functions .Jm (x) or the spherical Bessel functions .jm (x) of the first kind and the spherical harmonics:

w(x) =

⎧ ∞  ⎪ ⎪ αm Jm (kn|x|)eimθ , ⎨ m=−∞ ∞  m 

⎪ ⎪ ⎩

m=0 l=−m

.

v(x) =

d = 2,

l j (kn|x|)Y l (x), αm d = 3, m m ˆ

⎧ ∞  ⎪ ⎪ βm Jm (k|x|)eimθ , ⎨

d = 2,

⎪ ⎪ ⎩

d = 3,

m=−∞ ∞  m 

m=0 l=−m

l j (k|x|)Y l (x), βm m m ˆ

where .Yml is a spherical harmonic function of degree m and order l. Since  jm (|x|) =

.

π Jm+1/2 (|x|), 2|x|

we only consider the two-dimensional case and the three-dimensional case can be proved in a similar manner. For a fixed .m ∈ N+ , the solutions of .Δw + k 2 n2 w = 0 and .Δv + k 2 v = 0 in .Ω can be written as wk = αm Jm (kn|x|)eimθ , .

vk = βm Jm (k|x|)eimθ .

In order to guarantee that .w = v on the boundary .∂Ω, i.e., when .|x| = 1, we choose βm =

.

Jm (kn) αm . Jm (k)

Moreover, the transmission eigenvalues k’s are determined from the relation: .

∂v ∂w = ∂ν ∂ν

on ∂Ω.

288

6 Interior Transmission Resonance

Thus, with the help of the recurrence relation of the derivatives of the Bessel functions, the eigenvalues k’s are positive zeros of the following function fm (k) = Jm−1 (k)Jm (kn) − nJm (k)Jm−1 (kn),

m ≥ 1.

.

From (6.2.2) and (6.2.3), one can deduce that ' |x| ∈ [0, jm,1 ].

Jm (|x|) ≥ 0,

.

Next, we compute the following identity: .











fm jm,s fm jm,s ' = Jm−1 jm,s Jm njm,s Jm−1 (jm,s ' )Jm njm,s ' .

It can be shown [116] that the zeros .jm,s of the Bessel function .Jm (|x|) have the following sharp upper and lower bounds m−

.

as 1/3 as 1/3 3 2 21/3 m < j < m − m + , a m,s 20 s m1/3 21/3 21/3

(6.2.8)

where .as is the s-th negative zero of the Airy function and have the representation  as = −

.

3π (4s − 1) 8

2/3 (1 + σs ).

Here, .σs can be estimated by  0 ≤ σs ≤ 0.130

.

3π (4s − 1.051) 8

−2 .

By noting the choice of .s(m) and .s ' (m) in (6.3.20) for m sufficiently large, there holds jm,s = m(1 + C0 m2(γ1 −1)/3 + o(m2(γ1 −1)/3 )), .

jm,s ' = m(1 + C0 m2(γ2 −1)/3 + o(m2(γ2 −1)/3 )),

(6.2.9)

where .C0 is a positive constant. Note that the Bessel function admits the following asymptotic formula (see [84], p 129):  Jm (|x|) =





2

cos

|x|2 − m2 −

+m arcsin(m/|x|) −

 π  1 + o(1) , 4

.

π

|x|2

− m2

mπ 2 (6.2.10)

6.2 Scalar Case (Helmholtz Equations)

289

for .|x| > m and .m → ∞. By combing (6.3.24), (6.3.25) and some straightforward computations, one obtains    



π  1 π Jm njm,s = Cm,n cos m n2 − 1 − + arcsin + O(mς1 ) − ) 1+o(1) , 2 n 4

.

where .ς1 := 2(γ1 − 1)/3 and Cm,n

.

  −1/4  2  −1/4 2 2 2 2 n jm,s − m (n2 − 1)m := = + O(m2ς1 ). π π

And similarly,  

π  1 + o(1) , Jm−1 jm,s ' = Cm,1 cos mO(m2ς2 ) − 4

.

where .ς2 := 2(γ2 − 1)/3 and Cm,1

.

  −1/4  2 2 2 2 j ' −m O(m−1/2 m−ς2 /4 ). = := π m,s π

Without loss of generality we suppose that



Jm−1 jm,s Jm njm,s > 0.

.

We next show that there exists at least one choice of .s ' = mγ2 such that

Jm−1 (jm,s ' )Jm njm,s ' < 0,

.

that is,  .

cos mO(m

2ς2

  

π 1 π π ς2 2 cos m n − 1 − + arcsin + O(m ) − ) < 0. )− 4 2 n 4 (6.2.11)

Indeed, (6.2.11) can be easily realized by modifying .γ2 (noting that the two cosine functions above are always of different frequencies w.r.t .ς2 (γ2 )). Thus one has .km,s ∈ (jm,s(m) , jm,s ' (m) ) and by using (6.3.24) one has (6.3.21), which completes the proof. ⨆ ⨅ Remark 6.2.1 The parameters .γ1 and .γ2 in (6.3.20) are used to guarantee that there exists a transmission eigenvalue .km,s ∈ (jm,s(m) , jm,s ' (m) ) when m is sufficiently large. It is worth stating that the construction of .s(m) and .s ' (m) is important and technical for the proof of SLEs in the case of the ball.

290

6 Interior Transmission Resonance

With the help of Lemma 6.2.1, we are able to show the following important results about the existence of the SLEs. Theorem 6.2.1 Let .Ω = {x ∈ Rd : |x| < r0 ∈ R+ }, d = 2, 3, and .n > 1 be a constant. Consider the transmission eigenvalue problem (6.2.1). Then there exists a subsequence .{km,s } of the transmission eigenvalues .{k𝓁 } such that .∞ is the only accumulation point of the sequence .{km,s } and the corresponding eigenfunctions .vkm,s are boundary-localized around the boundary .∂Ω. Proof Without loss of generality, we assume that the radius of .Ω is .r0 = 1, we only consider the two-dimensional case and the three-dimensional case can be discussed similarly. We choose the sequence of .km,s satisfy (6.3.19) such that s and .s ' are chosen in (6.3.20), then such sequence .km,s → m, .m → ∞. We prove that the corresponding eigenfunctions .vkm,s are boundary-localized around the boundary .∂Ω. Let .Ωτ := {x : |x| < τ, τ < 1} with .ε := 1 − τ being a sufficiently small constant. We first prove that .Jm (km,s r) is monotonously increasing with respect to .r ∈ (0, 1). By using (6.2.2) and (6.3.23) one can show that ' jm,1 = m(1 + O(m−2/3 ))

.

as .m → ∞. Since we also have from (6.3.24) that km,s = m(1 + mς ),

.

ς ∈ [ς1 , ς2 ],

where .ςj = 2(γj − 1)/3, .j = 1, 2. One thus has ' km,s r = mr(1 + mς ) < jm,1 ,

.

for any fixed .r < 1 and sufficiently large m. One thus has .Jm' (km,s r) > 0 for ' .0 < r < 1, since .j m,1 is the first maximum of the Bessel function .Jm (km,s r). Furthermore, we note that there admits the following asymptotic formula (see [84], p. 129): Jm (km,s r) =

.

z=

√ 2 zm em 1−z

  1 + o(1) , √ (2π m)1/2 (1 − z2 )1/4 (1 + 1 − z2 )m km,s r , m

0 < r < 1.

6.2 Scalar Case (Helmholtz Equations)

291

Thus one can derive the following asymptotic expansion: ⎛ Jm (km,s |x|) < Jm (km,s τ ) = Cm

=Cm

−1/2

√

(1 − ε)

1+

e √

 1−

⎜ ⎜ ⎝ m

−1/2 ⎜ km,s τ



.



e

km,s τ m



1+

1−

2

⎞m

⎟   ⎟ 1 + o(1) ⎟   km,s τ 2 ⎠

2ε(1+o(1))+Cmς

2ε(1 + o(1)) + Cmς   =Cm−1/2 (1 − ε)m (1 + o(1))m 1 + o(1)

m

m

  1 + o(1)

(6.2.12) for .x ∈ Ωτ , where C is a positive constant. On the other hand, for m sufficiently large, one can choose .r1 by r1 =

.

' jm,1

km,s

,

where .τ < r1 < 1. Since ' m < jm,1 = km,s r1 ,

.

by using the asymptotic expansion (6.3.25), there holds  m1−ς/2 τ

1

Jm2 (km,s r) rdr

≥ m1−ς/2



1 r1

.

Jm2 (km,s r) rdr

2 m1−ς/2 ≥  π k 2 − m2 m,s =



 1 r1

2 − cos (r km,s 2

 m  π m2 mπ + m arcsin − )dr − 2 km,s r 4 r2

 2 m1−ς/2 1 − r1   1 + R(m) π k 2 − m2 2 m,s

  1  1  1 + R(m) , =√ 1 + o(1) + R(m) ≥ √ 2π 2π

(6.2.13)

292

6 Interior Transmission Resonance

where the remaining term .R(m) fulfils the following estimate as .m → ∞, 1 R(m) = 1 − r1 .



1

r1

    −2/3−ς 2/3ς 2 2 sin 2r km,s − m 1 + O(m +m ) dr

 √km,s 2 −m2 1  = √2 2 − m2 r1 km,s −m2 2(1 − r1 ) km,s    × sin r ' 1 + O(m−2/3−ς + m2/3ς ) dr ' → 0.

Note that ς

.

lim (1 − ε)m m− 2 = 0.

(6.2.14)

m→∞

Therefore, from (6.2.12) and (6.2.13), it holds that ‖vkm,s ‖2L2 (Ω

.

τ τ)

‖vkm,s ‖2L2 (Ω)

=

2 0 Jm (km,s r) rdr 1 2 0 Jm (km,s r) rdr

τ m1−ς/2 Jm2 (km,s τ ) 0 rdr ≤ → 0 as m → ∞. 1 m1−ς/2 r1 Jm2 (km,s r) rdr

Hence, the transmission eigenfunction .vkm,s is boundary-localized on the boundary ∂Ω. The proof is complete. ⨆ ⨅

.

Theorem 6.2.2 Under the same assumptions in Theorem 6.2.1 and suppose that {wkm,s } are the corresponding eigenfunctions with respect to w in (6.2.1). Then .{wkm,s } are not localized around .∂Ω, i.e., there exists a sufficiently small .ϵ0 ∈ R+ , which is independent of m, and .Ω1−ϵ0 := {x : |x| < 1 − ϵ0 }, such that .

‖wkm,s ‖2L2 (Ω .

1−ϵ0 )

‖wkm,s ‖2L2 (Ω)

m→∞

> 0,

(6.2.15)

Proof Since .wkm,s = βm Jm (km,s n|x|), one can immediately obtain that for .|x| = ' 1 jm,1 n km,s , .wkm,s

j'

attains its maximum value. Since . n1 km,1 < m,s

there exists .τ which is independent of m, such

that . n1

1 n

< 1 holds uniformly, thus

< τ < 1 (one can then choose

6.2 Scalar Case (Helmholtz Equations)

293

' /k ϵ0 = 1 − τ ). Noting that .jm,1 m,s < 1 and by using Theorem 6.2.1, one has

.



1 0

Jm2 (km,s nr) rdr =

1 n2

1 = 2 n

.

2 < 2 n



n

0

Jm2 (km,s r ' ) r ' dr '

' jm,1 km,s

 0



n ' jm,1 km,s

Jm2 (km,s r ' ) r ' dr '

Jm2 (km,s r ' ) r ' dr '

1 + 2 n 

=2



n ' jm,1 km,s

1 j' 1 m,1 n km,s

Jm2 (km,s r ' ) r ' dr '

Jm2 (km,s r) rdr. (6.2.16)

By following a similar arguments as in Theorem 6.2.1, one has for .τ ≤ t ≤ 1  Q(t) :=

t j' 1 m,1 n km,s

   π m 2 (r k 2 n2 − m2 /r 2 − mπ + m arcsin cos m,s km,s nr − 4 ) 2 2  dr = ' π n1 jkm,1 2 n2 − m2 /r 2 km,s m,s    1 2 n2 t 2 − m2 − j '2 − m2 . k ∼ m,s m,1 2 π km,s (6.2.17) 

.

Jm2 (km,s nr)rdr

t

Thus one has ‖wkm,s ‖2L2 (Ω

τ τ)

= 01

Jm2 (km,s nr) rdr

2 ‖wkm,s ‖2L2 (Ω) 0 Jm (km,s nr) rdr τ Jm2 (km,s nr)rdr j' 1 m,1 Q(τ ) n k ≥  1 m,s = 2 2Q(1) 2 j ' Jm (km,s nr) rdr .

1 m,1 n km,s

 2 n2 τ 2 − m2 − j '2 − m2 km,s m,1  ∼   '2 2 2 n2 − m2 − j 2 km,s m,1 − m √ n2 τ 2 − 1 > 0, ∼ √ 2 n2 − 1

which completes the proof.



⨆ ⨅

294

6 Interior Transmission Resonance

Theorems 6.2.1 and 6.2.2 indicate that we find a sequence of eigenfunction pairs {(wkm ,s , vkm ,s )} such that .{vkm ,s } are SLEs, whereas .{wkm ,s } are not SLEs. However, we would like to point out that we did not exclude the possibility of finding a sequence of eigenfunction pairs with both eigenfunctions being SLEs. Indeed, in what follows, we shall prove the existence of a sequence of eigenfunction pairs with both eigenfunctions being SLEs. Nevertheless, at this point, we can easily have the following result.

.

Theorem 6.2.3 Let .Ω = {x ∈ Rd : |x| < r0 ∈ R+ }, d = 2, 3, and .0 < n < 1 be a constant. Consider the transmission eigenvalue problem (6.2.1). Then there exists a subsequence .{km,s } of the transmission eigenvalues .{k𝓁 } such that .∞ is the only accumulation point of the sequence .{km,s } and the corresponding eigenfunctions .wkm,s are boundary-localized around the boundary .∂Ω. Proof Set  .k = k/n. It is directly verified that the first two equations in (6.2.1) become .(Δ +  k 2 )w = 0 and .(Δ +  k 2 n−2 )v = 0, while the transmission conditions on .∂Ω remain unchanged. Noting that .n−1 > 1, all of the previous results in Theorem 6.2.1 hold with v replaced by w. Thus, one has that .wk𝓁m is localized on the boundary .∂Ω. ⨆ ⨅ Let us consider the transmission eigenvalue problem (6.2.1) with B being a ball in .RN , .N = 2, 3, and n being a positive constant. By scaling and translation if necessary, we can assume that B is the unit ball, namely .B := {x ∈ RN ; |x| < 1}. In what follows, we set Bτ := {x ∈ RN ; |x| < τ },

τ ∈ (0, 1).

.

(6.2.18)

Definition 6.2.2 Consider a function .ψ ∈ L2 (B). It is said to be boundarylocalized if there exists .τ0 ∈ (0, 1), sufficiently close to 1, such that ‖ψ‖L2 (Bτ .

0)

‖ψ‖L2 (B)

⪡ 1.

(6.2.19)

It is easy to see that if .ψ is boundary-localized, then its .L2 -energy concentrate in a small neighbourhood of .∂B, namely .B\Bτ0 . The qualitative asymptotic smallness in (6.2.19) shall become more quantitatively definite in what follows. In fact, we can prove Theorem 6.2.4 Consider the transmission eigenvalue problem (6.2.1) and assume that B is the unit ball and .n /= 1 is a positive constant. Then for any given .τ ∈ (0, 1), there exists a sequence of transmission eigenfunctions .{wm , vm }m∈N associated to eigenvalues .km such that km → ∞ as m → ∞ and

.

lim

m→∞

‖ψm ‖L2 (Bτ ) ‖ψm ‖L2 (B)

= 0, ψm = wm , vm .

(6.2.20)

6.2 Scalar Case (Helmholtz Equations)

295

Remark 6.2.2 By (6.2.20), it is clear that both .{wm }m∈N and .{vm }m∈N are boundary-localized according to Definition 6.3.1. In our subsequent proof of Theorem 6.2.20, it can be seen that the transmission eigenmode corresponding to higher mode number is more localized around the surface. In other words, in (6.2.20), .τ can be very close to 1 provided .km is sufficiently large. It is interesting to point out that .1 − τ is actually the localizing radius of the eigenmode, which defines the super-resolution power of the wave imaging scheme proposed in [41]. Remark 6.2.3 It is sufficient for us to prove Theorem 6.2.4 only for the case .n > 1. In fact, let us suppose that Theorem 6.2.4 holds true for .n > 1, and instead consider −1 , .w the other case with .0 < n < 1. Set  .k = kn,  .n = n  = v and  .v = w. Then (6.2.1) can be recast as ⎧ ⎪ Δ w + k 2 n2 w  = 0 in D, ⎪ ⎨ 2  v=0 in D, Δ v+k  (6.2.21) . ⎪ ∂ v ∂w  ⎪ ⎩w on ∂D. = = v, ∂ν ∂ν Since  .n > 1, we readily have that there exist .(vm , wm ) = ( wm , vm ), .m ∈ N, associated to .km =  km /n → ∞, which are boundary-localized. In what follows, we shall derive a new sequence of transmission eigenfunctions which are both localized around the surface of the domain .Ω. At first, we prove Theorem 6.2.4 in the two-dimensional case. Let .x = (r cos θ, r sin θ ) ∈ R2 denote the polar coordinate. By Fourier expansion, the solutions to (6.2.1) have the following series expansions: w(x)) =

∞ 

.

αm Jm (kn|x|)eimθ ,

m=0

v(x)) =

∞ 

βm Jm (k|x|)eimθ ,

(6.2.22)

m=0

where .Jm is the m-th order Bessel function and .αm , βm ∈ C are the Fourier coefficients. Set wm (x)) = αm Jm (kn|x|)eimθ ,

.

vm (x)) = βm Jm (k|x|)eimθ .

(6.2.23)

In what follows, we shall construct the boundary-localized transmission eigenmodes of the form (6.2.23) to fulfil the requirement in Theorem 6.2.4. In order to make .(wm , vm ) in (6.2.23) transmission eigenfunctions of (6.2.1), one has by using the two transmission conditions on .∂D, together with straightforward calculations that βm =

.

Jm (kn) αm , Jm (k)

(6.2.24)

296

6 Interior Transmission Resonance

and k must be a root of the following function fm (k) = Jm−1 (k)Jm (kn) − nJm (k)Jm−1 (kn), m ≥ 1.

.

(6.2.25)

Next, we prove the existence of transmission eigenvalues by finding roots of .fm . In the sequel, we let .jm,s denote the s-th positive root of .Jm (t) (arranged according to ' denote the s-th positive root of .Jm' (t). Here, it is pointed the magnitude), and .jm,s out that both .Jm (t) and .Jm' (t) possess infinitely many positive roots, accumulating only at .∞ (cf. [105]). Lemma 6.2.2 Let .n > 1 and .s0 ∈ N be fixed. Then there exists .m0 (n, s0 ) ∈ N such that when .m > m0 (n, s0 ), one has .

jm,s0 ≤ m. n

(6.2.26)

Proof According to the formula (1.2) in [128], we know m−

.

3 2 21/3 as 1/3 as 1/3 m + , m < j < m − a m,s 20 s m1/3 21/3 21/3

where .as is the s-th negative zero of the Airy function and has the representation  as = −

.

3π (4s − 1) 8

2/3 (1 + σs ).

(6.2.27)

Here, .σs in (6.2.27) can be estimated by 

3π (4s − 1.051) .0 ≤ σs ≤ 0.130 8

−2 .

(6.2.28)

By combining the above estimates, one can show by straightforward calculations that .

The proof is complete.

jm,s0 ≤ m. n ⨆ ⨅

Lemma 6.2.3 Let .n > 1 and .s0 ∈ N be fixed. Then there exists .m0 (n, s0 ) ∈ N such that when .m > m0 (n, s0 ), the function .fm (k) in (6.2.25) possesses at least one zero jm,s jm,s +1 point in .( n 0 , . n0 ).

6.2 Scalar Case (Helmholtz Equations)

297

Proof According to the formula (9.5.2) in [1], we know ' ' ' m ≤ jm,1 < jm,1 < jm,2 < jm,2 < jm,3 < ··· .

.

jm,s0 +1 ' ≤ m ≤ jm,1 n jm,s0 jm,s0 +1 ( n , n ). We have

By Lemma 6.2.2, one has . consider .fm (k) for .k ∈

Jm (k) ≥ 0,

.

for .m > m0 (n, s0 + 1). Next we

! ' , k ∈ 0, jm,1

jm,s

jm,s

+1

and this implies .Jm (k) > 0 for .k ∈ ( n 0 , n0 ). Following Lemma 2.1 in [105], we know that the positive zeros of .Jm−1 are interlaced with those of .Jm , and hence Jm−1 (jm,s0 ) · Jm−1 (jm,s0 +1 ) < 0.

.

By using the above fact, one can show that    jm,s0 jm,s0 +1 · fm fm n n     jm,s0 +1 jm,s0 · Jm · Jm−1 (jm,s0 ) · Jm−1 (jm,s0 +1 ) = n2 Jm n n 

.

(6.2.29)

' ≤ n2 (Jm (jm,1 ))2 · Jm−1 (jm,s0 ) · Jm−1 (jm,s0 +1 )

< 0. which readily implies by Rolle’s theorem that there exists at least one zero point of jm,s jm,s +1 f (k) in .( n 0 , . n0 ). ⨆ ⨅ The proof is complete.

.

Clearly, Lemma 6.2.3 proves the existence of transmission eigenvalues. In what follows, for a fixed .s0 , we let the transmission eigenvalue be denoted by  klm := km,s0 ∈

.



jm,s0 jm,s0 +1 , n n

,

m = m0 +1, m0 +2, m0 +3, · · · ,

(6.2.30)

where .m0 = m0 (n, s0 ) is sufficiently large fulfilling the requirements in Lemmas 6.2.2 and 6.2.3. Lemma 6.2.4 There exists constants C and .γ such that .

Jm' (klm ) ⩽ Cmγ . Jm (klm )

(6.2.31)

298

6 Interior Transmission Resonance

Proof By Theorem 1 in [85], we have for .0 < x
0, .x ∈ (0, 1), .ϕ is monotonically increasing. 2 2

(1−x 2 +

1−x )

(1+

Noting that .

lim

m→∞

kl τ (1 − τ ) k lm − m = > 0, m m n

there exists .δ(τ, n) > 0 such that klm m ) klm τ ϕ( m )

ϕ(

.

< 1 − δ(τ, n).

(6.2.41)

6.2 Scalar Case (Helmholtz Equations)

303

Combining (6.2.39), (6.2.40) and (6.2.41), together with straightforward calculations, one can show that

.

‖vm ‖2L2 (B

τ) ‖vm ‖2L2 (B)

⩽ 144

n m4 τ 2 (1 − δ(τ, n))2m . (n − 1)2

That is lim

.

m→∞

‖vm ‖2L2 (B

τ)

‖vm ‖2L2 (B)

= 0. ⨆ ⨅

The proof is complete.

Theorem 6.2.6 Consider the same setup as that in Theorem 6.2.5. The corresponding transmission eigenfunctions .{wm }m∈N are also boundary-localized in the sense that .

lim

m→∞

‖wm ‖L2 (Bτ ) ‖wm ‖L2 (B)

= 0.

(6.2.42)

Proof Since .wm (x)) = Jm (nklm x), one has .

nklm > n

 s0 + 1 2  jm,s0 3 . >m 1+2 n m

' . Indeed, it can be Next, we show that for m sufficiently large, one has .nklm τ < jm,1 deduced that ' jm,1 m(1 + 3( m1 ) 3 + 2( m1 ) 3 ) m , < < . 2 nklm nklm m(1 + 2( s0m+1 ) 3 ) 2

4

(6.2.43)

where we make use of the following fact m .

jm,s0 +1

=

m n·

jm,s0 +1 n


τ. 2

>1−ε >

nklm

' . Next, by using the Carlini formula again, we have That is, .nklm τ < jm,1

‖wm ‖2L2 (B

τ)

‖wm ‖2L2 (B)

τ = 01 0

rJm2 (nklm r)dr rJm2 (nklm r)dr

τ

.

⩽

0



rJm2 (nklm r)dr

' jm,1 nklm

0

.

rJm2 (nklm r)dr

The rest of the proof is similar to that of Theorem 6.2.5, and by straightforward calculations one can show that

.



‖wm ‖2L2 (B

τ)

‖wm ‖2L2 (B)

⩽ 36nm τ

4 2

Jm (nklm τ ) ' ) Jm (jm,1

which readily implies (6.2.42). The proof is complete.

2 ,

⨆ ⨅

Remark 6.2.4 By (6.2.42), we readily see that for a given .τ ∈ (0, 1), there exists m0 ∈ N such that when .m > m0 ,

.

.

‖wm ‖L2 (Bτ ) ‖wm ‖L2 (B)

≤ ϵ0 ,

where .ϵ0 ∈ R+ is sufficiently small, and .wm , .m > m0 , are the transmission eigenfunctions in Theorem 6.2.6. Hence, by Definition 6.3.1, the transmission eigenfunctions are boundary-localised. We remark that it is clear that .m0 depends on the given .τ and .ϵ0 . The proof of Theorem 6.2.4 in three dimensions is similar to the two-dimensional case in Theorems 6.2.5 and 6.2.6. We only sketch the necessary modifications in what follows.

6.2 Scalar Case (Helmholtz Equations)

305

Theorem 6.2.7 Consider the same setup as that in Theorem 6.2.4 in .R3 and assume that .n > 1 and .τ ∈ (0, 1) is fixed. Then (6.2.20) holds true. Proof By Fourier expansion, the solutions to (6.2.1) in .R3 have the following series expansions: m ∞  

w(x) =

l αm jm (kn|x|)Yml (ˆx),

m=0 l=−m .

m ∞  

v(x) =

(6.2.45) l βm jm (k|x|)Yml (ˆx),

m=0 l=−m

where .xˆ := x/|x|, .Yml is the Spherical harmonic function of order m and degree l, and  π Jm+1/2 (|x|), (6.2.46) .jm (|x|) = 2|x| is known as the spherical Bessel function. In what follows, we shall look for boundary-localized transmission eigenfunctions of the following form: wl,m (x) = αm jm (kn|x|)Yml (ˆx), .

(6.2.47)

vl,m (x) = βm jm (k|x|)Yml (ˆx),

where .αm and .βm are constants. By using the two transmission conditions on .∂B, together with straightforward calculations, one can show that αm = 1,

.

1

βm = n− 2

Jm+ 1 (kn) 2

Jm+ 1 (k)

αm ,

2

and k should be a root of the following function fm+ 1 (k) = Jm− 1 (k)Jm+ 1 (kn) − nJm+ 1 (k)Jm− 1 (kn).

.

2

2

2

2

2

(6.2.48)

Next, we construct the desired transmission eigenvalues by showing that .fm+ 1 (k) 2 jm+ 21 ,s0 jm+ 21 ,s0 +1

has least one zero point in . , . Spherical Bessel function satisfies n n the same estimate in Lemma 6.2.3. Then for any fixed .s0 , there exists a sufficiently large .m0 (n, s0 ) ∈ N+ such that when .m > m0 (n, s0 ), we have .

jm+ 1 ,s0 +1 2

n

1 ≤m+ . 2

306

6 Interior Transmission Resonance

On the other hand, the spherical Bessel functions possess the following property [105]: m+

.

1 ' ' ' < jm+ 1 ,1 < jm+ < jm+ 1 ,2 < jm+ < ··· . ≤ jm+ 1 1 1 2 2 2 2 ,1 2 ,2 2 ,3

Consider the function .fm+ 1 (k) in .( 2 Bessel function, we have

jm+ 1 ,s

2 0

n

Jm+ 1 (k) ≥ 0,

.

2

,

jm+ 1 ,s

2 0 +1

n

). By the monotonicity of the

  ' . k ∈ 0, jm+ 1 ,1 2

According to the formula (9.5.2) in [1], we know

jm+ 1 ,s0

fm+ 1

= n Jm+ 1 2

.

· fm+ 1

n

2



2

jm+ 1 ,s0

n

· Jm+ 1

n

2

2

2

2

jm+ 1 ,s0 +1

jm+ 1 ,s0 +1

2

n

2

· Jm− 1 (jm+ 1 ,s0 ) · Jm− 1 (jm+ 1 ,s0 +1 ) 2

2

2

2

2 ' ≤ n2 (Jm+ 1 (jm+ 1 )) · Jm− 1 (jm+ 1 ,s0 ) · Jm− 1 (jm+ 1 ,s0 +1 ) ,1 2

2

2

2

2

2

< 0. Therefore by Rolle’s theorem, .fm+ 1 (k) has at least one zero point in 2  j jm+ 1 ,s +1 m+ 21 ,s0 2 0 . , . n n fixed .s0 ∈ N, we denote the aforementioned zero point in  j For any jm+ 1 ,s +1 m+ 21 ,s0 2 0 , as .klm (comparing to (6.2.30) in the two-dimensional case). . n n Let .(wm , vm ) be the transmission eigenfunctions associated to the eigenvalue .klm . Next we prove that .{vm }m∈N are boundary-localized on .∂B. By straightforward calculations, one has  " " "jm (kl |x|)"2 dx ‖vm ‖2L2 (B ) = m τ

 .

=2π

τ

0

 =2π

0

τ

Bτ

" " "jm (kl r)"2 r 2 dr = 2π m



τ 0

π 2 J 1 (kl r)dr = π 2 r 2r m+ 2 m

r 2 jm2 (klm r)dr



2

0

τ

2 rJm+ 1 (klm r)dr. 2

6.2 Scalar Case (Helmholtz Equations)

307

Hence, it holds that

.

τ

‖vm ‖2L2 (B

τ)

‖vm ‖2L2 (B)

0

rJ 2

(klm r)dr

0

rJ 2

(klm r)dr

= 1

Consider the function .ζ (r) = rJ 2

m+ 12

m+ 21

m+ 12

(6.2.49)

.

(klm r). It can be shown that .ζ is convex in

(0, 1). Using this fact, one can further estimate that

.

‖vm ‖2L2 (B

τ)

‖vm ‖2L2 (B)

τ 0

rJ 2

(klm r)dr

0

rJ 2

(klm r)dr

= 1

m+ 21

.

⩽ 2τ 2

m+ 12

Jm+ 1 (klm τ )

2 ⎛ ⎝1 + 2klm

2

Jm+ 1 (klm ) 2

J'

m+ 12

(klm )

Jm+ 1 (klm )

⎞ ⎠.

2

Similar to Lemma 6.2.4, one can show that there exists constants C and .γ such that k lm J ' .

m+ 12

(klm )

Jm+ 1 (klm )

< Cmγ .

2

Finally, by following a similar argument to the two-dimensional case and combining the above estimates, together with the use of the Carlini formula, one can show that .

lim

m→∞

‖vm ‖L2 (Bτ ) ‖vm ‖L2 (B)

= 0.

(6.2.50)

By following a completely similar argument to that of Theorem 6.2.6 in the twodimensional case, one can show that (6.2.50) also holds for .wm . The proof is complete. ⨆ ⨅ According to the boundary-localisation of the associated transmission eigenfunctions, the transmission eigenvalues are classified into three groups; see Fig. 6.3 for Fig. 6.3 Graphical illustration of the classification of transmission eigenvalues

308

6 Interior Transmission Resonance

a graphical illustration. For eigenvalues located in region I, both the v-part and wpart of the corresponding eigenfunctions are boundary-localised; for eigenvalues localised in region II, only the v-part is boundary-localised, whereas the w-part is not boundary-localised; and in region III, both the v-part and w-part are not boundary-localised. Next, we consider the case where .Ω is not necessarily of radial geometry and the refractive index n might be piecewise constant (but real). By extensive numerics, we shall verify that the spectral properties in Theorems 6.2.1 and 6.2.3 still hold in the general case. Moreover, through our numerical experiments, we can find more quantitative properties of the SLEs. In principle, we shall see that the SLEs are topologically robust against large deformation or even twisting of the material interface .∂Ω, and moreover the behaviours of the SLEs are mainly related to the refractive index n in a neighbourhood of .∂Ω. It is also observed that a pair of transmission eigenfunctions w and v cannot be SLEs simultaneously. Furthermore, the occurrence of SLEs can be more often if n is sufficiently large locally around .∂Ω (for v) or sufficiently small locally around .∂Ω (for w). To begin with, we note that if .(w, v) is a pair of eigenfunctions to (6.2.1), so is .α · (w, v) for any .α ∈ C\{0}. Hence, throughout the rest of this section, we shall normalize v (or w in some occasions). Moreover, we note the following scaling property of k in (6.2.1) with respect to the size of .Ω: for .ρ ∈ R+ , .ρ · k is an eigenvalue to (6.2.1) associated with .Ωρ := ρ1 Ω. Hence, we always assume that .diam(Ω) ∼ 1 in order to calibrate our study. Next, we present some typical numerical results to verify and demonstrate the SLEs in different scenarios. We mainly discuss the case .n > 1 and briefly remark the case .0 < n < 1. We shall also consider the SLEs for high-contrast mediums. First, we consider the scenario that n is sufficiently large, which corresponds to the case that a highcontrast medium is located inside .Ω (the medium outside .Ω possesses .n ≡ 1). In optics, medium with refraction index .n = 2 are considered as a high-contrast medium[40]. In Fig. 6.4, we calculate a transmission eigenvalue .k = 1.0080 for .Ω being a unit disk and plot the corresponding eigenfunctions w and v. It is clearly seen that v is an SLE. However, it is pointed out that the eigenfunction w is not a SLE. That is, w and v are not SLEs simultaneously. It is emphasized that this is only a numerical observation and there might exist a pair of transmission eigenfunctions which are SLEs simultaneously, which deserves further investigation. Moreover, in Fig. 6.4, we note that .diam(Ω), being 2, is much smaller than the underlying wavelength, being .2π/k ≈ 2π. Such an observation is critical for our subsequent development of the super-resolution imaging scheme. Figure 6.4c presents the SLE of a triangle and in particular, in d, e, and f we note significant localization phenomena at the concave part of .∂Ω, which is also a critical ingredient for our subsequent development of the super-resolution wave imaging. Next, we consider the case that n is relatively small, namely .n ∼ 1. Figure 6.5 presents several examples in both 2D and 3D. The existence of the SLEs is topologically very robust against large deformation or even twisting of the material interface .∂Ω; see Fig. 6.6.

6.2 Scalar Case (Helmholtz Equations)

(a) w

(d) k = 1.0007

309

(b) v

(c) k = 1.1932

(e) k = 1.1370

(f) k = 1.1370

Fig. 6.4 (a) and (b): eigenfunctions w and v to (6.2.1) associated with .n = 30, where .k = 1.0080; (c) and (d): eigenfunctions v’s to (6.2.1) of different shapes with .n = 30; (e) and (f): eigenfunction v to (6.2.1) with .n = 10

(a)

(b)

(c)

(d)

Fig. 6.5 Transmission eigenfunctions v’s to (6.2.1) associated with .n = 4, for different .Ω’s and k’s. (a) .k = 7.1925. (b) .k = 7.9604. (c) .k = 7.1023. (d) .k = 7.0818

310

6 Interior Transmission Resonance

(a)

(b)

(c)

(d)

Fig. 6.6 The existence of SLEs is topologically robust. Here, .n = 20 for all cases. (a) .k = 1.2802. (b) .k = 1.9604. (c) .k = 1.0007. (d) .k = 1.0133

(a) k = 2.4454

(b) k = 1.0001

Fig. 6.7 Transmission eigenfunctions v’s. (a): .n = 30 in the outside layer and .n = 4 in the inside triangle; (b): .n = 30 in the outside layer and the inside kite is a sound-soft obstacle

We also consider the SLEs for variable refractive inhomogeneities and coated objects. As remarked earlier, the SLEs also exist for variable refractive inhomogeneities. In Fig. 6.7a, the eigenfunction v is associated with .n = 30 in the outside thin layer and .n = 4 in the inside triangle. It is emphasized that the outside layer

6.2 Scalar Case (Helmholtz Equations)

311

is not required to be very thin in order to exhibit the SLEs. Indeed, as long as n is sufficiently large locally around .∂Ω, the SLEs can be found even for relatively small eigenvalues (cf. Fig. 6.4). This specific example shall be used again in our subsequent study. Figure 6.7b, corresponds to a coated object, where the inside kitedomain is a sound-soft obstacle. That is, in (6.2.1), w does not exist in the inside kite-domain and we impose a zero Dirichlet condition of w on the boundary of the inside kite-domain. Finally, we would like to remark that if .0 < n < 1, all the boundary localization results presented in the above numerical examples still hold with v replaced by w.

6.2.2 Super-Resolution Wave Imaging In this section, we consider an interesting and practically important application of the SLEs presented in the previous section. We propose an inverse scattering scheme that makes use of the longly neglected interior resonant modes to recover the unknown or inaccessible scatterer. It turns out that the proposed scheme can produce strikingly high imaging resolution compared to many existing imaging schemes. Let .Ω ⊂ Rd , d = 2, 3, be a bounded domain with a Lipschitz boundary d .∂Ω and a connected complement .R \Ω. Let .ν denote the unit outward normal to .∂Ω. Throughout the rest of this section, we shall assume that the refractive index 2 2 .n is a real-valued constant/piecewise constant function such that .n (x) ≡ 1 for 2 ∞ i d .x ∈ R \ Ω and .1/|n − 1| ∈ L (Ω). We take the incident field .u to be a timeharmonic plane wave of the form ui := ui (x, θ, k) = eikx·θ ,

.

x ∈ Rd ,

√ where .i = −1 is the imaginary unit, .k = ω/c the wavenumber, .ω ∈ R+ and d−1 the direction .c ∈ R+ the angular frequency and sound speed, respectively, .θ ∈ S d−1 d of propagation and .S := {x ∈ R : |x| = 1} is the unit sphere in .Rd . Clearly, the i incident field .u satisfies the Helmholtz equation Δui + k 2 ui = 0 in Rd .

.

Physically, the presence of the scatterer .Ω interrupts the propagation of the incident wave .ui , giving rise to the scattered field .us . Let .u := ui + us denote the total wave field. The forward scattering problem is modeled by the following system ⎧ Δu + k 2 n2 (x)u = 0 in Rd , ⎪ ⎪ ⎨ i s in Rd , u = u + u  . s ∂u ⎪ ⎪ lim r d−1 2 ⎩ − ikus = 0, r→∞ ∂r

(6.2.51)

312

6 Interior Transmission Resonance

where .r = |x| and the last limit in (6.2.51) characterizes the outgoing nature of the scattered wave field .us . The well-posedness of the scattering system (6.2.51) 2 (Rd ). is established [44], and in particular, there exists a unique solution .u ∈ Hloc Furthermore, the scattered field has the following asymptotic expansion: π

ei 4 .u (x, θ, k) = √ 8kπ s



 −i π4

e

k 2π

d−2

#  $ 1 ∞ u (ˆ x , θ, k) + O d−1 r r 2 eikr

as r → ∞,

which holds uniformly for all directions .xˆ := x/|x| ∈ Sd−1 . The complex-valued function .u∞ (ˆx, θ, k) = u∞ (ˆx, eikx·θ ), defined on the unit sphere .Sd−1 , is known as the far-field pattern of u, which encodes the information of the refractive index 2 .n . We are concerned with the inverse problem of imaging the support of the inhomogeneity, namely .Ω, by knowledge of .u∞ (ˆx, θ, k) for .xˆ , θ ∈ Sd−1 and .k ∈ I := (κ0 , κ1 ), which is an open interval in .R+ . It can be recasted as the following nonlinear operator equation F (Ω, n) = u∞ (ˆx, θ, k),

.

xˆ ∈ Sd−1 , θ ∈ Sd−1 , k ∈ I,

(6.2.52)

where .F is defined by the Helmholtz system (6.2.51). Such an inverse problem is a prototypical model for many industrial and engineering applications including medical imaging and nondestructive testing. There is the well-known Abbe diffraction limit for imaging the fine details of .∂Ω [106]. In fact, one has a minimum resolvable distance of .λ/(2N ), where .λ and .N stand for the wavelength and numerical aperture respectively. In modern optics, the Abbe resolution limit is roughly about half of the wavelength. Here, based on the use of the SLEs, we develop an imaging scheme that can break the Abbe resolution limit in recovering the fine details of .∂Ω for (6.2.52), independent of n, in certain scenarios of practical interest. The proposed imaging scheme consists of three phases. In Phase .I, we determine the transmission eigenvalues within the interval I by knowledge of the far-field data in (6.2.52), namely .u∞ (ˆx, θ, k) for .xˆ ∈ Sd−1 , θ ∈ Sd−1 and .k ∈ I . In Phase .II, we determine the corresponding transmission eigenfunctions associated to the transmission eigenvalues computed from Phase .I. Finally, in Phase .III, we make use the transmission eigenfunctions from Phase .II to design an imaging functional which can be used to determine the shape of the medium scatterer, namely .Ω. Let us first consider the determination of the transmission eigenvalues within the interval I by knowledge of the far-field data in (6.2.52). In fact, this problem has been addressed in [37]. Nevertheless, for completeness and self-containedness, we briefly discuss the main procedure as well as the rationale behind the method. To that end, for any given .z ∈ Rd , we let .Ψ (x, z, k) be the fundamental solution to the PDO .−Δ − k 2 : ⎧ i (1) ⎨ 4 H0 (k|x − z|), d = 2, .Ψ (x, z, k) = 1 eik|x−z| ⎩ , d = 3, 4π |x − z|

6.2 Scalar Case (Helmholtz Equations)

313

where .H0 is the first-kind Hankel function of zeroth-order. Let .Ψ ∞ (ˆx, z, k) signify the far-field pattern of .Ψ (x, z, k), which is given by (1)

⎧ iπ ⎪ 4 ⎪ ⎨ √e e−ik xˆ ·z , d = 2, ∞ 8kπ .Ψ (ˆx, z, k) = ⎪ ⎪ ⎩ 1 e−ik xˆ ·z , d = 3. 4π The determination of the transmission eigenvalues is based on the mechanism of the so-called linear sampling method (LSM), which is a qualitative method in inverse scattering theory to reconstruct the shape of the scatterer without a priori knowledge of either the physical properties or the number of disconnected components of the scatterer. The core of the LSM is the following far-field equation (Fk g)(ˆx) = Ψ ∞ (ˆx, z, k),

.

z ∈ Rd , g ∈ L2 (Sd−1 ),

(6.2.53)

where .Fk : L2 (Sd−1 ) ‫ →׀‬L2 (Sd−1 ) is the far-field operator defined by  (Fk g)(ˆx) :=

.

Sd−1

u∞ (ˆx, θ, k)g(θ ) ds(θ ),

xˆ ∈ Sd−1 .

(6.2.54)

While in a practical application, we can not get accurate far-field data for many reasons. We denote .u∞,ϵ (ˆx, θ, k) as the noisy measurement of the far-field data ∞ x, θ, k), where .0 < ϵ < 1 signifies the noise level. This could weigh .u (ˆ the difference between the ideal data .u∞ (ˆx, θ, k) and the noisy measurements ∞,ϵ (ˆ .u x, θ, k). Let .Fkϵ be the far-field operator corresponding to noisy measurement of the far-field data .u∞,ϵ (ˆx, θ, k), which means  ϵ .(Fk g)(ˆ x) := u∞,ϵ (ˆx, θ, k)g(θ ) ds(θ ), xˆ ∈ Sd−1 . Sd−1

Define the Herglotz wave function  vg,k (x) := Hk g(x) =

.

Sd−1

g(θ )eikx·θ ds(θ ),

x ∈ Rd ,

(6.2.55)

where .g(θ ) ∈ L2 (Sd−1 ) is called the Herglotz kernel of .vg,k (x). It is obvious that Herglotz wave functions are entire solutions to the Helmholtz equation. According to the strict mathematical justification for the effectiveness of the linear sampling method (see [37]), we have the following result: Lemma 6.2.5 If k is not a transmission eigenvalue, then there exists an approximate solution .gϵ (·, z) ∈ L2 (Sd−1 ) of the far-field Eq. (6.2.53) such that .Hk gϵ (·, z) converges in the .H 1 (Ω) norm as .ϵ → 0 when .z ∈ Ω.

314

6 Interior Transmission Resonance

However, if k is indeed a transmission eigenvalue, the statement in Lemma 6.2.5 is not true. To get the relevant result, we need the following assumption for all points .z ∈ Ω, .

lim ‖Fkϵ gϵ (·, z) − Ψ ∞ (·, z, k)‖L2 (Sd−1 ) = 0.

ϵ→0

(6.2.56)

The following lemma characterizes the solution to (6.2.56) when k is a transmission eigenvalue. Lemma 6.2.6 ([37]) Suppose that k is a transmission eigenvalue and assume that (6.2.56) holds. Then for almost every .z ∈ Ω, .‖Hk gϵ (·, z)‖L2 (Ω) can not be bounded as .ϵ → 0. Since .Hk is compact, there exists a constant .M > 0 such that ‖Hk gϵ (·, z)‖L2 (Ω) ≤ M‖gϵ (·, z)‖L2 (Sd−1 ) .

.

Thus, .‖gϵ (·, z)‖L2 (Sd−1 ) can not be bounded as .ϵ → 0 if k is a transmission eigenvalue. By the above two lemmas, we note that .‖gϵ (·, z)‖L2 (Sd−1 ) behaves quite differently when .k 2 is a transmission eigenvalue or not. Hence, one can use .‖gϵ (·, z)‖L2 (Sd−1 ) as an indicator to identify if k is a transmission eigenvalue or not. We formulate the following scheme, dubbed as Algorithm I, to determine the transmission eigenvalues. Algorithm I: Determination of transmission eigenvalues Step 1 Collect a family of far-field data .u∞,ϵ (ˆx, θ, k) for .(ˆx, θ, k) ∈ Sd−1 × Sd−1 × I , where I is an open interval in .R+ . Step 2 Pick a point .z ∈ Ω (a-priori information) and for each .k ∈ I , solve (6.2.53) to obtain the solution .gϵ (·, z). Step 3 Plot .‖gϵ (·, z)‖L2 (Sd−1 ) against .k ∈ I and find the transmission eigenvalues where peaks appear in the graph.

We note that .Ω is unknown, so we consider .‖gϵ (·, z)‖L2 (Sd−1 ) , instead of ‖Hk gϵ (·, z)‖L2 (Ω) in Step 3 of Algorithm I though .‖Hk gϵ (·, z)‖L2 (Ω) also behaves differently when k is a transmission eigenvalue or not. In the first phase, we determine the transmission eigenvalues within the interval I by knowledge of the far-field data in (6.2.52). We proceed to determine the corresponding transmission eigenfunctions. To that end, we first recall the following denseness result of the Herglotz wave introduced in (6.2.55).

.

6.2 Scalar Case (Helmholtz Equations)

315

Lemma 6.2.7 Let .Ω be a bounded domain of class .C α,1 , .α ∈ N ∪ {0}, in .Rd . Denote by .H the space of all Herglotz wave functions of the form (6.2.55). Define, respectively, H(Ω) := {u|Ω : u ∈ H},

.

and H(Ω) := {u ∈ C ∞ (Ω) : Δu + k 2 u = 0 in Ω}.

.

Then .H(Ω) is dense in .H(Ω) ∩ H α+1 (Ω) with respect to the .H α+1 (Ω)-norm. The following theorem states that if .k ∈ R+ is a transmission eigenvalue, then there exists a Herglotz wave function .vgϵ ,k such that the scattered field corresponding to this .vgϵ ,k as the incident field is nearly vanishing. Theorem 6.2.8 Suppose that .k ∈ R+ is a transmission eigenvalue in .Ω. For any sufficiently small .ϵ ∈ R+ , there exists .gϵ ∈ L2 (Sd−1 ) such that ‖Fk gϵ ‖L2 (Sd−1 ) = O(ϵ) and ‖vgϵ ,k ‖L2 (Ω) = O(1),

.

where .Fk is the far field operator defined by (6.2.54) and .vgϵ ,k is the Herglotz wave function defined by (6.2.55) with the kernel .gϵ . Proof Let .vk be a normalized transmission eigenfunction in .Ω associated to the transmission eigenvalue .k 2 , which means that .vk with .‖vk ‖L2 (Ω) = 1 is a solution of Δvk + k 2 vk = 0 in Ω.

.

By Lemma 6.2.7, for any sufficiently small .ϵ > 0, there exists .gϵ ∈ L2 (Sd−1 ) such that ‖vgϵ ,k − vk ‖L2 (Ω) < ϵ,

.

where .vgϵ ,k is the Herglotz wave function with the kernel .gϵ . Then, by the triangle inequality, ‖vgϵ ,k ‖L2 (Ω) ≤ ‖vgϵ ,k − vk ‖L2 (Ω) + ‖vk ‖L2 (Ω) < ϵ + ‖vk ‖L2 (Ω) ,

.

and ‖vgϵ ,k ‖L2 (Ω) ≥ ‖vk ‖L2 (Ω) − ‖vgϵ ,k − vk ‖L2 (Ω) > ‖vk ‖L2 (Ω) − ϵ.

.

Thus, one must have that .‖vgϵ ,k ‖L2 (Ω) = O(1).

316

6 Interior Transmission Resonance

Furthermore, from the definition of the far-field operator, .Fk gϵ is the far-field pattern produced by the incident wave .vgϵ ,k . According to Proposition 4.2 in [29], one has that ‖Fk gϵ ‖L2 (Sd−1 ) < Cϵ,

.

where .C = C(n, k) is a positive constant. The proof is complete.

⨆ ⨅

By Theorem 6.2.8 and normalization if necessary, we can say that the following optimization problem: .

min

g∈L2 (Sd−1 )

‖Fk g‖L2 (Sd−1 )

s.t. ‖vg,k ‖L2 (Ω) = 1

(6.2.57)

has at least one (approximate) solution .g0 ∈ L2 (Sd−1 ) when .k ∈ R+ is a transmission eigenvalue in .Ω. However, since .Ω is unknown, the constraint .‖vg,k ‖L2 (Ω) = 1 in the optimization formulation (6.2.57) is unpractical. Nevertheless, it is reasonable to address this issue by considering an alternative optimization problem: .

min

g∈L2 (Sd−1 )

‖Fk g‖L2 (Sd−1 )

s.t. ‖vg,k ‖L2 (BR ) = 1,

(6.2.58)

where .BR is an a-priori ball containing .Ω. Let .g0 be a “reasonable” solution to the optimization problem (6.2.58). Next, we show that the corresponding Herglotz wave .vg0 ,k is generically indeed an approximation to the transmission eigenfunction .vk associated to the transmission eigenvalue k. Theorem 6.2.9 Suppose .k ∈ R+ is a transmission eigenvalue in .Ω and .g0 is a solution to the optimization problem (6.2.58) satisfying ‖Fk g0 ‖L2 (Sd−1 ) ≤ ϵ ⪡ 1.

.

(6.2.59)

If we further assume that .Ω is of class .C 1,1 and .|n2 − 1| ≥ δ0 for a certain .δ0 ∈ R+ in a neighbourhood of .∂Ω, then the Herglotz wave .vg0 ,k is an approximation to a transmission eigenfunction .vk associated with the transmission eigenvalue k in the 2 .H (Ω)-norm. Proof Consider the scattering system (6.2.51). We let .ui = vg0 ,k , .usg0 ,k and u be respectively the incident, scattered and total wave fields. It is clear that one has ⎧ Δu + k 2 n2 (x)u = 0 ⎪ ⎪ ⎪ ⎪ ⎨ Δvg0 ,k + k 2 vg0 ,k = 0 . u = vg0 ,k + usg0 ,k ⎪ ⎪ ⎪ ∂us ⎪ ⎩ ∂u = ∂vg0 ,k + g0 ,k ∂ν ∂ν ∂ν

in Ω, in Ω, on ∂Ω, on ∂Ω.

(6.2.60)

6.2 Scalar Case (Helmholtz Equations)

317

According to our earlier discussion, .Fk g0 is the far-field pattern of .usg0 ,k . By virtue of (6.2.59) as well as the quantitative Rellich theorem established in [32], one has & s & & & ∂u & g0 ,k & s ≤ ψ(ϵ), (6.2.61) .‖ug ,k ‖H 3/2 (∂Ω) + & & 0 & ∂ν & 1/2 H

(∂Ω)

where .ψ is the stability function in [32], which is of double logarithmic type and satisfies .ψ(ϵ) → 0 as .ϵ → +0. This result illustrates that the smallness of the far-field pattern can ensure the scattered data on the boundary are also small. For more details about the quantitative Rellich theorem and the stability function .ψ, see references [32]. Consider the transmission eigenvalue problem (6.2.1). Setting .W = w − v, .V = k 2 v and .λ = −k 2 , (6.2.1) can be rewritten as (cf. [107, 120]): ⎧ 2 2 ⎪ ⎪ (Δ − λn (x))W + (n (x) − 1)V = 0 in Ω, ⎪ ⎪ ⎨ (Δ − λ)V = 0 in Ω, . W = 0 on ∂Ω, ⎪ ⎪ ⎪ ∂W ⎪ ⎩ =0 on ∂Ω. ∂ν Let .Δ00 denote the Laplacian with domain .H02 (Ω) and .Δ−− denote the Laplacian with domain .H 2 (Ω). By [107, 120], the squares of interior transmission eigenvalues are the spectrum of the generalized eigenvalue problem       2   W W Δ00 n2 − 1 W n 0 .(A − λIn ) := −λ = 0, V 0 Δ−− 0 1 V V

(6.2.62)

where .(W, V ) ∈ H02 (Ω) ⊕ H 2 (Ω). Now, we consider the PDE system (6.2.60). By the standard Sobolev extension as well as noting (6.2.61), we let .ζ ∈ H 2 (Ω) be such that ζ = usg0 ,k ,

.

∂usg0 ,k ∂ζ on ∂Ω, = ∂ν ∂ν

(6.2.63)

and ‖ζ ‖H 2 (Ω) ≤ Cψ(ϵ),

.

(6.2.64)

where C is a generic constant depending on .Ω. Introducing Υ1 = −(Δ+k 2 n2 )ζ,

.

Υ2 = 0,

Υ = (Υ1 , Υ2 )T ∈ L2 (Ω)⊕L2 (Ω),

(6.2.65)

318

6 Interior Transmission Resonance

and setting . u := u − ζ ∈ H 2 (Ω), (6.2.60) can be rewritten as ⎧ ⎪ Δ u + k 2 n2 (x) u = Υ1 ⎪ ⎪ ⎪ ⎨ Δvg0 ,k + k 2 vg0 ,k = Υ2 .  u = vg0 ,k ⎪ ⎪ ⎪ ∂vg0 ,k ∂ u ⎪ ⎩ = ∂ν ∂ν

in Ω, in Ω, on ∂Ω,

(6.2.66)

on ∂Ω.

Setting .Wg0 ,k =  u − vg0 ,k and .Vg0 ,k = k 2 vg0 ,k , we can rewrite the system (6.2.66) into the operator form (A − λIn )

.

  Wg0 ,k = Υ, Wg0 ,k ∈ H02 (Ω), Vg0 ,k ∈ H 2 (Ω). Vg0 ,k

(6.2.67)

Note that .λ = −k 2 is an eigenvalue to the operator Eq. (6.2.62). It is shown in [120] that .In−1 A possesses a UTC (upper triangular compact) resolvent. This enables one to apply the upper triangular analytic Fredholm theorem to the eigenvalue problem (6.2.62) as well as the operator Eq. (6.2.67), which enjoys the same properties as those within the analytic Fredholm theorem (in the current setup of our study). Next, we shall prove that the operator Eq. (6.2.67) solvable in the quotient space .(H02 (Ω) ⊕ H 2 (Ω))/(W ⊕ V), where .W ⊕ V is the finite-dimensional eigenspace to (6.2.62). In order to apply the Fredholm theorem, it is sufficient for us to show that .Υ ∈ Ker[(A − λIn )∗ ]⊥ . The kernel of .(A − λIn )∗ consists of functions 2 2 .(ξ, η) ∈ H (Ω) ⊕ H (Ω) satisfying 0  .

(Δ − λ)η + (n2 − 1)ξ = 0

in Ω,

(Δ − λ)ξ =

in Ω,

λ(n2

− 1)ξ

(6.2.68)

which, by introducing . η = ξ + λη ∈ H 2 (Ω), is equivalent to the following PDE system ⎧ ⎪ η=0 ⎪ ⎨(Δ − λ) . (Δ − λ)ξ = λ(n2 − 1)ξ ⎪ ⎪ ⎩ η = ξ, ∂  η=∂ ξ ν

ν

in Ω, in Ω,

(6.2.69)

on ∂Ω.

With the above fact and using (6.2.65), we have  Υ · (ξ, η)T =

Υ1 · ξ

.



Ω

 −(Δ + k n )ζ · ξ =

=

∂ ν ξ · ζ − ξ · ∂ν ζ .

2 2



Ω

= ∂Ω

(6.2.70)

∂Ω

∂ν ξ · usg0 ,k − ξ · ∂ν usg0 ,k ,

(6.2.71)

6.2 Scalar Case (Helmholtz Equations)

319

where in (6.2.70) we have made use of the fact .(Δ + k 2 n2 )ξ = 0 from (6.2.68); and in (6.2.71) we have made use of the fact in (6.2.63). From (6.2.60), we see that usg0 ,k = u − vg0 ,k , ∂ν usg0 ,k = ∂ν u − ∂ν vg0 ,k

.

on ∂Ω,

which readily yields that  .

∂Ω

∂ν ξ · usg0 ,k − ξ · ∂ν usg0 ,k

 

∂Ω

= ∂Ω



  ∂ν ξ · u − ξ · ∂ν u −

=

∂Ω

∂ν ξ · vg0 ,k − ξ · ∂ν vg0 ,k

.

 η · ∂ν vg0 ,k − ∂ν  η · vg0 ,k = 0,

(6.2.72) (6.2.73)

where from (6.2.72) to (6.2.73), we have made use of the transmission conditions on .∂Ω from (6.2.69). This implies that Υ ∈ Ker[(A − λIn )∗ ]⊥ .

.

Hence, (6.2.67) is solvable in .(H02 (Ω) ⊕ H 2 (Ω))/(W ⊕ V). Set

Wg∗0 ,k . = (A − λIn )−1 Υ in (H02 (Ω) ⊕ H 2 (Ω))/(W ⊕ V). Vg∗0 ,k

(6.2.74)

By (6.2.64), we obviously have ‖Υ ‖L2 (Ω)2 ≤ Cψ(ϵ),

.

(6.2.75)

with C a generic constant depending on .n, k and .Ω. Finally, by combining (6.2.74) and (6.2.75), one can show that ‖Vg0 ,k − Vk ‖H 2 (Ω) ≤ Cψ(ϵ) → 0 as ϵ → +0 for Vk ∈ V,

.

which readily implies that .vg0 ,k is an approximation to a transmission eigenfunction vk . The proof is complete. ⨆ ⨅

.

Remark 6.2.5 It is remarked that in Theorem 6.2.9, the .C 1,1 -regularity on .∂Ω is a technical condition. It is required in (6.2.61) and (6.2.63), and in particular according to [32], higher regularity might be required in deriving (6.2.61), which we choose not to explore further since it is not the focus here. Nevertheless, we believe that this regularity assumption on .∂Ω can be relaxed. In fact, according to

320

6 Interior Transmission Resonance

our numerical examples in what follows, even if .∂Ω is only Lipschitz continuous, one can still determine the approximate transmission eigenfunctions by solving the optimization problem (6.2.58). Remark 6.2.6 In the whole chapter, we assume that n is real and .|n2 − 1| ≥ δ0 . This assumption is mainly required for establishing the boundary-localisation of the transmission eigenfunctions. In fact, most of the results presented in this section, say Theorems 6.2.8 and 6.2.9, can be extended to the more general case where .n2 satisfies the more general assumptions of [120]. In the last two phases, using the far-field data in (6.2.52), we respectively determine the transmission eigenvalues within the interval I and the corresponding transmission eigenfunctions. In this part, we shall show that the transmission eigenfunctions can be used for the qualitative imaging of the shape of the medium scatterer .(Ω, n2 ), namely .∂Ω independent of .n2 . The basic idea can be described as follows. Let .vg0 ,k be determined from Phase .II which approximates a transmission eigenfunction within .Ω. Then according our study in Sect. 6.2.1, .vk can be a SLE if the conditions in Sect. 6.2.1 are fulfilled. In fact, we know from the numerical examples in Sect. 6.2.1, that if n is sufficiently large in a neighbourhood of .∂Ω, even for a relatively small transmission eigenvalues, the corresponding transmission eigenfunction is a SLE. Furthermore, in such a case, the SLEs occur very often in our extensive numerical experiments and a major part of the calculated transmission eigenfunctions are SLEs. This is a highly interesting phenomenon that is worth further theoretical investigation. Based on such an observation, it naturally leads to the following imaging functional for recovering .Ω: IkRes (z) := −ln|vg0 ,k (z)|.

.

If the underlying transmission eigenfunction .vk is a SLE, then .vg0 ,k is referred to as an approximate SLE. It can be easily seen that if .vg0 ,k an approximate SLE, then .IkRes (z) possesses a relative large value if z belongs to the interior of .Ω, or z is located at the corner/edge/highly-curved place on .∂Ω, whereas it possesses a relatively small value if z is located in the other places around .∂Ω. Since in Phase .II, multiple transmission eigenfunctions can be determined, we can superpose the imaging effects by introducing the following imaging functional: IKRes (z) := −ln L



.

|vg0 ,k (z)|,

(6.2.76)

k∈KL

where .KL = {k1 , k2 , · · · , kL } denotes the set of L distinct transmission eigenvalues determined in Phase .I. Based on the imaging functional (6.2.76), we then propose the following imaging scheme, which is referred to as imaging by interior resonant modes. Indeed, the proposed scheme is based on using the interior transmission eigen-modes, which are usually “discarded” or “avoided” in many existing inverse scattering schemes, say e.g. the factorization method [81].

6.2 Scalar Case (Helmholtz Equations)

321

Algorithm II: Imaging by interior resonant modes Step 1 For each resonant wavenumber k found in Algorithm I, solve the optimization problem (6.2.58) by the FTLS method or the GTLS method to obtain the Herglotz kernel .g0 . Step 2 Calculate the Herglotz wave function .vg0 ,k with the Herglotz kernel .g0 by the definition (6.2.55). Step 3 Plot the indicator function (6.2.76) in a proper domain containing the scatterer .Ω and identify the interior and corners (two dimension) or edges (three dimension) as bright points, and other boundary places as dark points in the graph to obtain the shape of the scatterer .Ω.

Finally, we would like to emphasize that according to our discussion above, the proposed imaging scheme should work for imaging a medium scatterer whose refractive index is highly-contrasted in a neighbourhood of its boundary. This clearly includes the case that the medium scatterer itself already possesses a high-contrast refractive index. On the other hand, for a regular refractive-indexed scatterer, one would need to first coat the scatterer by indirect means with a thin layer of highly refractive-indexed material, then our imaging scheme would work as well. Hence, it is unobjectionable to claim that the proposed method possesses a practical value for generic inverse scattering imaging. In what follows, we present several numerical examples to illustrate the effectiveness of the proposed imaging scheme. In order to simplify the situation, we only consider imaging a medium scatterer possessing a constant refractive index of a relatively high magnitude in two dimensions. It is remarked that the higher the refractive index is, the better imaging effect one can expect to achieve. Moreover, as emphasized above, this highly refractive-indexed medium can be located only in a neighbourhood of .∂Ω.

6.2.3 Numerical Examples In this section, we present the numerical experiments as mentioned above. To avoid the inverse crime, we use the finite element method to compute the scattering amplitudes .u∞ (ˆxi , dj ), i = 1, 2, · · · , M0 , j = 1, 2, · · · , N0 , where .xˆ i , .i = 1, 2, . . . , M0 denote the discrete observation directions and .dj .j = 1, 2, . . . , N0 denote the discrete incident directions. In the two dimensions, the observation and incident directions are equidistantly distributed on a unit circle. Then we consider the collected far-field matrix .F ∈ CM0 ×N0 such that ⎡

u∞ (ˆx1 , d1 ) u∞ (ˆx1 , d2 ) ⎢ u∞ (ˆx2 , d1 ) u∞ (ˆx2 , d2 ) ⎢ .F = ⎢ .. .. ⎣ . . u∞ (ˆxM0 , d1 ) u∞ (ˆxM0 , d2 )

⎤ · · · u∞ (ˆx1 , dN0 ) · · · u∞ (ˆx2 , dN0 ) ⎥ ⎥ ⎥. .. .. ⎦ . . ∞ · · · u (ˆxM0 , dN0 )

322

6 Interior Transmission Resonance

In order to test the sensibility of the proposed method, we further perturb F with random noise by setting F δ = F + δ‖F ‖

.

R1 + R2 i , ‖R1 + R2 i‖

where .δ > 0 represents the noise level, .R1 and .R2 are two .M0 × N0 matrixes containing pseudo-random values drawn from a normal distribution with the mean being zero and the standard deviation being one. In the following two examples, the noise level is given by .δ = 1%. In the first example, we let .Ω be a square with the side length being 2 and the refractive index being .n = 10. The synthetic far-field data are computed at 100 observation directions, 100 incident directions and 3000 wavenumbers within the interval .I = [0.6, 0.9], all equally distributed. Firstly, we use Algorithm I to determine four transmission eigenvalues, such as .k1 = 0.6219, k2 = 0.6896, k3 = 0.7858 and .k4 = 0.8370. Next, we can determine 4 approximate transmission eigenfunctions as well. Since n is relatively large, it turns out that the computed eigenfunctions are all approximate SLEs. We would like to emphasize again that we did not purposely design such a numerical example and indeed, as discussed earlier, the occurrence of the SLEs are very often. We present the reconstruction results in Fig. 6.8a–c, by using 1, 2 and 4 SLEs, respectively. One readily sees that the square is already finely reconstructed with 4 SLEs. For comparisons, we also present the reconstruction results by using a sampling type method by using

2

2

1

0.3

0

0.65 0.6

1

0.5

0.25

−2 −2

0.45

−1 0.2 −1

0

1

2

0.4 0.35

1

−1

0

1

0

0.25 0.2

−1

1

0.8 −1

2

0

1

2

0.7

(c) L = 4

2

0.8 0.7

1

2 1.5

1

0.6 0.5

0

0

1

0.4 −1

0.3

0.15 0

0.9 −1 −2 −2

2

0.3

(d) L = 1

1

(b) L = 2

2

−1

1.1

0

0.4 −2 −2

(a) L = 1

−2 −2

1.2

1

0.55 0

−1

2

−2 −2

0.2 −1

0

1

(e) L = 2

2

−1 −2 −2

0.5 −1

0

1

2

(f) L = 4

Fig. 6.8 Reconstructions of a square-object by multiple SLEs (top row) and multiple-frequency direct sampling method (bottom row), respectively, where .n = 10 for all cases. (a) .L = 1. (b) .L = 2. (c) .L = 4. (d) .L = 1. (e) .L = 2. (f) .L = 4

6.2 Scalar Case (Helmholtz Equations)

323

the multiple frequency scattering data in (6.2.52). The reconstruction results are presented in Fig. 6.8d–f. It can be seen that the reconstructions basically yield a spot without any resolution of the square-shape object. In fact, one can also implement the other popular imaging schemes including the linear sampling method or the factorization method, and the reconstruction effects shall remain almost the same. It is clear that the length of the square is much smaller than the underlying wavelength, .2π/k. This result illustrates that super-resolution reconstruction can be realized by the proposed method. This is unobjectionably expected since we make use of the interior resonant modes for the reconstruction. Finally, it is pointed out if one further performs standard imaging processing to the reconstructed image in Fig. 6.8c, one should be able to obtain a nearly-accurate reconstruction of the square-object. Figure 6.9 presents another example where the target domain is a kite-shaped object with .n = 10. The imaging frequencies are chosen within .[1, 3.1], and the computed transmission eigenvalues are .k1 = 1.2387, k2 = 1.4771, k3 = 3

3 0.1

1.5

0.08

0

1.5

0.04

0.4 0.3

0

0.06

−1.5

0.5

−1.5

0

1.5

−1.5 −3 −3

3

0.1 −1.5

(a) L = 1 0.4

1.5

0.3

0

0.2

−1.5

0.1 −1.5

0

1.5

0

1.5

0.5

1.5

1.5

0.6

0

0.4

0.3 0.2

−1.5

0.1

0.2 −1.5

0

1.5

3

0

1.5

3

3

1.4 1.2

1.5

1 0

0.8 0.6

−3 −3

0.4 −1.5

0

1.5

3

0.6 0.5

1.5

3

0.5

1.5

0.4

0.4

0

0.3 0.2

−1.5

0.1 −1.5

0

1.5

(h) L = 2

0.2

(f) L = 4

3

−3 −3

1.5

−1.5

−1.5

0.4 0

0

3

(e) L = 2 0.6

(g) L = 1

−1.5

(c) L = 4 0.8

(d) L = 1

−1.5

0.5

−1.5 −3 −3

3

3

−3 −3

3

3

−3 −3

1

0

(b) L = 2

3

−3 −3

1.5

1.5

0.2

0.02 −3 −3

3

3

0.3

0

0.2 −1.5 −3 −3

0.1 −1.5

0

1.5

3

(i) L = 4

Fig. 6.9 Reconstructions of a kite-object by multiple SLEs (top row) and multiple-frequency direct sampling method (middle row), respectively, where .n = 10 for all cases. Bottom row gives the corresponding reconstructions of combining the above two reconstruction means. (a) .L = 1. (b) .L = 2. (c) .L = 4. (d) .L = 1. (e) .L = 2. (f) .L = 4. (g) .L = 1. (h) .L = 2. (i) .L = 4

324

6 Interior Transmission Resonance

2.0513, k4 = 3.0013. The reconstructions by our proposed method are given by a–c, meanwhile the reconstructions by the sampling-type method are given by d–f. The sub-figures g–i present the combined results of the above two reconstructions. Clearly, Fig. 6.9i yields a very nice reconstruction of the kite-object, especially the concave part. Three remarks are in order. First, it can seen from the reconstructions in Fig. 6.9 that the sampling-type method tends to reconstruct a “larger” object while our proposed method tends to reconstruct a “smaller” object. This is physically reasonable since the sampling-type method as well as the other traditional inverse scattering schemes make use the measurement data away from the scatterer for the reconstruction, which amounts to “seeing” the scattering object from its outside, whereas our method makes use of the interior resonant modes, which amounts to “seeing” the scattering object its inside. Hence, hybridizing the two types of methods can yield a better reconstruction. Second, it is arguable that the superresolution effect comes from the high-contrast medium parameter n in this specific example. Indeed, as discussed earlier, a high-contrast n leads to a relatively small k that can induce the desired SLE for the reconstruction, which is a matter of fact. However, in practice, for a regular refractive inhomogeneity, one may first coat the object via indirect means with a thin layer of high-contrast medium (cf. Fig. 6.7), then apply the same reconstruction procedure as above. According to the results in Fig. 6.7, one would have the same super-resolution reconstructions as in Fig. 6.8a–c. Third, the super-resolution is achieved at the cost of a large amount of computations and a restrictive requirement on the high-precision of the measurement data. This is unobjectionable due to the increasing capabilities of physical apparatus nowadays.

6.2.4 Pseudo Surface Plasmon Resonances and Potential Applications Surface plasmon resonance (SPR) is the resonant oscillation of conducting electrons at the interface between negative and positive permittivity materials stimulated by incident light. It is a non-radiative electro-magnetic surface wave that propagates in a direction parallel to the negative permittivity/dielectric material interface [25, 66, 82]. Clearly, the SPR wave is a boundary-localized mode. It is in this sense that the SLE can be viewed as a certain SPR. Indeed, viewed from the inside of .Ω (this is unobjectionable since v is only supported in .Ω), the behaviour of a SLE is very much like a SPR. However, SPR usually occurs in the quasi-static regime (subwavelength scale), whereas SLE can occur in both the quasi-static regime and the high-frequency regime. Moreover, the SPR is usually generated from direct light incidence, whereas the generation of SLEs is rather indirect according to our earlier study. As is known that the SPR can have many industrial and engineering applications including color-based biosensors, different lab-on-a-chip sensors and diatom photosynthesis [82]. In what follows, we show that the SLEs can also be

6.2 Scalar Case (Helmholtz Equations)

325

generated through direct wave incidences. This will pave the way for the proposal of an interesting SLE sensing that is similar to the SPR sensing. First, we recall that assuming .Rd \Ω connected, the Herglotz waves of the form (6.2.55) are dense in the space .{v ∈ H 1 (Ω); (Δ + k 2 )v = 0 in Ω}. Hence, for any transmission eigenfunction v to (6.2.1), there exists .g ∈ L2 (Sd−1 ) such that .vgk ≈ v in .H 1 (Ω). Next, for a refractive inhomogeneity .n2 , .0 < n < 1, supported in .Ω with .Rd \Ω connected, we let .k0 be an eigenvalue to (6.2.1) with the eigenfunctions denoted as .(wΩ , vΩ ) such that .wΩ is a SLE. Let .vgk0 be a Herglotz wave function of the form (6.2.55) such that .vgk0 ≈ vΩ in .H 1 (Ω). Now, we consider the scattering problem (6.2.51) with the incident field .ui = vgk0 . It is straightforward to show that if .u∞ (x, ˆ vgk0 ) ≡ 0 (equivalent to .us (x, vgk0 ) = 0 in .Rd \Ω by Rellich’s Theorem), one then has the transmission eigenvalue problem (6.2.1) with .k = k0 , i .wΩ = u|Ω and .v = u |Ω , where u is the total field to (6.2.51). Conversely, noting i that .u ≈ vΩ from our earlier construction, one can show that .u∞ ≈ 0, and more importantly .u|Ω ≈ wΩ . Since .wΩ is a SLE, we see that .u|Ω is also a SLE (at least approximately). Set w -=

.

 u − ui u

in Rd \Ω, in Ω.

(6.2.77)

Clearly, .w - is generated from a direct incidence on the inhomogeneity .n2 . .w - ≈ 0 in d - ≈ wΩ . That is, .w - is localized around .∂Ω, which exhibits a similar .R \Ω and .w behaviour to the SPR oscillation. In what follows, we refer to .w - as a pseudo plasmon resonant (PSPR) mode. In Fig. 6.10a–c, we present a numerical illustration of the generation of a PSPR mode. We next propose a potential sensing application of the PSPR mode. Let .(Ω, n2 ) be an a-priori known inhomogeneity. Due to a certain reason, it is supposed that .∂Ω has some fine defects, namely, the support of the inhomogeneity actually becomes  Following the spirit of SPR sensing, one can detect the boundary defects as .∂ Ω. follows. Let .ui be an incident field that can generate a PSPR .w - associated with 2  n2 ), and we let .w .(Ω, n ) as above. The field impinges on .(Ω,  be the associated field according to (6.2.77). In Fig. 6.10e and f, we present the corresponding numerical results. It can be seen that the difference .w − w - is highly sensitive with respect to the  − ∂Ω. Hence, by the SPRS sensing, one can easily identify boundary defects .∂ Ω the existence of the fine boundary defects. It would be interesting to proceed further to recover such fine boundary defects by using the sensing data .w −w -.

6.2.5 Concluding Remarks and Discussions We have presented the discovery of a new type boundary localised modes of the transmission eigenfunctions for a special case that the underlying domain is radially

326

6 Interior Transmission Resonance

3

3

1.5

1

1.5

0

0

0

−1.5 −3 −3

−1 −1.5

0

1.5

(a) incident field ui

0 −0.5 −1 −1.5

0

1.5

3

3

1.5

1

1.5

0

0

0

−1.5

−1 −1.5

0

1.5

3

(d) incident field ui

1 0.5 0 −0.5

−1.5 −3 −3

−1 −1.5

0

1.5

(e) total field u

3

3

1

1.5

0.5 0

0

−0.5

−1.5 −3 −3

−1.5

0

1.5

3

−1

ˆ (c) generated PSPR mode w

(b) total field u

3

−3 −3

0.5

−1.5 −3 −3

3

1

3

1

1.5

0.5 0

0

−0.5

−1.5 −3 −3

−1 −1.5

0

1.5

3

(f) generated PSPR mode w

Fig. 6.10 Generation of the PSPR mode, where .k = 7.6548 and .n = 1/4. .(u, w -) and .( u, w ),  n2 ). (a) incident respectively, denote the corresponding fields associated with .(Ω, n2 ) and .(Ω, field .ui ; (b) total field u; (c) generated PSPR mode .w -; (d) incident field .ui ; (e) total field . u; (f) generated PSPR mode .w 

symmetric and the refractive index is constant. In what follows, we present some discussions and speculations about this intriguing topic. In the case that D is radially symmetric and n is constant, one can have the analytic derivation of the associated transmission eigenvalues and eigenfunctions for (6.2.1). Hence, though specific and relatively simpler, one can gain a much thorough and comprehensive understanding of the global structures of the transmission eigenfunctions, which can illuminate the direction for extending to more general cases, as shall be discussed in what follows. In the setup of our study, the transmission eigenvalues have been well studied [44, 88, 109], and we also refer to [37, 124, 125] for the distribution of transmission eigenvalues in general cases. Specifically, for a given Fourier mode of the transmission eigenfunctions, namely .wm and .vm in (6.2.23)–(6.2.24) with a given .m ∈ N ∪ {0}, there exist countably many (real) transmission eigenvalues .km,1 ≤ km,2 ≤ . . . ≤ km,l ≤ . . . → +∞. In [41], only the case with D being radially symmetric and n being constant is theoretically justified. In [41], extensive numerical experiments evidenced that the same property holds equally for generic cases with more general D and variable n. It is remarked that numerical experiments can also be used to show that the property unveiled here holds for the generic cases as well. However, it is unnecessary to repeat those numerical examples in [41]. To work along this line and in order to completely unveil such an intriguing global rigidity property of the transmission eigenfunctions, the natural extensions shall be to rigorously justify the generic cases with general D and variable n. Nevertheless, a more reasonable order is to

6.2 Scalar Case (Helmholtz Equations)

327

first consider the geometric effect by excluding the influence from the medium parameter. That is, one should first consider the case that D is a generic shape whereas n is still constant. In fact, a variable refractive index will make the situation radically more complicated. To illustrate this point, let us consider a simple case by assuming that D is the unit ball whereas n is variable. First, we assume that n is piecewise-constant which is .1/10 in .{x ∈ D; |x| < 1/2} and 10 in .{x ∈ D; 1/2 < |x| < 1}. It is not surprising that there still exist boundary-localised transmission eigenmodes, which are similar to the case with a uniform constant n. Indeed, this has been numerically verified in [41]. Second, we assume that the D is divided into two semi-balls, say a left one .D − and a right one .D + . Let n be .1/10 in .D − and 10 in .D + . It turns out that the localisation behaviours can be much more complicated, but more interesting. Indeed, some preliminary numerical experiments indicate that it can even happen that the transmission eigenfunctions are boundary-localised on the boundary of one of the semi-ball, but are un-patterned in the other semi-ball. The case that D is a generic domain and n is a constant has been theoretically justified in the recent work [42]. We would like to mention in passing that a highlytechnical quantized argument is developed in [42]. That means, it cannot provide accurate characterisation and comprehensive understanding as the special case with radially symmetric D and constant n. As discussed above, the coupling of general geometries with variable refractive indices, namely the case with general D and variable n, will make the corresponding study radically more complicated and challenging. The determination of the transmission eigenvalues as well as the associated transmission eigenfunctions from the corresponding scattering data has been considered in the literature, and moreover the transmission eigenvalues has been used to infer knowledge about the underlying scattering medium; see [37, 43, 109] as well as the references cited therein. The boundary-localisation property indicates that the transmission eigenfunctions carry the geometric information of the underlying scatterer .(D, n) (cf. Sect. ??), and hence can be used to recover the shape of the scatterer, which has been proposed in [41]. Here, we would like give two remarks regarding such a practical application. First, according to [41], the reconstruction mainly makes use of the boundary-localisation of the v-part of the transmission eigenfunctions, but not the w-part. Hence, one need not differentiate between the boundary-localisation property discovered here with the one in [41] from the point of view of the inverse problems. Second, according to the existing studies in the literature mentioned earlier, the determination of the large transmission eigenvalues from the scattering data is challenging and unpractical. However, it is numerically shown in [41] that if the scatterer .(D; n) possesses a high refractive index, namely n is sufficiently large, then the boundary-localisation of the transmission eigenfunctions can occur for transmission eigenvalues of very small amplitudes (this is partly evidenced by Lemma 6.2.3). More importantly from a practical viewpoint, this phenomenon is still true even if n is only large in a small neighbourhood of .∂D. Hence, it was proposed in [41] that before implementing the imaging scheme, one should coat the unknown scatterer (through indirect means) with a thin-layer of highly-refractive medium.

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6 Interior Transmission Resonance

6.3 Vectorial Case (Maxwell Equations) 6.3.1 Background We are concerned with the spectral geometry of the interior transmission eigenvalue problem arising in the time-harmonic electro-magnetic (EM) scattering described by the Maxwell system. Let D be a bounded Lipschitz domain in .R3 such that .R3 \D is connected. Let .ε and .μ be bounded positive functions such that .supp(1 − ε) ∪ supp(1 − μ) ⊂ D. In the physical context, .ε and .μ are the optical parameters of the space medium, and respectively signify the electric permittivity and magnetic permeability. Throughout, we assume that .μ ≡ 1. Consider a pair of incident EM waves .(Ei , Hi ) that are entire solutions to the following Maxwell system: ∇ × Ei − ikHi = 0,

.

∇ × Hi + ikEi = 0 in R3 ,

(6.3.1)

√ where .k ∈ R+ and .i := −1. Here, k signifies the wavenumber of the EM wave propagation. The EM inhomogeneity .(D, ε) interrupts the incident fields and leads to the scattered fields .Es and .Hs . Let .E(x) = Ei (x) + Es (x) and .H(x) = Hi (x) + Hs (x) denote the total electric and magnetic fields, respectively. The EM scattering is governed by the following Maxwell system: ⎧ ∇ × E − ikH = 0, in R3 , ⎪ ⎪ ⎪ ⎨ ∇ × H + ikεE = 0 in R3 , . ⎪

⎪ ⎪ ⎩ lim Hs × x − |x|Es = 0.

(6.3.2)

|x|→∞

The last limit in (6.3.2) is known as the Silver-Müller radiation condition, which characterizes the outgoing nature of the . scattered fields / and holds uniformly in the angular variable .xˆ := x/|x| ∈ S2 := x ∈ R3 ; |x| = 1 . The well-posedness of the scattering problem (6.3.2) can be conveniently found in [100, 112]. There exists a unique pair of solutions .(E, H) ∈ .Hloc (curl, R3 ) × Hloc (curl, R3 ) which admits the following asymptotic expansions as .|x| → ∞: Q(x) = Qi (x) +

.

eik|x| Q∞ (ˆx) + O |x|



1 |x|2

 ,

Q := E, H.

(6.3.3)

The functions .E∞ (ˆx) and .H∞ (ˆx) in (6.3.3) are respectively referred to as the electric and magnetic far field patterns, and satisfy the following one-to-one correspondence, E∞ (ˆx) = −ˆx × H∞ (ˆx)

.

and

H∞ (x) = xˆ × E∞ (ˆx),

∀ˆx ∈ S2 .

6.3 Vectorial Case (Maxwell Equations)

329

We next consider a particular case that non-scattering, a.k.a invisibility, occurs, namely .Q∞ ≡ 0. In such a case, by Rellich’s Theorem [44], one has .Qs = 0 in .R3 \D. With such an observation, one can directly verify that .Q1 = Q|D and i .Q2 = Q |D fulfil the following PDE system: ⎧ ∇ × E1 − ikH1 = 0, ⎪ ⎪ ⎨ ∇ × E2 − ikH2 = 0, . ⎪ ⎪ ⎩ ν × E1 = ν × E2 ,

∇ × H1 + ikεE1 = 0

in D,

∇ × H2 + ikE2 = 0

in D,

ν × H1 = ν × H2

on ∂D,

(6.3.4)

where and also in what follows, .ν signifies the exterior unit normal vector to .∂D. The system (6.3.4) is referred to as the Maxwell transmission eigenvalue problem. It is clear that .Ej = 0 and .Hj = 0, .j = 1, 2, are trivial solutions. If there exist nontrivial solutions, .(Ej , Hj )j =1,2 are called the transmission eigenfunctions and k is the associated transmission eigenvalue. We shall be mainly concerned with real transmission eigenvalues. Hence, if no scattering/invisibility occurs, then the restrictions of the incident and total EM fields are transmission eigenfunctions and the wave number is an eigenvalue. After eliminating .Hj , .j = 1, 2, in (6.3.4), we have the following reduced formulation of the transmission eigenvalue problem for .Ej ∈ H (curl, D), .j = 1, 2: ⎧ ∇ × (∇ × E1 ) − k 2 εE1 = 0, ⎪ ⎪ ⎨ . ∇ × (∇ × E2 ) − k 2 E2 = 0, ⎪ ⎪ ⎩ ν × E1 = ν × E2 ,

∇ · (εE1 ) = 0

in D,

∇ · E2 = 0

in D,

ν × (∇ × E1 ) = ν × (∇ × E2 ) on ∂D. (6.3.5)

For a convenient reference, if one considers the transverse EM scattering (cf. [38]), one can deduce the following transmission eigenvalue problem associated with the Helmholtz equation in .R2 : Δw + k 2 n2 w = 0, Δv + k 2 v = 0 in D; w = v, ∂ν w = ∂ν v on ∂D,

.

(6.3.6)

√ where .n := ϵ is known as the refractive index of the inhomogeneous medium, and .w, v correspond to the transverse parts of .E and .H.

6.3.2 Boundary-Localized Transmission Eigenmodes To begin with, we introduce a quantitative definition of boundary-localization. In what follows, for a sufficiently small .δ ∈ R+ , we set Nδ (∂D) := {x ∈ D; dist(x, ∂D) < δ}.

.

330

6 Interior Transmission Resonance

Definition 6.3.1 A function .Q ∈ L2 (D)3 is said to be boundary-localized if there exist sufficiently small .δ,  δ ∈ R+ such that .

‖Q‖L2 (D\Nδ (∂D))3 ‖Q‖L2 (D)3

n0 , .Q(n) is boundary-localized according to Definition 6.3.1. For terminological simplicity and convenience, we shall refer to .{Q(n) }n∈N+ as boundary-localized. Recalling the notion of mono-localization and bi-localization introduced earlier, our main finding in this section can be summarized in the following theorem. Theorem 6.3.1 Let D be a simply connected Lipschitz domain and .ε be a positive constant with .ε /= 1. Consider the Maxwell transmission eigenvalue problem (6.3.4). Then there exists a sequence of mono-localized eigenfunctions (n) (n) .(E 1 , E2 )n∈N+ , and there also exists a sequence of bi-localized eigenfunctions (n) (n) .(E 1 , E2 )n∈N+ , associated with the transmission eigenvalues .kn → ∞. In what follows, we shall rigorously prove Theorem 6.3.1 in the case that D is a ball, whereas in the general case, we only provide numerical verifications. In the numerics, we can actually consider variable .ε and the conclusion in Theorem 6.3.1 still holds true. We briefly discuss the transmission eigenvalues of the Maxwell system (6.3.5) for a ball associated with a constant refractive index. This has been derived in [113]. Nevertheless, some of the technical ingredients shall be needed in our subsequent analysis of the transmission eigenfunctions and for self-containedness and easy

6.3 Vectorial Case (Maxwell Equations)

331

reference, we present them in what follows. Without loss of generality, we assume 3 that .B√ R centres at the origin, namely, .BR = {x ∈ R : |x| < R ∈ R+ }. We also let .ϵ0 = ε ∈ R+ \{1}. Then (6.3.5) can be rewritten as ⎧ ⎨ ∇ × (∇ × E1 ) − k 2 ϵ02 E1 = 0, ∇ · (ϵ02 E1 ) = 0 . ∇ × (∇ × E2 ) − k 2 E2 = 0, ∇ · E2 = 0 ⎩ ν × E1 = ν × E2 , ν × (∇ × E1 ) = ν × (∇ × E2 )

in BR , in BR , on ∂BR . (6.3.9)

Since .ϵ0 is a positive constant, we can expand the solutions .E1 and .E2 of the system (6.3.9) into the Fourier series[44], E1 (x) =

n ∞  

anm Mnm (x) +

n=1 m=−n .

E2 (x) =

n ∞  

n ∞  

bnm Nnm (x),

n=1 m=−n

0m (x) +  anm M n

n=1 m=−n

n ∞  

m  bnm N0 n (x),

n=1 m=−n

where Mnm (x) = ∇ × {xjn (kϵ0 |x|)Ynm (ˆx)}, .

0m (x) = ∇ × {xjn (k|x|)Y m (ˆx)}, M n n

1 ∇ × Mnm (x), ik 1 m 0m (x). ∇ ×M N0 n n (x) = ik Nnm (x) =

(6.3.10)

Here, .jn (x) denotes the spherical Bessel functions of the first kind and .Ynm denotes a spherical harmonic function of degree n and order m. Moreover, one can verify that .Mnm and .Nnm are the solutions to ∇ × (∇ × E1 ) − k 2 ϵ02 E1 = 0,

.

∇ · (ϵ02 E1 ) = 0,

0m and .N0m are the solutions to and .M n n ∇ × (∇ × E2 ) − k 2 E2 = 0,

.

∇ · E2 = 0.

0m denote the TE (transverse electric) waves; .N m and In particular, .Mnm and .M n n m 0 .Nn denote the TM (transverse magnetic) waves.

332

6 Interior Transmission Resonance

For the TE mode, using the spherical coordinates .(ρ, θ, φ), .Mnm can be rewritten as ˆ Mnm = ∇ × {ρjn (kϵ0 ρ)Ynm (θ, φ) ρ} " " " ρˆ ρ θˆ ρ sin θ φˆ "" " 1 " " ∂ ∂ ∂ = 2 " " ∂ρ ∂θ ∂φ . " ρ sin θ " " ρjn (kϵ0 ρ)Ynm (θ, φ) 0 " 0

(6.3.11)

1 ∂Y m (θ, φ) ˆ ∂Y m (θ, φ) ˆ φ, jn (kϵ0 ρ) n θ − jn (kϵ0 ρ) n sin θ ∂φ ∂θ

=

where .ρ = |x| and ρˆ = sin θ cos φ e1 + sin θ sin φ e2 + cos θ e3 , .

θˆ = cos θ cos φ e1 + cos θ sin φ e2 − sin θ e3 , φˆ = − sin φ e1 + cos φ e2 .

Due to 1 ∂ . sin θ ∂θ



∂Y m (θ, φ) sin θ n ∂θ

 +

∂ 2 Ynm (θ, φ) + n(n + 1)Ynm (θ, φ) = 0, ∂φ 2 sin2 θ 1

we can derive that

∇ × Mnm

" " " " ρˆ ρ θˆ ρ sin θ φˆ " " 1 ∂ ∂ " " ∂ = 2 " " ∂ρ ∂θ ∂φ m m " ρ sin θ " " 0 ρ 1 jn (kϵ0 ρ) ∂Yn (θ,φ) −ρ sin θjn (kϵ0 ρ) ∂Yn (θ,φ) " sin θ ∂φ ∂θ     jn (kϵ0 ρ) 1 ∂ ∂Ynm (θ, φ) 1 ∂ 2 Ynm (θ, φ) =− ρˆ sin θ + ρ sin θ ∂θ ∂θ ∂φ 2 sin2 θ

.

=

+

1 ∂ ∂Y m (θ, φ) ˆ (ρjn (kϵ0 ρ)) n θ ρ ∂ρ ∂θ

+

∂ ∂Y m (θ, φ) ˆ 1 (ρjn (kϵ0 ρ)) n φ ρ sin θ ∂ρ ∂φ

1 ∂ 1 ∂Ynm (θ, φ) ˆ jn (kϵ0 ρ)n(n + 1)Ynm (θ, φ)ρˆ + (ρjn (kϵ0 ρ)) θ ρ ∂ρ ρ ∂θ +

1 ∂Ynm (θ, φ) ˆ ∂ (ρjn (kϵ0 ρ)) φ. ∂ρ ρ sin θ ∂φ (6.3.12)

6.3 Vectorial Case (Maxwell Equations)

333

Since the surface gradient in the spherical coordinates is given by ∇S f =

.

∂f ˆ 1 ∂f ˆ φ, θ+ sin θ ∂φ ∂θ

Eqs. (6.3.11) and (6.3.12) can be rewritten as ˆ Mnm = jn (kϵ0 ρ)∇S Ynm (θ, φ) × ρ, .

∇ × Mnm =

1 ∂ 1 (ρjn (kϵ0 ρ))∇S Ynm (θ, φ). jn (kϵ0 ρ)n(n + 1)Ynm (θ, φ)ρˆ + ρ ∂ρ ρ (6.3.13)

Similarly, one can deduce that 0m = jn (kρ)∇S Y m (θ, φ) × ρ, ˆ M n n .

0m = 1 jn (kρ)n(n + 1)Y m (θ, φ)ρˆ + 1 ∂ (ρjn (kρ))∇S Y m (θ, φ). ∇ ×M n n n ρ ρ ∂ρ (6.3.14)

According to the boundary condition to (6.3.9), it yields 0m , anm ρˆ × Mnm =  anm ρˆ × M n .

0m ). anm ρˆ × (∇ × Mnm ) =  anm ρˆ × (∇ × M n

Applying (6.3.13) and (6.3.14) with a straightforward calculation, one can obtain  .

 anm jn (kϵ0 ρ) −  anm jn (kρ) ∇S Ynm (θ, φ) = 0,   1 ∂ 1 ∂ anm (ρjn (kϵ0 ρ)) −  (ρjn (kρ)) ρˆ × ∇S Ynm (θ, φ) = 0, anm ρ ∂ρ ρ ∂ρ

ρ = R.

m Noting that .anm /= 0 and  .an /= 0, the eigenvalues k’s of the TE mode are positive zeros of the following function

  1 ∂ 1 ∂ (ρjn (kρ)) − jn (kρ) (ρjn (kϵ0 ρ)) fnTE (k) = jn (kϵ0 ρ) . ρ ∂ρ ρ ∂ρ ρ=R (6.3.15)

.

334

6 Interior Transmission Resonance

Using the recurrence relation of the derivatives of the spherical Bessel functions, 

 n jn (kρ) − jn+1 (kρ) , kρ .   n ∂jn (kϵ0 ρ) jn (kϵ0 ρ) − jn+1 (kϵ0 ρ) , = kϵ0 kϵ0 ρ ∂ρ ∂jn (kρ) =k ∂ρ

(6.3.16)

Eq. (6.3.15) can be rewritten as   fnTE (k) = k ϵ0 jn (kR)jn+1 (kϵ0 R) − jn+1 (kR)jn (kϵ0 R) ,

.

n ≥ 1.

(6.3.17)

On the other hand, for the TM mode, one can derive that 1 ∇ × Mnm (x) ik 1 ∂ 1 jn (kϵ0 ρ)n(n + 1)Ynm (θ, φ)ρˆ + (ρjn (kϵ0 ρ))∇S Ynm (θ, φ). = ikρ ikρ ∂ρ

Nnm (x) = .



Noting that .∇ × ∇ × Mnm − k 2 ϵ02 Mnm = 0, together with (6.3.10) and (6.3.13), one can derive that   1 m m ˆ ∇ × Mn = −ikϵ02 Mnm = −ikϵ02 jn (kϵ0 ρ)∇S Ynm (θ, φ) × ρ. .∇ × Nn = ∇ × ik m Similar results also hold for .N0 n . According to last equation of (6.3.9), it yields m bnm ρˆ × Nnm =  bnm ρˆ × N0 n , .

m bnm ρˆ × (∇ × Nnm ) =  bnm ρˆ × (∇ × N0 n ).

By following a similar argument as the TE mode case, one has

.

  1 ∂ 1 ∂ bnm (ρjn (kϵ0 ρ)) −  (ρjn (kρ)) ρˆ × ∇S Ynm (θ, φ) = 0, bnm ρ ∂ρ ρ ∂ρ   bnm jn (kρ) ∇S Ynm (θ, φ) = 0, ρ = R. − ik bnm ϵ02 jn (kϵ0 ρ) − 

1 ik

Noting that .bnm /= 0 and . bnm /= 0, the eigenvalues k’s for the TM mode are positive zeros of the following function   1 ∂ 1 ∂ (ρjn (kϵ0 ρ)) − ϵ02 jn (kϵ0 ρ) (ρjn (kρ)) . fnTM (k) = jn (kρ) ρ ∂ρ ρ ∂ρ ρ=R

.

6.3 Vectorial Case (Maxwell Equations)

335

Using the recurrence relation (6.3.16), the last equation can be rewritten as fnTM (k) = .

(1 − ϵ02 )(1 + n) jn (kR)jn (kϵ0 R) ρ   + kϵ0 ϵ0 jn (kϵ0 R)jn+1 (kR) − jn (kR)jn+1 (kϵ0 R) ,

n ≥ 1. (6.3.18)

To conclude, the transmission eigenvalues and the corresponding pair of eigenfunctions are presented in the following lemma. Lemma 6.3.1 Let .BR = {x ∈ R3 : |x| < R ∈ R+ } and .ϵ0 ∈ R+ be a positive constant with .ϵ0 /= 1. Let .(k, E1 , E2 ) denote the solution to the Maxwell system (6.3.9). It holds that: • if k solves the equation .fnTE (k) = 0, then k is a transmission eigenvalue of the TE 0m ) is the corresponding pair of eigenfunctions. mode and .(E1 , E2 ) := (Mnm , M n TM • if k solves the equation .fn (k) = 0, then k is a transmission eigenvalue m of the TM mode and .(E1 , E2 ) := (Nnm , N0 n ) is the corresponding pair of eigenfunctions. Moreover, from (6.3.17) and (6.3.18), one can find that the transmission eigenvalues depend on the parameter n, namely the order of the spherical Bessel function .jn . Next, we construct a sequence of transmission eigenvalues .{kn }n∈N+ and prove   (n) (n) is boundarythat one of the corresponding pair of eigenfunctions . E1 , E2 localized while the other is not. In what follows, for simplification, we only consider the case with .ϵ0 > 1 and the case of .0 < ϵ0 < 1 can be deduced by a similar argument. Lemma 6.3.2 Let .BR = {x ∈ R3 : |x| < R ∈ R+ } and .ϵ0 > 1 be a constant. Let .EBR denote the set of transmission eigenvalues of (6.3.9). Then there exists a sequence .{kn }n∈Z+ ⊂ EBR , such that for sufficiently large n , there holds kn ∈

.

r

n,s1 (n)

R

,

rn,s2 (n)  , R

(6.3.19)

where .rn,s denote the s-th positive root of the spherical Bessel function .jn (|x|) for a fixed order n, s1 (n) :=

.

   1 γ1 , n+ 2

s2 (n) =

   1 γ2 , n+ 2

0 < γ1 < γ2 < 1, (6.3.20)

336

6 Interior Transmission Resonance

and .[ t ] signifies the integer part of a real number t. Moreover, one has kn >

.

n + 12 , R

n = 1, 2, 3, · · · .

(6.3.21)

Proof For the TE mode, from Eq. (6.3.17), we have fnTE

.

r

n,s1



R

fnTE

r

n,s2



R

=

rn,s1 rn,s2 jn+1 (rn,s1 )jn (rn,s1 ϵ0 )jn+1 (rn,s2 )jn (rn,s2 ϵ0 ). R2

Similarly, for the TM mode, from Eq. (6.3.18), one has fnTM

r

.

n,s1



R

fnTM

r

n,s2



R

=

rn,s1 rn,s2 ϵ04 jn+1 (rn,s1 )jn (rn,s1 ϵ0 ) R2

jn+1 (rn,s2 )jn (rn,s2 ϵ0 ). Next, we will show that there exists .s1 and .s2 such that jn+1 (rn,s1 )jn (rn,s1 ϵ0 )jn+1 (rn,s2 )jn (rn,s2 ϵ0 ) < 0.

.

Based on the relationship between the Bessel and spherical Bessel function  jn (z) =

.

π Jn+1/2 (z), 2z

z > 0,

(6.3.22)

one can derive that the positive root .rn,s of spherical Bessel function .jn (z) has the following sharp upper and lower bounds[116] as 1 − 1/3 .n + 2 2



1 n+ 2

1/3 < rn,s +

as 1 < n + − 1/3 2 2

21/3 3 2 as  1/3 , 20 n + 12

  1 1/3 n+ 2 (6.3.23)

where .as is the s-th negative zero of the Airy function and has the representation 

3π (4s − 1) .as = − 8

2/3



(1 + σs ),

3π (4s − 1.051) 0 ≤ σs ≤ 0.130 8

−2 .

6.3 Vectorial Case (Maxwell Equations)

337

By noting the choice of .s1 and .s2 in (6.3.20) for sufficiently large n, it holds that    1  1 + C0 (n + 1/2)2(γ1 −1)/3 + o((n + 1/2)2(γ1 −1)/3 ) , rn,s1 = n + 2 .    1  1 + C0 (n + 1/2)2(γ2 −1)/3 + o((n + 1/2)2(γ2 −1)/3 ) , rn,s2 = n + 2 (6.3.24) where .C0 is a positive constant. Note that the Bessel function admits the following asymptotic formula [84, P. 129]  Jn (z) =

.

  π  2 nπ + n arcsin(n/z) − 1 + o(1) , z2 − n2 − cos √ 2 4 π z2 − n2 (6.3.25)

for .z > n and .n → ∞. Combing (6.3.22), (6.3.24), and (6.3.25), through a straightforward calculation, one obtains

jn rn,si ϵ0 =

.

  1 π ϵ02 − 1 − + arcsin cos (n + 1/2) (i) 2 ϵ 0 Cn,ϵ0   π    1 + o(1) , +O (n + 1/2)ςi − 4 1

where .ςi := 2(γi − 1)/3, i = 1, 2, and   −1/4 

 2 2 (i) ςi 1 + O . + 1/2) Cn,ϵ = n + 1/2 ϵ (ϵ − 1) (n 0 0 0

.

Correspondingly, one can derive that   π   

(i) 1 + o(1) , jn+1 rn,si = Cn,1 cos (n + 3/2)O (n + 1/2)ςi − 4

.

where (i) .C n,1

  1 1+ϵi /4 . =O n+ 2

Without loss of generality, we suppose that



jn−1 rn,s1 jn rn,s1 ϵ0 > 0.

.

We now show that there exists at least one choice of .s2 = (n + 1/2)γ2 such that



jn−1 rn,s2 jn rn,s2 ϵ0 < 0,

.

338

6 Interior Transmission Resonance

that is,    π cos (n + 3/2)O (n + 1/2)ς2 − 4   .   π  π 1 · cos (n + 1/2) ϵ02 − 1 − + arcsin + O (n + 1/2)ς2 < 0. − 2 ϵ0 4 (6.3.26) Noting that the above two cosine functions never have the same frequency, so it is easy to realize (6.3.26) by modifying .γ2 w.r.t .ς2 (γ2 ). Thus one has .kn ∈ (rn,s1 (n) /R, rn,s2 (n) /R). Equation (6.3.21) is then followed trivially from (6.3.24) and the proof is complete. ⨆ ⨅ From Lemma 6.3.2, we prove that there exists a sequences of transmission eigenvalues. Next, we show the  geometrical properties of the corresponding transmission (n) (n) eigenfunctions . E1 , E2 . Theorem 6.3.2 Let .BR = {x ∈ R3 : |x| < R ∈ R+ } and .ϵ0 > 1 be a constant. Let (n) (n) .(E 1 , E2 ) be the pair of transmission eigenfunctions associated with eigenvalue .kn in (6.3.19) for Maxwell’s transmission eigenvalue problem (6.3.9). Then it holds (n) (n) that the eigenfunction .E2 is boundary-localized but the other eigenfunction .E1 is not boundary-localized. Proof Let .Bτ := {x : |x| < τ, τ < R} and .kn be the eigenvalues defined in (6.3.19). For the TE mode, the eigenfunctions are given by m m ˆ . E(n) 1 = an ρ jn (kn ϵ0 ρ) ∇S Yn (θ, φ) × ρ,

.

(n)

ˆ E2 =  anm ρ jn (kn ρ) ∇S Ynm (θ, φ) × ρ.

(6.3.27) (6.3.28)

Correspondingly, for the TM mode, the eigenfunctions are given by (n) .E 1

(n)

E2

   ∂  m m jn (kn ϵ0 ρ)n(n + 1)Yn ρˆ + ρjn (kn ϵ0 ρ) ∇S Yn (θ, φ) , . ∂ρ (6.3.29)     bm ∂  = n (6.3.30) jn (kn ρ)n(n + 1)Ynm ρˆ + ρjn (kn ρ) ∇S Ynm (θ, φ) . ikn ρ ∂ρ bm = n ikn ρ

To begin with, for a fixed .τ , we prove that there exists sufficient large n such that kn τ < n + 1/2. Combining (6.3.19) and (6.3.24), we can derive that

.

1 .kn = R





1 n+ 1 + C0 (n + 1/2)ς , 2

ς ∈ [ς1 , ς2 ],

6.3 Vectorial Case (Maxwell Equations)

339

where .C0 is a positive constant and .ςi = 2(γi − 1)/3, .i = 1, 2. Thus, there exists a sufficiently large n, such that kn τ =

.

τ R

 n+



1 1 1 + C0 (n + 1/2)ς < n + , 2 2

(6.3.31)

for any fixed .τ < R. Next, we prove that the transmission eigenfunctions .E(n) 2 are boundary-localized around the boundary .∂BR . From [84, P.129], one has the following asymptotic formula: √ 2   zn en 1−z .Jn (nz) = 1 + o(1) , 0 < z < 1. √ (2π n)1/2 (1 − z2 )1/4 (1 + 1 − z2 )n (6.3.32) By a simple calculation, one can find that √ 2 ze 1−z . < 1, √ 1 + 1 − z2

0 < z < 1.

Therefore, from (6.3.32), we can derive the following asymptotic expansion:  jn (kn τ ) = =

π Jn+1/2 (kn τ ) 2kn τ 1

 √ 2 kn τ 1 −

kn τ n+1/2

⎛

.

⎜ k τ ⎜ n ×⎜ ⎝ n + 1/2



1/4

n+



2

 1−

e  1+

1−

kn τ n+1/2



1 2

kn τ n+1/2

−1/2

⎞n+1/2 ⎟ ⎟ 2 ⎟ ⎠



 1 + o(1)

   1 −1  < C1 n + 1 + o(1) , 2 where .C1 is a positive constant. According to [1, p.370], one has 1 ' ' ' < rn,1 < rn,2 < rn,2 < rn,3 < · · · ,. ≤ rn,1 2

√ π (z/2)n−1 ∞ z2 ' jn (z) = Π 1 − '2 , 4Γ (n + 1/2) s=1 rn,s n+

.

(6.3.33) (6.3.34)

340

6 Interior Transmission Resonance

' denotes the s-th positive zero of .j ' (z) that is derivative of .j (z). Using where .rn,s n n ' for sufficiently large n. Thus, (6.3.31) and (6.3.33), one can deduce that .kn τ < rn,1 by (6.3.34), we find that .jn (kn ρ) is a monotonically increasing with respect to .ρ ∈ (0, τ ). Hence, we obtain

 jn (kn ρ) < jn (kn τ ) < C1

.

1 n+ 2

−1 

 1 + o(1) ,

0 < ρ < τ.

(6.3.35)

One the other hand, for sufficiently large n , one can choose .τ1 τ1 =

.

' rn,1

kn

,

(6.3.36)

τ < τ1 < R.

Thus, we have n+

.

1 ' < rn,1 = kn τ1 . 2

Using the asymptotic expansion (6.3.25), one can deduce that 

R

τ

≥

jn2 (kn ρ) ρ dρ



R

τ1

≥ .

jn2 (kn ρ) ρ dρ 1



kn kn2 R 2 − (n + 12 )2   ⎛ 1 2   R n + 2 π 1 · cos2 ⎝ kn2 ρ 2 − n + − 2 2 τ1

(6.3.37)



 n + 12 1 π arcsin + n+ − dρ 2 kn ρ 4 

   1 −1−ς/2  R(R − τ1 ) n+ 1 + R(n) , = C2 2 2 where .C2 is a positive constant and the remaining term .R(n) has the approximation .

lim R(n)

n→∞

 = lim

R

n→∞ τ 1

⎛



1 sin 2 ⎝ kn2 ρ 2 − n + 2

2 −

  n + 21 π 2

6.3 Vectorial Case (Maxwell Equations)

341



 n + 12 1 + n+ arcsin dρ 2 kn ρ 

= 0. Thus, for the TE mode, using (6.3.28), (6.3.35) and (6.3.37), it holds that 2 ‖E(n) 2 ‖L2 (B

τ

)3

(n) ‖E2 ‖2L2 (B )3 R

.

"2  τ  π  2π "" " 2 m 0 0 0 "jn (kn ρ) ∇S Yn (θ, φ) × ρˆ " ρ sin θ dφdθdρ =   " "2 " 2 R π 2π " m 0 0 0 "jn (kn ρ) ∇S Yn (θ, φ) × ρˆ " ρ sin θ dφdθ dρ τ jn2 (kn τ ) 0 ρ 2 dρ ≤ R τ1 τ1 jn2 (kn ρ)ρ dρ  −2 C12 n + 12 τ 3 /3 <  −1−ς/2 C2 n + 12 τ1 R(R − τ1 )/2   2C12 τ 3 1 −1+ς/2 = → 0, n+ 3C2 τ1 R(R − τ1 ) 2

as

n → ∞.

Moreover, for the TM mode, according to the recurrence relation of spherical bessel function .

∂ jn−1 (ρ) − jn+1 (ρ) jn (ρ) = , ∂ρ 2

and formula (6.3.21), then the eigenfunction (6.3.30) can be rewritten as  bnm  n(n + 1) jn (kn ρ) jn (kn ρ)Ynm (θ, φ)ρˆ + ∇S Ynm (θ, φ) i kn ρ kn ρ  jn−1 (kn ρ) − jn+1 (kn ρ) + ∇S Ynm (θ, φ) 2  m  b n(n + 1) = n jn (kn ρ)Ynm (θ, φ)ρˆ i kn ρ     jn−1 (kn ρ) − jn+1 (kn ρ) + ∇S Ynm (θ, φ) 1 + O n−1 . 2

E(n) 2 =

.

(6.3.38)

Due to the Orthogonality property of the surface gradient to spherical harmonic function  ' . ∇S Ynm · ∇S Ynm' ds = n(n + 1)δnn' δmm' , S2

342

6 Interior Transmission Resonance

together with .∇S Ynm (θ, φ) = ρ∇Ynm (θ, φ) on .∂Bρ , and Eq. (6.3.37), it holds that  τ  n(n+1)

(n)

‖E2 ‖2L2 (B

3 τ) (n) 2 ‖E2 ‖L2 (B )3 R

=

 jn−1 (kn ρ)−jn+1 (kn ρ) 2 2 ρ dρ 0 2 kn ρ     2 2  R n(n+1) n+1 (kn ρ) ρ 2 + n(n + 1) jn−1 (kn ρ)−j ρ 2 dρ 0 kn ρ jn (kn ρ) 2 τ

≤ .

0

jn (kn ρ)

2

ρ 2 + n(n + 1)



n(n+1) 2 2 (k ρ) + j 2 (k ρ))/2 ρ 2 dρ jn (kn ρ) + (jn−1 n n+1 n kn2  R n(n+1) jn2 (kn ρ)ρ dρ/R τ1 kn2

 −2 C3 n + 12 τ <  −1−ς/2 (R − τ1 )/2 C2 n + 12

  1 −1+ς/2 2C3 τ n+ → 0, = C2 (R − τ1 ) 2

n → ∞,

as

where the constant .C3 is defined by C3 = 1 +

.

kn2 τ 2. n(n + 1)

Hence, the transmission eigenfunctions of .E(n) 2 for both TE and TM modes are boundary-localized on the boundary .∂BR . (n) Finally, it remains to prove that the eigenfunctions .E1 are not boundarylocalized around the boundary .∂BR . Without loss of generality, we only consider the TE mode case and the TM mode case can be proved in a similar manner. From (6.3.36), one can deduce that 

R 0

jn2 (kn ϵ0 ρ)ρ 2 dρ

=

= .

1



ϵ03 1 ϵ03 2

ϵ0 R 0 ' rn,1 kn

 0



ϵ0 R

jn2 (kn ρ ' )ρ '2 dρ '

jn2 (kn ρ ' )ρ '2 dρ ' +

jn2 (kn ρ ' )ρ '2 dρ ' ϵ03  R = 2 r ' jn2 (kn ϵ0 ρ)ρ 2 dρ.


0 ϵ02 − 1

n → ∞. ⨆ ⨅

344

6 Interior Transmission Resonance

Besides, we also construct a sequence of transmission  eigenvalues  .{kn }n∈N+ and (n)

(n)

prove that both corresponding pair of eigenfunctions . E1 , E2 localized.

are boundary-

Lemma 6.3.3 Let .BR = {x ∈ R3 : |x| < R ∈ R+ } and .ϵ0 > 1 be a constant. Let .EB' R denote the set of transmission eigenvalues of (6.3.9). For a fixed value ' .s0 ∈ N+ , there exists a subsequence .{kn }n∈N+ ⊂ E BR , such that for sufficient large n, there holds   rn,s0 rn,s0 +1 . , .kn ∈ (6.3.39) ϵ0 R ϵ0 R Moreover, one has .

n + 12 n + 12 < kn ≤ , ϵ0 R R

n = 1, 2, 3, · · · .

(6.3.40)

Proof For any fixed .s0 ∈ N+ , from (6.3.23) and (6.3.33), there exists a sufficiently large n such that .

rn,s0 +1 1 ' . ≤ n + ≤ rn,1 2 ϵ0

(6.3.41)

By the monotonicity of the Bessel function before the first local maximal, we have jn (k) ≥ 0,

.

' k ∈ (0, rn,1 ].

Noting that the positive root of .jn (|x|) are interlaced with those of .jn+1 (|x|) [105], hence we have jn+1 (rn,s0 ) · jn+1 (rn,s0 +1 ) < 0.

.

Therefore, from the last two equations and (6.3.17), we can derive that 





 rn,s0 +1 ϵ0 R     rn,s0 +1 rn,s0 r r . = n,s0 n,s0 +1 j jn+1 (rn,s0 ) jn jn+1 (rn,s0 +1 ) n ϵ0 ϵ0 R2 rn,s0 rn,s0 +1 2 '

jn rn,1 jn+1 (rn,s0 )jn+1 (rn,s0 +1 ) < 0. ≤ R2

Hence, there exists .kn ∈ rn,s0 /(ϵ0 R), rn,s0 +1 /(ϵ0 R) such that .fnTE (kn ) = 0. By a similar argument, one can also verify that .fnTM (kn ) = 0 and it proves (6.3.39). fnTE

rn,s0 ϵ0 R

fnTE

6.3 Vectorial Case (Maxwell Equations)

345

Furthermore, from (6.3.33) and (6.3.41), one can deduce (6.3.40) and it completes the proof. ⨆ ⨅ Theorem 6.3.3 Let .BR = {x ∈ R3 : |x| < R ∈ R+ } and .ϵ0 > 1 be a constant. Let (n) (n) .(E 1 , E2 ) be the pair of transmission eigenfunctions associated with eigenvalue .kn in (6.3.19) for Maxwell’s transmission eigenvalue problem (6.3.9). Then it holds (n) (n) that both eigenfunctions .E1 and .E2 are boundary-localized around the boundary .∂B. Proof Without loss of generality, we only consider the TE mode and the TM mode can be proved in a similar manner. Let .Bτ := {x : |x| < τ, τ < R} and .kn be the eigenvalues defined in (6.3.19). (n) We first prove that the transmission eigenfunctions .E2 are localized around the boundary .∂BR . Using (6.3.28) and (6.3.22), one has "2  τ  π  2π "" " 2 m (θ, φ) × ρ ˆ j (k ρ) ∇ Y " " ρ sin θ dφdθ dρ n n S n 0 0 0 =   " "2 " 2 R π 2π " m (θ, φ) × ρ ˆ j (k ρ) ∇ Y " " ρ sin θ dφdθ dρ n n S n 0 0 0 τ 2 0 Jn+ 1 (kn ρ)ρ dρ 2 . = R 2 (k ρ)ρ dρ J n 1 0

(n)

‖E2 ‖2L2 (B

3 τ) (n) 2 ‖E2 ‖L2 (B )3 R .

n+ 2

Set .f (ρ) = J 2

n+ 21

(kn ρ)ρ, through a straightforward calculation, one can verify that

f (ρ) is a convex function on .[0, R] for .kn ≤ (n + 1/2)/R. Therefore, the area of R the triangle under tangent of .f (R) is smaller than . 0 f (ρ)dρ, that is,

.

R2J 3 .

n+ 12



(kn R)

2Jn+ 1 (kn R) + 4kn RJ '

n+ 12

2

(kn R)

≤ 0

R

2 Jn+ 1 (kn ρ)ρ dρ. 2

Thus it holds that τ 2J 2

(n)

‖E2 ‖2L2 (B

.

3 τ) (n) 2 ‖E2 ‖L2 (B )3 R

≤

n+ 12

2

n+ 12

Jn+ 1 (kn τ ) 2

Jn+ 1 (kn R) 2



n+ 21

R2J 3

≤ τ2

 (kn τ ) 2Jn+ 1 (kn R) + 4kn RJ '

2 ⎛

(kn R)

(kn R) J'

n+ 12

(kn R)

⎞

⎝ 2 + 4kn · ⎠. R Jn+ 1 (kn R) R2 2

346

6 Interior Transmission Resonance

By Lemma 2.3 in [52], one can show that there exists positive constants .C3 and .γ such that   1 γ < C3 n + . Jn+ 1 (kn R) 2

J'

n+ 12

.

(kn R)

2

In addition, using the Calini formula, one can find that there exists positive constants C4 and .δ(τ, ϵ0 ) ∈ (0, 1), such that

.

J2

.

(k τ )

n+ 12 n J 2 1 (kn R) n+ 2

1

< C4 (1 − δ)n+ 2 .

Hence, combining the last three inequalities, one can obtain that (n)

.

‖E2 ‖2L2 (B

3 τ) (n) 2 ‖E2 ‖L2 (B )3 R

→ 0,

as n → ∞.

Next, we prove that the transmission eigenfunctions .E(n) are also surface 1 localized around the boundary .∂BR . For any fixed .s0 ∈ N+ , using (6.3.33) and (6.3.39), one can derive that

.

n + 12 n + 21 n + 12 R< < R < R. rn,s0 +1 k n ϵ0 rn,s0

Moreover, from inequality (6.3.23), one has .

n + 12 R = R, n→∞ rn,s +1 0 lim

n + 12 R = R. n→∞ rn,s 0 lim

Thus we obtain .

n + 21 = R. n→∞ kn ϵ0 lim

Hence, for .τ ∈ (0, R), there exists a sufficiently large .n0 ∈ N+ such that when n > n0 , one has

.

τ