Mathematical Analysis of Plasmonics Resonances for Nanoparticles and Applications Matias Ruiz

Mathematical analysis of plasmonics resonances for nanoparticles and applications Matias Ruiz

To cite this version:

Matias Ruiz. Mathematical analysis of plasmonics resonances for nanoparticles and applications. Plasmas. Université Paris sciences et lettres, 2017. English. NNT : 2017PSLEE054. tel-01827839

HAL Id: tel-01827839 https://tel.archives-ouvertes.fr/tel-01827839 Submitted on 2 Jul 2018

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE DE DOCTORAT de l’Université de recherche Paris Sciences Lettres – PSL Research University préparée à l’École Normale Supérieure

Analyse Mathématique de École doctorale n°386 Résonances Plasmoniques Spécialité: Mathématiques pour des Nanoparticules Appliquées et Applications Soutenue le 27.06.2017 Mathematical Analysis of Plasmonic Resonances for Nanoparticles and Applications Composition du Jury : par Matias Ruiz M. Habib Ammari ETH Zurich Directeur de thèse

M. Oscar Bruno California Institute of Technology Rapporteur

M. Gilles Lebeau Université de Nice - Sophia Antipolis Rapporteur

Mme. Liliana Borcea University of Michigan Examinatrice

M. Josselin Garnier École Polytechnique Examinateur

M. Bertrand Maury Université Paris Sud Examinateur

M. Gabriel Peyré École Normale Supérieure Président du jury

iii Acknowledgements

I would like to start by thanking the most important person in the achieve- ment of this Ph.D. Habib, I couldn’t have asked for a better advisor. I have learnt so much from you. You are not only an amazing scientist but also a great leader and person. I truly hope that our friendship and scientific collaboration will last long years. There is someone else who has been of key importance in the success of these three years of hard work. Hai Zhang, your enormous talent has and will be an inspiration for me. I thank you for our fruitful collaboration and for, more than once, setting me in the right path of mathematical discovery. I hope this is only the beginning of many works together. I would also like to greatly thank the members of my jury: Oscar Bruno and Gilles Lebeau, who accepted the time consuming mission of being my "rapporteurs" and Lilianna Borcea, Josselin Garnier, Bertrand Maury and Gabriel Peyré, who kindly agreed to be my "examinateurs". It is a great honor to have such world class researchers judging my work. Research is done better when working in groups. I would first like to thank Sanghyeon Yu, with whom I have already cosigned 4 research papers. You are an amazing mathematician and a very good friend. I hope our collaboration doesn’t stop with the end of this work. Pierre Millien and Francisco Romero, it has also been a privilege to work with you. I have really enjoyed the time we spent together, trying to solve a math problem, discussing about science or just visiting a city during a conference. I would also like to thank Mihai Putinar and Wei Wu, with whom I have cosigned one paper. Habib’s team is a big family where I have encountered wonderful people. Wenglong and Timothée, it has been a truly pleasure to have started and finished this experience along your side. I would also like to thank the other members (or former members) of Habib’s team that I had the fortune to met. Abdul, Brian, David, Alexander, Giovanni, Guanghui and Tomas. I thank you for all the moments we shared. The DMA is full of good people, starting by Bénédicte, Zaïna and Lara who always had the best disposition to help me with all the administrative issues during these three years. You were always there when I needed your help and for that, a thousand thanks. I had the chance to have my desk in the best office of the department, the "bureau de l’ambiance". Thank you Maxence, Jessica and Jeremy for making every day so pleasant and for considerably improving my level on baby foot. Of course, I cannot talk about the "bureau de l’ambiance" without mentioning its topological dual, the "bureau des plaisirs". Thank you Guillaume, Yichao and Wei for all the good moments. I also thank Charles, Jaime, Thibault and Rodolfo for your sympathy. When you are away from home, your friends become your family. I would like to thank all my friends who have been to my side in this process of expatriation. In particular, Jorge Gonazález, Sofia Olivares and Felipe Soto who arrived with me in France six and a half years ago. Thank you also Ignacio Atal, Francisco Vial (jedi master), Tomas Yany, Michel Lemaire and Juan Pablo de la Cruz. Special greetings to my good friend and bestman at my wedding, Arsalan Shoysoronov. I am proud to have you as a friend. iv

I also happen to have real family in France. Thank you Carla and Lolo for feeding me and sharing so many good moments. Nada de esto sería posible sin el apoyo de mi familia. Papá (Cesar), Mamá (Olga), ustedes lucharon incansablemente para que yo y mis hermanos tuviéramos siempre la mejor educación. Este trabajo es de ustedes. No puedo dejar de agradecer a dos personas cuyo apoyo ha sido fundamental. Carmen Luz Palacios y Fernando Salinas, les estaré eternamente agradecido. Este trabajo también se lo dedico a ustedes. And last but not least, I would like to thank Despi, my lovely and beau- tiful wife. You just make my life so happy. Thank you for having my back no matter what and for being my partner in life and crime. All of this would be meaningless if I couldn’t share it with you. v

Contents

Introduction 1

I Mathematical Analysis of Plasmonic Resonances 7

1 The Quasi-Static Limit 9 1.1 Introduction ...... 10 1.2 Preliminaries ...... 10 1.2.1 Layer potentials for the Laplace equation ...... 10 1.3 Layer potential formulation for the scattering problem . . . . 12 1.3.1 Contracted generalized polarization tensors ...... 13 1.4 Plasmonic resonances for the anisotropic problem ...... 15 1.5 A Maxwell-Garnett theory for plasmonic nanoparticles . . . . 19 1.6 Concluding remarks ...... 23

2 The Helmholtz Equation 25 2.1 Introduction ...... 26 2.2 Preliminaries ...... 26 2.2.1 Layer potentials for the Helmholtz equation ...... 26 2.3 Layer potential formulation for the scattering problem . . . . 28 2.3.1 Problem formulation and some basic results ...... 28 2.3.2 First-order correction to plasmonic resonances and field behavior at the plasmonic resonances ...... 32 2.4 Scattering and absorption enhancements ...... 37 2.4.1 Far-field expansion ...... 37 2.4.2 Energy flow ...... 38 2.4.3 Extinction, absorption, and scattering cross-sections and the optical theorem ...... 39 2.4.4 The quasi-static limit ...... 41 2.4.5 An upper bound for the averaged extinction cross-section 43 Bound for ellipses ...... 46 Bound for ellipsoids ...... 48 2.5 Link with the scattering coefficients ...... 48 2.5.1 The notion of scattering coefficients ...... 49 2.5.2 The leading-order term in the expansion of the scattering amplitude ...... 50 2.6 Concluding remarks ...... 53

3 The Full Maxwell Equations 55 3.1 Introduction ...... 56 3.2 Preliminaries ...... 57 3.3 Layer potential formulation for the scattering problem . . . . 60 3.3.1 First-order correction to plasmonic resonances and ﬁeld behavior at the plasmonic resonances ...... 70 vi

3.4 The extinction cross-section at the quasi-static limit . . . . . 76 3.5 Explicit computations for a spherical nanoparticle ...... 77 3.5.1 Vector spherical harmonics ...... 77 3.5.2 Explicit representations of boundary integral operators 79 3.5.3 Asymptotic behavior of the spectrum of (r) . . . . 81 WB 3.5.4 Extinction cross-section ...... 83 3.6 Explicit computations for a spherical shell ...... 86 3.6.1 Explicit representation of boundary integral operators 87 3.6.2 Asymptotic behavior of the spectrum of sh(r ) . . . 90 WB s 3.7 Concluding remarks ...... 92

II Applications 95

4 Heat Generation with Plasmonic Nanoparticles 97 4.1 Introduction ...... 98 4.2 Setting of the problem and the main results ...... 98 4.3 Heat generation ...... 101 4.3.1 Small volume expansion of the inner ﬁeld ...... 102 Rescaling ...... 102 Proof of Theorem 4.2.1 ...... 103 4.3.2 Small volume expansion of the temperature ...... 106 Re-scaling of the equations ...... 107 Proof of Theorem 4.2.2 ...... 107 4.3.3 Temperature elevation at the plasmonic resonance . . 109 4.3.4 Temperature elevation for two close-to-touching particles110 4.4 Numerical results ...... 111 4.4.1 Single-particle simulation ...... 112 Single-particle Helmholtz resonance ...... 112 Single-particle surface heat generation ...... 113 4.4.2 Two particles simulation ...... 115 Two particles Helmholtz resonance ...... 115 Two particles surface heat generation ...... 115 4.5 Concluding remarks ...... 116

5 Plasmonic Metasurfaces 119 5.1 Introduction ...... 120 5.2 Setting of the problem ...... 121 5.3 One-dimensional periodic Green function ...... 122 5.4 Boundary layer corrector and eﬀective impedance ...... 124 5.5 Numerical illustrations ...... 128 5.5.1 Setup and methods ...... 128 5.5.2 Results and discussion ...... 129 5.6 Concluding remarks ...... 129

6 Shape Recovery of Algebraic Domains 133 6.1 Introduction ...... 134 6.2 Plasmonic resonance for algebraic domains ...... 135 6.2.1 Algebraic domains of class ...... 135 Q 6.2.2 Explicit computation of the Neumann-Poincaré operator135 vii

6.2.3 Asymptotic behavior of the Neumann-Poincaré operator ...... 138 KD∗ 6.2.4 Generalized polarization tensors and their spectral representations ...... 140 First-order polarization tensor ...... 140 Second-order contracted generalized polarization tensors142 6.2.5 Classiﬁcation of algebraic domains in the class . . . 143 Q 6.3 Plasmonic resonances for two separated disks ...... 143 6.3.1 The bipolar coordinates and the boundary integral operators ...... 143 6.3.2 Neumann Poincaré-operator for two separated disks and its spectral decomposition ...... 145 6.3.3 The Polarization tensor ...... 147 6.3.4 Reconstruction of the separation distance ...... 148 6.4 Numerical illustrations ...... 148 6.5 Concluding remarks ...... 148

7 Super-Resolution with Plasmonic Nanoparticles 157 7.1 Introduction ...... 158 7.2 Multiple plasmonic nanoparticles ...... 158 7.2.1 Layer potential formulation in the multi-particle case . 158 7.2.2 Feld behavior at plasmonic resonances in the multi- particle case ...... 160 7.3 Super-resolution (super-focusing) by using plasmonic particles 166 7.3.1 Asymptotic expansion of the scattered ﬁeld ...... 166 7.3.2 Asymptotic expansion of the imaginary part of the Green function ...... 169 7.4 Concluding remarks ...... 170

8 Sensing Beyond the Resolution Limit 171 8.1 Introduction ...... 172 8.2 The forward problem ...... 173 8.2.1 The Green function in the presence of a small particle 174 8.2.2 Representation of the total potential ...... 175 8.2.3 Intermediate regime and asymptotic expansion of the scattered ﬁeld ...... 176 8.2.4 Representation of the shift using CGPTs ...... 179 Pj 8.3 The inverse problem ...... 181 8.3.1 CGPTs recovery algorithm ...... 181 8.3.2 Shape recovery from CGPTs ...... 183 8.4 Numerical Illustrations ...... 184 8.5 Concluding remarks ...... 187

A Layer Potentials for the Laplacian in two Dimensions 191

B Asymptotic Expansions 197

C Sum Rules for the Polarization Tensor 207

Bibliography 211

Notations

D ⋐ Rd denotes that D is an open, bounded and simply connected • subset of Rd;

∂D denotes the boundary of the open set D Rd; • ∈ We say that ∂D is of type 1,α, for 0 <α< 1, if ∂D is locally Lipschitz • C of order 0 <α< 1;

ν denotes the outward normal to ∂D and ∂ the outward normal deriva- • ∂ν tive;

ϕ (x) = limt 0+ ϕ(x tν), x ∂D; • ± → ± ∈ Id denotes the identity operator; • Hs(∂D) denotes the usual Sobolev space of order s on ∂D; • 1 1 2 2 ( , ) 1 , 1 denotes the duality pairing between H− (∂D) and H (∂D); • · · − 2 2 For any functional space F (∂D) deﬁned on ∂D, F (∂D) denotes its • 0 zero mean subspace;

(E,F ) denotes the set of bounded linear applications from E to F • L and (E) := (E,E); L L For α =(α ,α ) N2, ∂ := ∂α1 ∂α2 and α! := α !α !; • 1 2 ∈ α 1 d 1 2 χ(S), denotes the characteristic function of the set S; • z denotes the real part of z; • ℜ z denotes the imaginary part of z; • ℑ x denotes the norm of x Rd; • | | ∈ We denote by the Sommerfeld radiation condition for a function u in • dimension d = 2, 3, the following condition:

∂u (d+1)/2 ik u C x − ∂ x − m ≤ | |

| | as x + for some constant C independent of x. | | → ∞

to my parents, Cesar and Olga

Introduction

Light has been a major field of scientific curiosity and study since the beginning of science. Despite the age of the field, research in photonics is more active than ever, as evidenced by 2015 being proclaimed by the United Na- tions General Assembly as the "International Year of Light and Light-based Technologies". In the last decades, the field of photonics has seen a rev- olution due to the study of the anomalous properties of metallic particles, no bigger than some tens of nanometers, and their interaction with light. At this scale, and for some specific range of frequencies, this nanoparticles have the unique capability of enhancing the brightness and directivity of light, confining strong electromagnetic fields into advantageous directions. This phenomenon, called "plasmonic resonances for nanoparticles" or "surface plasmons", open a door for a wide range of applications, from novel healthcare techniques to efficient solar panels. To harvest such opportuni- ties, a deep mathematical understanding of the interactive effects between the particle size, shape and contrasts in the electromagnetic parameters is required. Although very significant experimental and modeling advances have been achieved in the field of nanoplasmonic during recent decades, very few properties have been introduced and analyzed in the mathematical literature. There is a clear lack of deep understanding of the theory of plasmonic resonance. The goal of this work is to fill some of these gaps - understand the mathematical structure of inverse problems arising in nanophotonics and propose, from a better mathematical basis, pertinent applications of plasmonic nanoparticles that will best meet the challenges of emerging nanotechnolo- gies.

Plasmonic nanoparticles

Plasmonic nanoparticles are particles, typically made of gold or silver, whose size range in the order of a few to a hundred nanometers. At this scale, they behave as metamaterials, meaning that their conductivity and/or pre- meabilitty has negative real part. When an external light wave is incident on the nanoparticle, the cloud of free electrons on the surface of the par- ticule oscilates at some specific frequencies, entering in a resonance mode; see Figure 1. These resonances depend on the electromagnetic parameters of the nanoparticle, those of the surrounding material, and the particle shape and size. High scattering and absorption cross sections (see [43] for precise definitions of these quantities) and strong near-fields are unique effects of plasmonic resonant nanoparticles; see Figure 2. Even though plasmonic nanoparticles have drawn the attention of scientists mainly in the 20th century, they have been first put into use thousands of years ago, when ancient civilizations made use of them for decoration and artistic purposes. Figure 3 shows the Lycurgus cup, a decorative Roman

Introduction 3

Figure 3: Lycurgus cup. Its galss contains gold-silver nanoparticles that resonates with red frequencies of incoming ight. (trustees of the British museum)

Driven by the search for new materials with interesting and unique optical properties, the field of plasmonic nanoparticles has grown immensely in the last decade [68]. Recent advances in nanofabrication techniques have made it possible to construct complex nanostructures such as arrays using plasmonic nanoparticles as components, allowing the design of new kinds of materials. Among this structures we find the so called "metasurfaces", con- sisting in a thin layer of periodically arranged nanoparticles mounted over a dielectric. This kind of composites are capable to control and transform optical waves in order to reduce scattering and make objects invisible or even trap electromagnetic waves in the goal of making efficient photovoltaic cells. Another thriving interest for optical studies of plasmon resonant nanoparticles is due to their recently proposed use in molecular biology, where the strong field enhancement can be used as efficient contrast for biological and cell imaging applications [48]. Nanoparticles are also being used in thermotherapy as nanometric heat- generators that can be activated remotely by external electromagnetic fields. Nanotherapy relies on a simple mechanism. First nanoparticles become attached to tumor cells using selective biomolecular linkers. Then heat generated by optically-simulated plasmonic nanoparticles destroys the tumor cells [51]. Scientists have long dreamt of an optical microscope that can be used to see, noninvasively and in vivo, the details of living matter and other materials. When attempting to image nanoscale structures with visible light, a fundamental problem arises: diffraction effects limit the resolution to a dimension of roughly half the wavelength. Recently, the use of plasmonics nanoparticles has been proposed in a number of emerging techniques that achieve resolution below the conventional resolution limit into what is called super-resolution techniques.

Contributions

It is important to understand the collective behavior of plasmonic nanoparticles to derive the macroscopic optical properties of materials with a dilute set of plasmonic inclusions. In this regard, we have obtained effective properties 4 Introduction of a periodic arrangement of arbitrarily-shaped nanoparticles and derived a condition on the volume fraction of the nanoparticles that insures the validity of the Maxwell-Garnett theory for predicting the effective optical properties of systems embedded in a dielectric host material at the plasmonic resonances. One of the most important parameters in the context of applications is the position of the resonances in terms of the wavelength or frequency. A longstanding problem is to tune this position by changing the particle size or the concentration of the nanoparticles in a solvent [49, 68]. It was experimentally observed, for instance, in [49, 89] that the scaling behavior of nanoparticles is critical. The question of how the resonant properties of plasmonic nanoparticles develops with increasing size or/and concentration is therefore fundamental. According to the quasi-static approximation for small particles, the surface plasmon resonance peak occurs when the particle’s polarizability is maximized. At this limit, since resonances are directly related to the Neumann- Poincaré integral operator, they are size-independent. However, as the particle size increases, a shift in the value of the resonances can be observed, for instance, in [49,81,89]. Using the Helmholtz equation to model light propagation we have precisely quantified the shift of the plasmonic resonance and the scattering absorption enhancement for a single nanoparticle. At the quasi-static limit, we gave a proof that the averages over the orientation of scattering and extinction cross-sections of a randomly oriented nanoparticle are given in terms of the imaginary part of the polarization tensor. Moreover, we have derived bounds in dimension two (optimal bounds) and three for the absorption and scattering cross-sections. Later on, we have generalized these results, providing the first mathematical study of the shift in plasmon resonance using the full Maxwell equations. Surprisingly, it turns out that in this case not only the spectrum of the Neumann-Poincaré operator plays a role in the resonance of the nanoparticles, but also its negative. We have explained how in the quasi-static limit, only the spectrum of the Neumann-Poincaré operator can be excited and that its negative can only be excited as in higher-order terms in the expansion of the electric field versus the size of the particle. Due to their high absorption enhancement, monitoring the temperature generated by the nanoparticles in the plasmonic resonance could be crucial for thermoterapy success. We have established an asymptotic expansion for the temperature in the border of arbitrary shaped particles, which turns out to be related, again, to the eigenvalues of the Neumann-Poincaré operator. If we consider the scattering by a layer of periodic plasmonic nanoparticles mounted on a perfectly conducting sheet, as the thickness of the layer, which is of the same order as the diameter of the individual nanoparticles, is negligible compared with the wavelength, it can be approximated by an impedance boundary condition. We have proved that at some resonant frequencies, the thin layer has anomalous reflection properties and can be viewed as a metasurface allowing the control and transformation of electromagnetic waves. We have also proved that using plasmonic resonances one can classify the shape of a class of domains with real algebraic boundaries and on the other hand recover the separation distance between two components of multiple Introduction 5 connected domains. These results have important applications in nanophotonics. They can be used in order to identify the shape and separation distance between plasmonic nanoparticles having known material parameters from measured plasmonic resonances, for which the scattering cross-section is maximized. The main objective of super-resolution is to create imaging approaches for objects significantly smaller than half the wavelenght, based on the use of resonant plasmonic nanoparticles. In a homogeneous space, particles smaller than half the wavelength cannot be resolved because the point spread function, which is the imaginary part of the Green function, has a width of roughly half the wavelenght. By following the methodology of [30], we have shown that super-resolution can be achieved when replacing the homogeneous media by a composite made of plasmonic nanoparticles. Moreover, we have shown that we can make use of plasmonic nanoparticles to recover fine details of a subwavelength non plasmonic nanoparticles, providing a mathematical foundation for plasmonic biosensing. These results open a door for the ill-posed inverse problem of reconstructing small objects from far-field measurements. The results obtained in this thesis have been published in [22–27].

Part I

Mathematical Analysis of Plasmonic Resonances

Chapter 1

The Quasi-Static Limit

Contents 1.1 Introduction ...... 10 1.2 Preliminaries ...... 10 1.2.1 Layer potentials for the Laplace equation . . . . . 10 1.3 Layer potential formulation for the scattering problem ...... 12 1.3.1 Contracted generalized polarization tensors . . . . 13 1.4 Plasmonic resonances for the anisotropic problem 15 1.5 A Maxwell-Garnett theory for plasmonic nanoparticles ...... 19 1.6 Concluding remarks ...... 23 10 Chapter 1. The Quasi-Static Limit

1.1 Introduction

Consider the scattering problem of iluminating a nanoparticle immersed in a homogeneous medium. When the size of the nanoparticle is significantly smaller than the wavelength of the incomming light, Maxwell equations can be approximated by the equation (1.3) [87]. We say that we are working in the quasi-static limit. This regime have been extensively used by the physics community to model the scattering of light by small nanoparticles such as plasmonic nanoparticles. In the mathematics community, the first efforts to give rigourous results on the plasmonic resonance phenomena have been done in this framwork [52]. In the first part of this chapter we give a brief review of the mathematical analysis of the plasmonic resonances for nanoparticles in the quasi-static regime. This analysis rely strongly in the use of layer potential techniques for the Laplace equation. Secondly, we investigate the overall optical properties of a collection of plasmonic nanoparticles. We treat a composite material in which plasmonic nanoparticles are embedded and isolated from each other. The Maxwell- Garnett theory provides a simple model for calculating the macroscopic optical properties of materials with a dilute inclusion of spherical nanoparticles [18]. Here, we extend the validity of the Maxwell-Garnett effective medium theory in order to describe the behavior of a system of arbitrary- shaped plasmonic resonant nanoparticles. We rigorously derive a condition on the volume fraction of the nanoparticles that insures its validity at the plasmonic resonances. To do so, we introduce the notion of plasmonic resonances for particles with anisotropic electromagnetic materials. This notion is introduced here for the first time. In section 1.4 we analyze the anisotropic quasi-static problem in terms of layer potentials and define the plasmonic resonances for anisotropic nanoparticles. Formulas for a small anisotropic perturbation of resonances of the isotropic formulas are derived. Section 1.5 is devoted to establish a Maxwell-Garnett type theory for approximating the plasmonic resonances of a periodic arrangement of arbitrary- shaped nanoparticles.

1.2 Preliminaries

In this section we recall important properties of the layer potentials for the Laplacian that will be of great use throughout this thesis.

1.2.1 Layer potentials for the Laplace equation Consider a domain D with boundary ∂D of type 1,α for 0 <α< 1. Let ν C denote the outward normal to ∂D. Deﬁne the single layer potential

[ϕ](x)= G(x,y)ϕ(y)dσ(y), x ∂D,x Rd ∂D, SD ∈ ∈ \ Z∂D where G(x,y) is the green function for the Laplacian. For d = 2 1 G(x,y)= log x y . 2π | − | 1.2. Preliminaries 11

For d = 3 1 G(x,y)= . −4π x y | − | We have the following lemma

1 1 d Lemma 1.2.1. 1. For ϕ H− 2 (∂D), [ϕ] H (R ∂D) is an Har- ∈ SD ∈ \ monic function on Rd ∂D; \

2. D[ϕ](x) + = D[ϕ](x) ; S S − 1 1 2 3. For d = 3, D[ϕ](x) = O( x 2 ) as x and D[ ] : H− (∂D) 1 S | | | | → ∞ S · → H 2 (∂D) is invertible, negative deﬁnite and self-adjoint for the duality 1 1 paring H− 2 (∂D),H 2 (∂D);

4. Same for d = 2 under condition that ∂D ϕdσ = 0. A more detailed analysis for the case d R= 2 is given in Appendix A. The Neumann-Poincaré operator (NP) associated with D is deﬁned KD∗ as follows:

1 x y,νx) ∗ [ϕ](x)= h − ϕ(y)dσ(y), x ∂D. KD 2π x y d ∈ Z∂D | − | It is related to the single layer potential by the following jump relation: SD

∂ D[ϕ] 1 1/2 S =( I + ∗ )[ϕ] for ϕ H− (∂D). (1.1) ∂ν ±2 KD ∈ ±

It can be shown that the operator λI : H 1/2(∂D) H 1/2(∂D) −KD∗ − → − is invertible for any λ > 1/2. Furthermore, is compact, its spectrum is | | KD∗ discrete and contained in ] 1/2, 1/2] with 0 being an accumulation point; − see for instance [18,32] for more details.

In general, D∗ is not symmetric for the pairing ( , ) 1 , 1 . Nevertheless, K · · 2 2 using Calderon’s identity −

= ∗ , KDSD SDKD can be symmetrized with the following inner product KD∗

(u,v) ∗ := ( D[v],u) 1 , 1 . H − S 2 − 2 3 It can be shown that in R , ( , ) ∗ deﬁnes a Hilbert space, equivalent to 1/2 2· · H H− (∂D) [12,32,61,65]. In R a similar analysis can be done to symmetrize . We refer the reader to Appendix A. KD∗ Let (λj,ϕj), j = 0, 1, 2,... be the eigenvalue and normalized eigenfunction pair of in (∂D). From the spectral theorem, we know that λ 0 KD∗ H∗ j → for j and ϕ form a base of (∂D). Therefore, the following repre- → ∞ j H∗ sentation formula holds: for any ϕ H 1/2(∂D), ∈ −

∞ D∗ [ϕ]= λj(ϕ,ϕj) ∗ ϕj. K H ⊗ Xj=0 12 Chapter 1. The Quasi-Static Limit

From the jump formula (1.1), we can see that 1/2 is always an eigenvalue of . If the D is simply connected, there is only one eigenvalue taking the KD∗ value 1/2. We denote this eigenvalue by λ0 and its corresponding eigenfunction ϕ0. 3 1 In R , let (∂D) be the space H 2 (∂D) equipped with the following H equivalent inner product

1 (u,v) = (( D)− [u],v) 1 , 1 . (1.2) H −S − 2 2 Then, is an isometry between (∂D) and (∂D). SD H∗ H A smilar result can be found in R2, see Appendix A.

1.3 Layer potential formulation for the scattering problem

We consider the scattering problem of a time-harmonic wave ui incident on a plasmonic nanoparticle. The homogeneous medium is characterized by its electric permittivity εm that we assume to be real and strictly positive. The particle occupying a bounded and simply connected domain D ⋐ Rd of class 1,α for some 0 <α< 1 is characterized by electric isotropic permittivity ε C c which may depend on the frequency of the incoming wave ω by the Drude model as 2 ωp εc = εc(ω)= 1 ε0. − ω(ω + iγ)!

Here, ωp is called the plasmon frequency, γ the damping parameter and ε0 is the permittivity of the free space. Assume that ε < 0, ε > 0, and deﬁne ℜ c ℑ c ε = ε χ(Rd D)+ ε χ(D). D m \ c where χ denotes the characteristic function. When the wavelength of the incoming wave is much larger than the particle’s size, the following is a good approximation of the Maxwell equations.

ε u = 0 in Rd ∂D, ∇· D∇ \ u+ u = 0 on ∂D,  ∂u− − ∂u (1.3)  εc εm = 0 on ∂D,  ∂ν + − ∂ν  u ui = O( 1 − ), x . x d −1 − | | | | → ∞   1 Here u corresponds to the electric potential. For some ϕ H− 2 (∂D), the ∈ solution u can be written as

u(x)= ui + [ϕ]. SD From Lemma 1.2.1 we can see that only the transmission conditions ∂u ∂u εc + εm = 0 on ∂D, ∂ν − ∂ν −

1.3. Layer potential formulation for the scattering problem 13 need to be satisﬁed. This translates into

i ∂ D[ϕ] ∂ D[ϕ] ∂u εc S (x) + εm S (x) =(εm εc) on ∂D. ∂ν − ∂ν − − ∂ν

From the jump formula for the single layer potential , i.e SD ∂ D[ϕ] 1 S (x) = Id + D∗ [ϕ](x), x ∂D. ∂ν ± ± 2 K ∈ we have ∂ui λ ∗ [ϕ]= , (1.4) −KD ∂ν with ε + ε λ = m c . 2(ε ε ) m − c Finally

i i 1 ∂u u = u + (λ ∗ )− [ ] SD −KD ∂ν ∂ui i ∞ ( ∂ν ,ϕj) ∗ = u + H D[ϕj]. (1.5) λ λj S Xj=1 − Recall that λ are eigenvalues and they satisfy λ < 1/2. In j KD∗ | j| the plasmonic case, ε (ω) can take negative values. Then it holds that ℜ c λ(ω) < 1/2 and 0 ε (ω) 1. So, for a certain frequency ω , the value |ℜ | ≤ℑ c ≪ j of λ(ωj) can be very close to an eigenvalue λj of the NP operator. Then, in ∂ui (1.5), the mode D[ϕj] will be ampliﬁed provided that ( ,ϕj) ∗ is non- S ∂ν H zero. As a result, the scattered ﬁeld u ui will show a resonant behavior. − This phenomenon is called the plasmonic quasi-static resonance.

1.3.1 Contracted generalized polarization tensors Decomposition (1.5) of u together with 1 ui(x)= ∂αui(0)xα α! α Nd X∈ and

α i 1 α i β 1 ∂x β (u u )(x)= ∂ u (0) y (λI ∗ )− [ ](y)dσ(y) ∂ G(x, 0)(1.6) − α!β! −KD ∂ν α , β 1 Z∂D | |X| |≥ 14 Chapter 1. The Quasi-Static Limit as x + . Introduce the generalized polarization tensors [4]: | | → ∞ α β 1 ∂x d M (λ,D) := y (λI ∗ )− [ ](y) dσ(y), α,β N . αβ −KD ∂ν ∈ Z∂D They will be of great use in chapter 8. We call M := M for α = β = 1 αβ | | | | the ﬁrst-order polarization tensor. Suppose that D = z + δB, where B has size of order 1. Then, from (1.6) we have

Theorem 1.3.1. In the far ﬁeld

δd+1 us(x)= ui(x) G(x, 0)M(λ,D) ui(0) + O . −∇y ∇ dist(λ,σ( )) KD∗ For a positive integer m, let Pm(x) be the complex-valued polynomial

m m α m β Pm(x)=(x1 + ix2) := aα x + i bβ x . (1.7) α =m β =m | X| | X| iθ m m Using polar coordinates x = re , the above coeﬃcients aα and bβ can also be characterized by

m α m m β m aα x = r cos mθ, and bβ x = r sin mθ. (1.8) α =m β =m | X| | X| We introduce the contracted generalized polarization tensors to be the following linear combinations of generalized polarization tensors using the co- eﬃcients in (1.7):

cc m n cs m n Mmn = aα aβMαβ, Mmn = aα bβMαβ, α =m β =n α =m β =n | X| |X| | X| |X| sc m n ss m n Mmn = bα aβMαβ, Mmn = bα bβMαβ. α =m β =n α =m β =n | X| |X| | X| |X| It is clear that

cc 1 ∂ (Pm) M = (P )(λI ∗ )− [ ℜ ] dσ, mn ℜ n −KD ∂ν Z∂D cs 1 ∂ (Pm) M = (P )(λI ∗ )− [ ℜ ] dσ, mn ℑ n −KD ∂ν Z∂D sc 1 ∂ (Pm) M = (P )(λI ∗ )− [ ℑ ] dσ, mn ℜ n −KD ∂ν Z∂D ss 1 ∂ (Pm) M = (P )(λI ∗ )− [ ℑ ] dσ. mn ℑ n −KD ∂ν Z∂D We refer to [18] for further details As recently shown [11, 18], the contracted generalized polarization tensors can eﬃciently be used for domain classiﬁcation. They provide a natural tool for describing shapes. In imaging applications, they can be stably reconstructed from the data by solving a least-squares problem. They capture 1.4. Plasmonic resonances for the anisotropic problem 15 high-frequency shape oscillations as well as topology. High-frequency oscillations of the shape of a domain are only contained in its high-order contracted generalized polarization tensors.

1.4 Plasmonic resonances for the anisotropic problem

In this section, we consider the scattering problem of a time-harmonic wave ui, incident on a plasmonic anisotropic nanoparticle. The homogeneous medium is characterized by its electric permittivity εm, while the particle occupying a bounded and simply connected domain Ω ⋐ R3 of class 1,α C for 0 <α< 1 is characterized by electric anisotropic permittivity A. We consider A to be a positive-deﬁnite symmetric matrix. In the quasi-static regime, the problem can be modeled as follows:

3 εmIdχ(R Ω)¯ + Aχ(Ω) u = 0, ∇· \ ∇ (1.9) i 2 u u = O( x − ), x + , | − | | | | | → ∞ where χ denotes the characteristic function and ui is a harmonic function in R3. We are interested in ﬁnding the plasmonic resonances for problem (1.9). First, introduce the fundamental solution to the operator A in dimension ∇· ∇ three 1 GA(x)= −4π det(A) A x | ∗ | 1 p A A with A = √A− . From now on, we denote G (x,y) := G (x y). ∗ − The single-layer potential associated with A is

A 1 1 [ϕ]: H− 2 (∂Ω) H 2 (∂Ω) SΩ −→ ϕ A[ϕ](x)= GA(x,y)ϕ(y)dσ(y), x R3. 7−→ SΩ ∈ Z∂Ω We can represent the unique solution to (1.9) in the following form [18]:

ui + [ψ], x R3 Ω¯, u(x)= SΩ ∈ \ A[φ], x Ω, SΩ ∈ 1 2 where (ψ,φ) H− 2 (∂Ω) is the unique solution to the following system ∈ of integral equations on ∂Ω: [ψ] A[φ] = ui, SΩ −SΩ − i (1.10)  ∂ Ω[ψ] A ∂u  εm S ν A Ω [φ] = εm .  ∂ν + − · ∇S − ∂ν −

 A 1 1 Lemma 1.4.1. The operator : H− 2 (∂Ω) H 2 (∂Ω) is invertible. More- SΩ → over, we have the jump formula

A 1 A ν A = Id +( )∗ · ∇SΩ ±2 KΩ ±

16 Chapter 1. The Quasi-Static Limit with

A x y,ν(x) ( )∗[ϕ](x)= − ϕ(y)dσ(y). Ω 3 K ∂Ω −4π det(A) A (x y) Z | ∗ − | s sp Proof. Let A∗ (H (∂Ω),H (∂Ω)) be such that A∗ [ϕ](x) = ϕ(A x) for s T ∈ L s T s ∗ ϕ H (∂Ω) and Ω = A Ω. Let rν (H (∂Ω),H (∂Ω)) be such that ∈ 1 ∗ ∈ L rν[ϕ](x) = A− ν(x) ϕ(x)e. It follows by the change of variables y = A y | ∗ | 1 ∗ that dσ(y)=e det√Ae A− ν(y) dσ(y). Thus, ∗| ∗ | A 1 1 e = A e − r− , e SΩ T ∗ SΩTA∗ ν A A 1 1 1 and in particular is invertible and its inverse ( )− = rν A e− − . SΩ SΩ T ∗ SΩ TA∗ Note that, for x ∂Ω, ∈ 1 A− ν(x) ν(x)= ∗ 1 , A− ν(x) | ∗ | where ν(x) is the outwarde normale to ∂Ω at x = A x. We have ∗ A 1 1 e e ν A Ω = ν A x e A∗ ee − rν− · ∇S · ∇ T SΩTA∗ ± ± 1 1 = ν AA A∗ xe e − rν− · ∗ T ∇ SΩTA∗ ± 1 1 1 = A ν ν e e r − A∗ x Ω A−∗ ν− | ∗ | · T ∇ S T ± 1 1 = Id +(r ) ∗e (r )− . (1.11) ±2 e νTA∗ KΩ νTA∗ The result follows from a change of variables in the expression of the operator

A 1 ( )∗ := (r ) ∗e (r )− . KΩ νTA∗ KΩ νTA∗

A Lemma 1.4.2. Ω is negative deﬁnite for the duality pairing ( , ) 1 , 1 and S · · 2 2 we can deﬁne a new inner product −

A (u,v) ∗ = (u, Ω [v]) 1 , 1 HA − S − 2 2

1 with which A∗ (∂Ω), the space induced by ( , ) ∗ , is equivalent to H− 2 (∂Ω). H · · HA 1 Proof. Let ϕ H− 2 (∂Ω). Using Lemma 1.4.1, we have ∈ A A ϕ = ν A Ω [ϕ] ν A Ω [ϕ] . · ∇S + − · ∇S −

1.4. Plasmonic resonances for the anisotropic problem 17

Thus

ϕ(x) A[ϕ](x)dσ(x) SΩ Z∂Ω A A A A = ν A Ω [ϕ] (x) Ω [ϕ](x)dσ(x) ν A Ω [ϕ] (x) Ω [ϕ](x)dσ(x) ∂Ω · ∇S + S − ∂Ω · ∇S S Z Z − A A A A = Ω [ϕ](x) A Ω [ϕ](x)dσ(x) Ω [ϕ](x) A Ω [ϕ](x)dσ(x) − R3 Ω¯ ∇S · ∇S − R3 Ω¯ S ∇· ∇S Z \ Z \ A[ϕ](x) A A[ϕ](x)dσ(x)+ A[ϕ](x) A A[ϕ](x)dσ(x) − ∇SΩ · ∇SΩ SΩ ∇· ∇SΩ ZΩ ZΩ A A = Ω [ϕ](x) A Ω [ϕ](x)dσ(x) 0, − R3 ∇S · ∇S ≤ Z where the equality is achieved if and only if ϕ = 0. Here we have used an integration by parts, the fact that A[ϕ](x) = O( x 1) as x , SΩ | |− | | → ∞ A A[ϕ](x) = 0 for x R3 ∂Ω and that A is positive-deﬁnite. ∇· ∇SΩ ∈ \ In the same manner, it is known that

2 2 ϕ ∗ = ϕ(x) Ω[ϕ](x)dσ(x)= Ω[ϕ](x) dσ(x). k kH − S R3 |∇S | Z∂Ω Z Since A is positive-deﬁnite we have

2 A 2 c ϕ ∗ ϕ(x) Ω [ϕ](x)dσ(x) C ϕ ∗ k kH ≤− S ≤ k kH Z∂Ω for some constants c,C > 0. 1 Using the fact that (∂Ω) is equivalent to H− 2 (∂Ω), we get the desired H∗ result.

From (1.10) we have φ =( A) 1( [ψ]+ ui), whereas, by Lemma 1.4.1, SΩ − SΩ the following equation holds for ψ:

[ψ]= F (1.12) QA with

1 A 1 A A 1 = ε Id +( )− + ε ∗ ( )∗( )− , (1.13) QA 2 m SΩ SΩ mKΩ − KΩ SΩ SΩ and

i ∂u A A 1 i F = ε + ν A [( )− u ] . − m ∂ν · ∇SΩ SΩ −

Propsition 1.4.1. A has a countable number of eigenvalues. Q A 1 1 Proof. It is clear that ( Ω)∗ : H− 2 (∂Ω) H− 2 (∂Ω) is a compact operator. A K A 1 → Hence, εm Ω∗ ( Ω)∗( Ω )− Ω is compact as well. Therefore, only the K −1 K S A S1 invertibility of 2 εmId +( Ω )− Ω needs to be proven. A S S 1 A 1 Since Ω is invertible, the invertibility of 2 εmId +( Ω )− Ω is equivalent S A S S to that of εm Ω + Ω. S S 1 2 Consider now, the bilinear form, for (ϕ,ψ) (H− 2 (∂Ω)) ∈ B(ϕ,ψ)= ε ϕ(x) A[ψ](x)dσ(x) ϕ(x) [ψ](x)dσ(x), − m SΩ − SΩ Z∂Ω Z∂Ω 18 Chapter 1. The Quasi-Static Limit

where εm > 0. From Lemma 1.4.2, we have

2 B(ψ,ψ) c ψ 1 ≥ k kH− 2 (∂Ω) for some constant c> 0. A It follows then, from the Lax-Milgram theorem that εm Ω + Ω is invertible 1 S S in H− 2 (∂Ω), whence the result.

Recall that the electromagnetic parameter of the problem, A, depends on the frequency, ω of the incident ﬁeld. Therefore the operator is frequency QA dependent and we should write (ω). QA We say that ω is a plasmonic resonance if

eig ( (ω)) 1 and is locally minimal for some j N, | j QA | ≪ ∈ where eig ( (ω)) stands for the j-th eigenvalue of (ω). j QA QA Equivalently, we can say that ω is a plamonic resonance if

1 ω = arg max A− (ω) ( ∗(∂Ω)). (1.14) ω kQ kL H From now on, we suppose that A is an anisotropic perturbation of an isotropic parameter, i.e., A = εc(Id + P ), with P being a symmetric matrix and P 1. k k ≪ Lemma 1.4.3. Let A = εc(Id + δR), with R being a symmetric matrix, R = O(1) and δ 1. Let Tr denote the trace of a matrix. Then, as k k ≪ δ 0, we have the following asymptotic expansions: → A 1 Ω = Ω + δ Ω,1 + o(δ) , S εc S S A 1 1 ( )− = ε − + δ + o(δ) , SΩ c SΩ BΩ,1 A ( )∗ = ∗ + δ ∗ + o(δ) KΩ KΩ KΩ,1 with

1 1 R(x y),x y Ω,1[ϕ](x) = Tr(R) Ω[ϕ](x) − 3− ϕ(y)dσ(y), S −2 S − 2 ∂Ω 4π x y 1 1 Z | − | = − − , BΩ,1 −SΩ SΩ,1SΩ 1 3 R(x y),x y x y,ν(x) ∗ = Tr(R) ∗ [ϕ](x) − − − ϕ(y)dσ(y). KΩ,1 −2 KΩ − 2 4π x y 5 Z∂Ω | −| Proof. Recall that, for δ small enough, δ (I + δR) 1 = Id R + O(δ2), − − 2 detp(I + δR)= = 1+ δTr(R)+ o(δ), (1 + δx + o(δ))s = 1+ δsx + o(δ), s R. ∈ 1 The results follow then from asymptotic expansions of , −4π det(A) A x β β = 1, 3 and the identity | ∗ | p A 1 1 1 1 ( )− = ε (Id + δ − + o(δ))− − . SΩ c SΩ SΩ,1 SΩ 1.5. A Maxwell-Garnett theory for plasmonic nanoparticles 19

Plugging the expressions above into the expression of we get the QA following result.

Lemma 1.4.4. As δ 0, the operator has the following asymptotic → QA expansion

= + δ + o(δ), QA QA,0 QA,1 where

εm + εc = Id +(ε ε ) ∗ , QA,0 2 m − c KΩ 1 = ε ( Id ∗ ) ∗ . QA,1 c 2 −KΩ BΩ,1SΩ −KΩ,1 We regard the operator as a perturbation of . As in section 3.3, QA QA,0 we use the standard perturbation theory to derive the perturbed eigenvalues and eigenvectors in (∂Ω). H∗ Let (λ ,ϕ ) be the eigenvalue and normalized eigenfunction pairs of j j KΩ∗ in (∂Ω) and τ the eigenvalues of . We have H∗ j QA,0 ε + ε τ = m c +(ε ε )λ . j 2 m − c j

For simplicity, we consider the case when λj is a simple eigenvalue of the operator . Deﬁne KΩ∗

Pj,l =( A,1[ϕj],ϕl) ∗ . Q H As δ 0, the perturbed eigenvalue and eigenfunction have the following → form:

τj(δ) = τj + δτj,1 + o(δ),

ϕj(δ) = ϕj + δϕj,1 + o(δ), where

τj,1 = Pjj, Pjl ϕj,1 = ϕl. ε ε (λ λ ) l=j m c j l X6 − − 1.5 A Maxwell-Garnett theory for plasmonic nanoparticles

In this subsection we derive eﬀective properties of a system of plasmonic nanoparticles. To begin with, we consider a bounded and simply connected domain Ω ⋐ R3 of class 1,α for 0 <α< 1, ﬁlled with a composite material C that consists of a matrix of constant electric permittivity εm and a set of periodically distributed plasmonic nanoparticles with (small) period η and electric permittivity εc. Let Y =] 1/2, 1/2[3 be the unit cell and denote δ = ηβ for β > 0. We set − 20 Chapter 1. The Quasi-Static Limit the (re-scaled) periodic function

γ = ε χ(Y D¯)+ ε χ(D), m \ c where D = δB with B ⋐ R3 being of class 1,α and the volume of B, B , is C | | assumed to be equal to 1. Thus, the electric permittivity of the composite is given by the periodic function

γη(x)= γ(x/η), which has period η. Now, consider the problem

γ u = 0 in Ω, (1.15) ∇· η∇ η with an appropriate boundary condition on ∂Ω. Then, there exists a homogeneous, generally anisotropic, permittivity γ∗, such that the replacement, as η 0, of the original equation (1.15) by →

γ∗ u = 0 in Ω ∇· ∇ 0 is a valid approximation in a certain sense. The coefficient γ∗ is called an effective permittivity. It represents the overall macroscopic material property of the periodic composite made of plasmonic nanoparticles embedded in an isotropic matrix. The (effective) matrix γ∗ =(γpq∗ )p,q=1,2,3 is defined by [18]

γ∗ = γ(x) u (x) u (x)dx, pq ∇ p ·∇ q ZY where up, for p = 1, 2, 3, is the unique solution to the cell problem

γ u = 0 in Y, ∇· ∇ p  up xp periodic (in each direction) with period 1, (1.16)  −  Y up(x)dx = 0.  Using Green’s R formula, we can rewrite γ∗ in the following form:

∂up γ∗ = ε u (x) (x)dσ(x). (1.17) pq m q ∂ν Z∂Y The matrix γ∗ depends on η as a parameter and cannot be written explicitly. The following lemmas are from [18].

Lemma 1.5.1. For p = 1, 2, 3, problem (1.16) has a unique solution up of the form

1 u (x)= x + C + (λ Id ∗ )− [ν ](x) in Y, p p p SD♯ ε −KD♯ p 1.5. A Maxwell-Garnett theory for plasmonic nanoparticles 21

where Cp is a constant, νp is the p-component of the outward unit normal to ∂D, λε is deﬁned by (3.17), and

[ϕ](x) = G (x,y)ϕ(y)dσ(y), SD♯ ♯ Z∂D ∂G♯(x,y) ∗ [ϕ](x) = ϕ(y)dσ(y) KD♯ ∂ν(x) Z∂D with G♯(x,y) being the periodic Green function deﬁned by

ei2πn (x y) G (x,y)= · − . ♯ − 4π2 n 2 n Z3 0 ∈X\{ } | | Lemma 1.5.2. Let and be the operators deﬁned as in Lemma 1.5.1. SD♯ KD♯∗ Then the following trace formula holds on ∂D

1 ∂ D♯[ϕ] ( Id + ∗ )[ϕ]= S . ±2 KD♯ ∂ν ±

For the sake of simplicity, for p = 1, 2, 3, we set

1 φ (y)=(λ Id ∗ )− [ν ](y) for y in ∂D. (1.18) p ε −KD♯ p Thus, from Lemma 1.5.1, we get

∂ yp + D♯[φp](y) γ∗ = ε y + C + [φ ](y) S dσ(y). pq m q q SD♯ q ∂ν Z∂Y Because of the periodicity of [φ ], we get SD♯ p

∂ D♯[φp] γpq∗ = εm δpq + yq S (y)dσ(y) . (1.19) ∂Y ∂ν Z In view of the periodicity of [φ ], the divergence theorem applied on Y D¯ SD♯ p \ and Lemma 1.5.2 yields (see [18])

∂ [φ ] y SD♯ p (y)= y φ (y)dσ(y). q ∂ν q p Z∂Y Z∂D Let

ψ (y)= φ (δy) for y ∂B. p p ∈ Then, by (1.19), we obtain

γ∗ = εm(Id + fP ), (1.20) where f = D = δ3(= η3β) is the volume fraction of D and P =(P ) | | pq p,q=1,2,3 is given by

Ppq = yqψp(y)dσ(y). (1.21) Z∂B To proceed with the computation of P we will need the following Lemma [18]. 22 Chapter 1. The Quasi-Static Limit

Lemma 1.5.3. There exists a smooth function R(x) in the unit cell Y such that 1 G (x,y)= + R(x y). ♯ −4π x y − | − | Moreover, the Taylor expansion of R(x) at 0 is given by 1 R(x)= R(0) (x2 + x2 + x2)+ O( x 4). − 6 1 2 3 | | Now we can prove the main result of this section, which shows the validity of the Maxwell-Garnett theory uniformly with respect to the frequency under the assumptions that

3/5 3 1 1 f dist(λε(ω),σ( ∗ )) and (Id δ R− T0)− = O(1), (1.22) ≪ KB − λε(ω) 1 where R− and T0 are to be deﬁned and dist(λε(ω),σ( ∗ )) is the distance λε(ω) KD between λ (ω) and the spectrum of . ε KB∗ Theorem 1.5.1. Assume that (1.22) holds. Then we have

8/3 f 1 f γ∗ = ε Id + fM(Id M)− + O , (1.23) m − 3 dist(λ (ω),σ( ))2 ε KB∗ uniformly in ω. Here, M = M(λε(ω),B) is the polarization tensor (3.36) associated with B and λε(ω). Proof. In view of Lemma 1.5.3 and (1.18), we can write, for x ∂D, ∈ ∂R(x y) (λ (ω)Id ∗ )[φ ](x) − φ (y)dσ(y)= ν (x), ε −KD p − ∂ν(x) p p Z∂D which yields, for x ∂B, ∈

2 ∂R(δ(x y)) (λ (ω)Id ∗ )[ψ ](x) δ − ψ (y)dσ(y)= ν (x). ε −KB p − ∂ν(x) p p Z∂B By virtue of Lemma 1.5.3, we get δ R(δ(x y)) = (x y)+ O(δ3) ∇ − −3 − uniformly in x,y ∂B. Since ψ (y)dσ(y) = 0, we now have ∈ ∂B p (R δ3TR+ δ5T )[ψ ](x)= ν (x), λε(ω) − 0 1 p p and so

3 1 5 1 1 (Id δ R− T0 + δ R− T1)[ψp](x)= R− [νp](x), (1.24) − λε(ω) λε(ω) λε(ω) where

R [ψ ](x) = (λ (ω)Id ∗ )[ψ ](x), λε(ω) p ε −KB p ν(x) T [ψ ](x) = yψ (y)dσ(y), 0 p 3 · p Z∂B T1 ( ∗(∂B)) = O(1). k kL H 1.5. A Maxwell-Garnett theory for plasmonic nanoparticles 23

Since is a compact self-adjoint operator in (∂B) it follows that [50] KB∗ H∗ 1 c (λε(ω)Id B∗ )− ( ∗(∂B)) (1.25) k −K kL H ≤ dist(λ (ω),σ( )) ε KB∗ for a constant c. It is clear that T0 is a compact operator. From the fact that the imaginary 3 1 part of Rλ (ω) is nonzero, it follows that Id δ R− T0 is invertible. ε − λε(ω) Under the assumption that

3 1 1 (Id δ R− T0)− = O(1), − λε(ω) 5 δ dist(λ (ω),σ( ∗ )), ≪ ε KB we get from (1.24) and (1.25)

3 1 5 1 1 1 ψp(x) = (Id δ R− T0 + δ R− T1)− R− [νp](x), − λε(ω) λε(ω) λε(ω) 5 3 1 1 1 δ = (Id δ R− T )− R− [ν ](x)+ O . λε(ω) 0 λε(ω) p − dist(λε(ω),σ( B∗ )) K Therefore, we obtain the estimate for ψp 1 ψp = O . dist(λε(ω),σ( B∗ )) K Now, we multiply (1.24) by yq and integrate over ∂B. We can derive from the estimate of ψp that

f δ5 P (Id M)= M + O 2 , − 3 dist(λε(ω),σ( B∗ )) K and therefore,

5 f 1 δ P = M(Id + M)− + O 2 3 dist(λε(ω),σ( B∗ )) K with P being deﬁned by (1.21). Since f = δ3 and

δ3 M = O , dist(λε(ω),σ( B∗ )) K it follows from (1.20) that the Maxwell-Garnett formula (1.23) holds (uniformly in the frequency ω) under the assumption (1.22) on the volume fraction f.

Remark 1.5.1. As a corollary of Theorem 1.5.1, we see that in the case when fM = O(1), which is equivalent to the scale f = O dist(λ (ω),σ( )) , the ε KB∗ matrix fM(Id f M) 1 may have a negative-deﬁnite symmetric real part. − 3 − This implies that the eﬀective medium is plasmonic as well as anisotropic.

Remark 1.5.2. It is worth emphasizing that Theorem 1.5.1 does not only prove the validity of the Maxwell-Garnett theory but it can also be used together with the results in section 1.4 in order to derive the plasmonic resonances of the eﬀective medium made of a dilute system of arbitrary-shaped 24 Chapter 1. The Quasi-Static Limit plasmonic nanoparticles, following (1.14)

1 ω = arg max γ−∗ (ω) ( ∗(∂Ω)). ω kQ kL H

1.6 Concluding remarks

In this chapter we have analyzed the plasmonic resonance phenomena as- suming a quasi-static approximation, which is valid for particles considerably smaller than the wavelength of the incoming wave. We have presented a rigorous mathematical framework for its analysis, given beforehand the necessary mathematical tools, relying mainly in layer potential techniques. The plasmonic resonances depend strongly in the spectral properties of the Neumann-Poincaré operator associated with D. We remark that this KD∗ operator is scale invariant. This imply that the quasi-static model cannot explain changes in the resonances given by the scaling of nanoparticiles. This problem is analyzed in chapter 2 and 3 with the study of Helmholtz and Maxwell equations, respectively We have also studied the anisotropic quasi-static problem in terms of layer potentials and defined the plasmonic resonances for anisotropic nanoparticles. Formulas for a small anisotropic perturbation of resonances of the isotropic formulas have been derived. The Maxwell-Garnett theory provides a simple model for calculating the macroscopic optical properties of materials with a dilute inclusion of spherical nanoparticles [18]. In this chapter we have rigorously obtained effective properties of a periodic arrangement of arbitrary-shaped nanoparticles and derived a condition on the volume fraction of the nanoparticles that insures the validity of the Maxwell-Garnett theory for predicting the effective optical properties of systems of embedded in a dielectric host material at the plasmonic resonances. 25

Chapter 2

The Helmholtz Equation

Contents 2.1 Introduction ...... 26 2.2 Preliminaries ...... 26 2.2.1 Layer potentials for the Helmholtz equation . . . . 26 2.3 Layer potential formulation for the scattering problem ...... 28 2.3.1 Problem formulation and some basic results . . . . 28 2.3.2 First-order correction to plasmonic resonances and field behavior at the plasmonic resonances . . . . . 32 2.4 Scattering and absorption enhancements . . . . . 37 2.4.1 Far-field expansion ...... 37 2.4.2 Energy flow ...... 38 2.4.3 Extinction, absorption, and scattering cross-sections and the optical theorem ...... 39 2.4.4 The quasi-static limit ...... 41 2.4.5 An upper bound for the averaged extinction cross- section ...... 43 2.5 Link with the scattering coefficients ...... 48 2.5.1 The notion of scattering coefficients ...... 49 2.5.2 The leading-order term in the expansion of the scattering amplitude ...... 50 2.6 Concluding remarks ...... 53 26 Chapter 2. The Helmholtz Equation

2.1 Introduction

As seen in chapter 1, plasmon resonances in nanoparticles can be treated at the quasi-static limit as an eigenvalue problem for the Neumann-Poincaré integral operator. This leads to direct calculation of resonance values of permittivity and optimal design of nanoparticles that resonate at specified frequencies. At this limit, they are size-independent. However, as the particle size increases, they are determined from scattering and absorption blow up and become size-dependent. This was experimentally observed, for instance, in [49, 81, 89]. In this chapter, using the Helmholtz equation to model light propagation, we first prove that, as the particle size increases and crosses its critical value for dipolar approximation which is justified in [9], the plasmonic resonances become size-dependent. The resonance condition is determined from absorption and scattering blow up and depends on the shape, size and electromagnetic parameters of both the nanoparticle and the surrounding material. Then, we precisely quantify the scattering absorption enhancements in plasmonic nanoparticles. We derive new bounds on the enhancement factors given the volume and electromagnetic parameters of the nanoparticles. At the quasi-static limit, we prove that the averages over the orientation of scattering and extinction cross-sections of a randomly oriented nanoparticle are given in terms of the imaginary part of the polarization tensor. More- over, we show that the polarization tensor blows up at plasmonic resonances and derive bounds for the absorption and scattering cross-sections. We also prove the blow-up of the first-order scattering coefficients at plasmonic resonances. The concept of scattering coefficients was introduced in [20] for scalar wave propagation problems and in [21] for the full Maxwell equations, rendering a powerful and efficient tool for the classification of the nanoparticle shapes. Using such a concept, we have explained in [6] the experimental results reported in [35]. The chapter is organized as follows. In section 2.3 we introduce a layer potential formulation for plasmonic resonances and derive asymptotic formulas for the plasmonic resonances and the near- and far-fields in terms of the size. In section 7.2 we consider the case of multiple plasmonic nanoparticles. Section 6.3 is devoted to the study of the scattering and absorption enhancements. The scattering coefficients are simply the Fourier coefficients of the scattering amplitude [20, 21]. In section 6.4 we investigate the behavior of the scattering coefficients at the plasmonic resonances.

2.2 Preliminaries 2.2.1 Layer potentials for the Helmholtz equation Let G be the Green function for the Helmholtz operator ∆+ k2 satisfying the Sommerfeld radiation condition. The Sommerfeld radiation condition can be expressed in dimension d = 2, 3, as follows: ∂u (d+1)/2 ik u C x − ∂ x − m ≤ | |

| | as x + for some constant C independent of x. | | → ∞ 2.2. Preliminaries 27

In R3, G is given by

eik x y G(x,y,k)= | − | . −4π x y | − | The single-layer potential and the Neumann-Poincaré integral operator for the Helmholtz equation are deﬁned as follows

k [ϕ](x)= G(x,y,k)ϕ(y)dσ(y), x R3, SD ∈ Z∂D k ∂G(x,y,k) ( )∗[ϕ](x)= ϕ(y)dσ(y), x ∂D, KD ∂ν(x) ∈ Z∂D Let us recall some well known properties [12]:

k 1 1 1 2 (i) : H− 2 (∂D) H 2 (∂D),H (R ∂D) is bounded; SD → loc \ 2 k 2 1 (ii) (∆ + k ) [ϕ](x) = 0 for x R ∂D, ϕ H− 2 (∂D); SD ∈ \ ∈ k 1 1 (iii) ( ) : H− 2 (∂D) H− 2 (∂D) is compact; KD ∗ → k 1 (iv) [ϕ], ϕ H− 2 (∂D), satisfies the Sommerfeld radiation condition at SD ∈ infinity; ∂ k [ϕ] (v) SD =( 1 I +( k ) )[ϕ]. ∂ν ± 2 KD ∗ ± 1 We have that, for any ψ,φ H− 2 (∂D), ∈ i km R2 u + D [ψ], x D, u := S ∈ \ (2.1) ( kc [φ], x D, SD ∈ 2 R2 ¯ with km = ω√εmµm and kc = ω√εcµc, satisfies ∆u + kmu = 0 in D, 2 i \ ∆u + kc u = 0 in D and u u satisfies the Sommerfeld radiation condition. − 1 To satisfy the boundary transmission conditions, ψ,φ H− 2 (∂D) need ∈ to satisfy the following system of integral equations on ∂D

km [ψ] kc [φ] = ui, SD −SD − i (2.2)  1 1 km 1 1 kc 1 ∂u  ε 2 I +( D )∗ [ψ]+ ε 2 I ( D )∗ [φ] = .  m K c − K −εm ∂ν The following result shows the existence of such a representation [19].

Theorem 2.2.1. The operator

1 2 1 1 : H− 2 (∂D) H 2 (∂D) H− 2 (∂D) T → × km kc 1 1 km 1 1 kc (ψ,φ) [ψ] [φ], I +( )∗ [ψ]+ I ( )∗ [φ] 7→ SD −SD ε 2 KD ε 2 − KD m c is invertible. 28 Chapter 2. The Helmholtz Equation

2.3 Layer potential formulation for the scattering problem 2.3.1 Problem formulation and some basic results We consider the scattering problem of a time-harmonic wave incident on a plasmonic nanoparticle. We use the Helmholtz equation instead of the full Maxwell equations. The homogeneous medium is characterized by its electric permittivity εm and its magnetic permeability µm, while the particle occupying a bounded and simply connected domain D ⋐ R3 (the two-dimensional case can be treated similarly using results from Appendix B.3) of class 1,α C for some 0 <α< 1 is characterized by electric permittivity εc and magnetic permeability µc, both of which may depend on the frequency. Assume that µ < 0, µ > 0, ε > 0, and deﬁne ℜ c ℑ c ℑ c

km = ω√εmµm, kc = ω√εcµc, and ε = ε χ(R3 D¯)+ ε χ(D¯), µ = ε χ(R3 D¯)+ ε χ(D), D m \ c D m \ c i ikmd x where χ denotes the characteristic function. Let u (x) = e · be the incident wave. Here, ω is the frequency and d is the unit incidence direction. Throughout this chapter, we assume that εm and µm are real and strictly positive and that k > 0. ℑ c Using dimensionless quantities, we assume that the particle D has size of order one and also the following condition holds.

Condition 2.1. We assume that the numbers εm,µm,εc,µc are dimensionless and are of order one. In addition, µ = o(1). We also assume that ω ℑ c is dimensionless and is of order o(1). It is worth emphasizing that in this section the variables ω refers to the ratio between the size of the particle and the incident wavelength. For real plasmonic nanoparticles made of noble metals such as silver and gold, their electric permittivity is only negative over a small range of frequencies in the optical regime. This is also the frequency range in which Condition 2.1 holds and also plasmonic resonance occurs. For the frequencies that are beyond that range, especially those near the origin, we shall assume that εc and µc are constant there. This assumption avoids complicated discussion on the dispersive property of electromagnetic parameters in that regime, and enables us to focus on the interesting frequency range when plasmonic resonance occurs. We also note that ω = o(1) implies that the plamsmonic nanoparticles have size much smaller than the incident wavelength. This is the case when plamsonic resonance occurs. The scattering problem can be modeled by the following Helmholtz equation

1 2 3 u + ω εDu = 0 in R ∂D, ∇· µD ∇ \  u u = 0 on ∂D,  +  − − (2.3)  1 ∂u 1 ∂u  = 0 on ∂D,  µ ∂ν − µ ∂ν m + c  −  us := u ui satisﬁes the Sommerfeld radiation condition.  −   2.3. Layer potential formulation for the scattering problem 29

Here, ∂/∂ν denotes the normal derivative and the Sommerfeld radiation condition. The model problem (2.3) is referred to as the transverse magnetic case. Note that all the results of this chapter hold true in the transverse electric case where εD and µD are interchanged. Let

i ikmd x F (x) = u (x)= e · , 1 − − i 1 ∂u i ikmd x F2(x) = (x)= kme · d ν(x) −µm ∂ν −µm · with ν(x) being the outward normal at x ∂D. ∈ By using the following single-layer potential and Neumann-Poincaré integral operator of section 2.2 we can represent the solution u in the following form ui + km [ψ], x R3 D,¯ u(x)= SD ∈ \ (2.4) kc [φ], x D, SD ∈ 1 where ψ,φ H− 2 (∂D) satisfy the following system of integral equations on ∈ ∂D [12]:

km kc D [ψ] D [φ] = F1, S −S (2.5) 1 1 km 1 1 kc ( Id +( )∗ [ψ]+ Id ( )∗ [φ] = F2, µm 2 KD µc 2 − KD where Id denotes the identity operator. In the sequel, we set 0 = . SD SD We are interested in the scattering in the quasi-static regime, i.e., for ω 1. Note that for ω small enough, kc is invertible [12]. We have ≪ SD φ =( kc ) 1 km [ψ] F , whereas the following equation holds for ψ SD − SD − 1 (ω)[ψ]= f, (2.6) AD where

1 1 km 1 1 kc kc 1 km D(ω) = Id +( D )∗ + Id ( D )∗ ( D )− D , (2.7) A µm 2 K µc 2 − K S S

1 1 kc kc 1 f = F2 + Id ( D )∗ ( D )− [F1]. (2.8) µc 2 − K S It is clear that 1 1 1 1 1 1 1 1 D(0) = D,0 = Id+ D∗ + Id D∗ = + Id D∗ , A A µm 2 K µc 2 −K 2µm 2µc − µc −µm K (2.9) where the notation =( 0 ) is used for simplicity. KD∗ KD ∗ We are interested in ﬁnding (ω) 1. We ﬁrst recall some basic facts AD − about the Neumann-Poincaré operator stated in chaper 1. See also [12, KD∗ 32,61,65]. Lemma 2.3.1. (i) The following Calderón identity holds: = ; KDSD SDKD∗ 1 (ii) The operator is self-adjoint in the Hilbert space H− 2 (∂D) equipped KD∗ with the following inner product

(u,v) ∗ = (u, D[v]) 1 , 1 (2.10) H − S − 2 2 30 Chapter 2. The Helmholtz Equation

1 1 2 2 with ( , ) 1 , 1 being the duality pairing between H− (∂D) and H (∂D), · · 2 2 which is− equivalent to the original one;

1 (iii) Let (∂D) be the space H− 2 (∂D) with the new inner product. Let H∗ (λj,ϕj), j = 0, 1, 2,... be the eigenvalue and normalized eigenfunction pair of in (∂D), then λ ( 1 , 1 ] and λ 0 as j ; KD∗ H∗ j ∈ − 2 2 j → → ∞ (iv) The following trace formula holds: for any ψ (∂D), ∈H∗

1 ∂ D[ψ] ( Id + ∗ )[ψ]= S ; −2 KD ∂ν −

(v) The following representation formula holds: for any ψ H 1/2(∂D), ∈ −

∞ D∗ [ψ]= λj(ψ,ϕj) ∗ ϕj. K H ⊗ Xj=0

It is clear that the following result holds.

1 Lemma 2.3.2. Let (∂D) be the space H 2 (∂D) equipped with the following H equivalent inner product

1 (u,v) = (( D)− [u],v) 1 , 1 . (2.11) H −S − 2 2 Then, is an isometry between (∂D) and (∂D). SD H∗ H We now present other useful observations and basic results. The following holds.

1 1 Lemma 2.3.3. (i) We have ( Id + ) − [1] = 0. − 2 KD∗ SD 1 (ii) Let λ0 = 2 . Then the corresponding eigenspace has dimension one and 1 is spanned by the function ϕ = c − [1] for some constant c such that 0 SD ϕ0 ∗ = 1. || ||H (iii) Moreover, (∂D) = (∂D) µϕ , µ C , where (∂D) is the H∗ H0∗ ⊕ { 0 ∈ } H0∗ zero mean subspace of (∂D) and ϕ (∂D) for j 1, i.e., H∗ j ∈ H0∗ ≥ (ϕj, 1) 1 , 1 = 0 for j 1. Here, ϕj j is the set of normalized eigen- 2 2 ≥ { } functions− of . KD∗ From (2.9), it is easy to see that

∞ D,0[ψ]= τj(ψ,ϕj) ∗ ϕj, (2.12) A H Xj=0 where 1 1 1 1 τj = + λj. (2.13) 2µm 2µc − µc − µm From (2.9), it is easy to see that

∞ D,0[ψ]= τj(ψ,ϕj) ∗ ϕj, (2.14) A H Xj=0 2.3. Layer potential formulation for the scattering problem 31 where 1 1 1 1 τj = + λj. (2.15) 2µm 2µc − µc − µm We now derive the asymptotic expansion of the operator (ω) as ω 0. A → Using the asymptotic expansions in terms of k of the operators k , ( k ) 1 SD SD − and ( k ) proved in Appendix B.1, we can obtain the following result. KD ∗ Lemma 2.3.4. As ω 0, the operator (ω): (∂D) (∂D) admits → AD H∗ →H∗ the asymptotic expansion

(ω)= + ω2 + O(ω3), AD AD,0 AD,2 where

εmµm εcµc 1 1 D,2 =(εm εc) D,2 + − ( Id D∗ ) D− D,2. (2.16) A − K µc 2 −K S S Proof. Recall that

1 1 km 1 1 kc kc 1 km D(ω)= Id +( D )∗ + Id ( D )∗ ( D )− D . (2.17) A µm 2 K µc 2 − K S S By a straightforward calculation, it follows that

kc 1 km 1 ( )− = Id + ω √ε µ + √ε µ − + SD SD c cBD,1SD m mSD SD,1 2 1 3 ω ε µ + √ε µ ε µ + ε µ − + O(ω ), c cBD,2SD c c m mBD,1SD,1 m mSD SD,2 1 = Id + ω √ε µ √ε µ − + m m − c c SD SD,1 2 1 1 1 ω (ε µ ε µ ) − + √ε µ (√ε µ √ε µ ) − − m m − c c SD SD,2 c c c c − m m SD SD,1SD SD,1 +O(ω3), where and are deﬁned by (B.5). Using the facts that BD,1 BD,2

1 1 Id ∗ − = 0 2 −KD SD SD,1 and 1 k 1 2 3 Id ( )∗ = Id ∗ k + O(k ), 2 − KD 2 −KD − KD,2 the lemma immediately follows.

We regard (ω) as a perturbation to the operator for small ω. AD AD,0 Using standard perturbation theory [85], we can derive the perturbed eigenvalues and their associated eigenfunctions. For simplicity, we consider the case when λ is a simple eigenvalue of the operator . j KD∗ We let

Rjl = D,2[ϕj],ϕl ∗ , (2.18) A H where is deﬁned by (2.16). AD,2 As ω goes to zero, the perturbed eigenvalue and eigenfunction have the following form:

2 3 τj(ω) = τj + ω τj,2 + O(ω ), (2.19) 2 3 ϕj(ω) = ϕj + ω ϕj,2 + O(ω ), (2.20) 32 Chapter 2. The Helmholtz Equation where

τj,2 = Rjj, (2.21) Rjl ϕj,2 = 1 1 ϕl. (2.22) (λj λ ) l=j µm µc l X6 − − 2.3.2 First-order correction to plasmonic resonances and field behavior at the plasmonic resonances We first introduce different notions of plasmonic resonance as follows. Definition 2.1. (i) We say that ω is a plasmonic resonance if

τ (ω) 1 and is locally minimal for some j. | j | ≪ (ii) We say that ω is a quasi-static plasmonic resonance if τ 1 and is | j| ≪ locally minimized for some j. Here, τj is defined by (2.15). (iii) We say that ω is a first-order corrected quasi-static plasmonic resonance if τ + ω2τ 1 and is locally minimized for some j. Here, the | j j,2| ≪ correction term τj,2 is defined by (2.21). Note that quasi-static resonances are size independent and is therefore a zero-order approximation of the plasmonic resonance in terms of the particle size while the first-order corrected quasi-static plasmonic resonance depends on the size of the nanoparticle (or equivalently on ω in view of the non- dimensionalization adopted herein). We are interested in solving the equation (ω)[φ]= f when ω is close AD to the resonance frequencies, i.e., when τj(ω) is very small for some j’s. In this case, the major part of the solution would be the contributions of the excited resonance modes ϕj(ω). We introduce the following definition. Definition 2.2. We call J N index set of resonance if τ ’s are close to ⊂ j zero when j J and are bounded from below when j J c. More precisely, ∈ ∈ we choose a threshold number η0 > 0 independent of ω such that

τ η > 0 for j J c. | j|≥ 0 ∈

Remark 2.3.1. Note that for j = 0, we have τ0 = 1/µm, which is of size one by our assumption. As a result, throughout this chapter, we always exclude 0 from the index set of resonance J. From now on, we shall use J as our index set of resonances. We assume throughout that the following conditions hold. Condition 2.2. Each eigenvalue λ for j J is a simple eigenvalue of the j ∈ operator . KD∗ Condition 2.3. Let µ + µ λ = m c . (2.23) 2(µ µ ) m − c We assume that λ = 0 or equivalently, µ = µ . 6 c 6 − m Condition 2.3, which is crucial to our analysis, implies that the set J is ﬁnite. Otherwise, inﬁnity resonance modes may be excited and the problem becomes unstable. We refer to [44,47,79] for detailed discussion on this case. 2.3. Layer potential formulation for the scattering problem 33

Remark 2.3.2. Note that in the ideal case when µ = 0, we know that ℑ c τj = 0 if λ defined in (2.23) is equal to λj. This the usual definition in the quasi-static limiting case when the wavelength is infinite. In the case µ = 0 ℑ c 6 but µ = o(1), one may neglect the imaginary part and still use the definition ℑ c to find the resonance frequency. The draw back of this definition is that the resonance frequency is independent of the size of the particle. Now, with the asymptotic expansion (8.11), we may find ω, the resonance frequency, according to the criterion in Definition 3.3 (i) in a small neighborhood of the resonant frequency of the quasi-static limiting case. The difference of the two frequency yields the shift of resonance frequency with respect to size of the particle.

We now deﬁne the projection PJ (ω) such that

ϕj(ω), j J, P (ω)[ϕj(ω)] = ∈ J 0, j J c. ∈ In fact, we have

1 1 P (ω)= P (ω)= (ξ (ω))− dξ, (2.24) J j 2πi −AD j J j J γj X∈ X∈ Z where γj is a Jordan curve in the complex plane enclosing only the eigenvalue τj(ω) among all the eigenvalues. To obtain an explicit representation of PJ (ω), we consider the adjoint operator (ω) . By a similar perturbation argument, we can obtain its AD ∗ perturbed eigenvalue and eigenfunction, which have the following form

τj(ω) = τj(ω), (2.25) 2 2 ϕj(ω) = ϕj + ω ϕj,2 + o(ω ). (2.26) e Using the eigenfunctionsf ϕj(ω), we can showe that

PJ (ωe)[x]= x, ϕj(ω) ∗ ϕj(ω). (2.27) j J H X∈ e Throughout this chapter, for two Banach spaces X and Y , by (X,Y ) we L denote the set of bounded linear operators from X into Y . We are now ready to solve the equation (ω)[ψ]= f. First, it is clear AD that

1 f, ϕj(ω) ∗ 1 ψ = D(ω)− [f]= H + D(ω)− [PJc (ω)[f]]. (2.28) A τj(ω) A j J X∈ f The following lemma holds.

1 Lemma 2.3.5. The norm D(ω)− PJc (ω) ( ∗(∂D), ∗(∂D)) is uniformly kA kL H H bounded in ω for ω suﬃciently small.

Proof. Consider the operator

(ω) c : P c (ω) ∗(∂D) P c (ω) ∗(∂D). AD |J J H → J H 34 Chapter 2. The Helmholtz Equation

η0 For ω small enough, we can show that dist(σ( (ω) c ), 0) , where AD |J ≥ 2 σ( (ω) c ) is the discrete spectrum of (ω) c . Then, it follows that AD |J AD |J

1 1 1 C1 c − c c D(ω)− (PJ (ω)f) = D(ω) PJc (PJ (ω)f) . exp( 2 ) PJ (ω)f , kA k k A | k η0 η0 k k where the notation A . B means that A CB for some constant C. ≤ On the other hand,

PJ (ω)f = f, ϕj(ω) ∗ ϕj(ω)= f,ϕj + O(ω) ∗ ϕj + O(ω) j J H j J H X∈ X∈ = f,ϕfj ∗ ϕj(ω)+ O(ω). j J H X∈ Thus, P c (ω) = (Id P (ω)) . (1 + O(ω)), k J k k − J k from which the desired result follows immediately.

Second, we have the following asymptotic expansion of f given by (2.8) with respect to ω.

Lemma 2.3.6. Let

ikmd z 1 1 1 1 f = i√ε µ e · [d ν(x)] + Id ∗ − [d (x z)] 1 − m m µ · µ 2 −KD SD · − m c and let z be the center of the domain D. In the space (∂D), as ω goes to H∗ zero, we have 2 f = ωf1 + O(ω ), in the sense that, for ω small enough,

2 f ωf1 ∗ Cω k − kH ≤ for some constant C independent of ω.

Proof. A direct calculation yields

1 1 kc kc 1 f = F2 + Id ( D )∗ ( D )− [F1] µc 2 − K S

i ikmd z 2 = ω √εmµme · [d ν(x)] + O(ω )+ − µm ·

1 1 1 2 ikmd z Id D∗ ( D)− + ω D,1 + O(ω ) [ e · 1+ iω√εmµm[d (x z)] µc 2 −K S B − · − +O(ω2)]

ikmd z ikmd z e · 1 1 ωe · 1 = Id D∗ D− [1] Id D∗ D,1[1] − µc 2 −K S − µc 2 −K B − ikmd z 1 1 1 1 2 ωi√ε µ e · [d ν(x)] + Id ∗ − [d (x z)] + O(ω ) m m µ · µ 2 −KD SD · − m c ikmd z 1 1 1 1 = ωi√ε µ e · [d ν(x)] + Id ∗ − [d (x z)] − m m µ · µ 2 −KD SD · − m c +O(ω2), 2.3. Layer potential formulation for the scattering problem 35 where we have made use of the facts that

1 1 Id ∗ − [1] = 0 2 −KD SD and 1 [χ(∂D)] = c − [χ(∂D)] BD,1 SD for some constant c; see again Appendix B.1.

Finally, we are ready to state our main result in this section.

Theorem 2.3.1. Let D has size of order one. Under Conditions 2.1, 2.2, and 2.3 the scattered ﬁeld us = u ui due to a single plasmonic particle D − has the following representation:

us = km [ψ], SD where

ω f1, ϕj(ω) ∗ ϕj(ω) ψ = H + O(ω), τj(ω) j J X∈ e ik eikmd z d ν(x),ϕ ϕ + O(ω2) m · · j ∗ j = H2 + O(ω) λ λj + O(ω ) j J X∈ − with λ being given by (2.23).

Proof. We have

f, ϕj(ω) ∗ ϕj(ω) 1 ψ = H + D(ω)− (PJc (ω)f), τj(ω) A j J X∈ e 2 ω f1,ϕj ∗ ϕj + O(ω ) = + O(ω). 1 1 1H 1 2 + λj + O(ω ) j J 2µm 2µc µc µm X∈ − − We now compute f1,ϕj ∗ with f1 given in Lemma 2.3.6. We only need to show that H 1 1 Id D∗ D− [d (x z)] ,ϕj =(d ν(x),ϕj) ∗ . (2.29) 2 −K S · − ∗ · H H 36 Chapter 2. The Helmholtz Equation

Indeed, we have

1 1 1 1 ( Id ∗ ) − [d (x z)],ϕj = − [d (x z)], Id D D[ϕj] D D ∗ D 1 1 2 −K S · − − S · − 2 −K S 2 , 2 H − 1 1 = D− [d (x z)], D Id D∗ [ϕj] − S · − S 2 −K 1 , 1 − 2 2 1 = d (x z), Id D∗ [ϕj] − · − 2 −K 1 , 1 − 2 2 ∂ [ϕ ] = d (x z), D j S 1 1 − · − − ∂ν 2 , 2 −− ∂[d (x z)] = · − [ϕ ]dσ ∂ν SD j Z∂D ∆[d (x z)] D[ϕj] ∆ D[ϕj][d (x z)] dx − D · − S − S · − Z = d ν(x),ϕj , − · ∗ H where we have used the fact that [ϕ ] is harmonic in D. This proves the SD j desired identity and the rest of the theorem follows immediately.

Corollary 2.3.1. Assume the same conditions as in Theorem 3.3.2. Under the additional condition that

3 min τj(ω) ω , (2.30) j J | | ≫ ∈ we have

ik eikmd z d ν(x),ϕ ϕ + O(ω2) m · · j ∗ j ψ = H 1 + O(ω). 2 1 1 − j J λ λj+ ω µ µ τj,2 X∈ − c − m More generally, under the additional condition that

m+1 min τj(ω) ω , j J ≫ ∈ for some integer m> 2, we have

ik eikmd z d ν(x),ϕ ϕ + O(ω2) m · · j ∗ j ψ = 1 H 1 + O(ω). 2 1 1 − m 1 1 − j J λ λj + ω µ µ τj,2 + + ω µ µ τj,m X∈ − c − m ··· c − m Re-scaling back to original dimensional variables, we suppose that the magnetic permeability µc of the nanoparticle is changing with respect to the operating angular frequency ω while that of the surrounding medium, µm, is independent of ω. Then we can write

µc(ω)= µ′(ω)+ iµ′′(ω). (2.31) 2.4. Scattering and absorption enhancements 37

Because of causality, the real and imaginary parts of µc obey the following Kramer–Kronig relations:

+ 1 ∞ 1 µ′′(ω)= p.v. µ′(s)ds, −π ω s Z−∞ + − (2.32) 1 ∞ 1 µ′(ω)= p.v. µ′′(s)ds, π ω s Z−∞ − where p.v. stands for the principle value. The magnetic permeability µc(ω) can be described by the Drude model; see, for instance, [86]. We have

ω2 µ (ω)= µ (1 F ), (2.33) c 0 − ω2 ω2 + iτ 1ω − 0 − 1 where τ > 0 is the nanoparticle’s bulk electron relaxation rate (τ − is the damping coeﬃcient), F is a ﬁlling factor, and ω0 is a localized plasmon resonant frequency. When

2 2 2 2 2 2 2 2 (1 F )(ω ω ) Fω (ω ω )+ τ − ω < 0, − − 0 − 0 − 0 the real part of µc(ω) is negative. We suppose that D = z + δB. The quasi-static plasmonic resonance is deﬁned by ω such that

µ + µ (ω) m c = λ ℜ2(µ µ (ω)) j m − c for some j, where λj is an eigenvalue of the Neumann-Poincaré operator (= ). It is clear that such deﬁnition is independent of the nanoparti- KD∗ KB∗ cle’s size. In view of (8.11), the shifted plasmonic resonance is deﬁned by

1 1 1 1 argmin + λ + ω2δ2τ , 2µ 2µ (ω) − µ (ω) − µ j j,2 m c c m where τj,2 is given by (2.21) with D replaced by B.

2.4 Scattering and absorption enhancements

In this section we analyze the scattering and absorption enhancements. We prove that, at the quasi-static limit, the averages over the orientation of scattering and extinction cross-sections of a randomly oriented nanoparticle are given by (2.43) and (2.44), where M given by (2.40) is the polarization tensor associated with the nanoparticle D and the magnetic contrast µc(ω)/µm. In view of (2.48), the polarization tensor M blows up at the plasmonic resonances, which yields scattering and absorption enhancements. A bound on the extinction cross-section is derived in (2.50). As shown in (2.53) and (2.55), it can be sharpened for nanoparticles of elliptical or ellipsoidal shapes.

2.4.1 Far-ﬁeld expansion For simplicity, we assume throughout this section that D contains the origin. We ﬁrst prove the following representation for the scattering amplitude. 38 Chapter 2. The Helmholtz Equation

x x 1 ikm y A ,d = e− |x| · ψ(y)dσ(y) (2.35) ∞ x −4π | | Z∂D being the scattering amplitude and ψ being deﬁned by (2.5).

Proof. We recall that the scattered ﬁeld us can be represented as follows:

us(x) = km [ψ](x) SD 1 eikm x y = | − | ψ(y)dσ(y). −4π x y Z∂D | − | From x y 1 x y = x 1 · + O( ) , | − | | | − x 2 x 2 | | | | it follows that

2.4.2 Energy flow The following definitions are from [43]. We include them here for the sake of completeness. The analogous quantity of the Poynting vector in scalar wave theory is the energy flux vector; see [43]. We recall that for a real monochromatic field

iωt U(x,t)= u(x)e− , ℜ the averaged value of the energy ﬂux vector, taken over an interval which is long compared to the period of the oscillations, is given by

F (x)= iC [u(x) u(x) u(x) u(x)] , − ∇ − ∇ where C is a positive constant depending on the polarization mode. In the transverse electric case, C = ω/µm while in the transverse magnetic case C = ω/εm. Assume that the particle is contained in the ball BR of radius R and center the origin. We now consider the outward ﬂow of energy through the sphere ∂BR:

= F (x) ν(x)dσ(x), W · Z∂BR where ν(x) is the outward normal at x ∂B . ∈ R 2.4. Scattering and absorption enhancements 39

As the total ﬁeld can be written as U = us + ui, the ﬂow can be decomposed into three parts:

i s = + + ′, W W W W where

i = iC ui(x) ui(x) ui(x) ui(x) ν(x) dσ(x), W − ∂B ∇ − ∇ · Z R h i s = iC [us(x) us(x) us(x) us(x)] ν(x) dσ(x), W − ∇ − ∇ · Z∂BR i s s i i s s i ′ = iC u (x) u (x) u (x) u (x) u (x) u (x)+ u (x) u (x) ν(x) dσ(x). W − ∂B ∇ − ∇ − ∇ ∇ · Z R h i It is straightforward to check that , i, a, and in the above deﬁni- W W W W′ tions are independent of the radius R as long as the particle is contained in B . In the case where ui is a plane wave, we can see that i = 0: R W i = iC ui(x) ui(x) ui(x) ui(x) dσ(x), W − ∂B ∇ − ∇ Z R h i ikmd x ikmd x ikmd x ikmd x = iC e− · ikmde · + e · kmde− · ν(x) dσ(x), − ∂B · Z R h i = 2Ck d ν(x) dσ(x), m · Z∂BR = 0.

In a non absorbing medium with non absorbing scatterer, is equal to zero W because the electromagnetic energy would be conserved by the scattering process. However, if the scatterer is an absorbing body, the conservation of energy gives the rate of absorption as

a = . W −W Therefore, we have

a s + = ′. W W −W Here, is called the extinction rate. It is the rate at which the energy is W′ removed by the scatterer from the illuminating plane wave, and it is the sum of the rate of absorption and the rate at which energy is scattered.

2.4.3 Extinction, absorption, and scattering cross-sections and the optical theorem

Denote by U i the quantity U i(x) = ui(x) ui(x) ui(x) ui(x) . In the ∇ − ∇ case of a plane wave illumination, U i(x ) is independent of x and is given by i U = 2km. 40 Chapter 2. The Helmholtz Equation

Deﬁnition 2.3. The scattering cross-section Qs, the absorption cross-section Qa and the extinction cross-section are deﬁned by

s a Qs = W , Qa = W , Qext = −W′ . U i U i U i Note that these quantities are independent of x for a plane wave illumination.

i ikmd x Theorem 2.4.1 (Optical theorem). If u (x) = e · , where d is a unit direction, then 4π Qext =Qs + Qa = [A (d,d)] , (2.36) km ℑ ∞ Qs = A (ˆx,d) 2dσ(ˆx) (2.37) S2 | ∞ | Z with A being the scattering amplitude deﬁned by (2.35). ∞ Proof. The Sommerfeld radiation condition gives, for any x ∂B , ∈ R us(x) ν(x) ik us(x). (2.38) ∇ · ∼ m Hence, from (2.34) we get

2ik x 2 us(x) us(x) ν(x) us(x) us(x) ν(x) m A ,d , ∇ · − ∇ · ∼− x 2 ∞ x | | | | which yields (2.37). We now compute the extinction rate. We have

i ikmd x u (x) ν(x)= ik d ν(x)e · . (2.39) ∇ · m · Therefore, using 2.38 and 2.39, it follows that

ikm( x d x) ikm( x d x) i s s i e | |− · e | |− · x u (x) u (x) ν(x) u (x) u (x) ν(x) ikm d ν + ikm A ,d ∇ · − ∇ · ∼ x · x ∞ x | | | | | | ik eikm x d ν(x) x = m | |− · (d ν(x) + 1) A ,d . x · ∞ x | | | | For x ∂B , we can write ∈ R ik e ikmRν(x) (d ν(x)) x ui(x) us(x) ν(x) us(x) ui(x) ν(x) m − · − (d ν(x) + 1) A ,d . ∇ · − ∇ · ∼ R · ∞ x | | We now use Jones’ lemma (see, for instance, [43, Chapter 13.3]) to write the following asymptotic expansion as R → ∞

1 ikmd ν(x) 2πi ikmR ikmR (ν(x))e− · dσ(x) (d)e− ( d)e , R ∂B G ∼ km G −G − Z R to obtain

ui(x) us(x) us(x) ui(x) ν(x) 4πA (d,d) as R . ∂B ∇ − ∇ · ∼− ∞ → ∞ Z R h i 2.4. Scattering and absorption enhancements 41

Therefore,

′ = i4πC A (d) A (d) = 8πC [A (d)] . W − ∞ − ∞ ℑ ∞ Since

ui(x) ui(x) ui(x) ui(x) = 2k , ∇ − ∇ m

we get the result.

2.4.4 The quasi-static limit We start by recalling the small volume expansion for the far-ﬁeld. Let λ be deﬁned by (2.23) and let

1 M(λ,D) := (λId ∗ )− [ν]xdσ(x) (2.40) −KD Z∂D be the polarization tensor. The following asymptotic expansion holds. It can be proved by exactly the same arguments as those in [9].

Propsition 2.4.2. Assume that D = δB + z. As δ goes to zero the scattered ﬁeld us can be written as follows:

ε us(x) = k2 c 1 D G(x,z,k )ui(z) G(x,z,k ) M(λ,D) ui(z) − m ε − | | m −∇z m · ∇ m δ4 +O dist(λ,σ( )) D∗ K (2.41) for x away from D. Here, dist(λ,σ( )) denotes min λ λ with λ being KD∗ j | − j| j the eigenvalues of . KD∗ s We denote the ﬁrst term in the right hand side of (2.41) by u1 and the s s s second term by u2. It is clear that u1 represent monopole radiation and u2 the dipole radiation. We explicitly compute the scattering amplitude A in i ikmd x ∞ (2.34). Take u (x) = e · and assume again for simplicity that z = 0. Note that

eikm x ε us(x)= k2 | | c 1 D . 1 m 4π x ε − ·| | | | m Throughout the chapter, we are interested in the case when the frequency is near the plasmonic resonant frequency, then the polarization ten- s sor M(λ,D) blow up and hence the magnitude of the dipole part u2 is much 42 Chapter 2. The Helmholtz Equation

s greater than that of the monopole part u1. Therefore, the leading term in the scattered ﬁeld (2.41) is given by the dipole part, i.e.

eikm x x us(x) k2 | | M(λ,D)d . (2.42) ≈− m 4π x x · | | | | In the next proposition we write the extinction and scattering cross- sections in terms of the polarization tensor.

Propsition 2.4.3. Near plasmonic resonant frequency, the leading-order term (as δ goes to zero) of the average over the orientation of the extinction cross-section of a randomly oriented nanoparticle is given by 4πk Qext = m [TrM(λ,D)] , (2.43) m 3 ℑ where Tr denotes the trace of a matrix. The leading-order term of the average over the orientation scattering cross-section of a randomly oriented nanoparticle is given by

k4 Qs = m TrM(λ,D) 2 . (2.44) m 9π | | Proof. Remark from (2.42) that the scattering amplitude A in the case of ∞ a plane wave illumination is given by

x k2 x A ,d = m M(λ,D)d. (2.45) ∞ x − 4π x · | | | | Using Theorem 2.4.1, we can see that for a given orientation

Qext = 4πk [d M(λ,D)d] . − mℑ · Therefore, if we integrate Qext over all illuminations we ﬁnd that

ext Qm = km d M(λ,D)ddσ(d) . − ℑ S2 · Z Since M(λ,D) is symmetric, it can be written as M(λ,D) = P tN(λ)P ℑ ℑ where P is unitary and N is diagonal and real. Then, by the change of variables d = P tx and using spherical coordinates, it follows that

ext Qm = km x N(λ)xdσ(x) , − S2 · Z and therefore, 4πk 4πk Qext = m [TrN(λ)] = m [TrM(λ,D)] . (2.46) m − 3 − 3 ℑ 2.4. Scattering and absorption enhancements 43

Now, we compute the averaged scattering cross-section. Let M(λ,D) = ℜ P tN(λ)P where P is unitary and N is diagonal and real. We have

4 e se e km e e 2 Qm = 2 x M(λ,D)d dσ(x) dσ(d), 16π S2 S2 | · | ZZ × 4 2 2 km = 2 x N(λ)d dσ(x)dσ(d)+ x N(λ)d dσ(x) dσ(d) . 16π S2 S2 · S2 S2 · ZZ × ZZ ×

Then a straightforward computation e e in sphericale e coordinates e givese e e e

k4 Qs = m TrM(λ,D) 2 , m 9π | | which completes the proof.

From Theorem 2.4.1, we obtain that the averaged absorption cross-section is given by

4πk k4 Qa = m [TrM(λ,D)] m TrM(λ,D) 2 . m − 3 ℑ − 9π | | a Therefore, under the condition (2.30), Qm blows up at plasmonic resonances.

2.4.5 An upper bound for the averaged extinction cross-section The goal of this section is to derive an upper bound for the modulus of the ext averaged extinction cross-section Qm of a randomly oriented nanoparticle. Recall that the entries Ml,m(λ,D) of the polarization tensor M(λ,D) are given by 1 M (λ,D) := x (λI ∗ )− [ν ](x) dσ(x). (2.47) l,m l −KD m Z∂D For a 1,α domain D in Rd, is compact and self-adjoint in . Thus, we C KD∗ H∗ can write 1 ∞ (ψ,ϕj) ∗ ϕj (λId D∗ )− [ψ]= H ⊗ , −K λ λj Xj=0 − with (λ ,ϕ ) being the eigenvalues and eigenvectors of in (see Lemma j j KD∗ H∗ 2.3.1). Hence, the entries of the polarization tensor M can be decomposed as (j) ∞ αl,m Ml,m(λ,D)= , (2.48) λ λj Xj=1 − (j) where αl,m := (νm,ϕj) ∗ (ϕj,xl) 1 , 1 . Note that (νm,χ(∂D)) 1 , 1 = 0. So, H − 2 2 − 2 2 (0) considering the fact that λ0 = 1/2, we have (νm,ϕ0) ∗ = 0 and so, α = 0. H l,m The following lemmas are useful for us.

Lemma 2.4.1. We have

α(j) 0, j 1. l,l ≥ ≥ 44 Chapter 2. The Helmholtz Equation

Proof. For d = 3, we have

1 1 1 (ϕj,xl) 1 1 = λj − Id D∗ [ϕj],xl 2 , 2 1 1 − 2 − 2 −K 2 , 2 − 1 ∂ [ϕ ] = D j ,x − S l 1 1 1/2 λj ∂ν 2 , 2 − − − ∂xl = D[ϕj]dσ ∆xl D[ϕj] xl∆ D[ϕj] dx ∂D ∂ν S − D S − S Z Z (νl,ϕj) ∗ = H , 1/2 λ − j where we used the fact that [ϕ ] is harmonic in D. The same result holds SD j for d = 2 if we change by (see Appendix B.3). Since λ < 1/2 for SD SD | j| j 1, we obtain the result. ≥ e Lemma 2.4.2. Let (j) ∞ αl,m Ml,m(λ,D)= λ λj Xj=1 − be the (l, m)-entry of the polarization tensor M associated with a 1,α domain C D ⋐ Rd. Then, the following properties hold:

(i)

∞ α(j) = δ D ; l,m l,m| | Xj=1 (ii)

d ∞ (d 2) λ α(j) = − D ; i l,l 2 | | Xj=1 Xl=1 (iii)

d d ∞ 2 (j) (d 4) 2 λj αl,l = − D + D[νl] dx. 4 | | D |∇S | Xj=1 Xl=1 Xl=1 Z Proof. The proof can be found in Appendix C.

Let λ = λ′ + iλ′′. We have

d (j) ∞ λ′′ α (Tr(M(λ,D))) = | | l=1 l,l . (2.49) ℑ (λ λ )2 + λ 2 j=1 ′ Pj ′′ X −

For d = 2 the spectrum σ( ) 1/2 is symmetric. For d = 3 this is no KD∗ \{ } longer true. Nevertheless, for our purposes, we can assume that σ( D∗ ) 1/2 (j) K \{ } is symmetric by deﬁning αl,l = 0 if λj is not in the original spectrum. Without loss of generality we assume for ease of notation that Conditions 2.2 and 2.3 hold. Then we deﬁne the bijection ρ : N+ N+ such that → 2.4. Scattering and absorption enhancements 45

λ = λ and we can write ρ(j) − j

1 ∞ λ β ∞ λ β(ρ(j)) (Tr(M(λ,D))) = | ′′| j + | ′′| ℑ 2  (λ λ )2 + λ 2 (λ + λ )2 + λ 2  j=1 ′ j ′′ j=1 ′ j ′′ X − X  2 2 2 (j) (ρ(j)) (j) (ρ(j)) λ ∞ (λ′ + λ′′ + λ )(β + β ) + 2λ′λj(β β ) = ′′ j − , | | 2 2 2 2 2 (λ′ λj) + λ′′ (λ′ + λj) + λ′′ Xj=1 − d (j) where βj = αl,l . l=1 From LemmaX 2.4.1 it follows that

2 2 2 (j) (ρ(j)) (j) (ρ(j)) (λ′ + λ′′ + λ )(β + β ) + 2λ′λj(β β ) j − 0. (λ λ )2 + λ 2 (λ + λ )2 + λ 2 ≥ ′ − j ′′ ′ j ′′ Moreover,

(λ 2 + λ 2 + λ2)(β(j) + β(ρ(j))) + 2λ λ (β(j) β(ρ(j))) ′ ′′ j ′ j − (λ λ )2 + λ 2 (λ + λ )2 + λ 2 ≤ ′ − j ′′ ′ j ′′ 2 2 2 (j) (ρ(j)) (j) (ρ(j)) 2 (λ′ + λ′′ + λ )(β + β ) + 2λ′λj(β β ) λ j − ′′ 2 2 2 + O( 2 2 ). λ′′ (4λ′ + λ′′ ) 4λ′ + λ′′ Hence,

(Tr(M(λ,D))) ℑ ≤ 2 2 2 (j) (ρ(j)) (j) (ρ(j)) 2 λ′′ ∞ (λ′ + λ′′ + λj )(β + β ) + 2λ′(λjβ + λρ(j)β ) λ′′ | | 2 2 2 + O( 2 2 ). 2 λ′′ (4λ′ + λ′′ ) 4λ′ + λ′′ Xj=1 Using Lemma 2.4.2 we obtain the following result.

Theorem 2.4.2. Let M(λ,D) be the polarization tensor associated with a 1,α domain D ⋐ Rd with λ = λ + iλ such that λ 1 and λ < 1/2. C ′ ′′ | ′′| ≪ | ′| Then,

d λ′′ D (Tr(M(λ,D))) |2 || |2 + ℑ ≤ λ′′ + 4λ′ d 1 2 (d 4) 2 (d 2) dλ′ D + − D + [ν ] dx + 2λ′ − D + λ (λ 2 + 4λ 2) | | 4 | | |∇SD l | 2 | | | ′′| ′′ ′ l=1 ZD ! 2 X λ′′ O( 2 2 ). 4λ′ + λ′′ The bound in the above theorem depends not only on the volume of the particle but also on its geometry. Nevertheless, we remark that, since λ < 1 , | j| 2 d ∞ d D λ2 α(j) < | |. j l,l 4 Xj=1 Xl=1 Hence, we can ﬁnd a geometry independent, but not optimal, bound. 46 Chapter 2. The Helmholtz Equation

Corollary 2.4.1. We have

2 1 2 1 (d 2) d λ′′ D λ′′ (Tr(M(λ,D))) d D λ′ + + 2λ′ − D + | || | +O( ). ℑ ≤ λ (λ 2 + 4λ 2) | | 4 2 | | λ 2 + 4λ 2 4λ 2 + λ 2 | ′′| ′′ ′ ′′ ′ ′ ′′ (2.50)

Bound for ellipses

If D is an ellipse whose semi-axes are on the x1- and x2- axes and of length a and b, respectively, then its polarization tensor takes the form [12]

D | 1 |a b 0 λ 2 a+−b M(λ,D)=  −  . (2.51) D  0 | |   λ + 1 a b   2 a+−b    On the other hand, it is known that in (∂D) [65] H∗ 1 a b j σ( ∗ ) 1/2 = − , j = 1, 2,... . KD \{ } ±2 a + b ( ) Then, from (2.48), we also have

(j) (j) ∞ α1,1 ∞ α1,2 j j 1 a b 1 a b  j=1 λ − j=1 λ −  M(λ,D)= X − 2 a+b X − 2 a+b .  (j) (j)   ∞ α1,2 ∞ α2,2   j j   1 a b 1 a b   j=1 λ − j=1 λ −   X − 2 a+b X − 2 a+b    1 a b Let λ = − and (λ ) = i N such that [ϕ ] = λ ϕ . It is clear 1 2 a + b V j { ∈ KD∗ i j i} now that

α(i) = α(i) = D , α(i) = α(i) = 0 (2.52) 1,1 2,2 | | 1,1 2,2 i (λ1) i ( λ1) i (λ ) i ( λ ) ∈VX ∈VX− ∈VXj ∈VX− j for j 2 and ≥ (i) α1,2 = 0 i (λ ) ∈VXj for j 1. ≥ In view of (2.52), we have

β(j) β(ρ(j)) 4λ 2β(j) + λ 2(β(j) + β(j)) λ 2 + ′ ′′ + O( ′′ ). (λ λ )2 + λ 2 (λ + λ )2 + λ 2 ≤ λ 2(4λ 2 + λ 2) 4λ 2 + λ 2 ′ − j ′′ ′ j ′′ ′′ ′ ′′ ′ ′′ Hence,

2 (j) 2 (j) (j) 2 λ′′ ∞ 4λ′ β + λ′′ (β + β ) λ′′ (Tr(M(λ,D))) | | 2 2 2 + O( 2 2 ). |ℑ |≤ 2 λ′′ (4λ′ + λ′′ ) 4λ′ + λ′′ Xj=1 2.4. Scattering and absorption enhancements 47

Note that for for any ellipse D of semi-axes of length a and b, (Tr(M(λ, D))) = ℑ (Tr(M(λ,D))). Then using Lemma 2.4.2 we obtain the following result. ℑ e e Corollary 2.4.2. For any ellipse D of semi-axes of length a and b, we have

e 2 2 D 4λ′ 2 λ′′ D λ′′ (Tr(M(λ, D))) | |2 2 + |2 || |2 + O( 2 2 ). (2.53) |ℑ |≤ λ′′ (λ′′ + 4λ′ ) λ′′ + 4λ′ 4λ′ + λ′′ | | e e Figure 2.1e shows (2.53) and the average extinction of two ellipses of semi- axis a and b, where the ratio a/b = 2 and a/b = 4, respectively.

25 Bound a/b = 2 a/b = 4 20

10 Averaged extinction

0 2 3 4 5 Wavelength of the incoming plane wave 10 7 · − Figure 2.1: Optimal bound for ellipses.

We can see from (2.49), Lemma 2.4.1 and the ﬁrst sum rule in Lemma 2.4.2 that for an arbitrary shape B, (Tr(M(λ,B))) is a convex combination λ′′ |ℑ | of (λ′ λ| )2|+λ′′2 for λj σ( B∗ ) 1/2 . Since ellipses put all the weight of − j ∈ K \{ } 1 a b this convex combination in λ = − , we have for any ellipse D and any ± 1 ± 2 a+b shape B such that B = D , | | | | e (Tr(M(λe∗,B))) (Tr(M(λ∗, D))) |ℑ |≤|ℑ | 1 a b with λ∗ = − + iλ′′. e ± 2 a+b Thus, bound (2.53) applies for any arbitrary shape B in dimension two. This implies that, for a given material and a given desired resonance frequency ω∗, the optimal shape for the extinction resonance (in the quasi-static 1 a b limit) is an ellipse of semi-axis a and b such that λ (ω )= − . ′ ∗ ± 2 a+b 48 Chapter 2. The Helmholtz Equation

Bound for ellipsoids Let D be an ellipsoid given by

2 2 2 x1 x2 x3 2 + 2 + 2 = 1. (2.54) p1 p2 p3 The following holds [12].

Lemma 2.4.3. Let D be the ellipsoid deﬁned by (2.54). Then, for x D, ∈ [ν ](x)= s x , l = 1, 2, 3, SD l l l where

p1p2p3 ∞ 1 sl = ds. − 2 (p2 + s) (p2 + s)(p2 + s)(p2 + s) Z0 l 1 2 3 Then we have p

3 2 2 2 2 D[νl] dx =(s1 + s2 + s3) D . D |∇S | | | Xl=1 Z For a rotated ellipsoid D = D with being a rotation matrix, we have R R M(λ, D) = M(λ,D) T and so Tr(M(λ, D)) = Tr(M(λ,D)). Therefore, R R for any ellipsoid D of semi-axese of length p1,p2 and p3 the following result holds. e e e Corollary 2.4.3. For any ellipsoid D of semi-axes of length p1,p2 and p3, we have e 2 1 2 2 2 2 D 3λ′ + λ′ 4 +(s1 + s2 + s3) 3 λ′′ D λ′′ (Tr(M(λ, D))) | | − 2 2 + |2 || |2 +O( 2 2 ), ℑ ≤ λ′′ (λ′′ + 4λ′ ) λ′′ + 4λ′ 4λ′ + λ′′ e | | e (2.55) e where for j = 1, 2, 3,

p p p 1 s = 1 2 3 ∞ ds. j 2 2 2 2 − 2 0 (p + s) (p + s)(p + s)(p + s) Z j 1 2 3 p 2.5 Link with the scattering coeﬃcients

Our aim in this section is to exhibit the mechanism underlying plasmonic resonances in terms of the scattering coefficients corresponding to the nanoparticle. The concept of scattering coefficients was first introduced in [20]. It plays a key role in constructing cloaking structures. It was extended in [21] to the full Maxwell equations. The scattering coefficients are simply the Fourier coefficients of the scattering amplitude A . In Theorem 2.5.1 we provide an ∞ asymptotic expansion of the scattering amplitude in terms of the scattering coefficients of order 1. Our formula shows that, under physical conditions, ± the scattering coefficients of orders 1 are the only scattering coefficients ± inducing the scattering cross-section enhancement. For simplicity we only consider here the two-dimensional case. 2.5. Link with the scattering coefficients 49

2.5.1 The notion of scattering coeﬃcients From Graf’s addition formula [12] and (2.4) the following asymptotic formula holds as x | | → ∞

s i i (1) inθx inθy u (x)=(u u )(x)= H (k x )e J (k y )e− ψ(y)dσ(y), − −4 n m| | n m| | n Z Z∂D X∈ (1) where x = ( x ,θ ) in polar coordinates, Hn is the Hankel function of the | | x ﬁrst kind and order n, Jn is the Bessel function of order n and ψ is the solution to (2.6). i ikmd x For u (x)= e · we have

ui(x)= a (ui)J (k x )eimθx , m m m| | m Z X∈ π i im( θd) where am(u )= e 2 − . By the superposition principle, we get

i ψ = am(u )ψm, m Z X∈ where ψm is solution to (2.6) replacing f by

(m) (m) 1 1 kc kc 1 (m) f := F2 + Id ( D )∗ ( D )− [F1 ] µc 2 − K S with

F (m)(x) = J (k x )eimθx , 1 − m m| | imθx (m) 1 ∂Jm(km x )e F2 (x) = | | . −µm ∂ν We have

π s i i (1) inθx im( θ ) u (x)=(u u )(x)= H (k x )e W e 2 − d , − −4 n m| | nm n Z m Z X∈ X∈ where inθy W = J (k y )e− ψ (y)dσ(y). (2.56) nm n m| | m Z∂D The coeﬃcients Wnm are called the scattering coeﬃcients. Lemma 2.5.1. In the space (∂D), as ω goes to zero, we have H∗ f (0) = O(ω2), ( 1) ( 1) 2 f ± = ωf1± + O(ω ), f (m) = O(ωm), m > 1, | | where ε µ ( 1) √ m m 1 i θν 1 1 1 i θx f1± = e ± + ( Id D∗ ) D− [ x e ± ] . ∓ 2 µm µc 2 −K S | | e 50 Chapter 2. The Helmholtz Equation

2 Proof. Recall that J0(x)=1+ O(x ). By virtue of the fact that

1 kc kc 1 2 Id ( )∗ ( )− [χ(∂D)] = O(ω ), 2 − KD SD we arrive at the estimate for f (0) (see Appendix B.3). Moreover,

x 3 J 1(x)= + O(x ) ± ± 2 together with the fact that

1 kc kc 1 1 1 2 Id ( )∗ ( )− =( Id ∗ ) − + O(ω log ω) 2 − KD SD 2 −KD SD

( 1) gives the expansion of f ± in terms of ω (see Appendixe B.3). m (m) Finally, Jm(x)= O(x ) immediately yields the desired estimate for f .

From Theorem 3.3.2, we can see that

(m) f , ϕj(ω) ∗ ϕj(ω) 1 ψm = H + D(ω)− (PJc (ω)f). (2.57) τj(ω) A j J X∈ e Hence, from the deﬁnition of the scattering coeﬃcients,

(m) inθx f , ϕj(ω) ∗ ϕj(ω),Jn(km x )e− 1 , 1 H | | 2 2 inθy Wnm = − + Jn(km y )e− O(ω)dσ(y). τj(ω) | | j J e ∂D X∈ Z (2.58) Since

m 1 ex | | Jm(x) ∼ (2π m ) 2 m | | | | as m , we have p → ∞ C m f (m) | | . | |≤ m m | || | Using the Cauchy-Schwarz inequality and Lemma 2.5.1, we obtain the following result.

2.5.2 The leading-order term in the expansion of the scattering amplitude In the following, we analyze the first-order scattering coefficients. 2.5. Link with the scattering coefficients 51

Lemma 2.5.2. Assume that Conditions 1 and 2 hold. Then,

O(ω2) ψ0 = + O(ω), τj(ω) j J X∈ √εmµm 1 1 iθν 3 ω (e± ,ϕj) ∗ ϕj + O(ω log ω) ± 2 µm − µc H ψ 1 = + O(ω). ± τj(ω) j J X∈

Proof. The expression of ψ0 follows from (2.57) and Lemma 2.5.1. Chang- 1 1 iθx ing D by D in Theorem 3.3.2 gives ( Id ∗ ) − [ x e ],ϕj = S S 2 − KD SD | | ∗ iθν H (e ,ϕj) ∗ . Using now Lemma 2.5.1 in (2.57) yields the expression of − He e ψ 1. ± Recall that in two dimensions,

1 1 1 1 2 τj(ω)= + λj + O(ω log ω), 2µm 2µc − µc − µm where λ is an eigenvalue of and λ = 1/2. Recall also that for 0 J we j KD∗ 0 ∈ need τ 0 and so µ , which is a limiting case that we can ignore. In j → m → ∞ practice, PJ (ω)[ϕ0(ω)] = 0. We also have (ϕj,χ(∂D)) 1 , 1 = 0 for j = 0. − 2 2 6 It follows then from the above lemmas and the expression (2.58) of the scattering coeﬃcients that

O(ω4 log ω) W00 = + O(ω), τj(ω) j J X∈ O(ω3 log ω) W0 1 = + O(ω), ± τj(ω) j J X∈ 3 O(ω ) 2 W 10 = + O(ω ). ± τj(ω) j J X∈

Note that W 1 1 has a special structure. Indeed, from Lemma 2.5.2 and ± ± equation (2.58), we have

√εmµm 1 1 iθx iθν 4 ω ϕ ,J (k x )e 1 1 e ,ϕ + O(ω log ω) 2 µm µc j 1 m ∓ , ± j ∗ ±± − | | 2 2 W 1 1 = − H ± ± τj(ω) j J X∈ +O(ω2),

2 εmµm 1 1 iθx iθν 4 ω ϕ , x e 1 1 e ,ϕ + O(ω log ω) 4 µm µc j ∓ , ± j ∗ = ±± − | | − 2 2 H + O(ω2), τj(ω) j J X∈ iθx iθν 2 k2 ϕj, x e∓ 1 , 1 e± ,ϕj ∗ + O(ω log ω) m ±± | | − 2 2 H = 2 + O(1) , 4  λ λj + O(ω log ω)  j J X∈ −   52 Chapter 2. The Helmholtz Equation

2 where λ is deﬁned by (2.23). Now, assume that minj J τj(ω) ω log ω. ∈ | | ≫ Then,

iθx iθν 2 ϕj, x e∓ 1 , 1 e± ,ϕj ∗ km ±± | | 2 2 W 1 1 = − H + O(1) . (2.59) ± ± 4  λ λj  j J X∈ −   Deﬁne the contracted polarization tensors by

iθx 1 iθν N , (λ,D) := x e± (λI D∗ )− [e± ](x) dσ(x). ± ± | | −K Z∂D It is clear that

N (λ,D) = M (λ,D) M (λ,D)+ i2M (λ,D), +,+ 1,1 − 2,2 1,2 N+, (λ,D) = M1,1(λ,D)+ M2,2(λ,D), − N ,+(λ,D) = M1,1(λ,D)+ M2,2(λ,D), − N , (λ,D) = M1,1(λ,D) M2,2(λ,D) i2M1,2(λ,D), − − − − where Ml,m(λ,D) is the (l, m)-entry of the polarization tensor given by (2.40). Finally, considering the above we can state the following result.

Theorem 2.5.1. Let A be the scattering amplitude in the far-ﬁeld deﬁned ∞ i ikmd x in (2.35) for the incoming plane wave u (x)= e · . Assume Conditions 1 and 2 and 2 min τj(ω) ω log ω. j J | | ≫ ∈ Then, A admits the following asymptotic expansion ∞ T x x 2 A = W1d + O(ω ), ∞ x x | | | | where

W 11 + W1 1 2W11 i W1 1 W 11 W1 = − − − − − − . i W1 1 W 11 W 11 W1 1 2W11 − − − − − − − − Here, Wnm are the scattering coeﬃcients deﬁned by (2.56). Proof. From (2.45), we have

T x 2 x A = km M(λ,D)d. ∞ x − x | | | | Since is compact and self-adjoint in , we have KD∗ H∗

iθx iθν ∞ ϕj, x e± 1 , 1 e± ,ϕj ∗ | | 2 2 N , (λ,D) = − H ± ± λ λj Xj=1 − iθx iθν ϕj, x e± 1 , 1 e± ,ϕj ∗ = | | − 2 2 H + O(1). λ λj j J X∈ − 2.6. Concluding remarks 53

We have then from (2.59) that

2 km 2 N+,+(λ,D) = W 11 + O(ω ), − 4 − 2 km 2 N+, (λ,D) = W11 + O(ω ), − 4 − − 2 km 2 N ,+(λ,D) = W11 + O(ω ), − 4 − − 2 km 2 N , (λ,D) = W1 1 + O(ω ). − 4 − − − In view of 1 M11 = (N+,+ + N , + 2N+, ) , 4 − − − 1 M22 = ( N+,+ N , + 2N+, ) , 4 − − − − − i M12 = − (N+,+ N , ) , 4 − − − we get the result.

2.6 Concluding remarks

In this chapter, based on perturbation arguments, we studied the scattering by plasmonic nanoparticles when the frequency of the incoming light is close to a resonant frequency. We have derived the shift and broadening of the plasmon resonance with changes in size. The localization algorithms developed in [12, 41] can be extended to the problem of imaging plasmonic nanoparticles. We have precisely quantified the scattering and absorption cross-section enhancements and gave optimal bounds on the enhancement factors. We have also linked the plasmonic resonances to the scattering coefficients and showed that the leading-order term of the scattering amplitude can be expressed in terms of the -one order of the scattering coefficients. ± The generalization to the full Maxwell equations of the methods and results of the chapter are the subject of chapter 3.

Chapter 3

The Full Maxwell Equations

Contents 3.1 Introduction ...... 56 3.2 Preliminaries ...... 57 3.3 Layer potential formulation for the scattering problem ...... 60 3.3.1 First-order correction to plasmonic resonances and ﬁeld behavior at the plasmonic resonances . . . . . 70 3.4 The extinction cross-section at the quasi-static limit ...... 76 3.5 Explicit computations for a spherical nanoparticle 77 3.5.1 Vector spherical harmonics ...... 77 3.5.2 Explicit representations of boundary integral operators ...... 79

3.5.3 Asymptotic behavior of the spectrum of B(r) . . 81 W 3.5.4 Extinction cross-section ...... 83 3.6 Explicit computations for a spherical shell . . . . 86 3.6.1 Explicit representation of boundary integral operators ...... 87 sh 3.6.2 Asymptotic behavior of the spectrum of (rs) . 90 WB 3.7 Concluding remarks ...... 92 56 Chapter 3. The Full Maxwell Equations

3.1 Introduction

The optical response of plasmon resonant nanoparticles is dominated by the appearance of plasmon resonances over a wide range of wavelengths [68]. For individual particles or very low concentrations in a solvent of non-interacting nanoparticles, separated from one another by distances larger than the wavelength, these resonances depend on the electromagnetic parameters of the nanoparticle, those of the surrounding material, and the particle shape and size. High scattering and absorption cross sections and strong near-fields are unique effects of plasmonic resonant nanoparticles. One of the most important parameters in the context of applications is the position of the resonances in terms of wavelength or frequency. A longstanding problem is to tune this position by changing the particle size or the concentration of the nanoparticles in a solvent [49, 68]. It was experimentally observed, for instance, in [49,89] that the scaling behavior of nanoparticles is critical. The question of how the resonant properties of plasmonic nanoparticles develops with increasing size or/and concentration is therefore fundamental. At the quasi-static limit, plasmon resonances in nanoparticles can be treated as an eigenvalue problem for the Neumann-Poincaré integral operator [9,52,72,73]. Unfortunately, at this limit, they are size-independent. This chapter provides the first mathematical study of the shift in plasmon resonance using the full Maxwell equations. It generalizes to the full Maxwell equations the results obtained in chapter 2 where the Helmholtz equation was used to model light propagation. Theorem 3.3.1 gives an asymptotic expansion of the plasmonic resonances in terms of the size of the nanoparticle. Theorem 3.3.2 provides the near field behavior of the electromagnetic fields near the plasmonic resonant frequencies. The far-field behavior is described in Theorem 3.4.1. Theorem 3.4.2 shows the blow up rate of the extinction cross section (the sum of the absorption and scattering cross sections) at the plasmonic resonance. Theorem 3.5.1 in section 3.5 considers the special case of spherical nanoparticles. The chapter is organized as follows. In section 3.2 we first review com- monly used function spaces. Then we introduce layer potentials associated with the Laplace operator and recall their mapping properties. Of particular interest is the Neumann-Poincaré operator associated with the particle KD∗ D defined in (3.4). We state some of its important properties in Lemma 3.2.1. In section 3.3 we first derive a layer potential formulation for the scattering problem for the full Maxwell equations in (3.11). Then we obtain a first-order correction to plasmonic resonances in terms of the size of the nanoparticle in Theorem 3.3.1. This enables us to analyze the shift and broadening of the plasmon resonance with changes in size and shape of the nanoparticles. The resonance condition is determined from absorption and scattering blow up and depends on the shape, size and electromagnetic parameters of both the nanoparticle and the surrounding material. Surprisingly, it turns out that in this case not only the spectrum of the Neumann-Poincaré operator plays a role in the resonance of the nanoparticles, but also its negative, i.e., σ( ). We explain − KD∗ how in the quasi-static limit, only the spectrum of the Neumann-Poincaré operator can be excited. This is an important finding in our chapter. Note that it is not clear for what kind of geometries in R3 the spectrum of the 3.2. Preliminaries 57

Neumann-Poincaré operator has symmetry, that is, if λ σ( ) so does ∈ KD∗ λ. For instance, such symmetry is not present in the case of a spheri- − cal nanoparticle while for a spherical shell the spectrum of the associated Neumann-Poincaré operator is symmetric around zero. When the particle size increases and deviates from the dipole approximation, the resonances become size-dependent. Moreover, a part of the spectrum of negative of the Neumann-Poincaré operator can be excited as in higher-order terms in the expansion of the electric field versus the size of the particle. In section 3.4, using the quasi-static limit for the electromagnetic fields, we derive a formula for the enhancement of the extinction cross-section. Finally, in seccion 3.5 we provide calculations for the case of spherical nanoparticles and explicitly compute the shift in the spectrum of the Neumann-Poincaré operator and the extinction cross-section. In section 3.6 we consider the case of a spherical shell and apply degenerate perturbation theory since the eigenvalues associated with the corresponding Neumann- Poincaré operator are not simple. The explicit results obtained in sections 3.5 and 3.6 illustrate our main findings in sections 3.3 and 3.4.

3.2 Preliminaries

Here and throughout this chapter, we assume that D is bounded, simply connected, and of class 1,α for 0 <α< 1. We note by the curl operator C ∇× for a vector ﬁeld in R3. We denote by Hs(∂D) the usual Sobolev space of order s on ∂D and

Hs (∂D)= ϕ Hs(∂D) 3,ν ϕ = 0 . T ∈ · n o 1 R3 1 R3 We also need the space Hloc( ) of functions locally in H ( ). We introduce the surface gradient, surface divergence and Laplace-Beltrami operator and denote them by , and ∆ , respectively. We de- ∇∂D ∇∂D· ∂D ﬁne the vectorial and scalar surface curl by curl~ ∂Dϕ = ν ∂Dϕ for 1 −1 × ∇ ϕ H 2 (∂D) and curl ϕ = ν ( ϕ) for ϕ H− 2 (∂D), respec- ∈ ∂D − · ∇∂D × ∈ T tively. We remind that

= ∆ , ∇∂D ·∇∂D ∂D curl curl~ = ∆ , ∂D ∂D − ∂D curl~ = 0, ∇∂D · ∂D curl = 0. ∂D∇∂D We introduce the following functional space:

1 1 1 2 2 H− (div,∂D) = ϕ H− (∂D), ϕ H− 2 (∂D) . T ∈ T ∇∂D · ∈ Let G be the Green function for the Helmholtz operator ∆+ k2, that is,

2 (∆ + k )G(x,y,k)= δy, 58 Chapter 3. The Full Maxwell Equations

eik x y G(x,y,k)= | − | , x = y. (3.1) −4π x y 6 | − | Deﬁne the following boundary integral operators and refer to [18,78] for their mapping properties:

1 1 ~k [ϕ]: H− 2 (∂D) H 2 (∂D) or H1 (R3)3 (3.2) SD T −→ T loc ϕ ~k [ϕ](x)= G(x,y,k)ϕ(y)dσ(y), x ∂D or x R3; 7−→ SD ∈ ∈ Z∂D

k 1 1 1 3 [ϕ]: H− 2 (∂D) H 2 (∂D) or H (R ) (3.3) SD −→ loc ϕ k [ϕ](x)= G(x,y,k)ϕ(y)dσ(y), x ∂D or x R3; 7−→ SD ∈ ∈ Z∂D

1 1 ∗ [ϕ]: H− 2 (∂D) H− 2 (∂D) (3.4) KD −→ ∂G(x,y, 0) ϕ ∗ [ϕ](x)= ϕ(y)dσ(y), x ∂D; 7−→ KD ∂ν(x) ∈ Z∂D

1 1 k [ϕ]: H− 2 (div,∂D) H− 2 (div,∂D) (3.5) MD T −→ T ϕ k [ϕ](x)= ν(x) G(x,y,k)ϕ(y)dσ(y), x ∂D; 7−→ MD ×∇x × ∈ Z∂D

1 1 k [ϕ]: H− 2 (div,∂D) H− 2 (div,∂D) (3.6) LD T −→ T ϕ k [ϕ](x)= ν(x) k2 ~k [ϕ](x)+ k [ ϕ](x) , x ∂D. 7−→ LD × SD ∇SD ∇∂D · ∈ Throughout this chapter, we denote ~0 , 0 , 0 by ~ , , , respec- SD SD MD SD SD MD tively. We also denote D by the ( , ) 1 1 -adjoint of ∗ , where ( , ) 1 1 is 2 , 2 D 2 , 2 K 1 · · − 1 K · · − the duality pairing between H− 2 (∂D) and H 2 (∂D). We recall now some useful results on the operator . See chapter 1 KD∗ and [12, 32,61,65].

Lemma 3.2.1. (i) The following Calderón identity holds: = ; KDSD SDKD∗ 1 (ii) The operator is compact self-adjoint in the Hilbert space H− 2 (∂D) KD∗ equipped with the following inner product

(u,v) ∗ = (u, D[v]) 1 , 1 , (3.7) H − S − 2 2 3.2. Preliminaries 59

1 with which ∗(∂D), the space induced by ( , ) ∗ , is equivalent to H− 2 (∂D); H · · H (iii) Let (λj,ϕj), j = 0, 1, 2,... be the eigenvalue and normalized eigenfunction pair of in (∂D). Then, λ ( 1 , 1 ], λ = 1/2 for j 1, KD∗ H∗ j ∈ − 2 2 j 6 ≥ λ 0 as j and ϕ (∂D) for j 1, where (∂D) is the j → → ∞ j ∈ H0∗ ≥ H0∗ zero mean subspace of (∂D); H∗ (iv) The following representation formula holds: for any ψ H 1/2(∂D), ∈ −

∞ D∗ [ψ]= λj(ψ,ϕj) ∗ ϕj; K H ⊗ Xj=0 (v) The following trace formula holds: for any ψ (∂D), ∈H∗

1 ∂ D[ϕ] ( Id + ∗ )[ϕ]= S . ±2 KD ∂ν ±

1 (vi) Let (∂D) be the space H 2 (∂D) equipped with the following equivalent H inner product 1 (u,v) = ( D− [u],v) 1 , 1 . (3.8) H − S − 2 2 Then, is an isometry between (∂D) and (∂D). SD H∗ H In (vi) in Lemma 3.2.1, we refer to [18] for the invertibility of the single- layer potential in three dimensions. SD The following result holds.

Lemma 3.2.2. The following Helmholtz decomposition holds [38]:

1 3 1 H− 2 (div,∂D)= H 2 (∂D) curl~ H 2 (∂D). T ∇∂D ⊕ ∂D 3 1 2 − 2 Remark 3.2.1. The Laplace-Beltrami operator ∆∂D : H0 (∂D) H0 (∂D) 3 1 → 2 − 2 is invertible. Here H0 (∂D) and H0 (∂D) are the zero mean subspaces of 3 1 H 2 (∂D) and H− 2 (∂D) respectively.

The following results on the operator are of great importance. We MD refer to [78] for a proof of the following compactness property of . MD 1 1 Lemma 3.2.3. The operator : H− 2 (div,∂D) H− 2 (div,∂D) is a MD T −→ T compact operator.

Lemma 3.2.4. The following identities hold [9, 53]:

1 [curl~ ϕ] = curl~ [ϕ], ϕ H 2 (∂D), MD ∂D ∂DKD ∀ ∈ 1 3 [ ϕ] = ∆− ∗ [∆ ϕ]+ curl~ [ϕ], ϕ H 2 (∂D), MD ∇∂D −∇∂D ∂DKD ∂D ∂DRD ∀ ∈ where 1 = ∆− curl . (3.9) RD − ∂D ∂DMD∇∂D 60 Chapter 3. The Full Maxwell Equations

3.3 Layer potential formulation for the scattering problem

We consider the scattering problem of a time-harmonic electromagnetic wave incident on a plasmonic nanoparticle. The homogeneous medium is characterized by electric permittivity εm and magnetic permeability µm, while the particle occupying a bounded and simply connected domain D ⋐ R3 of class 1,α for 0 <α< 1 is characterized by electric permittivity ε and magnetic C c permeability µc, both of which depend on the frequency. Deﬁne

km = ω√εmµm, kc = ω√εcµc, and ε = ε χ(R3 D¯)+ ε χ(D), µ = ε χ(R3 D¯)+ ε χ(D), D m \ c D m \ c where χ denotes the characteristic function. For a given incident plane wave (Ei,Hi), solution to the Maxwell equations in free space

Ei = iωµ Hi in R3, ∇× m Hi = iωε Ei in R3, ∇× − m the scattering problem can be modeled by the following system of equations

E = iωµ H in R3 ∂D, ∇× D \ H = iωε E in R3 ∂D, (3.10) ∇× − D \ ν E + ν E = ν H + ν H = 0 on ∂D, × − × − × − × − subject to the Silver-Müller radiation condition:

i x i lim x √µm(H H )(x) √εm(E E )(x) = 0 x | | − × x − − | |→∞ | | uniformly in x/ x . Here and throughout the chapter, the subscripts indi- | | ± cate, as said before, the limits from outside and inside D, respectively. Using the boundary integral operators (3.2) and (3.5), the solution to (3.10) can be represented as [90]

i ~km ~km R3 ¯ E (x)+ µm D [ψ](x)+ D [φ](x), x D, E(x)= ∇× S ∇×∇× S ∈ \ ( µ ~kc [ψ](x)+ ~kc [φ](x), x D, c∇× SD ∇×∇× SD ∈ (3.11) and i H(x)= ( E)(x) x R3 ∂D, (3.12) −ωµD ∇× ∈ \ 3.3. Layer potential formulation for the scattering problem 61

1 2 where the pair (ψ,φ) H− 2 (div,∂D) is the unique solution to ∈ T µ + µ c m Id + µ kc µ km kc km 2 cMD − mMD LD −LD ψ  k2 k2 k2 k2  kc km c + m Id + c kc m km φ LD −LD 2µ 2µ µ MD − µ MD  c m c m   ν Ei  = × i . iων H × ∂D (3.13)

Let D = z + δB where B contains the origin and B = O(1). For any x z | | x ∂D, let x = − ∂B and define for each function f defined on ∂D, a ∈ δ ∈ corresponding function defined on B as follows e η(f)(x)= f(z + δx). (3.14)

Throughout this chapter, for two Banach spaces X and Y , by (X,Y ) we e e L denote the set of bounded linear operators from X into Y . We will also denote by (X) the set (X,X). L L 1 Lemma 3.3.1. For ϕ H− 2 (div,∂D), the following asymptotic expansion ∈ T holds

∞ k [ϕ](x)= [η(ϕ)](˜x)+ δj k [η(ϕ)](˜x), MD MB MB,j Xj=2 where

j k (ik) j 1 [η(ϕ)](x)= − ν(˜x) x˜ y˜ − η(ϕ)(˜y)dσ(˜y). MB,j 4πj! ×∇x˜ ×| − | Z∂B k Moreover, e 1 is uniformly bounded with respect to j. In B,j − 2 kM k HT (div,∂B) L 1 particular, the convergence holds in H− 2 (div,∂B) and k is analytic L T MD in δ. Proof. We can see, after a change of variables, that

k [ϕ](x)= ν(˜x) G(˜x, y,δk˜ )η(ϕ)(˜y)dσ(˜y). MD ×∇x˜ × Z∂B A Taylor expansion of G(˜x, y,δk˜ ) yields

j j ∞ (iδk x˜ y˜ ) 1 ∞ j (ik) j 1 G(˜x, y,δk˜ )= | − | = + δ x˜ y˜ − . − j!4π x˜ y˜ −4π x˜ y˜ 4πj! | − | Xj=0 | − | | − | Xj=1 Hence,

j k ∞ j (ik) j 1 D[ϕ](x)= B[η(ϕ)](˜x)+ δ − ν(˜x) x˜ x˜ y˜ − η(ϕ)(˜y)dσ(˜y). M M ∂B 4πj! ×∇ ×| − | Xj=2 Z j 1 k Note that from the regularity of x˜ y˜ − , j 2, [η(ϕ)] 1 is B,j − 2 | − | ≥ kM kHT (div,∂B) k uniformly bounded with respect to j, and therefore, B,j − 1 kM k H 2 (div,∂B) L T 62 Chapter 3. The Full Maxwell Equations is uniformly bounded with respect to j as well.

1 Lemma 3.3.2. For ϕ H− 2 (div,∂D), the following asymptotic expansion ∈ T holds

∞ ( kc km )[ϕ](x)= δjω [η(ϕ)](˜x), LD −LD LB,j Xj=1 where

[η(ϕ)](˜x)= LB,j j 2 j 2 x˜ y˜ − (˜x y˜) Cjν(˜x) x˜ y˜ − η(ϕ)(˜y)dσ(˜y) | − | − ∂B η(ϕ)(˜y)dσ(˜y) , × ∂B | − | − ∂B j + 1 ∇ · Z Z and

j j+1 j+1 i (kc km ) C = − . j ω4π(j 1)! −

Moreover, B,j 1 is uniformly bounded with respect to j. In − 2 kL k HT (div,∂B) L 1 particular, the convergence holds in H− 2 (div,∂B) and k is analytic in L T LD δ. Proof. The proof is similar to that of Lemma 3.3.1.

Using Lemma 3.3.1 and Lemma 3.3.2, we can write the system of equations (3.13) as follows:

η(ν Ei) × η(ψ) µm µc B(δ) =  − i  , (3.15) W ωη(φ) η(iν H ) × ∂B  εm εc   −    where

µ km µ kc 2 m B,2 c B,2 1 2 λµId B + δ M − M (δ B,1 + δ B,2)  −M µm µc µm µc L L  B(δ) = − − km kc + W 1 εm εc  (δ + δ2 ) λ Id + δ2 MB,2 − MB,2   ε ε LB,1 LB,2 ε −MB ε ε   m − c m − c  +O(δ3), (3.16) and the material parameter contrasts λµ and λε are given by µ + µ ε + ε λ = c m , λ = c m . (3.17) µ 2(µ µ ) ε 2(ε ε ) m − c m − c It is clear that

λµId B 0 B(0) = B,0 = −M . W W 0 λεId B −M Moreover,

(δ)= + δ + δ2 + O(δ3), WB WB,0 WB,1 WB,2 3.3. Layer potential formulation for the scattering problem 63 in the sense that

(δ) δ δ2 Cδ3 kWB −WB,0 − WB,1 − WB,2k≤ for a constant C independent of δ. Here A = sup Ai,j 1 for i,j − 2 k k k kHT (div,∂B) any operator-valued matrix A with entries Ai,j. 1 We are now interested in ﬁnding − (δ). For this purpose, we ﬁrst consider WB solving the problem (λId )[ψ]= ϕ (3.18) −MB 1 2 for (ψ,ϕ) H− 2 (div,∂B) and λ σ( ), where σ( ) is the spectrum ∈ T 6∈ MB MB of B. M 1 − 2 Using the Helmholtz decomposition of HT (div,∂B) in Lemma 3.2.2, we can reduce (3.18) to an equivalent system of equations involving some well known operators.

1 − 2 (1) (2) Deﬁnition 3.1. For u HT (div,∂B), we denote by u and u any two 3 ∈ 1 2 functions in H0 (∂B) and H 2 (∂B), respectively, such that

u = u(1) + curl~ u(2). ∇∂B ∂B Note that u(1) is uniquely deﬁned and u(2) is deﬁned up to a constant function.

Lemma 3.3.3. Assume λ = 1 , then problem (3.18) is equivalent to 6 2 ψ(1) ϕ(1) (λId ) = , (3.19) − MB ψ(2) ϕ(2)

3 f 1 (1) (2) 2 where (ϕ ,ϕ ) H (∂B) H 2 (∂B) and ∈ 0 × 1 ∆∂B− B∗ ∆∂B 0 B = − K . M RB KB Here, is deﬁnedf by (3.9) with D replaced with B. RB 3 1 (1) (2) 2 Proof. Let (ψ ,ψ ) H0 (∂B) H 2 (∂B) be a solution (if there is any) ∈ 3 × 1 (1) (2) 2 to (3.19) where (ϕ ,ϕ ) H (∂B) H 2 (∂B) satisﬁes ∈ 0 × ϕ = ϕ(1) + curl~ ϕ(2). ∇∂B ∂B We have

1 (1) (1) λId +∆− ∗ ∆ [ψ ] = ϕ , (3.20) ∂BKB ∂B λψ(2) [ψ(1)] [ψ(2)] = ϕ(2). (3.21) −RB −KB Taking in (3.20), curl~ in (3.21), adding up and using the identities of ∇∂B ∂B Lemma 3.2.4 yields

(λId )[ ψ(1) + curl~ ψ(2)]= ϕ(1) + curl~ ϕ(2). −MB ∇∂B ∂B ∇∂B ∂B 64 Chapter 3. The Full Maxwell Equations

Therefore

ψ = ψ(1) + curl~ ψ(2), ∇∂B ∂B is a solution of (3.18). 3 (1) (2) 2 Conversely, let ψ be the solution to (3.18). There exist (ψ ,ψ ) H0 (∂B) 1 3 1 ∈ × (1) (2) 2 H 2 (∂B) and (ϕ ,ϕ ) H (∂B) H 2 (∂B) such that ∈ 0 × ψ = ψ(1) + curl~ ψ(2), ∇∂B ∂B ϕ = ϕ(1) + curl~ ϕ(2), ∇∂B ∂B and we have

(λId )[ ψ(1) + curl~ ψ(2)]= ϕ(1) + curl~ ϕ(2). (3.22) −MB ∇∂B ∂B ∇∂B ∂B Taking in the above equation and using the identities of Lemma 3.2.4 ∇∂B· yields

1 (1) (1) ∆ λId +∆− ∗ ∆ [ψ ]=∆ ϕ . ∂B ∂BKB ∂B ∂B 3 Since (ψ(1),ϕ(1)) (H 2 (∂B))2 we get ∈ 0 1 (1) (1) λId +∆− ∗ ∆ [ψ ]= ϕ . ∂BKB ∂B Taking curl∂B in (3.22) and using the identities of Lemma 3.2.4 yields

∆ (λψ(2) [ψ(1)] [ψ(2)])=∆ ϕ(2). ∂B −RB −KB ∂B Therefore, there exists a constant c such that

λψ(2) [ψ(1)] [ψ(2)]= ϕ(2) + cχ(∂B). −RB −KB 1 Since (χ(∂B)) = χ(∂B) we have KB 2 c c λ ψ(2) [ψ(1)] ψ(2) = ϕ(2). − λ 1/2 −RB −KB − λ 1/2 − h − i 3 1 (1) (2) c 2 Hence, ψ ,ψ H (∂B) H 2 (∂B) is a solution to (3.19) − λ 1/2 ∈ 0 × − Let us now analyze the spectral properties of in MB 3 1 H(∂B) := H 2 (∂B) H 2 (∂Bf), (3.23) 0 × equipped with the inner product

(1) (1) (2) (2) (u,v)H(∂B) = (∆∂Bu , ∆∂Bv ) ∗ +(u ,v ) , H H

3 1 which is equivalent to H 2 (∂B) H 2 (∂B). 0 × By abuse of notation we call u(1) and u(2) the ﬁrst and second components of any u H(∂B). ∈ We will assume for simplicity the following condition. 3.3. Layer potential formulation for the scattering problem 65

Condition 3.1. The eigenvalues of are simple. KB∗ Recall that and are compact and self-adjoint in (∂B) and KB∗ KB H∗ (∂B), respectively. Since B is the ( , ) 1 , 1 adjoint of B∗ , we have H K · · − 2 2 K σ( ) = σ( ), where σ( ) (resp. σ( )) is the (discrete) spectrum KB KB∗ KB KB∗ of (resp. ). KB KB∗ Deﬁne 1 σ = σ( ∗ ) σ( ) , 1 −KB \ KB ∪ {−2} σ = σ( ) σ( ∗ ), (3.24) 2 KB \ −KB σ = σ( ∗ ) σ( ). 3 −KB ∩ KB

Let λj,1 σ1, j = 1, 2 ... and let ϕj,1 be an associated normalized eigenfunc- ∈ 1 tion of as deﬁned in Lemma 3.2.1. Note that ϕ H− 2 (∂B) for j 1. KB∗ j,1 ∈ 0 ≥ Then,

1 ∆∂B− ϕj,1 ψj,1 = 1 1 (λ Id ) [∆− ϕ ] j,1 −KB − RB ∂B j,1 satisﬁes

[ψ ]= λ ψ . MB j,1 j,1 j,1

Let λj,2 σ2 and let ϕj,2 bef an associated normalized eigenfunction of B. ∈ K Then,

0 ψ = j,2 ϕ j,2 satisﬁes

[ψ ]= λ ψ . MB j,2 j,2 j,2 Now, assume that Conditionf 3.1 holds. Let λ σ , let ϕ(1) be the associ- j,3 ∈ 3 j,3 ated normalized eigenfunction of and let ϕ(2) be the associated normal- KB∗ j,3 ized eigenfunction of . Then, KB 0 ψj,3 = (2) ϕj,3 ! satisﬁes

[ψ ]= λ ψ , MB j,3 j,3 j,3 and λj,3 has a ﬁrst-order generalizedf eigenfunction given by

1 (1) c∆∂B− ϕj,3 ψj,3,g = 1 1 (1) (3.25) (λ Id ) (2) [c∆ ϕ ]  j,3 B − span ϕ ⊥ B ∂B− j,3  −K P { j,3 } R   1 (1) (2) (2) for a constant c such that (2) [c∆ ϕ ]= ϕ . Here, span ϕ span ϕ B ∂B− j,3 j,3 j,3 P { j,3 }R − { } is the vector space spanned by ϕ(2), span ϕ(2) is the orthogonal space to j,3 { j,3 }⊥ 66 Chapter 3. The Full Maxwell Equations

(2) span ϕ in (∂B) (Lemma 3.2.1 ), and (2) (resp. (2) is j,3 span ϕ span ϕ ⊥ { } H P { j,3 } P { j,3 } the orthogonal (in (∂B)) projection on span ϕ(2) (resp. span ϕ(2) ). H { j,3 } { j,3 }⊥ We remark that the function ψj,3,g is determined by the following equation

[ψ ]= λ ψ + ψ . MB j,3,g j,3 j,3,g j,3 Consequently, the followingf result holds.

Propsition 3.3.1. The spectrum σ( B) = σ1 σ2 σ3 = σ( B∗ ) 1 M ∪ ∪ −K ∪ σ( ∗ ) in H(∂B). Moreover, under Condition 3.1, B has eigen- KB \{−2} f M functions ψ associated to the eigenvalues λ σ for j = 1, 2,... and j,i j,i ∈ i i = 1, 2, 3, and generalized eigenfunctions of order one ψj,3,gfassociated to λ σ , all of which form a non-orthogonal basis of H(∂B) (deﬁned by j,3 ∈ 3 (3.23)).

Proof. It is clear that λ is bijective if and only if λ / σ( ) σ( ) −MB ∈ −KB∗ ∪ KB∗ \ 1 . {− 2 } It is only left to show thatf ψj,1,ψj,2,ψj,3,ψj,3,g, j = 1, 2,... form a non- orthogonal basis of H(∂B). Indeed, let

ψ(1) ψ = H(∂B). ψ(2) ∈ (1) (1) Since ψj,1 ψj,3,g, j = 1, 2,... form an orthogonal basis of 0∗(∂B), which ∪ 1 H is equivalent to H− 2 (∂B), there exist α ,κ I := (j, 1) (j, 3,g) : j = 0 κ ∈ 1 { ∪ 1, 2,... such that } (1) 1 (1) ψ = ακ∆∂B− ψκ , κ I1 X∈ and

α 2 . | κ| ≤ ∞ κ I1 X∈ (2) It is clear that ψκ 1 is uniformly bounded with respect to κ I1. k kH 2 (∂B) ∈ Then

(2) 1 h := α ψ H 2 (∂B). κ κ ∈ κ I1 X∈ (2) (2) Since ψj,2 ψj,3 , j = 1, 2,... form an orthogonal basis of (∂B), which is ∪ 1 H equivalent to H 2 (∂B), there exist α ,κ I := (j, 2) (j, 3) : j = 1, 2,... κ ∈ 2 { ∪ } such that

ψ(2) h = α ψ(2), − κ κ κ I2 X∈ 3.3. Layer potential formulation for the scattering problem 67 and

α 2 . | κ| ≤ ∞ κ I2 X∈ Hence, there exist α ,κ I I such that κ ∈ 1 ∪ 2

ψ = ακψκ,

κ I1 I2 ∈X∪ and

α 2 . | κ| ≤ ∞ κ I1 I2 ∈X∪

To have the compactness of , we need the following condition. MB Condition 3.2. σ3 is finite. f Indeed, if σ is not finite we have ( ψ ; j 1 ) = λ ψ + 3 MB { j,3,g ≥ } { j,3 j,g,3 ψ ; j 1 whose adherence is not compact. However, if σ is finite, j,3 ≥ } 3 using Proposition 3.3.1 we can approximatef by a sequence of finite-rank MB operators. Throughout this chapter, we assume that Conditionf 3.2 holds, even though an analysis can still be done for the case where σ3 is infinite; see section 3.6. Definition 3.2. Let be the basis of H(∂B) formed by the eigenfunctions B and generalized eigenfunctions of as stated in Lemma 3.3.1. For ψ MB ∈ H(∂B), we denote by α(ψ,ψ ) the projection of ψ into ψ such that κ κ ∈B f ψ = α(ψ,ψκ)ψκ. κ X The following lemma follows from the Fredholm alternative.

Lemma 3.3.4. Let

ψ(1) ψ = H(∂B). ψ(2) ∈ Then,

(ψ, ψκ)H(∂B) , κ =(j,i), i = 1, 2,  (ψκ, ψκ)H(∂B)  e  (ψ, ψ ′ )  κ H(∂B) α(ψ,ψκ)=  e , κ =(j, 3,g),κ′ =(j, 3),  (ψκ, ψκ′ )H(∂B)  e (ψ, ψκg )H(∂B) α(ψ,ψκg )(ψκg , ψκg )H(∂B)  − e , κ =(j, 3),κg =(j, 3,g),  (ψ , ψ )  κ κg H(∂B)  e e  where ψ Ker(λ¯ ) for κe= (j,i), i = 1, 2, 3; ψ Ker(λ¯ )2 κ ∈ κ − MB∗ κ ∈ κ − MB∗ for κ =(j, 3,g) and B∗ is the H(∂B)-adjoint of B. e Mf M e f The following remarkf is in order. g 68 Chapter 3. The Full Maxwell Equations

(1) Remark 3.3.1. Note that, since ϕj,1 and ϕj,3 form an orthogonal basis of 1 (∂B), equivalent to H− 2 (∂B), we also have H0∗ 0 (1) (∆∂Bψ ,ϕj,1) ∗ , κ =(j, 1), H α(ψ,ψκ)= 1 (1) (1) (∆∂Bψ ,ϕj,3 ) ∗ , κ =(j, 3,g), ( c H where c is deﬁned in (3.25).

Remark 3.3.2. For i = 1, 2, 3, and j = 1, 2,...,

1 ψj,i (λId )− [ψ ] = , − MB j,i λ λ − j,i 1 ψj,3,g ψj,3 (λId f)− [ψ ] = + . − MB j,3,g λ λ (λ λ )2 − j,3 − j,3 Now we turn to thef original equation (3.13). The following result holds.

Lemma 3.3.5. The system of equations (3.13) is equivalent to

η(ν Ei)(1) × µm µc (1)   η(ψ) η(ν −Ei)(2) (2) × η(ψ)  µm µc  WB(δ)   =   , (3.26) ωη(φ)(1)  η(iν −Hi)(1)   ×  ∂B  ωη(φ)(2)       εm εc     η(iν −Hi)(2)     ×   εm εc   −  where

2 3 WB(δ) = WB,0 + δWB,1 + δ WB,2 + O(δ ) with

λµId B O WB,0 = − M , O λ Id ε − MB ! f 1 O f B,1 µm µc L WB,1 =  1 −  , O ε ε LB,1 e  m − c   1 µ 1  e B,2 B,2 µm µc M µm µc L WB,2 = − − ,  1 1 ε  fB,2 eB,2  εm εc L εm εc M   − −  and e f

1 ∆∂B− B∗ ∆∂B 0 B = − K , M RB KB f 1 km kc 1 km kc ~ µ ∆∂B− ∂B (µm B,2 µc B,2) ∂B ∆∂B− ∂B (µm B,2 µc B,2)curl∂B B,2 = 1∇ · M km− M kc ∇ 1∇ · M km− M kc , M ∆− curl (µ µ ) ∆− curl (µ µ )curl~ − ∂B ∂B mMB,2 − cMB,2 ∇∂B − ∂B ∂B mMB,2 − cMB,2 ∂B ! f 3.3. Layer potential formulation for the scattering problem 69

1 km kc 1 km kc ~ ε ∆∂B− ∂B (εm B,2 εc B,2) ∂B ∆∂B− ∂B (εm B,2 εc B,2)curl∂B B,2 = 1∇ · M km− M kc ∇ 1∇ · M km− M kc , M ∆− curl (ε ε ) ∆− curl (ε ε )curl~ − ∂B ∂B mMB,2 − cMB,2 ∇∂B − ∂B ∂B mMB,2 − cMB,2 ∂B ! f 1 1 ~ ∆∂B− ∂B B,s ∂B ∆∂B− ∂B B,scurl∂B B,s = 1∇ ·L ∇ 1∇ ·L , L ∆− curl ∆− curl curl~ − ∂B ∂BLB,s∇∂B − ∂B ∂BLB,s ∂B ! e for s = 1, 2. 2 Moreover, the eigenfunctions of WB,0 in H(∂B) are given by

ψ Ψ = j,i , j = 0, 1, 2,... ; i = 1, 2, 3, 1,j,i O O Ψ = , j = 0, 1, 2,... ; i = 1, 2, 3, 2,j,i ψ j,i associated to the eigenvalues λ λ and λ λ , respectively, and gener- µ − j,i ε − j,i alized eigenfunctions of order one

ψ Ψ = j,3,g , 1,j,3,g O O Ψ = , 2,j,3,g ψ j,3,g associated to eigenvalues λ λ and λ λ , respectively, all of which µ − j,3 ε − j,3 form a non-orthogonal basis of H(∂B)2.

Proof. The proof follows directly from Lemmas 3.3.1 and 3.3.3.

We regard the operator WB(δ) as a perturbation of the operator WB,0 for small δ. Using perturbation theory, we can derive the perturbed eigenvalues and their associated eigenfunctions in H(∂B)2. We denote by Γ = (k,j,i) : k = 1, 2; j = 1, 2,... ; i = 1, 2, 3 the set of indices for the eigenfunctions of W and by Γ = (k,j, 3,g): k = 1, 2; j = B,0 g 1, 2,... the set of indices for the generalized eigenfunctions. We denote by γ the generalized eigenfunction index corresponding to eigenfunction index g γ and vice-versa. We also denote by

λµ λj,i, k = 1, τγ = − (3.27) λ λ , k = 2. ε − j,i Condition 3.3. λ = λ . µ 6 ε In the following we will only consider γ Γ with which there is no ∈ generalized eigenfunction index associated. In other words, we only consider γ = (k,i,j) Γ such that λ σ σ (see (3.24) for the deﬁnitions). ∈ j,i ∈ 1 ∪ 2 We call this subset Γsim. Note that Conditions 3.1 and 3.3 imply that the eigenvalues of W indexed by γ Γsim are simple. B,0 ∈ Theorem 3.3.1. As δ , the perturbed eigenvalues and eigenfunctions in- → dexed by γ Γsim have the following asymptotic expansions: ∈ 2 3 τγ(δ) = τγ + δτγ,1 + δ τγ,2 + O(δ ), (3.28) 2 Ψγ(δ) = Ψγ + δΨγ,1 + O(δ ), 70 Chapter 3. The Full Maxwell Equations where

(WB,1Ψγ, Ψγ)H(∂B)2 τγ,1 = = 0, (Ψγ, Ψγ)H(∂B)2 e (WB,2Ψγ, Ψγ)H(∂B)2 (WB,1Ψγ,1, Ψγ)H(∂B)2 τγ,2 = e − (3.29), (Ψγ, Ψγ)H(∂B)2 e e (τγ WB,0)Ψγ,1 = WB,1Ψγ. − − e 2 Here, Ψ ′ Ker(¯τ ′ W ) and W is the H(∂B) adjoint of W . γ ∈ γ − B,∗ 0 B,∗ 0 B,0 Usinge Lemma 3.3.4 and Remark 3.3.2 we can solve Ψγ,1. Indeed,

′ α( WB,1Ψγ, Ψγ′ )Ψγ′ Ψγg Ψγ′ ′ Ψγ,1 = − + α( WB,1Ψγ, Ψγg ) + ′ ′ ′ 2 ′ τγ τγ ′ − τγ τγ (τγ τγ ) γ ∈Γ γg∈Γg ′ − − − γX6=γ γX′6=γ + α( W Ψ , Ψ )Ψ . − B,1 γ γ γ By abuse of notation,

α(x ,ψ ) γ = (1,j,i), κ =(j,i), α(x, Ψ )= 1 κ (3.30) γ α(x ,ψ ) γ = (2,j,i), κ =(j,i), 2 κ for x x = 1 H(∂B)2 x2 ∈ with α being introduced in Deﬁnition 3.2.

Consider now the degenerate case γ Γ Γsim =: Γdeg = γ =(k,i,j) Γ ∈ \ { ∈ s.t λ σ . It is clear that, for γ Γdeg, the algebraic multiplicity of the j,i ∈ 3} ∈ eigenvalue τγ is 2 while the geometric multiplicity is 1. In this case every eigenvalue τγ and associated eigenfunction Ψγ will split into two branches, as δ goes to zero, represented by a convergent Puiseux series as [28]:

τ (δ) = τ +( 1)hδ1/2τ +( 1)2hδ2/2τ + O(δ3/2), h = 0,(3.31)1, γ,h γ − γ,1 − γ,2 Ψ (δ) = Ψ +( 1)hδ1/2Ψ +( 1)2hδ2/2Ψ + O(δ3/2), h = 0, 1, γ,h γ − γ,1 − γ,2 where τγ,j and Ψγ,j can be recovered by recurrence formulas. We refer to [62] for more details.

3.3.1 First-order correction to plasmonic resonances and ﬁeld behavior at the plasmonic resonances

Recall that the electric and magnetic parameters, εc and µc, depend on the frequency of the incident ﬁeld, ω, following the Drude model [9]. Therefore, the eigenvalues of the operator WB,0 and perturbation in the eigenvalues depend on the frequency as well, that is,

2 3 τ (δ, ω) = τ (ω)+ δ τ (ω)+ O(δ ), γ Γsim, γ γ γ,2 ∈ 1/2 h 2/2 2h 3/2 τ (δ, ω) = τ + δ ( 1) τ (ω)+ δ ( 1) τ (ω)+ O(δ ), γ Γdeg, h = 0, 1. γ,h γ − γ,1 − γ,2 ∈ 3.3. Layer potential formulation for the scattering problem 71

In the sequel, we will omit frequency dependence to simplify the notation. However, we will keep in mind that all these quantities are frequency dependent. We first recall different notions of plasmonic resonance, see chapter 2. Definition 3.3. (i) We say that ω is a plasmonic resonance if τ (δ) 1 | γ | ≪ and is locally minimized for some γ Γsim or τ (δ) 1 and is ∈ | γ,h | ≪ locally minimized for some γ Γdeg, h = 0, 1. ∈ (ii) We say that ω is a quasi-static plasmonic resonance if τ 1 and is | γ| ≪ locally minimized for some γ Γ. Here, τ is defined by (3.27). ∈ γ (iii) We say that ω is a first-order corrected quasi-static plasmonic resonance 2 if τγ + δ τγ,2 1 and is locally minimized for some γ Γsim or | 1/2 |h ≪ ∈ τ + δ ( 1) τ 1 and is locally minimized for some γ Γdeg, | γ − γ,1| ≪ ∈ h = 0, 1. Here, the correction terms τγ,2 and τγ,1 are defined by (3.29) and (3.31). Note that quasi-static resonance is size independent and is therefore a zero-order approximation of the plasmonic resonance in terms of the particle size while the first-order corrected quasi-static plasmonic resonance depends on the size of the nanoparticle. We are interested in solving equation (3.26)

WB(δ)Ψ = f, where

η(ν Ei)(1) × µm µc (1)   η(ψ) η(ν −Ei)(2) (2) × η(ψ)   Ψ=   ,f =  µm µc  ωη(φ)(1)  η(iν −Hi)(1)   ×  ∂B  ωη(φ)(2)       εm εc     η(iν −Hi)(2)     ×   εm εc   −  for ω close to the resonance frequencies, i.e., when τγ(δ) is very small for some γ’s Γsim or τ (δ) is very small for some γ’s Γdeg, h = 0, 1. In this ∈ γ,h ∈ case, the major part of the solution would be the contributions of the excited resonance modes Ψγ(δ) and Ψγ,h(δ).

We introduce the following deﬁnition. Deﬁnition 3.4. We call J Γ index set of resonances if τ ’s are close to ⊂ γ zero when γ Γ and are bounded from below when γ Γc. More precisely, ∈ ∈ we choose a threshold number η0 > 0 independent of ω such that

τ η > 0 for γ J c. | γ|≥ 0 ∈ From now on, we shall use J as our index set of resonances. For simplicity, we assume throughout this chapter that the following condition holds. Condition 3.4. We assume that λ = 0, λ = 0 or equivalently, µ = µ , µ 6 ε 6 c 6 − m ε = ε . c 6 − m 72 Chapter 3. The Full Maxwell Equations

It follows that the set J is finite. Consider the space = span Ψ (δ), Ψ (δ); γ J, h = 0, 1 . Note that, EJ { γ γ,h ∈ } under Condition 3.4, J is finite dimensional. Similarly, we define Jc as the E c E spanned by Ψγ(δ), Ψγ,h(δ); γ J , h = 0, 1 and eventually other vectors to ∈ 2 complete the base. We have H(∂B) = c . EJ ⊕EJ We define PJ (δ) and PJc (δ) as the (non-orthogonal) projection into the finite-dimensional space and infinite-dimensional space c , respectively. EJ EJ It is clear that, for any f H(∂B)2 ∈

f = PJ (δ)[f]+ PJc (δ)[f].

Moreover, we have an explicit representation for PJ (δ)

PJ (δ)[f]= αδ(f, Ψγ(δ))Ψγ(δ)+ αδ(f, Ψγ,h(δ))Ψγ,h(δ).

γ J Γsim γ∈J∩Γdeg ∈X∩ hX=0,1 (3.32) Here, as in Lemma 3.3.4, ˜ (f, Ψγ(δ))H(∂B)2 α (f, Ψ (δ)) = , γ J Γ , δ γ ˜ sim (Ψγ(δ), Ψγ(δ))H(∂B)2 ∈ ∩ ˜ (f, Ψγ,h(δ))H(∂B)2 α (f, Ψ (δ)) = , γ J Γ , h = 0, 1, δ γ,h ˜ deg (Ψγ,h(δ), Ψγ,h(δ))H(∂B)2 ∈ ∩ where Ψγ Ker(¯τγ,h(δ) WB∗ (δ)), Ψγ,h Ker(¯τγ,h(δ) WB∗ (δ)) and WB∗ (δ) ∈ 2 − ∈ − is the H(∂B) -adjoint of WB(δ). Wee are now ready to solve the equatione WB(δ)Ψ = f. In view of Remark 3.3.2,

1 αδ(f, Ψγ(δ))Ψγ(δ) αδ(f, Ψγ,h(δ))Ψγ,h(δ) 1 Ψ= WB− (δ)[f]= + +WB− (δ)PJc (δ)[f]. τγ(δ) τγ,h(δ) γ J Γsim γ∈J∩Γdeg ∈X∩ hX=0,1 (3.33) The following lemma holds. A similar result was proved for δ = 0 in [6].

1 Lemma 3.3.6. The norm WB− (δ)PJc (δ) (H(∂B)2,H(∂B)2) is uniformly bounded k kL in ω and δ.

Proof. Consider the operator

2 2 W (δ) c : P c (δ)H(∂B) P c (δ)H(∂B) . B |J J → J η0 We can show that for every ω and δ, dist(σ(W (δ) c ), 0) , where B |J ≥ 2 σ(W (δ) c ) is the discrete spectrum of W (δ) c . Here and throughout B |J B |J the chapter, dist denotes the distance. Then, it follows that

1 1 1 C1 1 C1 c c c c WB− (δ)PJ (δ)[f] = WB− (δ) J PJ (δ)[f] . exp( 2 ) PJ (δ)[f] . exp( 2 ) f , k k k | k η0 η0 k k η0 η0 k k where the notation A . B means that A CB for some constant C inde- ≤ pendent of A and B.

Finally, we are ready to state our main result in this section. 3.3. Layer potential formulation for the scattering problem 73

Theorem 3.3.2. Let η be deﬁned by (3.14). Under Conditions 3.1, 3.2, 3.3 and 3.4, the scattered ﬁeld Es = E Ei due to a single plasmonic particle − has the following representation:

Es = µ ~km [ψ](x)+ ~km [φ](x) x R3 D,¯ m∇× SD ∇×∇× SD ∈ \ where

1 (1) (2) ψ = η− ψ + curl~ ψ , ∇∂B ∂B 1 1 (1) (2) φ = η− φ + curl~ φ , ω ∇∂Be ∂Be e e ψ(1) ψ(2) Ψ =   φe(1)    φe(2)     e  1/2 α(f, Ψγ)Ψγ + O(δ) ζ1(f)Ψγ + ζ2(f)Ψγ,1 + O(δ ) = e + + O(1), τγ(δ) τγ,0(δ)τγ,1(δ) γ J Γsim γ J Γdeg ∈X∩ ∈X∩ and

a2 (f, Ψ˜ ) 2 τ (f, Ψ˜ ) 2 (τ + τ ) γ,1 H(∂B) γ γ H(∂B) γ,1 γ a1 ζ1(f) = − , a1 ˜ (f, Ψγ)H(∂B)2 ζ2(f) = , a1 ˜ ˜ a1 = (Ψγ, Ψγ,1)H(∂B)2 + (Ψγ,1, Ψγ)H(∂B)2 , ˜ ˜ ˜ a2 = (Ψγ, Ψγ,2)H(∂B)2 + (Ψγ,2, Ψγ)H(∂B)2 + (Ψγ,1, Ψγ,1)H(∂B)2 .

Proof. Recall that

αδ(f, Ψγ(δ))Ψγ(δ) αδ(f, Ψγ,h(δ))Ψγ,h(δ) 1 Ψ= + + WB− (δ)PJc (δ)[f]. τγ(δ) τγ,h(δ) γ J Γsim γ∈J∩Γdeg ∈X∩ hX=0,1

1 By Lemma 3.3.6, we have WB− (δ)PJc (δ)[f]= O(1). If γ J Γsim, an asymptotic expansion on δ yields ∈ ∩

αδ(f, Ψγ(δ))Ψγ(δ)= α(f, Ψγ)Ψγ + O(δ).

If γ J Γdeg then (Ψ , Ψ˜ ) 2 = 0. Therefore, an asymptotic expansion ∈ ∩ γ γ H(∂B) on δ yields

h ˜ ( 1) (f, Ψγ)H(∂B)2 Ψγ α (f, Ψ (δ))Ψ (δ) = − + δ γ,h γ,h 1/2 δ− a1 1 a2 (f, Ψ˜ ) 2 (f, Ψ˜ ) 2 Ψ +(f, Ψ˜ ) 2 Ψ a γ,1 H(∂B) − γ H(∂B) a γ γ H(∂B) γ,1 1 1 +O(δ1/2) 74 Chapter 3. The Full Maxwell Equations with ˜ ˜ a1 = (Ψγ, Ψγ,1)H(∂B)2 + (Ψγ,1, Ψγ)H(∂B)2 , ˜ ˜ ˜ a2 = (Ψγ, Ψγ,2)H(∂B)2 + (Ψγ,2, Ψγ)H(∂B)2 + (Ψγ,1, Ψγ,1)H(∂B)2 .

Since τ (δ) = τ + δ1/2( 1)hτ + O(δ), the result follows by adding the γ,h γ − γ,1 terms α (f, Ψ (δ))Ψ (δ) α (f, Ψ (δ))Ψ (δ) δ γ,0 γ,0 and δ γ,1 γ,1 . τγ,0(δ) τγ,1(δ) The proof is then complete.

Corollary 3.3.1. Assume the same conditions as in Theorem 3.3.2. Under the additional condition that

3 min τγ(δ) δ , min τγ(δ) δ, (3.34) γ J Γsim | | ≫ γ J Γdeg | | ≫ ∈ ∩ ∈ ∩ we have

1/2 α(f, Ψγ)Ψγ + O(δ) ζ1(f)Ψγ + ζ2(f)Ψγ,1 + O(δ ) Ψ= 2 + 2 2 + O(1). τγ + δ τγ,2 τγ δτγ,1 γ J Γsim γ J Γdeg ∈X∩ ∈X∩ − Corollary 3.3.2. Assume the same conditions as in Theorem 3.3.2. Under the additional condition that

2 1/2 min τγ(δ) δ , min τγ(δ) δ , (3.35) γ J Γsim | | ≫ γ J Γdeg | | ≫ ∈ ∩ ∈ ∩ we have

α(f, Ψγ)Ψγ + O(δ) α(f, Ψγ)Ψγ Ψγ,g Ψγ Ψ= + + α(f, Ψγ,g) + 2 + O(1). τγ τγ τγ τγ γ J Γsim γ J Γdeg ∈X∩ ∈X∩ Proof. We have

1 αδ(f, Ψγ,0(δ))Ψγ,0(δ) αδ(f, Ψγ,1(δ))Ψγ,1(δ) lim WB− (δ)Pspan Ψγ,0(δ),Ψγ,1(δ) [f] = lim + δ 0 { } δ 0 τ (δ) τ (δ) → → γ,0 γ,1 1 = WB,−0(δ)Pspan Ψγ ,Ψγ [f] { g } α(f, Ψ )Ψ Ψ Ψ = γ γ + α(f, Ψ ) γ,g + γ , τ γ,g τ τ 2 γ γ γ 2 where γ J Γdeg, f H(∂B) and )Pspan is the projection into the linear ∈ ∩ ∈ E space generated by the elements in the set E.

Remark 3.3.3. Note that for γ J, ∈

τ min dist λ ,σ( ∗ ) σ( ∗ ) , dist λ ,σ( ∗ ) σ( ∗ ) . γ ≈ µ KB ∪− KB ε KB ∪− KB It is clear, from Remark 3.3.3, that resonances can occur when exciting the spectrum of or/and that of . We substantiate in the following KB∗ −KB∗ that only the spectrum of can be excited to create the plasmonic reso- KB∗ nances in the quasi-static regime. 3.3. Layer potential formulation for the scattering problem 75

Recall that

η(ν Ei)(1) × µ µ  m c  η(ν −Ei)(2) ×   f =  µm µc  ,  η(iν −Hi)(1)    ∂B  ×   εm εc   η(iν −Hi)(2)     ×   εm εc   −  and therefore,

i (1) 1 i η(ν E ) ∆− ∂B η(ν E ) f := × = ∂B∇ · × . 1 µ µ µ µ m − c m − c Now, suppose γ = (1,j, 1) J (recall that J is the index set of resonances). ∈ Then τ = λ λ , where λ σ = σ( ) σ( ). From Remark γ µ − 1,j 1,j ∈ 1 −KB∗ \ KB∗ 3.3.1,

1 i α(f, Ψγ)=(∆∂Bf1,ϕj,1) ∗ = α(f, Ψγ)= ( ∂B η(ν E ),ϕj,1) ∗ , H µ µ ∇ · × H m − c where ϕ (∂B) is a normalized eigenfunction of (∂B). j,1 ∈H0∗ KB∗ A Taylor expansion of Ei gives, for x ∂D, ∈ ∞ (x z)β∂βEi(z) Ei(x)= − . β ! β N3 X∈ | | Thus,

η(ν Ei)(˜x)= η(ν)(˜x) Ei(z)+ O(δ), × × and

η(ν Ei)(˜x) = η(ν)(˜x) Ei(z)+ O(δ) ∇∂B · × − ·∇× = O(δ).

Therefore, the zeroth-order term of the expansion of η(ν Ei) in δ is ∇∂B · × zero. Hence,

α(f, Ψγ) = 0.

In the same way, we have

α(f, Ψγ) = 0,

α(f, Ψγg ) = 0 for γ = (2,j, 1) J and γ such that γ J. ∈ g ∈ As a result we see that the spectrum of is not excited in the zeroth- −KB∗ order term. However, we note that σ( ) can be excited in higher-order −KB∗ terms. 76 Chapter 3. The Full Maxwell Equations

3.4 The extinction cross-section at the quasi-static limit

The aim of this section is to derive an expression of the extinction cross section and estimate its blow up at the plasmonic resonances. We first recall the quasi-static limit of the electric field at plasmonic resonances. The formula was first obtained in [9], but it can be derived by pursuing further computations in Corollary 3.3.2. In this formula, the polarization tensor is a key ingredient. It allows to express the quasi-static limit or zeroth-order approximation of the electromagnetic fields far away from the particle. The polarization tensor is given by [18]

1 M(λ,D)= (λId ∗ )− [ν](x)xdσ(x), (3.36) −KD Z∂D where λ C ( 1/2, 1/2). In view of Lemma 3.2.1, we have ∈ \ − ∞ 1 M(λ,D)= (ν,ϕj) ∗ (ϕj,x) 1 , 1 , (3.37) λ λj H − 2 2 Xj=1 − since (ν,ϕ0) ∗ = 0; H The following result follows from [9]. Theorem 3.4.1. Let d = min dist λ ,σ( ) σ( ) , dist λ ,σ( ) σ( ) . σ µ KD∗ ∪− KD∗ ε KD∗ ∪− KD∗ Then, for D = z+δB ⋐ R3 of class 1,α for 0 <α< 1, the following uniform C far-ﬁeld expansion holds

4 s iωµm i 2 i δ E = Gd(x,z,km)M(λµ,D)H (z) ω µmGd(x,z,km)M(λε,D)E (z)+ O( ), − εm ∇× − dσ where

1 2 Gd(x,z,km)= εm G(x,z,km)Id + 2 DxG(x,z,km) km is the Dyadic Green (matrix valued) function for the full Maxwell equations. In order to express the extinction cross section we need to write the far-field behavior of the electric and magnetic fields. We first recall the representation for the scattering amplitude [78]. It is well-known that the solution (E,H) to the system (3.10) has the following far-field expansion as x + : | | → ∞ eikm x 1 Es(x)= | | A (ˆx)+ O , − 4π x ∞ x 2 | | | | and

ikmxˆ y 2 ikmxˆ y A (ˆx)= iµmkmxˆ e− · ψ(y)dσ(y) kmxˆ xˆ e− · φ(y)dσ(y), ∞ − × − × × Z∂D Z∂D 3.5. Explicit computations for a spherical nanoparticle 77

x and xˆ = x . We also| | need the optical cross-section theorem for the scattering of electromagnetic waves [43], which can be stated as follows. Assume that the incident ﬁelds are plane waves given by

i ikmd x E (x) = pe · , i ikmd x H (x) = d pe · , × where p R3 and d R3 with d = 1 are such that p d = 0. Then, the ∈ ∈ | | · extinction cross-section is given by

ext 4π p A (d) Q = · ∞ , k ℑ p 2 m | | where A is the scattering amplitude. ∞ From Taylor expansions on the formula of Theorem 3.4.1, it follows that the following far-ﬁeld asymptotic expansion holds:

Es = ikm x e | | ikm(d xˆ) z 2 ikm(d xˆ) z t ωµ k e − · xˆ Id M(λ ,D)(d p) k e − · Id xˆxˆ M(λ ,D)p − 4π x m m × µ × − m − ε | | 1 δ4 +O( )+ O( ), x 2 d | | σ 4 where xˆ = x/ x . Therefore, up to an error term of order O( δ ), we have | | dσ

ikm(d xˆ) z 2 ikm(d xˆ) z t A (ˆx)= ωµmkme − · xˆ Id M(λµ,D)(d p) kme − · Id xˆxˆ M(λε,D)p. ∞ × × − − (3.38) Formula (3.38) allows us to compute the extinction cross-section Qext in terms of the polarization tensors associated with the particle D and the material parameter contrasts. Moreover, an estimate for the blow up of Qext at the plasmonic resonances follows immediately from (3.37).

Theorem 3.4.2. We have 4π Qext = p ωµ k d Id M(λ ,D)(d p) k2 Id ddt M(λ ,D)p , k p 2 ℑ · m m × µ × − m − ε m| | where M(λµ,D) and M(λε,D) are the polarization tensors associated with D and λ = λµ and λ = λε, respectively.

3.5 Explicit computations for a spherical nanoparticle

In this section we consider a spherical nanoparticle and explicitly compute the ﬁrst order correction in terms of radius of its plasmonic resonances. We also derive and explicit formula for the extinction cross section.

3.5.1 Vector spherical harmonics

x m Let xˆ = x . For m = n,...,n and n = 1, 2,..., set Yn to be the spherical | | − harmonics deﬁned on the unit sphere S = x R3, x = 1 . For a wave { ∈ | | } 78 Chapter 3. The Full Maxwell Equations number k > 0, the function

v (k; x)= h(1)(k x )Y m(ˆx) n,m n | | n satisﬁes the Helmholtz equation ∆v + k2v = 0 in R3 0 together with the \{ } Sommerfeld radiation condition

∂vn,m lim x (k; x) ikvn,m(k; x) = 0. x | | ∂ x − | |→∞ | | Similarly, let vn,m(x) be deﬁned by

v (x)= j (k x )Y m(ˆx), e n,m n | | n where jn is the spherical Bessel function of the first kind. Then the function e 3 vn,m satisfies the Helmholtz equation in R . Next, define the vector spherical harmonics by e 1 m Un,m = SYn (ˆx) and Vn,m =x ˆ Un,m n(n + 1)∇ × for m = n,...,n andp n = 1, 2,.... Here, xˆ S and denote the sur- − ∈ ∇S face gradient on the unit sphere S. The vector spherical harmonics form a 2 complete orthogonal basis for LT (S). Using the vectorial spherical harmonics, we can separate the solutions of Maxwell’s equations into multipole solutions; see [78, Section 5.3]. Define the exterior transverse electric multipoles, i.e., E x = 0, as · ETE (x)= n(n + 1)h(1)(k x )V (ˆx), n,m − n | | n,m TE i (1) (3.39)  Hn,m(x)= p n(n + 1)hn (k x )Vn,m(ˆx) ,  −ωµ∇× − | | p and the exterior transverse magnetic multipoles, i.e., H x = 0, as · i ETM (x)= n(n + 1)h(1)(k x )V (ˆx) , n,m ωǫ∇× − n | | n,m  (3.40) TM p (1)  H (x)= n(n + 1)h (k x )Vn,m(ˆx). n,m − n | | The exterior electricp and magnetic multipoles satisfy the Sommerfeld radiation condition. In the same manner, one defines the interior multipoles TE TE TM TM (1) (En,m, Hn,m) and (En,m , Hn,m ) with hn replaced by jn, i.e.,

e e Ee TE (xe)= n(n + 1)j (k x )V (ˆx), n,m − n | | n,m (3.41)  TE pi TE Hen,m(x)= En,m(x),  −ωµ∇× and  e e TM Hn,m (x)= n(n + 1)jn(k x )Vn,m(ˆx), −i | | (3.42)  ETM (x)= p HTM (x).  en,m ωǫ∇× n,m  e e 3.5. Explicit computations for a spherical nanoparticle 79

Note that one has

TE n(n + 1) n(n + 1) (1) m En,m(k; x)= n(k x )Un,m(ˆx)+ hn (k x )Yn (ˆx)ˆx ∇× p x H | | x | | | | | | (3.43) and

(1) (1) (t)= j (t)+ tj′ (t), (t)= h (t)+ t(h )′(t). Jn n n Hn n n For x > y , the following addition formula holds: | | | | n ∞ ik ǫ T G(x,y,k)I = ETM (x)ETM (y) − n(n + 1) µ n,m n,m n=1 m= n − X nX ∞ ik e T ETE (x)ETE (y) − n(n + 1) n,m n,m n=1 m= n − X n X i ∞ e T v (x) v (y) . (3.45) − k ∇ n,m ∇ n,m n=1 m= n X X− Alternatively, for x < y , we have e | | | | n ∞ ik ǫ G(x,y,k)I = ETM (x)ETM (y)T − n(n + 1) µ n,m n,m n=1 m= n − X nX ∞ ik e ETE (x)ETE (y)T − n(n + 1) n,m n,m n=1 m= n − X n X i ∞ e v (x) v (y)T . (3.46) − k ∇ n,m ∇ n,m n=1 m= n X X− e 3.5.2 Explicit representations of boundary integral operators Let D be a sphere of radius r> 0. We have the following results.

Lemma 3.5.1. Let ∂D = x = r . Then, for r >r, we have {| | } ′ ~k + (1) ν SD[Un,m] x =r′ =( ikr)hn (kr′) n(kr)Un,m, (3.47) ×∇× | | − J 2 ~k + r ν SD[Vn,m] x =r′ = ik jn(kr) n(kr′)Vn,m, (3.48) ×∇× | | r′ H

~k + r ν SD[Un,m] x =r′ = ik n(kr) n(kr′)Vn,m, (3.49) ×∇×∇× | | − r′ J H

~k + 2 (1) ν SD[Vn,m] x =r′ = ik(kr) jn(kr)hn (kr′)Un,m. (3.50) ×∇×∇× | |

80 Chapter 3. The Full Maxwell Equations

For r′

~k + r ν SD[Un,m] x =r′ = ik n(kr′) n(kr)Vn,m, (3.53) ×∇×∇× | | − r′ J H

~k + 2 (1) ν SD[Vn,m] x =r′ = ik(kr) jn(kr′)hn (kr)Un,m. (3.54) ×∇×∇× | |

Proof. We only consider (3.47). The other formulas can be proved in a similar way. TE TM TE TM From (3.43), (3.44), and the deﬁnitions of En,m,En,m , En,m and En,m , we have e e G(x,y,k)U (ˆy) ∇x × n,m n ∞ ik ǫ = ETM (x)ETM (y) U (ˆy) − n(n + 1) µ ∇× n,m n,m · p,q n=1 m= n − X nX ∞ ik e + ETE (x)ETE (y) U (ˆy) n(n + 1) ∇× n,m n,m · p,q n=1 m= n − X X n ∞ ik ǫ e i 1 = ETM (x)− (kr)U (ˆy) U (ˆy) − µ ∇× n,m ωε r Jn n,m · p,q n=1 n(n + 1) m= n − X nX ∞ p ik + ETE (x)( 1)j (kr)V (ˆy) U (ˆy) ∇× n,m − n n,m · p,q n=1 n(n + 1) m= n X X− p for y = r and x > y . Therefore, we get on x = r | | | | | | | | S~k [U ] = G(x,y,k)U (ˆy) ∇× D n,m + ∇x × n,m Z y =r | | kr 1 TM = n(kr)( En,m (x)) x =r. (3.55) n(n + 1) ωµJ ∇× || |

Since p i i ETM = ETE = k2ETE, ∇× p,q ωε∇×∇× p,q ωε p,q we obtain

~k ikr TE xˆ SD[Un,m] = n(kr)(ˆx En,m(x)) x =r ×∇× + n(n + 1)J × || |

=( ikr)h(1)(kr) (kr)U on x = r, p− n Jn n,m | | which completes the proof. Note that

~k 1 k ν D[φ] =( I + D)[φ] on ∂D, ×∇× S ± ∓2 M and recall the following identity, which was proved in [90],

ν ~k [φ]= k [φ] on ∂D. ×∇×∇× SD LD 3.5. Explicit computations for a spherical nanoparticle 81

For m = n,...,n and n = 1, 2, 3,..., let H (∂D) be the subspace of − n,m H(∂D) deﬁned by

H (∂D)= span U ,V . n,m { n,m n,m} Let us represent the operators k and k explicitly on the subspace H (∂D). MD LD m,n Using Un,m,Vn,m as basis vectors, we obtain the following matrix representations for k and k on the subspace H (∂D): MD LD n,m

1 (1) ikrhn (kr) n(kr) 0 k = 2 − J , (3.56) MD  1  0 + ikrjn(kr) n(kr)  2 H    and

0 ik(kr)2j (kr)h(1)(kr) k = n n . (3.57) LD ik (kr) (kr) 0 − Jn Hn ! 3.5.3 Asymptotic behavior of the spectrum of (r) WB Now we consider the asymptotic expansions of the operator (r) and its WB spectrum when r 1. ≪ It is well-known that, as t 0, → tn 1 j (t)= 1 t2 + O(t4) , n (2n + 1)!! − 2(2n + 3) (1) n 1 1 2 4 h (t)= i((2n 1)!!)t− − 1+ t + O(t ) . (3.58) n − − 2(2n 1) − By making use of these asymptotics of the spherical Bessel functions, we obtain that

n n n+1 (1) n + 1 t 1 n + 1 t n + 3 t 3 i n(t)h (t˜)= + t˜ t + O(t ), J n 2n + 1 t˜ t˜ 2(2n 1)(2n + 1) t˜ − 2(2n + 1)(2n + 3) t˜ n − n n+1 n t 1 n + 2 t n t 3 ijn(t) n(t˜)= − + − t˜+ t + O(t ), H 2n + 1 t˜ t˜ 2(2n 1)(2n + 1) t˜ 2(2n + 1)(2n + 3) t˜ n − n n+1 (1) 1 t 1 1 t 1 t 3 ijn(t)h (t˜)= + t˜ t + O(t ), n 2n + 1 t˜ t˜ 2(2n 1)(2n + 1) t˜ − 2(2n + 1)(2n + 3) t˜ − ( n)(n+ 1) t n 1 (n + 1)( n + 2) t n n(n + 3) t n+1 i n(t) n(t˜)= − + − t˜+ t J H 2n + 1 t˜ t˜ 2(2n 1)(2n + 1) t˜ 2(2n + 1)(2n + 3) t˜ − + O(t3), (3.59) for small t, t˜ 1 with t t˜. ≪ ≈ So, we have

( 1) 2 − +(kr) rn 0 k 2(2n + 1) 4 D = + O(r ), (3.60) M  1 2  0 +(kr) sn  2(2n + 1)    82 Chapter 3. The Full Maxwell Equations and 2 0 k rpn k = n(n + 1) 1 + O(r3), (3.61) LD  + k2rq 0  2n + 1 r n where   1 p = , n 2n + 1 (n + 1)(n 2) n(n + 3) q = − , n 2(2n 1)(2n + 1) − 2(2n + 1)(2n + 3) − n + 1 (n + 3) r = + , n −2(2n 1)(2n + 1) 2(2n + 1)(2n + 3) − n 2 n s = − + . (3.62) n −2(2n 1)(2n + 1) 2(2n + 1)(2n + 3) − Therefore, we can obtain

(r)= + r + r2 + O(r3), WB WB,0 WB,1 WB,2 where ( 1) λ − 0 0 0 µ − 2(2n + 1)  1  0 λµ 0 0  − 2(2n + 1)  B,0 =   , W  ( 1)   0 0 λε − 0   − 2(2n + 1)   1   0 0 0 λ   ε   − 2(2n + 1)  (3.63) 

0 0 0 ωCµpn 0 0 ωCµqn 0 B,1 =   , (3.64) W 0 ωCεpn 0 0 ωC q 0 0 0   ε n   2  ω Dµrn 0 0 0 2 0 ω Dµsn 0 0 B,2 =  2  , (3.65) W 0 0 ω Dεrn 0  0 0 0 ω2D s   ε n   and µ ε µ ε µ ε µ ε C = c c − m m , C = c c − m m , (3.66) µ µ µ ε ε ε m − c m − c ε µ2 ε µ2 ε2µ ε2 µ D = c c − m m , D = c c − m m . (3.67) µ µ µ ε ε ε m − c m − c 3.5. Explicit computations for a spherical nanoparticle 83

By applying the standard perturbation theory, the asymptotics of eigenvalues of (r) are obtained as follows: up to an error term of order O(r3), WB ( 1) p q λ − +(rω)2 C C n n + D r + O(r3), µ − 2(2n + 1) ε µ λ λ + p µ n µ − ε n 1 p q λ +(rω)2 C C n n + D s + O(r3), µ − 2(2n + 1) ε µ λ λ p µ n µ − ε − n ( 1) p q λ − +(rω)2 C C n n + D r + O(r3), ε − 2(2n + 1) ε µ λ λ + p ε n ε − µ n 1 p q λ +(rω)2 C C n n + D s + O(r3), ε − 2(2n + 1) ε µ λ λ p ε n ε − µ − n and the asymptotics of the associated eigenfunction are given by C q [1, 0, 0, 0]T + rω ε n [0, 0, 0, 1]T + O(r2), λ λ + p µ − ε n C 1 [0, 1, 0, 0]T + rω ε [0, 0, 1, 0]T + O(r2), 2n + 1 λ λ p µ − ε − n C q [0, 0, 1, 0]T + rω µ n [0, 1, 0, 0]T + O(r2), λ λ + p ε − µ n C 1 [0, 0, 0, 1]T + rω µ [1, 0, 0, 0]T + O(r2). 2n + 1 λ λ p ε − µ − n 3.5.4 Extinction cross-section In this subsection, we compute the extinction cross-section Qext. We need the following lemma. Lemma 3.5.2. Let D be a sphere with radius r > 0 and suppose that Ei is given by

n i ∞ TE TE TM TM E (x)= αnl En,l (x; km)+ αnl En,l (x; km), n=1 l= n X X− e e TE TM for some coeﬃcients αnl ,αnl . Then the scattered wave can be represented as follows: for x >r, | | n s ∞ TE TE TE TM TM TM E (x)= αnl Sn En,l (x; km)+ αnl Sn En,l (x; km), n=1 l= n X X− TE TM where Sn and Sn are given by µ j (k r) (k r) µ j (k r) (k r) STE = c n c n m m n m n c , n J (1) − J µ (k r)hn (k r) µ j (k r) (k r) mJn c m − c n c H m ε j (k r) (k r) ε j (k r) (k r) STM = c n c n m m n m n c . n J (1) − J ε (k r)hn (k r) ε j (k r) (k r) mJn c m − c n c H m 84 Chapter 3. The Full Maxwell Equations

i TE Proof. Let E = En,l (x; km). We look for a solution of the following form:

e TE a En,l (x; kc), x r. e | | Then, from the boundarye condition on ∂D, we easily see that

(1) jn(kmr) jn(kcr) hn (kmr) a 1 = 1 −1 . (3.68) n(kmr) n(kcr) n(kmr) b µm J µc J − µm H ! Therefore, the coeﬃcient a and b can be obtained as follows:

1 (1) − (1) 1/a jn(kmr) hn (kmr) jn(kcr) hn (kcr) 1 = 1 1 1 1 , b/a n(kmr) n(kmr) n(kcr) n(kcr) 0 µm J µm H ! µc J µc H ! 1 (1) µmkmr (k r) h (k r) jn(kcr) µm n m n m = 1H − 1 , i n(kmr) jn(kmr) µ n(kcr) − µm J ! c J µm (1) n(kmr)jn(kcr) hn (kmr) n(kcr) H − µc J = ikmr , (3.69) −  µm  n(kmr)jn(kcr)+ jn(kmr) n(kcr) −J µc J    where we have used the following Wronskian identity for the spherical Bessel function:

(1) (1) (1) i j (t) (t) h (t) (t)= t j (t)(h )′(t) j′ (t)h (t) = . n Hn − n Jn n n − n n t Therefore, we immediately see that

µ j (k r) (k r) µ j (k r) (k r) b = c n c n m m n m n c . J (1) − J µ (k r)hn (k r) µ j (k r) (k r) mJn c m − c n c H m i TM Now suppose that E = En,l (x; km). We look for a solution in the following form: e TM c En,l (x; kc), x < r, E = TM TM | | (En,l (x; km)+ dEn,l (x; km), x > r. e | | Then, from the boundarye conditions on x = r, we obtain | | 1 1 1 1 n(kcr) n(kcr) c n(kmr) n(kmr) 1 εc J εc H = εm J εm H . j (k r) h(1)(k r) 0 j (k r) h(1)(k r) d n c n c ! n m n m ! (3.70)

By solving (3.70), we get

ε j (k r) (k r) ε j (k r) (k r) d = c n c n m m n m n c . J (1) − J ε (k r)hn (k r) ε j (k r) (k r) mJn c m − c n c H m By the principle of superposition, the conclusion immediately follows. 3.5. Explicit computations for a spherical nanoparticle 85

We also need the following lemma concerning the scattering amplitude A . ∞ Lemma 3.5.3. Suppose that the scattered electric ﬁeld Es is given by

n s ∞ TE TE TM TM E (x)= βnl En,l (x; km)+ βnl En,l (x; km) n=1 l= n X X− for R3 D. Then the scattering amplitude A can be represented as follows: \ ∞ n n ∞ 4π( i) TE µm TM A (ˆx)= − n(n + 1) βnl Vn,l(ˆx)+ βnl Un,l . ∞ ikm εm n=1 l= n r X X− p Proof. It is well-known that

(1) 1 it i n+1 π h (t) e e− 2 as t , n ∼ t → ∞ and (1) 1 it i n π (h )′(t) e e− 2 as t . n ∼ t → ∞ Then one can easily see that as x , | | → ∞

ikm x TE e | | i n+1 π En,m(x; km) e− 2 n(n + 1)Vn,l(ˆx) ∼− km x | | p and ikm x TM e | | µm i n+1 π En,m (x; km) e− 2 n(n + 1)Un,l(ˆx). ∼− km x εm | | r p By applying these asymptotics to the series expansion of Es, the conclusion follows.

A plane wave can be represented as a series expansion. The following lemma is proved in [66]. Lemma 3.5.4. Let Ei be a plane wave, that is, Ei(x)= peikmd x with d S · ∈ and p d = 0. Then we have the following series representation for a plane · wave as follows:

n i ∞ pw,TE TE pw,TM TM E (x)= αnl En,l (x; km)+ αnl En,l (x; km), n=1 l= n X X− e e where ( 1)4πin αpw,TE = − i V (d) p , nl n,l ·  n(n + 1) n  pw,TM p( 1)4πi εm αnl = − Un,l(d) p . n(n + 1) µm · r  Now we are ready to computep the extinction cross-section Qext. Theorem 3.5.1. Assume that Ei(x) = peikmd x with d S and p d = 0. · ∈ · Let D be a sphere with radius r. Then the extinction cross-section is given 86 Chapter 3. The Full Maxwell Equations by

n ∞ (4π)3 Qext = ( 1)STE(V (d) p)2 + iSTM (U (d) p)2 . k2 p 2 ℑ − n n,l · n n,l · n=1 l= n m| | X X− Moreover, for small r> 0, we have

1 3 ext ( 1)(4πkmr) 2 µc µm 2 2 εc εm 2 Q = − 2 2 i − (V1,l(d) p) + − (U1,l(d) p) k p ℑ 3 2µm + µc · 3 2εm + εc · l= 1 m X− | | 4 + O((kmr) ).

Proof. Let us ﬁrst compute the scattering amplitude A when Ei is a plane ∞ wave. From n ∞ 4π( i)n A (ˆx)= − n(n + 1) ∞ ikm n=1 l= n X X− p µ αpw,TESTEV (ˆx)+ m αpw,TM STM U × nl n n,l ε nl n n,l r m n 2 ∞ (4π) TE TM = ( 1)Sn (Vn,l(d) p)Vn,l + iSn (Un,l(d) p)Un,l . km − · · n=1 l= n X X− Therefore, we have

ext 4π p A (d) ∞ Q = · 2 km ℑ p n | | ∞ (4π)3 = ( 1)STE(V (d) p)2 + iSTM (U (d) p)2 . k2 p 2 ℑ − n n,l · n n,l · n=1 l= n m| | X X− Now we assume that r 1. By applying (3.58), one can easily see that ≪ 3 TE 2 (µc µm)(kmr) 4 S1 = i − + O(r ), 3 2µm + µc 3 TM 2 (εc εm)(kmr) 4 S1 = i − + O(r ), 3 2εm + εc STE,STM = O(r4), for n 2. n n ≥ Therefore, we obtain, up to an error term of order O(r4),

1 3 3 3 ext ( 1)(4π) 2 (µc µm)(kmr) 2 2 (εc εm)(kmr) 2 Q = − 2 2 i − (V1,l(d) p) + − (U1,l(d) p) . k p ℑ 3 2µm + µc · 3 2εm + εc · l= 1 m X− | | The proof is complete.

3.6 Explicit computations for a spherical shell

In this section we consider a spherical shell. Since, in this case, the eigenvalues associated with the corresponding Neumann-Poincaré operator are not 3.6. Explicit computations for a spherical shell 87 simple, we apply degenerate perturbation theory in order to compute the ﬁrst order eﬀect of the size.

3.6.1 Explicit representation of boundary integral operators

Let Ds and Dc be a spherical shell with radius rs and rc with rs > rc > 0. Let (εm,µm) in Dc, (ε,µ)= (ε ,µ ) in D D¯ ,  s s s \ c (ε ,µ ) in R3 D¯ . m m \ s Let   r ρ = c . rs The solution to the transmission problem can be represented as follows

µ ~kc [ψ ](x)+ ~kc [φ ](x) c Ds s Ds s ∇× S k∇×∇×c S kc +µc ~ [ψc](x)+ ~ [φc](x) x Dc,  ∇× SDc ∇×∇× SDc ∈  ks ks  µs ~ [ψs](x)+ ~ [φs](x) E(x)=  ∇× SDs ∇×∇× SDs  ~ks ~ks ¯  +µs [ψc](x)+ [φc](x) x Ds Dc,  ∇× SDc ∇×∇× SDc ∈ \ Ei + µ ~km [ψ ](x)+ ~km [φ ](x) m Ds s Ds s  ∇× S km ∇×∇× S km 3  +µm ~ [ψc](x)+ ~ [φc](x) x R D¯s,  ∇× SDc ∇×∇× SDc ∈ \  (3.71)  and  i H(x)= ( E)(x) x R3 ∂D, (3.72) −ωµD ∇× ∈ \ 1 2 1 2 where the pair (ψ ,φ ,ψ ,φ ) H− 2 (div,∂D ) H− 2 (div,∂D ) is the s s c c ∈ T s × T c unique solution to i ψs ψs ν E sh sh × i φs W W φs iων H W sh   := 11 12   =  ×  ψ W sh W sh ψ 0 c 21 22 c  φ   φ   0   c   c          with

µs + µm ks km ks km Id + µs D µm D D D sh 2 M s − M s L s −L s W11 =  k2 k2 k2 k2  , ks km s + m Id + s ks m km Ds Ds Ds Ds  L −L 2µs 2µm µs M − µm M   (3.73) 

~ks ~km ~ks ~km µsν D µmν D ν D ν D sh ×∇× S c − ×∇× S c ×∇×∇× S c − ×∇×∇× S c W = k2 k2 , 12  ν ~ks ν ~km s ν ~ks m ν ~km  Dc Dc Dc Dc ×∇×∇× S − ×∇×∇× S µs ×∇× S − µm ×∇× S  (3.74)  88 Chapter 3. The Full Maxwell Equations

~kc ~ks ~kc ~ks µcν D + µsν D ν D + ν D sh − ×∇× S s ×∇× S s − ×∇×∇× S s ×∇×∇× S s W = k2 k2 , 21  ν ~kc + ν ~ks c ν ~kc + s ν ~ks  Ds Ds Ds Ds − ×∇×∇× S ×∇×∇× S −µc ×∇× S µs ×∇× S  (3.75) 

µc + µs kc ks kc ks Id µc D + µs D D + D sh − 2 − M c M c −L c L c W22 =  k2 k2 k2 k2  . kc + ks c + s Id c kc + s ks Dc Dc Dc Dc  −L L − 2µc 2µs − µc M µs M   (3.76)  sh sh Note that W11 and W22 are similar to the operator in left-hand side of (3.13). In the previous section for the sphere case, we have already obtained the matrix representation of this operator and its asymptotic expansion. ~k By Lemma 3.5.1, we can represent ν D x =r′ and ν ×∇× S || | ×∇×∇× ~k D x =r′ in a matrix form as follows(using Un,m,Vn,m as basis): S || | (i) For r′ >r,

(1) ~k ( ikr) n(kr)hn (kr′) 0 ν D x =r′ = − J r2 , ×∇× S || | 0 ik ′ jn(kr) n(kr′) ! r H (3.77)

2 (1) ~k 0 ik(kr) jn(kr)hn (kr′) ν D x =r′ = r ; ×∇×∇× S || | ik (kr) (kr ) 0 − r′ Jn Hn ′ ! (3.78)

(ii) For r′

~k ( ikr)jn(kr′) n(kr) 0 ν D x =r′ = − H r2 (1) , ×∇× S || | 0 ik ′ (kr ) n (kr) r Jn ′ h ! (3.79)

2 (1) ~k 0 ik(kr) jn(kr′)hn (kr) ν D x =r′ = r . ×∇×∇× S || | ik (kr ) (kr) 0 − r′ Jn ′ Hn ! (3.80)

sh Using the above formulas, the matrix representation of the operators W12 sh and W21 can be easily obtained. We now consider scaling of W sh. First, we need some definitions. Let Ds = z + rsBs where Bs contains the origin and Bs = O(1). Let Bc be | | x z defined in a similar way. For any x ∂Ds (or ∂Dc), let x = − ∂Bs (or ∈ rs ∈ ∂Bc with rs replaced by rc) and define for each function f defined on ∂Ds (or ∂Dc), a corresponding function defined on B as followse

ηs(f)(x)= f(z + rsx), ηc(f)(x)= f(z + rcx). (3.81)

e e e e 3.6. Explicit computations for a spherical shell 89

Then, in a similar way to the sphere case, let us write

η(ν Ei) × ηs(ψs) µm µs η(iν −Hi) sh ωηs(φs)  ×  WB (rs)   = εm εs . ηc(ψc) 0−    ωηc(φc)   0          Using (U ,V ,U ,V ) (U ,V ,U ,V ) as basis, we can n,m n,m n,m n,m × n,m n,m n,m n,m represent W sh(r ) in a 8 8 matrix form in a subspace H (∂B ) H (∂B ). B s × n,m s × n,m c Then, by using (3.59), their asymptotic expansion can also be obtained. sh Here, the resulting asymptotics of the matrix WB are given as follows. Write sh(r )= sh + r sh + r2 sh + O(r3), (3.82) WB s WB,0 sWB,1 s WB,2 s where

sh Λµ,ε P0,n Q0,n B,0 = + , (3.83) W Λµ,ε R0,n P0,n −

sh P1,n Q1,n sh P2,n Q2,n B,1 = , B,2 = . W R1,n P1,n W R2,n P2,n − − Here, the matrix Pj,n,Qj,n and Rj,n are given by

λµ pn λµ pn Λµ,ε =   ,P0,n =  −  , λε pn  λ   p   ε  − n    

gn fn 2 fn gn Q0,n = ρ   , R0,n =   , gn fn  f   g   n  n    

Cµpn Dµrn Cµqn 2 Dµsn P1,n = ω   ,P2,n = ω   , Cεpn Dεrn C q   D s   ε n   ε n    

Cµpñ Dµrñ Cµqñ 2 Dµsñ Q1,n = ωρ   , Q2,n = ω ρ   , Cεpñ Dεrñ  C q˜   D s˜   ε n   ε n     

Cµpñ 1 Cµqñ R1,n = ωρ−   , − Cεpñ  C q˜   ε n    90 Chapter 3. The Full Maxwell Equations

Dµsñ 2 1 Dµrñ R2,n = ω ρ−  −  . Dεsñ  D r˜   − ε n    Here, pn,qn,rn,sn are defined as (3.62) and pñ, qñ, rñ, sñ,Dµ and Dε are defined as follows:

n n n 1 n + 1 f = ρ , g = ρ − , (3.84) n 2n + 1 n 2n + 1 1 p˜ = ρn+1, (3.85) n 2n + 1 (n + 1)(n 2) n(n + 3) q˜ = − ρn ρn+2, (3.86) n 2(2n 1)(2n + 1) − 2(2n + 1)(2n + 3) − n + 1 (n + 3) r˜ = ρn + ρn+2, (3.87) n −2(2n 1)(2n + 1) 2(2n + 1)(2n + 3) − n 2 n s˜ = − ρn+1 + ρn+3, (3.88) n −2(2n 1)(2n + 1) 2(2n + 1)(2n + 3) − and ε µ2 ε µ2 ε2µ ε2 µ D = s s − m m , D = s s − m m . (3.89) µ µ µ ε ε ε m − s m − s 3.6.2 Asymptotic behavior of the spectrum of sh(r ) WB s Let us deﬁne 1 λsh = 1 + 4n(n + 1)ρ2n+1. n 2(2n + 1) p Note that λsh are eigenvalues of the Neumann-Poincaré operator on the ± n shell. sh It turns out that the eigenvalues of WB,0 are as follows

λ + λsh, λ λsh, λ + λsh, λ λsh, µ n µ − n ε n ε − n for n = 0, 1, 2,..., and their multiplicities is 2. Their associated eigenfunctions are as follows:

λ + λsh E0 := (λsh + p )e + f e , E0 := (λsh p )e + g e , µ n −→ 1 n n 1 n 5 2 n − n 2 n 6 λ λsh E0 := ( λsh + p )e + f e , E0 := ( λsh p )e + g e , µ − n −→ 3 − n n 1 n 5 4 − n − n 2 n 6 λ + λsh E0 := (λsh + p )e + f e , E0 := (λsh p )e + g e , ε n −→ 5 n n 3 n 7 6 n − n 4 n 8 λ λsh E0 := ( λsh + p )e + f e , E0 := ( λsh p )e + g e , ε − n −→ 7 − n n 3 n 7 8 − n − n 4 n 8 where e 8 is standard unit basis in R8. { i}i=1 To derive asymptotic expansions of the eigenvalues, we apply degenerate eigenvalue perturbation theory (since the multiplicity of each of these 3.6. Explicit computations for a spherical shell 91 eigenvalues is 2). To state the result, we need some deﬁnitions. Let

(λsh p )a b ( λsh p )a b T = C n − n 1,n − 1,n , T = C − n − n 1,n − 1,n , 16,n ε E0 E0 18,n ε E0 E0 | 1 || 6 | | 1 || 8 | (λsh + p )a b ( λsh + p )a b T = C n n 2,n − 2,n , T = C − n n 2,n − 2,n , 25,n ε E0 E0 27,n ε E0 E0 | 2 || 5 | | 2 || 7 | (λsh p )a b ( λsh p )a b T = C n − n 3,n − 3,n , T = C − n − n 3,n − 3,n , 36,n ε E0 E0 38,n ε E0 E0 | 3 || 6 | | 3 || 8 | (λsh + p )a b ( λsh + p )a b T = C n n 4,n − 4,n , T = C − n n 4,n − 4,n , 45,n ε E0 E0 47,n ε E0 E0 | 4 || 5 | | 4 || 7 | Cµ Cµ Cµ Cµ T52,n = T16,n, T54,n = T18,n, T61,n = T25,n, T63,n = T27,n, Cε Cε Cε Cε Cµ Cµ Cµ Cµ T72,n = T36,n, T74,n = T38,n, T81,n = T45,n, T83,n = T47,n, Cε Cε Cε Cε where

sh a1,n =(λn + pn)qn + ρfnq˜n, a =(λsh p )p + ρg p˜ , 2,n n − n n n n a =( λsh + p )q + ρf q˜ , 3,n − n n n n n a =( λsh p )p + ρg p˜ , 4,n − n − n n n n and

1 sh b1,n = fngnqn + ρ− (λn + pn)gnq˜n, 1 sh b = f g p + ρ− (λ p )f p˜ , 2,n n n n n − n n n 1 sh b = f g q + ρ− ( λ + p )g q˜ , 3,n n n n − n n n n 1 sh b = f g p + ρ− ( λ p )f p˜ . 4,n n n n − n − n n n We also deﬁne (λsh + p )((λsh + p )r + ρf r˜ )+ f ((λsh + p )ρ 1s˜ f r ) K = D n n n n n n n n n n − n − n n , 1,n µ E0 2 | 1 | g (( λsh + p )ρ 1r˜ g s )+(λsh p )((λsh p )s + ρg s˜ ) K = D n − n n − n − n n n − n n − n n n n , 2,n µ E0 2 | 2 | ( λsh + p )(( λsh + p )r + ρf r˜ )+ f (( λsh + p )ρ 1s˜ f r ) K = D − n n − n n n n n n − n n − n − n n , 3,n µ E0 2 | 3 | g ((λsh + p )ρ 1r˜ g s )+( λsh p )(( λsh p )s + ρg s˜ ) K = D n n n − n − n n − n − n − n − n n n n , 4,n µ E0 2 | 4 | Dε Dε Dε Dε K5,n = K1,n, K6,n = K2,n, K7,n = K3,n, K8,n = K4,n. Dµ Dµ Dµ Dµ 92 Chapter 3. The Full Maxwell Equations

Now we are ready to state the result. The followings are asymptotics of sh eigenvalues of WB (rs) T T T T λ + λ +(r ω)2 16,n 61,n + 18,n 81,n + K + O(r3), µ ε s λ λ λ λ + 2λsh 1,n s µ − ε µ − ε n T T T T λ + λ +(r ω)2 16,n 61,n + 18,n 81,n + K + O(r3), µ ε s λ λ λ λ + 2λsh 2,n s µ − ε µ − ε n T T T T λ λ +(r ω)2 36,n 63,n + 38,n 83,n + K + O(r3), µ − ε s λ λ 2λsh λ λ 3,n s µ − ε − n µ − ε T T T T λ λ +(r ω)2 36,n 63,n + 38,n 83,n + K + O(r3), µ − ε s λ λ 2λsh λ λ 4,n s µ − ε − n µ − ε T T T T λ + λ +(r ω)2 52,n 25,n + 54,n 45,n + K + O(r3), ε µ s λ λ λ λ + 2λsh 5,n s ε − µ ε − µ n T T T T λ + λ +(r ω)2 52,n 25,n + 54,n 45,n + K + O(r3), ε µ s λ λ λ λ + 2λsh 6,n s ε − µ ε − µ n T T T T λ λ +(r ω)2 72,n 27,n + 74,n 47,n + K + O(r3), ε − µ s λ λ 2λsh λ λ 7,n s ε − µ − n ε − µ T T T T λ λ +(r ω)2 72,n 27,n + 74,n 47,n + K + O(r3). ε − µ s λ λ 2λsh λ λ 8,n s ε − µ − n ε − µ We also have the following asymptotic expansions of the eigenfunctions:

T T E0 + r ω 16,n E0 + 18,n E0 + O(r2), 1 s λ λ 6 λ λ + 2λsh 8 s µ − ε µ − ε n T T E0 + r ω 25,n E0 + 27,n E0 + O(r2), 2 s λ λ 5 λ λ + 2λsh 7 s µ − ε µ − ε n T T E0 + r ω 36,n E0 + 38,n E0 + O(r2), 3 s λ λ 2λsh 6 λ λ 8 s µ − ε − n µ − ε T T E0 + r ω 45,n E0 + 47,n E0 + O(r2), 4 s λ λ 2λsh 5 λ λ 7 s µ − ε − n µ − ε T T E0 + r ω 52,n E0 + 54,n E0 + O(r2), 5 s λ λ 2 λ λ + 2λsh 4 s µ − ε µ − ε n T T E0 + r ω 61,n E0 + 63,n E0 + O(r2), 6 s λ λ 1 λ λ + 2λsh 3 s µ − ε µ − ε n T T E0 + r ω 72,n E0 + 74,n E0 + O(r2), 7 s λ λ 2λsh 2 λ λ 4 s µ − ε − n µ − ε T T E0 + r ω 81,n E0 + 83,n E0 + O(r2). 8 s λ λ 2λsh 1 λ λ 3 s µ − ε − n µ − ε Interestingly, the ﬁrst-order term (of order δ) is still zero in the asymptotic expansions of the eigenvalues. This is due to the fact that degenerate eigenfunctions does not interact with each other.

3.7 Concluding remarks

In this chapter, we have given the ﬁrst rigorous detailed description of the scaling behavior of plasmonic resonances for the full Maxwell equations, improving our understanding of light scattering by plasmonic nanoparticles. 3.7. Concluding remarks 93

The particle dimension and interparticle distances are considered to be in- ﬁnitely small compared with the wavelength of the interacting light. We have also shown formulas indicating the blow up rate of the extinction cross section at the plasmonic resonance and give explicit formulas for the case of spherical and spherical shell nanoparticles.

Part II

Applications

Chapter 4

Heat Generation with Plasmonic Nanoparticles

Contents 4.1 Introduction ...... 98 4.2 Setting of the problem and the main results . . . 98 4.3 Heat generation ...... 101 4.3.1 Small volume expansion of the inner ﬁeld . . . . . 102 4.3.2 Small volume expansion of the temperature . . . . 106 4.3.3 Temperature elevation at the plasmonic resonance 109 4.3.4 Temperature elevation for two close-to-touching particles ...... 110 4.4 Numerical results ...... 111 4.4.1 Single-particle simulation ...... 112 4.4.2 Two particles simulation ...... 115 4.5 Concluding remarks ...... 116 98 Chapter 4. Heat Generation with Plasmonic Nanoparticles

4.1 Introduction

Our aim in this chapter is to provide a mathematical and numerical framework for analyzing photothermal effects using plasmonic nanoparticles. At or near the plasmonic resonant frequencies, strong enhancement of scattering and absorption occurs, see chapter 2, 3 and [86]. This translates into an efficient heat generation in the presence of electromagnetic radiation. More- over, plasmonic nanoparticles biocompatibility makes them suitable for use in nanotherapy [36]. Nanotherapy relies on a simple mechanism. First nanoparticles become attached to tumor cells using selective biomolecular linkers. Then heat generated by optically-simulated plasmonic nanoparticles destroys the tumor cells [51]. In this nanomedical application, the temperature increase is the most important parameter [71,83]. It depends in a highly nontrivial way on the shape, the number, and the arrangement of the nanoparticles. Moreover, it is challenging to measure it at the surface of the nanoparticles [51]. In this chapter, we derive an asymptotic formula for the temperature at the surface of plasmonic nanoparticles of arbitrary shapes. Our formula holds for clusters of simply connected nanoparticles. It allows to estimate the collective response of plasmonic nanoparticles. In particular, it shows that the total amount of heat generated by two interacting nanoparticles is significantly different from the heat created by two single nanoparticles. The more interacting nanoparticles, the stronger the temperature increase. Our results in this chapter formally explain the experimental observations reported in [51]. The chapter is organized as follows. In section 4.2 we describe the mathematical setting for the physical phenomena we are modeling. To this end, we use the Helmholtz equation to model the propagation of light which we couple to the heat equation. Later on, we present our main results in this chapter which consist on original asymptotic formulas for the inner field and the temperature on the boundaries of the nanoparticles. In section 4.3 we prove Theorems 4.2.1 and 4.2.2. These results clarify the strong dependency of the heat generation on the geometry of the particles as it depends on the eigenvalues of the associated Neumann-Poincaré operator. In section 4.4 we present numerical examples of the temperature at the boundary of single and multiple particles.

4.2 Setting of the problem and the main results

In this chapter, we use the Helmholtz equation for modeling the propagation of light. This can be thought of as a special case of Maxwell’s equations, when the incident wave ui is a transverse electric or transverse magnetic (TE or TM) polarized wave. This approximation, also called paraxial approximation [54], is a good model for a laser beam which are used, in particular, in full- ﬁeld optical coherence tomography. We will therefore model the propagation of a laser beam in a host domain (tissue), hosting a nanoparticle. Let the nanoparticle occupy a bounded domain D ⋐ R2 of class 1,α for C some 0 <α< 1. Furthermore, let D = z + δB, where B is centered at the origin and B = O(1). | | We denote by ε (x) and µ (x), x D, the electric permittivity and c c ∈ magnetic permeability of the particle, respectively, both of which may depend 4.2. Setting of the problem and the main results 99

on the frequency ω of the incident wave. Assume that εc(x)= ε0εc′ , µc(x)= µ µ and that ε < 0, ε > 0, µ < 0, µ > 0. Here and throughout, ε 0 c′ ℜ c′ ℑ c′ ℜ c′ ℑ c′ 0 and µ0 are the permittivity and permeability of vacuum. Similarly, we denote by ε (x) = ε ε and µ (x) = µ µ , x R2 D m 0 m′ m 0 m′ ∈ \ the permittivity and permeability of the host medium, both of which do not depend on the frequency ω of the incident wave. Assume that εm and µm are real and strictly positive. The index of refraction of the medium (with the nanoparticle) is given by

n(x)= ε µ χ(D)(x)+ ε µ χ(R2 D)(x), c′ c′ m′ m′ \ where χ denotes thep indicator function.p The scattering problem for a TE incident wave ui is modeled as follows:

c2 u + ω2u = 0 in R2 ∂D, ∇· n2 ∇ \   u+ u = 0 on ∂D,  − −   1 ∂u 1 ∂u  = 0 on ∂D, εm ∂ν + − εc ∂ν −  s i  u := u u satisfies the Sommerfeld radiation condition at infinity,  −  (4.1) where ∂ denotes the outward normal derivative and c = 1 is the speed ∂ν √ε0µ0 ∂ of light in vacuum. We use the notation ∂ν indicating ± ∂u (x)= lim u(x tν(x)) ν(x), ∂ν t 0+ ∇ ± · ± → with ν being the outward unit normal vector to ∂D. The interaction of the electromagnetic waves with the medium produces a heat flow of energy which translates into a change of temperature governed by the heat equation [37]

∂τ ω ρC γ τ = (ε) u 2 in (R2 ∂D) (0,T ), ∂t −∇· ∇ 2π ℑ | | \ ×  τ τ = 0 on ∂D,  +  − − (4.2)  ∂τ ∂τ  γ γ = 0 on ∂D,  m ∂ν − c ∂ν + −  τ(x, 0) = 0,    whereρ = ρ χ(D)+ρ χ(R2 D) is the mass density, C = C χ(D)+C χ(R2 D) c m \ c m \ is the thermal capacity, γ = γ χ(D)+ γ χ(R2 D) is the thermal conductiv- c m \ ity, T R is the ﬁnal time of measurements and ε = ε χ(D)+ ε χ(R2 D). ∈ c m \ We further assume that ρc, ρm,Cc,Cm,γc,γm are real positive constants. Note that (ε) = 0 in R2 D and so, outside D, the heat equation is ℑ \ homogeneous. The coupling of equations (4.1) and (4.2) describes the physics of our problem. 100 Chapter 4. Heat Generation with Plasmonic Nanoparticles

We remark that, in general, the index of refraction varies with temperature; hence, a solution to the above equations would imply a dependency on time for the electric field u, which contradicts the time-harmonic assumption leading to model (4.1). Nevertheless, the time-scale on the dynamics of the index of refraction is much larger than the time-scale on the dynamics of the interaction of the electromagnetic wave with the medium. Therefore, we will not integrate a time-varying component into the index of refraction. Let G( ,k) be the Green function for the Helmholtz operator ∆ + k2 · satisfying the Sommerfeld radiation condition. In dimension two, G is given by i G(x,k)= H(1)(k x ), −4 0 | | (1) where H0 is the Hankel function of first kind and order 0. We denote G(x,y,k) := G(x y,k). − Recall the definition of the single-layer potential and Neumann-Poincaré integral operator for the Helmholtz equation

k [ϕ](x)= G(x,y,k)ϕ(y)dσ(y), x ∂D or x R2, SD ∈ ∈ Z∂D and k ∂G(x,y,k) ( )∗[ϕ](x)= ϕ(y)dσ(y), x ∂D. KD ∂ν(x) ∈ Z∂D Our main results in this chapter are the following. Theorem 4.2.1. For an incident wave ui 2(R2), the solution u to (4.1), ∈C inside a plasmonic particle occupying a domain D = z+δB, has the following asymptotic expansion as δ 0 in L2(D), → 3 i 1 i δ u = u (z)+ δ(x z)+ λ Id ∗ − [ν] u (z)+O , − SD ε −KD ·∇ dist(λ ,σ( )) ε KD∗ where ν is the outward normal to D, σ( D∗ ) denotes the spectrum of D∗ in 1 K K H− 2 (∂D) and ε + ε λ := c m . ε 2(ε ε ) c − m Theorem 4.2.2. Let u be the solution to (4.1). The solution τ to (4.2) on the boundary ∂D of a plasmonic particle occupying the domain D = z + δB has the following asymptotic expansion as δ 0, uniformly in (x,t) ∂D → ∈ × (0,T ),

4 bc 1 ∂FD( , ,bc) δ log δ τ(x,t)= F (x,t,b ) (λ Id ∗ )− [ · · ](x,t)+ O , D c −VD γ −KD ∂ν dist(λ ,σ( ))2 ε KD∗ 4.3. Heat generation 101 where ν is the outward normal to D and γ + γ λ := c m , γ 2(γ γ ) c − m ρcCc bc := , γc |x−y|2 t 4bc(t−t′) ω e− 2 F (x,t,b ) := (ε ) u (y)dydt′, D c 2πγ ℑ c 4πb (t t )| | c Z0 ZD c − ′ |x−y|2 t − 4bc(t−t′) bc e [f](x,t) := f(y,t′)dydt′. VD 4πb (t t ) Z0 Z∂D c − ′ Remark 4.2.1. We remark that Theorem 4.2.1 and Theorem 4.2.2 are independent. A generalization of Theorem 4.2.2 to R3 is straightforward and the same type of small volume approximation can be found using the techniques presented in this chapter. In fact, in R3, the operators involved in the ﬁrst term of the temperature small volume expansion are

|x−y|2 t 4bc(t−t′) ω e− 2 FD(x,t,bc) := (εc) 3 E (y)dydt′, 2πγc ℑ 0 D 4πb (t t ) 2 | | Z Z c − ′ |x−y|2 t − 4bc(t−t′) bc e D [f](x,t) := 3 f(y,t′)dydt′. V 0 ∂D 4πb (t t ) 2 Z Z c − ′ Here E is the vectorial electric ﬁeld as a result of Maxwell equations. A small volume expansion for E inside the nanoparticle for the plasmonic case can be found using the same techniques of chapter 3.

4.3 Heat generation

In this section we consider the coupling of equations (4.1) and (4.2), that is,

c2 u + ω2u = 0 in R2 ∂D, ∇· n2 ∇ \   u+ u = 0 on ∂D,  − −   1 ∂u 1 ∂u  = 0 on ∂D,  εm ∂ν + − εc ∂ν  −  s i  u := u u satisﬁes the Sommerfeld radiation condition at inﬁnity,  −   ρcCc ∂τ ω 2  ∆τ = (εc) u in D (0,T ), (4.3)  γc ∂t − 2πγc ℑ | | ×  ρ C ∂τ m m ∆τ = 0 in (R2 D) (0,T ),  γm ∂t − \ ×    τ+ τ = 0 on ∂D,  − −  ∂τ ∂τ   γm γc = 0 on ∂D,  ∂ν + − ∂ν  −   τ(x, 0) = 0.    102 Chapter 4. Heat Generation with Plasmonic Nanoparticles

Under the assumption that the index of refraction n does not depend on the temperature, we can solve equation (4.1) separately from equation (4.2). Our goal is to establish a small volume expansion for the resulting temperature at the surface of the nanoparticle as a function of time. To do so, we ﬁrst need to compute the electric ﬁeld inside the nanoparticle as a result of a plasmonic resonance. The results of the following sections rely heavily on the use of layer potentials for the Helmholtz equation. We refer to chapter 2 for a summary.

4.3.1 Small volume expansion of the inner ﬁeld We proceed in this section to prove Theorem 4.2.1.

Rescaling Since we are working with nanoparticles, we want to rescale equation (2.2) to study the solution for a small volume approximation by using representation (2.1). x z Recall that D = z + δB. For any x ∂D, x := − ∂B and for each ∈ δ ∈ function f defined on ∂D, we introduce a corresponding function defined on ∂B as follows e η(f)(x)= f(z + δx). (4.4) It follows that e e k δk D[ϕ](x) = δ B [η(ϕ)](x), S S (4.5) k δk ( D)∗[ϕ](x) = ( B )∗[η(ϕ)](x), K K e so system (2.2) becomes e η(ui) δkm [η(ψ)] δkc [η(φ)] = , SB −SB − δ  1 ∂ui  1 1 δkm 1 1 δkc  ε 2 Id +( B )∗ [η(ψ)] + ε 2 Id ( B )∗ [η(φ)] = η( ). m K c − K −εm ∂ν (4.6)   Note that the system is defined on ∂B. For δ small enough δkm is invertible (see Appendix B.3). Therefore, SB i δkm 1 δkc δkm 1 η(u ) η(ψ)=( )− [η(φ)] ( )− [ ]. SB SB − SB δ Hence, we have the following equation for η(φ):

I (δ)[η(φ)] = f I , AB where

I 1 1 δkm δkm 1 δkc 1 1 δkc (δ) = I +( )∗ ( )− + Id ( )∗ , AB εm 2 KB SB SB εc 2 − KB i i (4.7) I 1 ∂u 1 1 δkm δkm 1 η(u ) f = η( )+ ε 2 Id +( B )∗ ( B )− [ ]. −εm ∂ν m K S δ 4.3. Heat generation 103

Proof of Theorem 4.2.1 To express the solution to (4.1) in D, asymptotically on the size of the nanoparticle δ, we make use of the representation (2.1). We derive an asymptotic expansion for η(φ) on δ to later compute δ δkc [η(φ)] and scale back to SB D. We divide the proof into three steps. Step 1. We first compute an asymptotic for I (δ) and f I . AB Let (∂B) be defined by (A.3) with D replaced by B. In ( (∂B)), H∗ L H∗ we have the following asymptotic expansion as δ 0 (see Appendix B.3) → δk 1 δk 2 m c ∗ ( )− = + δkm ( B +Υδkc )+ O(δ log δ), SB SB PH0 U S 1 δk 1 2 Id ( )∗ = Id ∗ + O(δ log δ). 2 ± KB 2 ±KB e Let ϕ be an eigenfunction of associated to the eigenvalue 1/2 (see 0 KB∗ Appendix A) and let be defined by (B.12) with k replaced with δk . Uδkm m Then it follows that 1 Id + ∗ = . 2 KB Uδkm Uδkm Therefore, in ( (∂B)), L H∗

I 1 1 1 1 1 2 ∗ B(δ)= + Id + B∗ + δkm ( B +Υδkc )+ O(δ log δ), A 2ε 2ε ε − ε K PH0 ε U S m c m c m and from the deﬁnition of we get e Uδkm

I 1 1 1 1 1 B[ϕ0]+ τδkc 2 B(δ)= + Id + B∗ ∗ + S ( ,ϕ0) ∗ ϕ0+O(δ log δ). A 2ε 2ε ε − ε K PH0 ε [ϕ ]+ τ · H m c m c m SB 0 δkm (4.8) In the same manner, in the space (∂B), H∗ i i i I 1 ∂u 1 1 η(u ) η(u ) 2 ∗ f = η( )+ Id + B∗ − [ ]+ δkm [ ]+ O(δ log δ) . ε − ∂ν 2 K PH0 SB δ U δ m We can further develop f I . Indeed, fore every x˜ ∂B, a Taylor expansion ∈ yields

∂ui η( )(˜x) = ν(˜x) ui(δx˜ + z) = ν(˜x) ui(z)+ O(δ), ∂ν ·∇ ·∇ η(ui) ui(δx˜ + z) ui(z) (˜x) = = +x ˜ ui(z)+ O(δ). δ δ δ ·∇ The regularity of ui ensures that the previous formulas hold in (∂B). H∗ The fact that x˜ ui(z) is harmonic in B and Lemma A.0.4 imply that ·∇

i 1 1 i ν u (z)=( Id B∗ ) ∗ − [˜x u (z)] − ·∇ 2 −K PH0 SB ·∇ in (∂B). e H∗ Thus, in (∂B), H∗ i I 1 1 i u (z) i ∗ f = − [˜x u (z)] + δkm [ +x ˜ u (z)] + O(δ) . ε PH0 SB ·∇ U δ ∇ m e 104 Chapter 4. Heat Generation with Plasmonic Nanoparticles

From the deﬁnition of we get Uδkm i 1 i I 1 1 i u (z)ϕ0 ( B− [˜x u (z)],ϕ0) ∗ ϕ0 f = ∗ − [˜x u (z)] + S ·∇ H + O(δ) . H0 B εm P S ·∇ δ( B[ϕ0]+ τδkm ) − B[ϕ0]+ τδkm ! S e S e (4.9) Step 2. We compute ( I (δ)) 1f I . AB − We begin by computing an asymptotic expansion of ( I (δ)) 1. AB − I 1 1 1 1 The operator := + Id + ∗ maps ∗ into ∗. A0 2εm 2εc εm − εc KB H0 H0 Hence, the operator deﬁned by (which appears in the expansion of I (δ)) AB

I I 1 B[ϕ0]+ τδkc B,0 := 0 ∗ + S ( ,ϕ0) ∗ ϕ0, A A PH0 ε [ϕ ]+ τ · H m SB 0 δkm is invertible of inverse

I 1 I 1 B[ϕ0]+ τδkm ( B,0)− =( 0)− ∗ + εm S ( ,ϕ0) ∗ ϕ0. A A PH0 [ϕ ]+ τ · H SB 0 δkc Therefore, we can write

I 1 I 1 2 1 I 1 ( )− (δ)= Id +( )− O(δ log δ) − ( )− . AB AB,0 AB,0 Since is a compact self-adjoint operator in (∂B) it follows that KB∗ H∗ I 1 c ( 0)− ( ∗(∂B)) , (4.10) k A kL H ≤ dist(0,σ( I )) A0 for a constant c. Therefore, for δ small enough, we obtain

I 1 I I 1 2 1 I 1 I ( (δ))− f = Id +( )− O(δ log δ) − ( )− f AB AB,0 AB,0 i 1 i I 1 2 1 u (z)ϕ0 ( B− [˜x u (z)],ϕ0) ∗ ϕ0 = Id +( )− O(δ log δ) − S ·∇ H + AB,0 δ( [ϕ ]+ τ ) − [ϕ ]+ τ SB 0 δkc SB 0 δkc e I 1 1 1 i δ ( 0)− ∗ − [˜x u (z)] + O A ε PH0 SB ·∇ dist(0,σ( I )) m A 0 ! 1 i i e ∗ u (z)ϕ0 ( B− [˜x u (z)],ϕ0) ϕ0 I 1 1 1 i = S ·∇ H +( 0)− ∗ − [˜x u (z)] + δ( [ϕ ]+ τ ) − [ϕ ]+ τ A ε PH0 SB ·∇ SB 0 δkc SB 0 δkc m δ e O . e dist(0,σ( I )) A 0 4.3. Heat generation 105

Using the representation formula of described in Lemma A.0.2 we can KB∗ further develop the third term in the above expression to obtain

1 i I 1 1 i ∞ ( B− [˜x u (z)],ϕj) ∗ ϕj ( 0)− ∗ − [˜x u (z)] = S ·∇ H H0 B 1 εm εm A P S ·∇ 2 + 2ε ε 1 λj Xj=1 e c − c − e 1 i ∞ ( B− [˜x u (z)],ϕj) ∗ ϕj 1 i = S ·∇ H ( − [˜x u (z)],ϕj) ∗ ϕj 1 εm εm B H 2 + 2ε ε 1 λj − S ·∇ ! Xj=1 e c − c − 1 i + ∗ − [˜x u (z)] e PH0 SB ·∇ 1 i 1 i ∞ 1 ( B− [˜x u (z)],ϕj) ∗ ϕj = ∗ [˜x u (z)] + (λ ) S ·∇ H . 0 B− e j PH S ·∇ − 2 λ λj Xj=1 e − e Using the same arguments as those in the proof of Lemma A.0.4, we have

i 1 1 i (ν u (z),ϕj) ∗ (λj )( − [˜x u (z)],ϕj) ∗ = ·∇ H , − 2 SB ·∇ H λ 1 j − 2 and consequently, e

I 1 1 1 i 1 i 1 i ( ) ∗ [˜x u (z)] = ∗ [˜x u (z)]+(λ Id ) [ν] u (z). 0 − 0 B− 0 B− ε B∗ − A εm PH S ·∇ PH S ·∇ −K ·∇ Therefore, e e

i 1 i I 1 I u (z)ϕ0 ( B− [˜x u (z)],ϕ0) ∗ ϕ0 1 i ( B(δ))− f = S ·∇ H + ∗ − [˜x u (z)] + A δ( [ϕ ]+ τ ) − [ϕ ]+ τ PH0 SB ·∇ SB 0 δkc SB 0 δkc e δ (λ Id ) 1[ν] ui(z)+ O e . ε −KB∗ − ·∇ dist(0,σ( I )) A 0

Step 3. Finally, we compute η(u)= δ δkc ( I (δ)) 1f I . SB AB − From Appendix B.3, the following holds when δkc is viewed as an oper- SB ator from the space (∂B) to (∂B): H∗ H δkc = +Υ + O(δ2 log δ). SB SB δkc In particular, we have e

δkc [ϕ ]= [ϕ ]+ τ + O(δ2 log δ). SB 0 SB 0 δkc It can be veriﬁed that the same expansion holds when viewed as an operator from (∂B) into L2(B). H∗ Note that the following identity holds

1 i ( B− [˜x u (z)],ϕ0) ∗ ϕ0 1 i S ·∇ H + ∗ − [˜x u (z)] = − [ϕ ]+ τ PH0 SB ·∇ SB 0 δkc e 1 i Υδkc B− [˜x u (z)] ϕ0 1e i S ·∇ + − [˜x u (z)]. − [ϕ ]+ τ SB ·∇ B 0 δkc Se e 106 Chapter 4. Heat Generation with Plasmonic Nanoparticles

Straightforward calculations and the fact that is harmonic in B yields SB 2 δkc I 1 I i 1 i δ δ ( (δ))− f = u (z)+ δ x˜ + λ Id ∗ − [ν] u (z)+ O SB AB SB ε −KB ·∇ dist(λ ,σ( )) ε KB∗ in L2(B). Using Lemma A.0.3 to scale back the estimate to D leads to the desired result.

4.3.2 Small volume expansion of the temperature We proceed in this section to prove Theorem 4.2.2. To do so, we make use of the Laplace transform method [46, 58, 69]. Consider equation (4.3) and deﬁne the Laplace transform of a function g(t) by

∞ st L(g)(s)= e− g(t)dt. Z0 Taking the Laplace transform of the equations on τ in (4.3) we formally obtain the following system:

ρcCc s τˆ( ,s) ∆ˆτ( ,s)= L(gu)( ,s) in D, γc · − · ·   ρmCm 2  s τˆ( ,s) ∆ˆτ( ,s) = 0 in R D,  γm · − · \    τˆ+( ,s) τˆ ( ,s) = 0 on ∂D,  · − − ·  ∂τˆ ∂τˆ γm γc = 0 on ∂D,  ∂ν + − ∂ν  −   τˆ( ,s) satisfies the Sommerfeld radiation condition at infinity,  ·  (4.11)  where τˆ( ,s) and L(gu)( ,s) are the Laplace transforms of τ and gu := ω · 2 · (εc) u , respectively, and s C ( , 0]. 2πγc ℑ | | ∈ \ −∞ A rigorous justification for the derivation of system (4.11) and the validity of the inverse transform of the solution can be found in [58]. 1 Using layer potential techniques we have that, for any p,ˆ qˆ H− 2 (∂D), ∈ τˆ defined by

βγm [ˆp], x R2 D, τˆ := −SD ∈ \ (4.12) ˆ βγc ( FD( ,y,βγc ) [ˆq], x D, − · −SD ∈ satisﬁes the diﬀerential equations in (4.11) together with the Sommerfeld radiation condition. Here β := i s ρmCm , β := i s ρcCc and γm γm γc γc q q Fˆ ( ,β ) := G( ,y,β )L(g )(y)dy. D · γc · γc u ZD 4.3. Heat generation 107

1 To satisfy the boundary transmission conditions, pˆ and qˆ H− 2 (∂D) ∈ should satisfy the following system of integral equations on ∂D:

βγm [ˆp]+ βγc [ˆq] = Fˆ ( ,β ), −SD SD − D · γc  β β ∂FˆD( ,βγ )  γ 1 Id +( γm ) [ˆp]+ γ 1 Id +( γc ) [ˆq] = γ · c .  − m 2 KD ∗ c − 2 KD ∗ − c ∂ν (4.13)  Re-scaling of the equations x z Recall that D = z + δB, for any x ∂D, x := − ∂B, for each function ∈ δ ∈ f defined on ∂D, η is such that η(f)(x)= f(z + δx) and e k [ϕ](x) = δ δk[η(ϕ)](x), SD eSB e ( k ) [ϕ](x) = ( δk) [η(ϕ)](x). KD ∗ KB ∗ e We can also verify that e ˆ 2 ˆ FD(x,βγc ) = δ FB(ˆx,δβγc ), ∂Fˆ ∂Fˆ D (x,β ) = δ B (ˆx.δβ ). ∂ν γc ∂ν γc Note that in the above identity, in the left-hand side we differentiate with respect to x while in the right-hand side we differentiate with respect to x˜. To simplify the notation, we will use Fˆ to refer to Fˆ ( ,δβ ). B B · γc We rescale system (4.13) to arrive at

δβγm [η(ˆp)] + δβγc [η(ˆq)] = δFˆ , −SB SB − B  δβ δβ ∂FˆB  γ 1 Id +( γm ) [η(ˆp)] + γ 1 Id +( γc ) [η(ˆq)] = γ δ .  − m 2 KB ∗ c − 2 KB ∗ − c ∂ν   For δ small enough, δβγc is invertible (see Appendix B.3). Therefore, it SB follows that

δβγm 1 δβγc δβγm 1 η(ˆp)=( )− [η(ˆq)]+( )− δFˆ . SB SB SB B h i Hence, we have the following equation for η(ˆq):

h (δ)[η(ˆq)] = f h, AB where

h (δ) = γ 1 Id +( δβγm ) ( δβγm ) 1 δβγc + γ 1 Id +( δβγc ) , AB − m 2 KB ∗ SB − SB c − 2 KB ∗ ∂Fˆ f h = γ δ B + γ 1 Id +( δβγm ) ( δβγm ) 1 δFˆ . − c ∂ν m 2 KB ∗ SB − B h i (4.14)

Proof of Theorem 4.2.2 To express the solution of (4.2) on ∂D (0,T ), asymptotically on the size of × the nanoparticle δ, we make use of the representation (4.12). We will compute 108 Chapter 4. Heat Generation with Plasmonic Nanoparticles an asymptotic expansion for η(ˆq) on δ to later compute δ δβγc [η(ˆq)] on ∂B, SB scale back to D and take Laplace inverse. Using the asymptotic expansions of Appendix B.3 the following asymptotic for h (δ) holds in ( (∂B)) AB L H∗ h (δ)= h+O(δ2 log δ), AB A0 where

h 1 = γ + γ Id γ γ ∗ . A0 − 2 c m − c − m KB In the same manner, in (∂B), H∗ ˆ 5 h ∂FB 1 1 δ log δ f = γ δ + γ Id + ∗ − [δFˆ ]+ O − c ∂ν m 2 KB SB B dist(λ ,σ( ))2 ε KD∗ ˆ 5 ∂FB 1 e 1 1 δ log δ = γ δ γ Id ∗ − [δFˆ ]+ γ − [δFˆ ]+ O . − c ∂ν − m 2 −KB SB B mSB B dist(λ ,σ( ))2 ε KD∗ e e ˆ δ2 Here the remainder comes from the fact that F = O ∗ 2 . B dist(λε,σ( )) KD Note that ∆Fˆ = η(L(g )) δ2β2 Fˆ in B and ∆Fˆ = 0 in R2 D¯. We B u − γc B B \ can further verify that FˆB satisﬁes the assumption required in Lemma A.0.4. Thus we have ˆ 5 1 1 ∂FB 1 δ Id ∗ − [δFˆ ]= δ + C ϕ + γ − [δFˆ ]+ O , 2 −KB SB B − ∂ν u 0 mSB B dist(λ ,σ( ))2 ε KD∗ e e δ3 where C is a constant such that C = O ∗ 2 . u u dist(λε,σ( )) KD After replacing the above in the expression of f h we ﬁnd that

h 1 h η(ˆq) = ( B(δ))− f A ˆ 5 1 ∂FB Cuγm δ log δ = (λγId ∗ )− [δ ]+ ϕ0 + O , −KB ∂ν (γ γ )(λ 1 ) dist(λ ,σ( ))2 c m γ 2 ε KD∗ − − (4.15) where γ + γ λ = c m . γ 2(γ γ ) c − m Finally, in (∂B), H∗ ˆ 6 2 δβγc 1 ∂δFB Cuγm δβγc δ log δ η(ˆτ)= δ FˆB δ (λγId ∗ )− [ ] δ [ϕ0]+O . − − SB −KB ∂ν −(γ γ )(λ 1 ) SB dist(λ ,σ( ))2 c m γ 2 ε KD∗ − − (4.16) It can be shown, from the regularity of the remainders, that the previous identity also holds in L2(∂B). Using Holder’s inequality we can prove that

δβγc [ϕ] ∞ C ϕ 2 , kSB kL (∂B) ≤ k kL (∂B) for some constant C. Hence, we ﬁnd that identity (4.16) also holds true 4 δβγcf δ log δ uniformly on ∂B and C δ [ϕ ](˜x) = O ∗ 2 , uniformly in u B 0 dist(λε,σ( )) S KD 4.3. Heat generation 109

∂B. Scaling back to D gives

ˆ 4 βγc 1 ∂FD( ,βγc ) δ log δ τˆ(x,s)= Fˆ (x,β ) (λ Id ∗ )− [ · ]+ O (4.17). − D γc −SD γ −KD ∂ν dist(λ ,σ( ))2 ε KD∗ Before we take the inverse Laplace transform to (4.17) we note that (see [69])

L K(x, ,b ) = G(x,β ), · c − γc

ρcCc where bc := and K(x, ,bc) is the fundamental solution of the heat γc · equation. In dimension two, K is given by

|x|2 e− 4bct K(x,t,γ)= . 4πbct We denote K(x,y,t,t ,b ) := K(x y,t t ,b ). By the properties of the ′ c − − ′ c Laplace transform, we have

· Fˆ (x,β )= G(x,y,β )L(g )(y)dy = L K(x,y, ,t′,b )g (y)dydt′ . − D γc − γc u · c u ZD Z0 ZD

We deﬁne FD as follows

t FD(x,t,bc) := K(x,y,t,t′,bc)gu(y)dydt′. (4.18) Z0 ZD Similarly, we have that for a function f

· G(x,y,β )L(f)(y)dy = L K(x,y, ,t′,b )f(y,t′)dydt′ . − γc · c Z∂D Z0 Z∂D We deﬁne bc as follows VD t bc [f](x,t) := K(x,y,t,t′,b )f(y,t′)dydt′. (4.19) VD c Z0 Z∂D Finally, using Fubini’s theorem and taking Laplace inverse we ﬁnd that

4 bc 1 ∂FD( , ,bc) δ log δ τ(x,t)= F (x,t,b ) (λ Id ∗ )− [ · · ](x,t)+ O , D c −VD γ −KD ∂ν dist(λ ,σ( ))2 ε KD∗ uniformly in (x,t) ∂D (0,T ). ∈ × 4.3.3 Temperature elevation at the plasmonic resonance

i ikmd x Suppose that the incident wave is u (x) = e · , where d is a unit vector. For a nanoparticle occupying a domain D = z +δB, the inner ﬁeld u solution to (4.1) is given by Theorem 4.2.1, which states that, in L2(D),

ikmd z 1 u e · 1+ ik λ Id ∗ − [ν] d , ≈ mSD ε −KD · 110 Chapter 4. Heat Generation with Plasmonic Nanoparticles and hence

2 2 1 1 u 1+2k i λ Id ∗ − [ν] d + k λ Id ∗ − [ν] d . | | ≈ mℜ SD ε −KD · mSD ε −KD · (4.20) Using Lemma A.0.2, we can write

1 ∞ (ν d,ϕj) ∗ D[ϕj] λ Id ∗ − [ν] d = · H S , SD ε −KD · λ λ j=1 ε − j X and therefore, for a given plasmonic frequency ω, we have

1 (ν d,ϕj∗ ) ∗ D[ϕj∗ ] D λεId D∗ − [ν] d · H S . S −K · ≈ λ (ω) λ ∗ ε − j Here j is such that λ ∗ = (λ (ω)) and the eignevalue λ ∗ is assumed to be ∗ j ℜ ε j simple. If this was not the case, (ν d,ϕj∗ ) ∗ D[ϕj∗ ] should be replaced by · H S the corresponding sum over an orthonormal basis of eigenfunctions for the eigenspace associated to λj∗ . Replacing in (4.20) we ﬁnd

2 (ν d,ϕj∗ ) ∗ 2 (ν d,ϕj∗ ) ∗ 2 H ∗ ∗ FD[gu] FD[1] + 2km · FD[ D[ϕj ]] + km · H 2 FD[ D[ϕj ] ] , ≈ λ (ω) λ ∗ S λ (ω) λ ∗ S ε j ε j | − | | 2 − | 2 ∂FD (ν d,ϕj∗ ) ∗ ∂FD[ D[ϕj∗ ]] 2 (ν d,ϕj∗ ) ∗ ∂FD[ D[ϕj∗ ] ] H 2km · S + km · H 2 S . ∂ν ≈ λ (ω) λ ∗ ∂ν λ (ω) λ ∗ ∂ν | ε − j | | ε − j | Then, the temperature on the boundary of a nanoparticle at the plasmonic resonance can be estimated by plugging the above approximations of FD and ∂F (x,t,b ) D c into ∂ν

4 bc 1 ∂FD( , ,bc) δ log δ τ(x,t)= F (x,t,b ) (λ Id ∗ )− [ · · ](x,t)+ O . D c −VD γ −KD ∂ν dist(λ ,σ( ))2 ε KD∗ 4.3.4 Temperature elevation for two close-to-touching particles Lemma A.0.4 implies that

4 ∂FD(x,t,bc) 1 1 δ log δ = Id ∗ − [F ](x,t)+ O . ∂ν − 2 −KD SD D dist(λ ,σ( ))2 ε KD∗ Therefore, we can write the temperaturee on the boundary of the nanoparticle as

4 bc 1 ∂FD( , ,bc) δ log δ ∗ τ(x,t)= FD(x,t,bc)+ D (λγId D∗ )− E 1 [ · · ](x,t)+O 2 , V −K PH \ 2 ∂ν dist(λ ,σ( )) ε D∗ (4.21) K 4.4. Numerical results 111

∗ 1 where E 1 is the projection into ∗ E : the complement in ∗(∂ ) of PH \ 2 H \ 2 H D the eigenspace associated to the eigenvalue 1 of . This implies that, even 2 KD∗ 1 1 ∂FD( , ,bc) ∗ if λγ is close to 2 , the quantity (λγId D∗ )− E 1 [ · · ](x,t) − K PH \ 2 ∂ν δ2 will remain of order O ∗ 2 , provided that the second largest dist(λε,σ( )) KD eigenvalue of is not close to 1 . KD∗ 2 Even if this is in general the case for smooth boundaries ∂D, it turns out that for nanoparticles with two connected close-to-touching subparts with 1 contact of order m, a family of eigenvalues of D∗ in ∗ E 1 approaches 2 as K H \ 2 (see [42])

ζ 1 1 1 1 1 λ c ζ − m + o(ζ − m ), n ∼ 2 − n where ζ is the distance between connected subparts and cn is an increasing sequence of positive numbers. Now, λ 1 is the kind of situations encountered for metallic nanoparti- γ ≈ 2 cles immersed in water or some biological tissue. As an example, the thermal W W conductivity of gold is γc = 318 mK and that of pure water is γm = 0.6 mK . This gives λ 0.5019. γ ≈ In view of this, the second term in (4.21) may increase considerably for some type of close-to-touching particles. We stress, nevertheless, that this is not the general case. For a more re- ﬁned analysis, asymptotics of the eigenfunctions of should be also studied. KD∗ 4.4 Numerical results

The numerical experiments for this work can be divided into two parts. The ﬁrst one is the Helmholtz equation solution approximation, which is obtained by using Theorem 4.2.1. The second part is the Heat equation solution computation, which is obtained using Theorem 4.2.2. The major tasks surrounding the numerical implementation of these formulas are integrating against a singular kernel. The numerical computations of the operators F [ ] and ∂ F [ ] can be achieved by meshing the domain D · ν D · D and integrating semi-analytically inside the triangles that are close to the singularities. We used the following formula to avoid numerical diﬀerentia- tion:

2 ∂FD(x,t,bc) 1 x y y x,νx = exp −| − | h − 2 igu(y)dy, x ∂D. ∂ν 2πbc D 4bct x y ∈ Z | − | (4.22) For all the presented simulations, we considered an incident plane wave given by i ikmd x u (x)= e · , where d = (1, 1)/√2 R2 is the illumination direction and k = 2π/750 109 ∈ m · is the frequency (in the red range). The considered nanoparticles are ellipses with semi-axes 30nm and 20nm, respectively. It is worth noticing that the illumination direction d is relevant solely in the asymptotic formula in Theorem 4.2.1. Its role is to define the coefficients 112 Chapter 4. Heat Generation with Plasmonic Nanoparticles of a linear combination of both components of (λ Id ) 1[v] R2. We SD ǫ −KD∗ − ∈ will see from the numerical simulations that this is fundamental if we wish to maximize the produced electromagnetic field, and therefore the generated heat inside the nanoparticles. With respect to the asymptotic formula established in Theorem 4.2.1, besides the nanoparticle’s shape D, the sole parameter that is left is λǫ. For all the following simulations we will consider this as a free parameter that we will use to excite the eigenvalues of the Neumann-Poincaré operator and hence to generate resonances. The physical justification that allows us to do this is based on the Drude model [9]. Whenever we mention that we approach a particular eigenvalue λ of , we will adopt λ = λ + 0.001i. j KD∗ ǫ j With respect to the heat equation coefficients, we use realistic values of gold for nanoparticles, and water for tissues.

4.4.1 Single-particle simulation We consider one elliptical nanoparticle D ⋐ R2 centered at the origin, with its semi-major axis aligned with the x-axis.

Single-particle Helmholtz resonance Resonance is achieved by approaching the eigenvalues of the Neumann-Poincaré operator with λ , and afterwards applying it to each of the components KD∗ ǫ of the normal ν to ∂D. It turns out that for some eigenfunctions of , the KD∗ normal of the shape is almost orthogonal, in (∂D), to them. Therefore, H∗ we cannot observe resonance for their associated eigenvalues. In Figure 4.1 we can see values of the inner product between the eigenfunctions of KD∗ and the components νx and νy of ν. Figure 4.1 suggests us which are the available resonant modes with the respective strength of each coordinate. In

10 x 8 y

0 0 10 20 30 40 50

Figure 4.1: Inner product in ∗(∂D) between the eigen- ∗ H functions of D and the components νx and νy of the normal K ν to ∂D.

Figure 4.2 we present the absolute value of the inner field for the first three resonant modes, corresponding to the second, third and sixth eigenvalue of , respectively. In Figure 4.3 we decompose the inner field into the zeroth- KD∗ order and the first-order terms respectively given by ui(z)+ δ(x z) ui(z) 1 − ∇ and λ Id − [ν] ui(z). Figure 4.4 shows the components of the SD ε −KD∗ ·∇ vector (λ Id ) 1[v]. SD ǫ −KD∗ − From Figure 4.3, we can see that when we excite the nanoparticle at its resonant mode, the largest contribution to the electromagnetic field comes 4.4. Numerical results 113

First resonance mode Second resonance mode Third resonance mode

Figure 4.2: Absolute value of the electromagnetic ﬁeld inside the nanoparticle at the ﬁrst resonant modes, being those when λǫ approaches the second, third and sixth eigenvalue of ∗ . KD Zeroth-order component First-order component

Figure 4.3: First resonant mode of the nanoparticle decomposed in its ﬁrst- and second-order term in the formula given by Theorem 4.2.1. Both images are absolute values of the respective component.

The x-component The y-component

Figure 4.4: Absolute value of the vectorial components of the first-order term for the first resonant mode. from the first-order term of the small volume expansion formula established in Theorem 4.2.1. Observing the vectorial components of the first-order term in Figure 4.4 tells us how important is the illumination direction as the x-component is significantly stronger than the y-component. If we wish to maximize the electromagnetic field and therefore the generated heat, the recommended illumination direction would be around d = (1, 0)t (with t being the transpose), as it was initially suggested by Figure 4.1.

Single-particle surface heat generation Considering the electromagnetic field inside the nanoparticle given by the first resonant mode presented in Figure 4.2, following the formula given by Theo- rem 4.2.2, we compute the generated heat on the surface of the nanoparticle. In Figure 4.5 we plot the generated heat in three dimensions and present a two dimensional plot obtained by parameterizing the boundary. In Fig- ure 4.6 we decompose the heat in its first- and second-order terms given by 114 Chapter 4. Heat Generation with Plasmonic Nanoparticles

∂F ( , ,b ) formula 4.2.2, being F (x,t,b ) and bc (λ Id ) 1[ D · · c ](x,t) D c −VD γ −KD∗ − ∂ν respectively. In Figure 4.7, we integrate the total heat on the boundary and plot it as a function of time, for each component.

3D plot of generated heat at time 2D plot of generated heat at time T =1 T =1 10 -14

10 -14 1.5 1.6

1.4 1.4

1.3 1.2

1.2 1 2 2 - - /2 0 /2 0 0 -8 10 -8 10 -2 -2

Figure 4.5: At the left-hand side, we can see a three- dimensional plot of the nanoparticle heat, the red shape is a reference value to show where the nanoparticle is located. At the right-hand side we can see a two-dimensional plot of the generated heat, where the boundary was parametrized following p(θ)=(a cos(θ),b sin(θ)), θ [ π, π], with a and b being the semi-major and semi-minor∈ axes,− respectively.

Two-dimensional plot of the Two-dimensional plot of the zeroth-order term at T =1 ﬁrst-order term at T =1 -14 10 -15 10 3.3 1.4

3.28 1.2

3.26 1

3.24 0.8 - - /2 0 /2 - - /2 0 /2

Figure 4.6: Two-dimensional plots of the zeroth- and ﬁrst- order components of the heat on the boundary when time is equal to one. As time goes on, each point of the graph increases, but the general shape is preserved.

Integrated zeroth-order Integrated ﬁrst-order Integrated heat over time component of heat over component of heat over time time -21 10 -20 10 10 -15 2.5 5.8 10 -20 1.25 5 4.2 0 10 -25 10 -5 10 -3 10 0 10 -5 10 -3 10 0 10 -5 10 -3 10 0 time time time

Figure 4.7: Time-logarithmic plots showing the total heat on the boundary for each component of the heat. The values were obtained for each fixed time, by integrating over the boundary the computed heat. From left- to right-hand side: The total heat, the zeroth-order and its first-order, according to formula given by Theorem 4.2.2. Notice that the first- order term is plotted in a log-log scale.

We can observe that the heat is not symmetric, this can be noticed from the total inner ﬁeld for the ﬁrst resonance mode in Figure 4.2. The reason behind this non symmetry is because we are illuminating with direction d = 4.4. Numerical results 115

(1, 1)t/√2 over an ellipse. From Figure 4.7 we can notice that the ﬁrst-order term converges, while the zeroth-order term increases logarithmically, as it is expected from the known solution of the heat equation for constant source in two dimensions that the heat increases logarithmically.

4.4.2 Two particles simulation We consider two elliptical nanoparticles D , D , D = D D , with the same 1 2 1 ∪ 2 shape and orientation as the nanoparticle considered in the above example. The particle D is centered at the origin and D is centered at (0, 4.1 10 9), 1 2 · − resulting in a separation distance of 0.1nm between the two particles.

Two particles Helmholtz resonance Following the same analysis as the one for one particle, in Figure 4.8 we present the inner product between the eigenfunctions of with each com- KD∗ ponent of the normal of D. We can observe that there are more available resonant modes. In particular we can see that when λǫ approaches the 36th or 37th eigenvalues, we achieve strong resonant modes.

12 x

10 y

0 0 10 20 30 40 50

Figure 4.8: inner product in ∗∂D between the eigenval- ∗ H ues of and each component of the normal of ∂D, νx and KD νy.

In Figure 4.2 we present the absolute value of the inner field for the resonant modes corresponding to the 6th, 37th and 38th eigenvalues of . KD∗ Similarly to the case with one particle, the dominant term in the electromagnetic field for each case is the first-order term. In Figure 4.10 we decompose the first-order term in its x-component and y-component. As suggested by Figure 4.8, for the resonant mode associated to the 38th eigenfunction of , the stronger component is the one on the y direction, KD∗ meaning that if we wish to maximize the electromagnetic field, and therefore the generated heat, it is suggested to consider the illumination vector d = (0, 1)t.

Two particles surface heat generation Similarly to the analysis carried out for one particle, we now compute the generated heat for these two particles while undergoing resonance for the resonant mode associated to the 38th eigenvalue of . In Figure 4.11 we KD∗ plot generated heat in the boundary of the two nanoparticles. In Figure 4.12 we decompose the generated heat in its zeroth and ﬁrst-order component, for each of the two nanoparticles. 116 Chapter 4. Heat Generation with Plasmonic Nanoparticles

Resonant mode associated Resonant mode associated Resonant mode associated to the 6th eigenvalue of to the 37th eigenvalue of to the 38th eigenvalue of ∗ ∗ ∗ KD KD KD

Figure 4.9: Absolute value of the electromagnetic ﬁeld inside the nanoparticle at the resonant modes associated to the 6th, 37th and 38th resonant modes, obtained when λǫ approaches the respective eigenvalues of ∗ . KD The x-component The y-component

Figure 4.10: Absolute value of the vectorial components of the ﬁrst-order term for the 38th resonant mode.

Similarly to the single nanoparticle case, there is no symmetry on the heat values on the boundary, which is due to the illumination. We have not provided the plots of the heat integrated along the boundary, as the conclusions are the same as the ones in the single nanoparticle case: The total heat on the boundary increases logarithmically, initially on time the dominant term is the ﬁst-order one, but as time increases the zeroth-order term becomes the predominant one.

4.5 Concluding remarks

In this chapter we have derived an asymptotic formula for the temperature elevation due to plasmonic nanoparticles. We have considered thermal coupling within close-to-touching nanoparticles, where the temperature ﬁeld deviates signiﬁcantly from the one generated by single nanoparticles. Com- bined with the methods developed in [13,14], our results can be used for the optical and thermal detection and localization of plasmonic nanoparticles. As reported in [77], the detection and localization of nanoparticles in highly scattering media such as biological tissue remains a challenge. They can also be used for monitoring temperature elevation due to plasmonic nanoparticles 4.5. Concluding remarks 117

3D plot of heat on ∂(D1 D2) at time Heat on ∂D1 at time T =1 ∪ -11 T =1. 10 10 -12 1.5

13 1 12

11 0.5 10

9 0 - - /2 0 /2 8

7 Heat on ∂D2 at time T =1 6 10 -11 1.5 5

4 1

6 4 0.5 -8 10 2 2 0 0 10 -8 0 -2 -2 - - /2 0 /2

Figure 4.11: Generated heat on the boundary of the nanoparticles for time equal to 1. On the left we can see a three dimensional view of the heat, the red shapes are ref- erential to show the location of the nanoparticles. On the right-hand side we can see the two dimensional heat plots corresponding to each nanoparticle. To obtain these plots we parameterized the boundary of each nanoparticle with p(θ)=(a cos(θ),b sin(θ)) + z, θ [ π, π], where z R2 corresponds to the center of each∈ nanoparticle.− On the∈ top we can see nanoparticle D2 and on the bottom nanoparticle D1. based on the photoacoustic signal recently analyzed in [91]. Thermoacous- tic signals generated by nanoparticle heating can be computed numerically based on the successive resolution of the thermal diffusion problem considered in this chapter and a thermoelastic problem, taking into account the size and shape of the nanoparticle, thermoelastic and elastic properties of both the particle and its environment, and the temperature-dependence of the thermal expansion coefficient of the environment. For sufficiently high illumination fluences, this temperature dependence yields a nonlinear rela- tionship between the photoacoustic amplitude and the fluence [82]. 118 Chapter 4. Heat Generation with Plasmonic Nanoparticles

Heat zeroth component on ∂D Heat ﬁrst component on ∂D2 at time T =1 2 -11 10 -12 10 3.3 1.5

3.28 1

3.26 0.5

3.24 0 - - /2 0 /2 - - /2 0 /2

Heat zeroth component on ∂D1 at time Heat ﬁrst component on ∂D1 at time T =1 T =1 -12 10 -12 10 3.3 15

10 3.28 5 3.26 0

3.24 -5 - - /2 0 /2 - - /2 0 /2

Figure 4.12: Two-dimensional plots of the zeroth and ﬁrst component of the heat at time 1, for each nanoparticle. On the left column we have the zeroth component of the heat, on the right-hand side column we have the ﬁrst component of the heat. On top we show the values for nanoparticle D2, on the bottom we show the values for nanoparticle D1. 119

Chapter 5

Plasmonic Metasurfaces

Contents 5.1 Introduction ...... 120 5.2 Setting of the problem ...... 121 5.3 One-dimensional periodic Green function . . . . . 122 5.4 Boundary layer corrector and eﬀective impedance124 5.5 Numerical illustrations ...... 128 5.5.1 Setup and methods ...... 128 5.5.2 Results and discussion ...... 129 5.6 Concluding remarks ...... 129 120 Chapter 5. Plasmonic Metasurfaces

5.1 Introduction

In this chapter we consider the scattering by a layer of periodic plasmonic nanoparticles mounted on a perfectly conducting sheet. We design the layer in order to control and transform waves. Since the thickness of the layer, which is of the same order of the diameter of the individual nanoparticles, is negligible compared to the wavelength, it can be approximated by an impedance boundary condition. Our main result is to prove that at some resonant frequencies, which are fully characterized in terms of the periodicity, the shape and the material parameters of the nanoparticles, the thin layer has anomalous reflection properties and can be viewed as a metasurface. Since the period of the array is much smaller than the wavelength, the resonant frequencies of the array of nanoparticles differ significantly from those of single nanoparticles. As shown in this chapter, they are associated with eigenvalues of a periodic Neumann-Poincaré type operator. In contrast with quasi-static plasmonic resonances of single nanoparticles, they depend on the particle size. For simplicity, only one-dimensional arrays embedded in R2 are considered in this chapter. The extension to the two-dimensional case is straightforward and the dependence of the plasmonic resonances on the parameters of the lattice is easy to derive. The array of plasmonic nanoparticles can be used to efficiently reduce the scattering of the perfectly conducting sheet. We present numerical results to illustrate our main findings in this chapter, which open a door for a mathematical and numerical framework for realizing full control of waves using metasurfaces [2, 76, 92]. Our approach applies to any example of periodic distributions of resonators having resonances in the quasi-static regime. It provides a framework for explaining the observed extraordinary or meta properties of such structures and for optimizing these properties. The results presented in this chapter hold for arbitrary-shaped nanoparticles. Simula- tions with disks, ellipses, and rings are shown. In this connection, we refer to the recent works [59, 70, 80, 93]. It is also worth highlighting that at optical frequencies, a perfectly conducting approximation breaks down and needs to be replaced by a proper material response. In this chapter, the perfectly conducting boundary condition is used only for simplicity of the presentation. Similar effective boundary conditions can be obtained by using exactly the same approach presented here for penetrable half-space. The chapter is organized as follows. We first formulate the problem of approximating the effect of a thin layer with impedance boundary conditions and give useful results on the one-dimensional periodic Green function. Then we derive the effective impedance boundary conditions and give the shape derivative of the impedance parameter. In doing so, we analyze the spectral properties of the one-dimensional periodic Neumann-Poincaré operator defined by (5.10) and obtain an explicit formula for the equivalent boundary condition in terms of its eigenvalues and eigenvectors. Finally, we illustrate with a few numerical experiments the anomalous change in the equivalent impedance boundary condition due to the plasmonic resonances of the periodic array of nanoparticles. For simplicity, we only consider the scalar wave equation and use a two-dimensional setup. The results of this chapter can be readily generalized to higher dimensions as well as to the full Maxwell equations. 5.2. Setting of the problem 121

5.2 Setting of the problem

We use the Helmholtz equation to model the propagation of light. This approximation can be viewed as a special case of Maxwell’s equations, when the incident wave ui is transverse magnetic (TM) or transverse electric (TE) polarized. Consider a particle occupying a bounded domain D ⋐ R2 of class 1,α C for some 0 <α< 1 and with size of order δ 1. The particle is char- ≪ acterized by electric permittivity εc and magnetic permeability µc, both of which may depend on the frequency of the incident wave. Assume that mε > 0, eµ < 0, mµ > 0 and deﬁne ℑ c ℜ c ℑ c

km = ω√εmµm, kc = ω√εcµc, where εm and µm are the permittivity and permeability of free space respectively and ω is the frequency. Throughout this chapter, we assume that εm and µm are real and positive and km is of order 1. We consider the conﬁguration shown in Figure 5.1, where a particle D is repeated periodically in the x1-axis with period δ, and is at a distance of order δ from the boundary x = 0 of the half-space R2 := (x ,x ) R2, x > 0 . 2 + { 1 2 ∈ 2 } We denote by this collection of periodically arranged particles and Ω := D R2 . + \ D

Figure 5.1: Thin layer of nanoparticles in the half space.

i ikmd x Let u (x) = e · be the incident wave. Here, d is the unit incidence direction. The scattering problem is modeled as follows

1 2 R2 u + ω ε u = 0 in + ∂ , ∇· µ ∇ D \ D  D u u = 0 on ∂ ,  +  − − D  1 ∂u 1 ∂u  = 0 on ∂ , (5.1)  µ ∂ν − µ ∂ν D  m + c − u ui satisﬁes an outgoing radiation condition at inﬁnity,  −  2  u = 0 on ∂R = (x1, 0), x1 R ,  + { ∈ }  where

ε = εmχ(Ω) + εcχ( ), µ = εmχ(Ω) + εcχ( ), D D D D and ∂/∂ν denotes the outward normal derivative on ∂ . D 122 Chapter 5. Plasmonic Metasurfaces

Following [1], under the assumption that the wavelength of the incident wave is much larger than the size of the nanoparticle, a certain homogenization occurs, and we can construct z C such that the solution to ∈ 2 R2 ∆uapp + kmuapp = 0 in +,  u + δz ∂uapp = 0 on ∂R2 , (5.2) app ∂x2 +   u ui satisﬁes outgoing radiation condition at inﬁnity, app −  gives the leading order approximation for u. We will refer to u +δz ∂uapp = app ∂x2 0 as the equivalent impedance boundary condition for problem (5.1). A proof of existence and uniqueness of a solution to (5.2) follows immediately from [45].

5.3 One-dimensional periodic Green function

Consider the function G : R2 C satisfying ♯ →

∆G♯(x)= δ(x +(n, 0)). (5.3) n Z X∈ 2 We call G♯ the 1-d periodic Green function for R .

Lemma 5.3.1. Let x =(x1,x2), then 1 G (x)= log sinh2(πx )+sin2(πx ) , ♯ 4π 2 1 satisﬁes (5.3).

Proof. We have

∆G♯(x) = δ(x +(n, 0)) n Z X∈ = δ(x2)δ(x1 + n) n Z X∈ i2πnx1 = δ(x2)e , (5.4) n Z X∈ where we have used the Poisson summation formula n Z δ(x1 + n) = i2πnx ∈ Z e 1 . n P ∈On the other hand, since G is periodic in x of period 1, we have P ♯ 1

i2πnx1 G♯(x)= βn(x2)e , n Z X∈ therefore ′′ 2 i2πnx1 ∆G♯(x)= (βn(x2)+(i2πn) βn)e . (5.5) n Z X∈ Comparing (5.4) and (5.5) yields

′′ 2 βn(x2)+(i2πn) βn = δ(x2). 5.3. One-dimensional periodic Green function 123

A solution to the previous equation can be found by using standard techniques for ordinary diﬀerential equations. We have 1 β = x + c, 0 2| 2|

1 2π n x2 β = − e− | || |, n = 0, n 4π n 6 | | where c is a constant. Subsequently,

1 1 2π n x2 i2πnx1 G (x) = x + c e− | || |e ♯ 2| 2| − 4π n n Z 0 ∈X\{ } | | 1 1 2πn x2 = x + c e− | | cos(2πnx ) 2| 2| − 2πn 1 n N 0 ∈X\{ } 1 = log sinh2(πx )+sin2(πx ) , 4π 2 1 where we have used the summation identity (see, for instance, [57, pp. 813- 814])

1 2πn x2 1 log(2) e− | | cos(i2πnx )= x 2πn 1 2| 2|− 2π n N 0 ∈X\{ } 1 log sinh2(πx )+sin2(πx ) , −4π 2 1 log(2) and deﬁned c = . − 2π Let us also denote by G (x,y) := G (x y). In the following we deﬁne ♯ ♯ − the 1-d periodic single layer potential and 1-d periodic Neumann-Poincaré 1 1 operator, respectively, for a bounded domain B ⋐ , R which we − 2 2 × assume to be of class 1,α for some 0 <α< 1. Let C 1 1 2 1 : H− 2 (∂B) H (R ),H 2 (∂B) SB♯ −→ loc ϕ [ϕ](x)= G (x,y)ϕ(y)dσ(y) 7−→ SB,♯ ♯ Z∂B for x R2,x ∂B and let ∈ ∈ 1 1 ∗ : H− 2 (∂B) H− 2 (∂B) KB♯ −→ ∂G♯(x,y) ϕ ∗ [ϕ](x)= ϕ(y)dσ(y) 7−→ KB,♯ ∂ν(x) Z∂B for x ∂B. As in [65], the periodic Neumann-Poincaré operator can be ∈ symmetrized. The following lemma holds. 1 Lemma 5.3.2. (i) For any ϕ H− 2 (∂B), B♯[ϕ] is harmonic in B and 1 1 ∈ S in , R B; − 2 2 × \ 124 Chapter 5. Plasmonic Metasurfaces

1 (ii) The following trace formula holds: for any ϕ H− 2 (∂B), ∈

1 ∂ B♯[ϕ] ( Id + ∗ )[ϕ]= S ; −2 KB♯ ∂ν −

(iii) The following Calderón identity holds: = , where KB♯SB♯ SB♯KB♯∗ KB♯ is the L2-adjoint of ; KB♯∗ 1 1 (iv) The operator : H− 2 (∂B) H− 2 (∂B) is compact self-adjoint KB♯∗ 0 → 0 equipped with the following inner product

(u,v) ∗ = (u, [v]) 1 1 (5.6) 0 B♯ , H − S − 2 2

1 1 2 2 with ( , ) 1 1 being the duality pairing between H− (∂B) and H (∂B), 2 , 2 0 0 · · − 1 which makes equivalent to H− 2 (∂B). Here, by E we denote the H0∗ 0 0 zero-mean subspace of E.

(v) Let (λj,ϕj), j = 1, 2,... be the eigenvalue and normalized eigenfunction pair of in (∂B), then λ ( 1 , 1 ) and λ 0 as j . KB♯∗ H0∗ j ∈ − 2 2 j → → ∞ 2 2 Proof. First, note that a Taylor expansion of sinh (πx2)+sin (πx1) yields

log x G (x)= | | + R(x), ♯ 2π where R is a smooth function such that 1 R(x)= log(1 + O(x2 x2)). 4π 2 − 1 We can decompose the operators and on (∂B) accordingly. We SB♯ KB♯∗ H0∗ have

= + , ∗ = ∗ + , SB♯ SB GB KB♯ KB FB where and are the single layer potential and Neumann-Poincaré op- SB KB∗ erator (see [18]), respectively, and , are smoothing operators. Using GB FB this fact, the proof of the Lemma follows the same arguments as those given in [12, 18].

5.4 Boundary layer corrector and eﬀective impedance

In order to compute z, we introduce the following asymptotic expansion [1,3]:

(0) (0) (1) (1) u = u + uBL + δ(u + uBL)+ ... (5.7) where the leading-order term u(0) is solution to

(0) 2 (0) R2 ∆u + kmu = 0 in +, (0) R2  u = 0 on ∂ +,   u(0) ui satisfies an outgoing radiation condition at infinity. −   5.4. Boundary layer corrector and effective impedance 125

(0) (1) The boundary-layer correctors uBL and uBL have to be exponentially de- (0) caying in the x2-direction. Note that according to [1, 3], uBL is introduced in order to correct (up to the first-order in δ) the transmission condition on the boundary of the nanoparticles, which is not satisfied by the leading-order (0) (1) term u in the asymptotic expansion of u, while uBL is a higher-order correction term and does not contribute to the first-order equivalent boundary condition in (5.2). (0) We next construct the corrector uBL. We first introduce a function α and a complex constant α such that they satisfy the rescaled problem: ∞ ∆α = 0 in R2 , +\B ∪B  α + α = 0 on ∂ ,  | − |− B   1 ∂α 1 ∂α 1 1  = ν2 on ∂ , (5.8)  µ ∂ν − µ ∂ν µ − µ B  m + c c m − α = 0 on ∂R2 ,  +   α α is exponentially decaying as x2 + .  − ∞ → ∞   Here, ν =(ν1,ν2) and B = D/δ is repeated periodically in the x1-axis with period 1 and is the collection of these periodically arranged particles. (0) B Then uBL is defined by

(0) (0) ∂u x uBL(x) := δ (x1, 0) α( ) α . ∂x2 δ − ∞ The corrector u(1) can be found to be the solution to (1) 2 (1) R2 ∆u + kmu = 0 in +, (0)  u(1) = α ∂u on ∂R2 , ∂x2 +  ∞  u(1) satisﬁes an outgoing radiation condition at inﬁnity.  By writing (0) (1) uapp = u + δu , (5.9) we arrive at (5.2) with z = α , up to a second order term in δ. We − ∞ summarize the above results in the following theorem.

Theorem 5.4.1. The solution uapp to (5.2) with z = α approximates − ∞ pointwisely (for x > 0) the exact solution u to (5.1) as δ 0, up to a 2 → second order term in δ. In order to compute α , we derive an integral representation for the ∞ solution α to (5.8). We make use of the periodic Green function G♯ deﬁned by (5.3). Let

G+(x,y)= G (x y ,x y ) G (x y , x y ) , ♯ ♯ 1 − 1 2 − 2 − ♯ 1 − 1 − 2 − 2 126 Chapter 5. Plasmonic Metasurfaces which is the periodic Green’s function in the upper half space with Dirichlet boundary conditions, and deﬁne

+ 1 1 2 1 : H− 2 (∂B) H (R ),H 2 (∂B) SB♯ −→ loc ϕ + [ϕ](x)= G+(x,y)ϕ(y)dσ(y) 7−→ SB,♯ ♯ Z∂B for x R2 ,x ∂B and ∈ + ∈ + 1 1 ( ∗ ) : H− 2 (∂B) H− 2 (∂B) KB♯ −→ ∂G+(x,y) (5.10) + ♯ ϕ ( ∗ ) [ϕ](x)= ϕ(y)dσ(y) 7−→ KB,♯ ∂ν(x) Z∂B for x ∂B. ∈ It can be easily proved that all the results of Lemma 5.3.2 hold true for + + 1 and ( ) . Moreover, for any ϕ H− 2 (∂B), we have SB♯ KB♯∗ ∈ + [ϕ](x) = 0 for x ∂R2 . SB,♯ ∈ + + Now, we can readily see that α can be represented as α = B,♯[ϕ], where 1 S ϕ H− 2 (∂B) satisﬁes ∈ + + 1 ∂ [ϕ] 1 ∂ [ϕ] 1 1 SB,♯ SB,♯ = ν on ∂B. µ ∂ν − µ ∂ν µ − µ 2 m + c c m −

Using the jump formula from Lemma 5.3.2, we arrive at

+ λ Id ( ∗ ) [ϕ]= ν , µ − KB♯ 2 where µ + µ λ = c m . µ 2(µ µ ) c − m Therefore, using item (v) in Lemma 5.3.2 on the characterization of the spectrum of and the fact that the spectra of ( )+ and are the KB♯∗ KB♯∗ KB♯∗ same, we obtain that

+ + 1 α = λ Id ( ∗ ) − [ν ]. SB,♯ µ − KB♯ 2 Lemma 5.4.1. Let x =(x ,x ). Then, for x + , the following asymp- 1 2 2 → ∞ totic expansion holds:

x2 α = α + O(e− ), ∞ with

+ 1 α = y2 λµId ( B♯∗ ) − [ν2](y)dσ(y). ∞ − − K Z∂B 5.4. Boundary layer corrector and eﬀective impedance 127

+ Proof. The result follows from an asymptotic analysis of G♯ (x,y). Indeed, suppose that x + , we have 2 → ∞ G+(x,y)= 1 log sinh2(π(x y )) + sin2(π(x y )) ♯ 4π 2 − 2 1 − 1 1 log sinh2(π(x + y )) + sin2(π(x y )) −4π 2 2 1 − 1 1 = log sinh2(π(x y )) 4π 2 − 2 1 log sinh2(π(x + y )) −4π 2 2 1 +O log 1+ sinh2(x ) 2 1 eπ(x2 y2) e π(x2+y2) = log − − − 2π 2 π(x2+y2) π(x2 y2) e e− − x2 log − + O log 1+ e− 2 − 2 x2 = y + O(e− ), − 2 which yields the desired result.

Finally, it is important to note that α depends on the geometry and ∞ size of the particle B. Since ( )+ : is a compact self-adjoint operator, where is KB♯∗ H0∗ →H0∗ H0∗ deﬁned as in Lemma 5.3.2, we can write

+ 1 α = y2 λµId ( B♯∗ ) − [ν2](y)dσ(y), ∞ − − K Z∂B ∞ (ϕ ,ν ) ∗ ϕ(y) j 2 0 j = y2 H dσ(y), − ∂B λµ λj Z Xj=1 − (ϕ ,ν ) ∗ (ϕ ,y ) 1 1 ∞ j 2 0 j 2 , = H − 2 2 , λµ λj Xj=1 − where λ ,λ ,... are the eigenvalues of ( )+ and ϕ ,ϕ ,... is a corre- 1 2 KB♯∗ 1 2 sponding orthornormal basis of eigenfunctions. On the other hand, by integrating by parts we get 1 (ϕ ,y ) 1 1 = (ϕ ,ν ) ∗ . j 2 , 1 j 2 0 − 2 2 λ H 2 − j This together with the fact that mλ < 0 (by the Drude model [9]), yield ℑ µ the following lemma. Lemma 5.4.2. We have mα > 0. ℑ ∞ Finally, we give a formula for the shape derivative [16] of α . This ∞ formula can be used to optimize α , for a given frequency ω, in terms of | ∞| the shape B of the nanoparticle. Let Bη be an η-perturbation of B; i.e., let 128 Chapter 5. Plasmonic Metasurfaces h 1(∂B) and ∂B be given by ∈C η

∂B = x + ηh(x)ν(x),x ∂B . η ∈ Following [17] (see also [12]), we can prove that

µm α (Bη) = α (B)+ η( 1) ∞ ∞ µc − ∂v ∂w µ ∂v ∂w h + c dσ, × ∂ν ∂ν µ ∂τ ∂τ Z∂B − − m − − where ∂/∂τ is the tangential derivative on ∂B, v and w periodic with respect to x1 of period 1 and satisfy

∆v = 0 in R2 , +\B ∪B   v + v = 0 on ∂ ,  | − |− B  ∂v µ ∂v  m  = 0 on ∂ , ∂ν + − µc ∂ν B −   v x2 0 as x2 + ,  − → → ∞  and  ∆w = 0 in R2 , +\B ∪B  µm w + w = 0 on ∂ ,  µc  | − |− B   ∂w ∂w  = 0 on ∂ , ∂ν + − ∂ν B −   w x2 0 as x2 + ,  − → → ∞  respectively. Therefore, the following lemma holds.

Lemma 5.4.3. The shape derivative dSα (B) of α is given by ∞ ∞

µm ∂v ∂w µc ∂v ∂w dSα (B)=( 1) + . ∞ µ − ∂ν ∂ν µ ∂τ ∂τ c − − m − −

If we aim to maximize the functional J := 1 α 2 over B, then it can 2 | ∞| be easily seen that J is Fréchet diﬀerentiable and its Fréchet derivative is given by edSα (B)α (B). As in [11], in order to include cases where ℜ ∞ ∞ topology changes and multiple components are allowed, a level-set version of the optimization procedure described below can be developed.

5.5 Numerical illustrations 5.5.1 Setup and methods We use the Drude model [9] to model the electromagnetic properties of the materials of our problem. We use water for the half space and gold for the metallic nanoparticles. We recall that, from the Drude model, the properties of the materials depend on the frequency of the incoming wave, or equivalently, on the wavelength. To compute α and the integral (geometry | ∞| 5.6. Concluding remarks 129 dependent) operators involved on its expression, we make a simple uniform discretization with 200 points of the corresponding geometric figures and use a standard quadrature midpoint rule. Figure 5.2 shows α as a function of the wavelength for disks of different | ∞| sizes, all centered at (0, 0.5). Figure 5.3 shows α as a function of the wavelength for two disks of the | ∞| same fixed radius equal to 0.2 but centered at two different distances from x2 = 0. In Figures 5.4 and 5.5 we plot α as a function of the wavelength for a | ∞| disk and a group of three well-separated disks. We can see that a disk can be excited roughly at one single frequency whereas three disks can be excited at different frequencies but with lower values of α . | ∞| The previous results consist only of nanodisks. Here we give a few other examples to confirm how general are the conclusions obtained. Figure 5.6 shows the blow up of α for an ellipse. In Figure 5.7 we consider a triangle | ∞| with rounded corners. In Figure 5.8, values of α are computed for a | ∞| circular ring.

5.5.2 Results and discussion An important conclusion is that the spectrum of the periodic Neumann- Poincaré operator defined by (5.10) varies with the position and size of the particles. Our results hold for arbitrary-shaped nanoparticles. The resonances of the effective impedance α depend not only on the geometry of ∞ the particle B but also on its size and position. One can see (Figures 5.2 and 5.3) a change in the magnitude and a shift of the resonances. The plasmonic resonances shift to smaller wavelengths and the magnitude of the peak value increases with increasing volume. We remark that this is not particular to the examples considered here. In fact, this is the case for any particle. These two phenomena are due to the strong interaction between the particles and the ground that appears as their sizes increase while the period of the arrangement is fixed. Note also that in our analysis we did not assume the particles to be simply connected. In fact, the theory is still valid for particles which have two or more components. This allows for more possibilities when choosing a particular geometry for the optimization of the effective impedance. For instance, one may want to design a geometry such that a single frequency is excited with a very pronounced peak or, on the other hand, to excite not only a specific frequency but rather a group of them.

5.6 Concluding remarks

In this chapter we have considered the scattering by an array of plasmonic nanoparticles mounted on a perfectly conducting plate and showed both analytically and numerically the significant change in the boundary condition induced by the nanoparticles at their periodic plasmonic frequencies. We have also proposed an optimization approach to maximize this change in terms of the shape of the nanoparticles. Our results in this chapter can be generalized in many directions. Different boundary conditions on the plate as well as curved plates can be considered. Our approach can be easily extended 130 Chapter 5. Plasmonic Metasurfaces to two-dimensional arrays embedded in R3 and the lattice effect can be included. Full Maxwell’s equations to model the light propagation can be used. The observed extraordinary or meta properties of periodic distributions of subwavelength resonators can be explained by the approach proposed in this chapter.

90 0.1 80 0.2 0.3 70 0.4

1 , | 40

0 1.5 2 2.5 3 3.5 -7 wavelength #10

Figure 5.2: α∞ as a function of the wavelength for disks of diﬀerent| | radii, ranging from 0.1 to 0.4.

25 0.25 0.45

, | 10

0 1.5 2 2.5 3 3.5 -7 wavelength #10

Figure 5.3: α∞ as a function of the wavelength for a disk | | centered respectively at distance 0.25 and 0.45 from x2 = 0. 5.6. Concluding remarks 131

70 1

60 0.5

50 0 -0.5 0 0.5

, | 30

0 1.5 2 2.5 3 3.5 -7 wavelength #10

Figure 5.4: Well localized resonance for a disk.

5 1 4.5 0.5 4 0 3.5 -0.5 0 0.5

3 |

1 2.5

, | 2

1.5

0.5

0 1.5 2 2.5 3 3.5 -7 wavelength #10

Figure 5.5: Delocalized resonances for three well- separated disks.

30 1

25 0.5

20 -0.5 0 0.5 |

1 15

, |

0 1.5 2 2.5 3 3.5 -7 wavelength #10

Figure 5.6: Well localized resonance for an ellipse. 132 Chapter 5. Plasmonic Metasurfaces

18 1

16 0.5 14 0 12 -0.5 0 0.5

1 , | 8

0 1.5 2 2.5 3 3.5 -7 wavelength #10

Figure 5.7: Delocalized resonances for a triangle with rounded corners.

3 1

2.5 0.5

2 -0.5 0 0.5 |

1 1.5

, |

0.5

0 1.5 2 2.5 3 3.5 -7 wavelength #10

Figure 5.8: Wide resonance for a ring. 133

Chapter 6

Shape Recovery of Algebraic Domains

Contents 6.1 Introduction ...... 134 6.2 Plasmonic resonance for algebraic domains . . . . 135 6.2.1 Algebraic domains of class ...... 135 Q 6.2.2 Explicit computation of the Neumann-Poincaré operator ...... 135 6.2.3 Asymptotic behavior of the Neumann-Poincaré operator ∗ ...... 138 KD 6.2.4 Generalized polarization tensors and their spectral representations ...... 140 6.2.5 Classiﬁcation of algebraic domains in the class . 143 Q 6.3 Plasmonic resonances for two separated disks . . 143 6.3.1 The bipolar coordinates and the boundary integral operators ...... 143 6.3.2 Neumann Poincaré-operator for two separated disks and its spectral decomposition ...... 145 6.3.3 The Polarization tensor ...... 147 6.3.4 Reconstruction of the separation distance . . . . . 148 6.4 Numerical illustrations ...... 148 6.5 Concluding remarks ...... 148 134 Chapter 6. Shape Recovery of Algebraic Domains

6.1 Introduction

In this chapter, we prove that based on plasmonic resonances we can on one hand classify the shape of a class of domains with real algebraic boundaries and on the other hand recover the separation distance between two components of multiple connected domains. These results have important applications in nanophotonics. They can be used in order to identify the shape and separation distance between plasmonic nanoparticles having known material parameters from measured plasmonic resonances, for which the scattering cross-section is maximized. A real algebraic curve is the zero level set of a bivariate polynomial. Domains enclosed by real algebraic curves (henceforth simply called algebraic domains) are dense, in Hausdorff metric among all planar domains. On a simpler note, every smooth curve can be approximated by a sequence of algebraic curves. This observation turns algebraic curves into an efficient tool for describing shapes [64, 88, 94]. Note that an algebraic domain which is the sub level set of a polynomial of degree n can uniquely be determined from its set of two-dimensional moments of order less than or equal to 3n [55, 67]. In this chapter we consider a class of algebraic curves determined via conformal mappings by two parameters m and δ, with m being the order of the polynomial parametrizing the curve and δ being a shape parameter, see (6.1) and (6.2). One can think of algebraic domains as non-generic, but dense, among all planar domains, as much as polynomials are non-generic, but dense among all continuous functions on a compact set. In either case, the identifications/reconstructions have to be complemented by a fine analysis of the rate of convergence. The main results of the present chapter are: (i) Algebraic domains described by (6.2) have only two plasmonic resonances asymptotically (in δ). Based on these two plasmonic resonances, one can classify them; (ii) Two nearly touching disks have an infinite number of plasmonic resonances and the separating distance can be determined from the measurement of the first plasmonic resonance. The chapter is organized as follows. In section 6.2 we give explicit calculations of the Neumann-Poincaré operator associated with an algebraic domain. Moreover, we analyze its asymptotic behavior as δ approaches zero. We compute the first- and second-order contracted polarization tensors, and show how to use them to determine the two parameters describing the algebraic boundaries. In section 6.3 we consider two nearly touching disks. We use the bipolar coordinates to compute the spectrum of the associated Neumann-Poincaré operator. We show that all the eigenvalues of the associated Neumann-Poincaré operator contribute to the set of plasmonic resonances. From the first-order polarization tensor, we show that we can recover the separating distance between the disks. In section 6.4 we illustrate our main findings in this chapter with several numerical examples. 6.2. Plasmonic resonance for algebraic domains 135

6.2 Plasmonic resonance for algebraic domains 6.2.1 Algebraic domains of class Q Let Ω be the unit disk in C. For m N and a R, define Φ : C Ω C ∈ ∈ m,a \ → by a Φ (ζ)= ζ + . m,a ζm Assume that Φ is injective on C Ω. We introduce the class as the m,a \ Q collection of all bounded domains D C bounded by the curves ⊂ ∂D = Φ (ζ): ζ = r for some r > 1, m N and a R. { m,a | | 0} 0 ∈ ∈ Note that Φ is a conformal mapping from ζ > r onto C D. In m,a {| | 0} \ what follows, we shall suppress the subscript m, a from Φm,a for the ease of notation. Conformal images of the unit disc by rational functions are also called quadrature domains. We refer to [56, 84] for details and ramifications of the theory of quadrature domains. In particular, up to the inversion z 1/z, 7→ the complements of the domains in class are quadrature domains. We ρ+iθ Q ρ0 write for convenience ζ = e . Let ρ0 be such that r0 = e . Let J be the Jacobian defined by J = ∂ Φ(eξ) . ξ |ξ=ρ+iθ In the (ρ, θ) plane, the normal derivative ∂/∂ν on ∂D is represented as

∂ 1 ∂ = . ∂ν J ∂ρ Moreover, the boundary ∂D is parametrized by

ρ0+iθ ρ0+iθ mρ0 imθ θ Φ(e )= e + ae− − . 7→

If we ﬁx the constant a and change ρ0, then the size and the shape of ∂D will change accordingly. In order to leave the shape unchanged, we need to represent the constant a in a diﬀerent way. We write

a = e(m+1)ρ0 δ. (6.1)

Then the boundary ∂D can be represented as

ρ0+iθ ρ0 iθ imθ θ Φ(e )= e (e + δe− ). (6.2) 7→

Now, if we ﬁx the constant δ and change ρ0, then it is clear that only the size changes and the shape stays unaﬀected. The parameter eρ0 can be considered as a generalized radius of D because it determines the size. In conclusion, the shape of D is determined by the two parameters m and δ, while the size by the parameter ρ0.

6.2.2 Explicit computation of the Neumann-Poincaré operator In this section, we compute the Neumann-Poincaré operator on ∂D explicitly. We need to compute [J 1 cos nθ] and [J 1 sin nθ] explicitly. Our KD∗ − KD∗ − 136 Chapter 6. Shape Recovery of Algebraic Domains strategy is as follows. Let u = [J 1 cos nθ] and v = [J 1 sin nθ]. If SD − SD − u,v can be obtained explicitly, then [J 1 cos nθ] and [J 1 sin nθ] are KD∗ − KD∗ − immediately derived by using the following identity:

1 ∂ [ϕ] ∂ [ϕ] D∗ [ϕ]= S + S , (6.3) K 2 ∂ν + ∂ν − which follows from (1.1). For simplicity, we consider only u. By using the continuity of the single layer potential and the jump relation (1.1), we can see that the function u is the solution to the following problem:

∆u = 0 in C ∂D, \  u = u + on ∂D, |− |  (6.4)  ∂u ∂u 1  = J − cos nθ on ∂D,  ∂ν + − ∂ν − u = O( z 1 ) as z .  −  | | | | → ∞  Let u(ρ, θ)=(u Φ)(eρ+iθ). Since Φ(ζ) is conformal on ζ >eρ0 , the above ◦ | | problem can be rewritten as follows: e ∆u = 0 for ρ < ρ0,

 ∆u = 0 for ρ > ρ0,   u = u on ρ = ρ ,  + 0 (6.5)  |−e |  ∂u ∂u  = cos nθ on ρ = ρ0, e∂ρ + −e ∂ρ −  ρ  u e= O(e− e) as ρ .  → ∞  Note that in (6.5), the first equation for u is not represented in terms of u. e |D This is due to the singularity of Φ(ζ) near ζ = 0. Hence, we need to consider u more carefully. If a = 1 and m = 1, then D becomes an ellipse and (ρ, θ) |D e are called the elliptic coordinates. In this case, equation (6.5) for u can be easily solved by imposing some appropriate conditions on ρ = ρ0 and ρ = 0. However, for general shaped domains, this is not easy. e Fortunately, we can overcome this difficulty by the fact that the shape of the domain D is defined by a rational function Φ(ζ) = ζ + a/ζm. Our strategy is to seek a solution to (6.5) such that

u(z)= a polynomial of degree n in z for z D. ℜ{ } ∈ We can show that, for 1 n m, u in equation (6.5) can be explicitly ≤ ≤ |D solved by using the following ansatz: a n u (z) zn = ζ + |D ∝ ℜ{ } ℜ ζm nn o k n n k a ρ+iθ = ζ − (ζ = e ) ℜ k ζm k=0 X n nρ k n tmnρ mn = e cos nθ + a e k cos t θ, (6.6) k − k Xk=1 6.2. Plasmonic resonance for algebraic domains 137

mn where the constant tk is deﬁned by

tmn =(m + 1)k n, 0 k n. k − ≤ ≤ As will be seen later, for the purpose of computing the polarization tensor, we consider only the case where 1 n m. (If n > m, u (z) turns out to ≤ ≤ |D be more complicated polynomial than zn but is still a polynomial of degree n.) Let us assume 1 n m. In view of (6.6), we deﬁne ≤ ≤ n n mn nρ k tk ρ mn e cos nθ + a e− cos tk θ, ρ < ρ0,  k=1 k! w(ρ, θ):=  X  n  n mn  n(ρ 2ρ0) k tk ρ mn e− − cos nθ + a e− cos tk θ, ρ > ρ0. k=1 k!  X  Note that w is harmonic in ρ < ρ and ρ > ρ and w = O(e ρ) as { 0} { 0} − ρ . Moreover, → ∞

w + = w on ρ = ρ0, | |− (6.7) ∂w ∂w nρ0  =( 2)ne cos nθ on ρ = ρ0.  ∂ρ + − ∂ρ − −

Therefore, the function w is equal to u up to a multiplicative constant. More precisely, we have

1 nρ0 u(ρ, θ)= e e− w(ρ, θ). (6.8) −2n Now we are ready to compute [J 1 cos nθ]. We can check that e KD∗ − n 1 ∂w + ∂w n mn − mn k tk ρ0 mn + = tk a e− cos tk θ. (6.9) 2 ∂ρ ρ=ρ0 ∂ρ ρ=ρ0 − k k=1 X Then it follows from (6.3) and (6.8) that

n mn 1 1 k tk n mn ∗ [J − cos nθ]= δ cos t θ (6.10) KD J 2n k k Xk=1 for 1 n m. In exactly the same manner, we can show that ≤ ≤ n mn 1 1 k tk n mn ∗ [J − sin nθ]= δ sin t θ. (6.11) KD −J 2n k k Xk=1 It is worth mentioning that we can also compute the single layer potentials 1 1 for J − cos nθ and J − sin nθ: n 1 1 1 k n mn [J − cos nθ]= cos nθ δ cos t θ, (6.12) SD −2n − 2n k k Xk=1 138 Chapter 6. Shape Recovery of Algebraic Domains and n 1 1 1 k n mn [J − sin nθ]= sin nθ + δ sin t θ. (6.13) SD −2n 2n k k Xk=1 6.2.3 Asymptotic behavior of the Neumann-Poincaré operator ∗ KD If δ is small enough, then the shape of ∂D is close to a circle. Next we investigate the asymptotic behavior of the Neumann-Poincaré operator and its spectrum for small δ. From (6.10), we infer

1 (m + 1 n) 1 2 ∗ [J − cos nθ]= δ − J − cos(m + 1 n)θ + O(δ ), KD 2 − 1 (m + 1 n) 1 2 ∗ [J − sin nθ]= δ − J − sin(m + 1 n)θ + O(δ ) (6.14) KD − 2 − for small δ and 1 n m. One can verify the decay ≤ ≤ 1 1 2 ∗ [J − cos nθ], ∗ [J − sin nθ]= O(δ ) (6.15) KD KD for small δ and n m + 1. ≥ Let us denote by

c 1 s 1 vn = J − cos nθ, vn = J − sin nθ, and let Vc and Vs be the subspaces deﬁned by

V = span vc,vc,...,vc ,...,vc ,... and V = span vs,vs,...,vs ,...,vs ,... . c { 1 2 n m } s { 1 2 n m } In view of (6.14) and (6.15), we can easily see that the Neunamm-Poincaré operator can be approximated by a finite rank operator for small δ. To KD∗ state this fact, we define a finite rank operator c by Fm c c c (m + 1 n)vm+1 n, 1 n m, m[vn]= − − ≤ ≤ F (0, n m + 1. ≥ c [vs ] = 0, n 1. Fm n ≥ Similarly, we define s by Fm s s s (m + 1 n)vm+1 n, 1 n m, m[vn]= − − ≤ ≤ F (0, n m + 1. ≥ s [vc ] = 0, n 1. Fm n ≥

Then, on the subspace Vc, we have

δ c 2 ∗ = + O(δ ). KD 2Fm

Similarly, on the subspace Vs, we have

δ s 2 ∗ = + O(δ ). (6.16) KD −2Fm 6.2. Plasmonic resonance for algebraic domains 139

Here, O(δ2) is with respect to the operator norm. Since the operators c and s are of ﬁnite rank, they have matrix rep- Fm Fm resentations. Using vc m as basis, c can be represented as the following { n}n=1 Fm matrix MD,m:

0 0 ... 1 0 ... 2   MD,m := ...... (6.17)  m 1 ... 0   −   m ... 0 0      Clearly, s has the same matrix representation M using vs m as basis. Fm D,m { n}n=1 Let us now consider the eigenvalues and the associated eigenvectors of the matrix MD,m. The following lemma can be easily proven. Lemma 6.2.1. (i) If m is odd, that is, m = 2k 1 for some k N, then − ∈ the matrix MD,m has the following eigenvalues:

k, √1 m, 2 (m 1),..., (k 1) (k + 1), ± · ± · − ± − · and the associated eigenvectorsp are given byp

ek, e1 √m em, e2 √m 1 em 1, ..., √k 1 ek 1 √k + 1 ek+1, ± ± − − − − ±

where ei is the unit vector in the i-th direction. (ii) If m is even, that is, m = 2k for some k N, then the matrix M ∈ D,m has the following eigenvalues:

√1 m, 2 (m 1),..., k (k + 1), ± · ± · − ± · and the associated eigenvectorsp are given by p

e1 √m em, √2 e2 √m 1 em 1, ..., √k ek √k + 1 ek+1. ± ± − − ± Using (6.16), Lemma 6.2.1 and the perturbation theory [62], we get the following asymptotic result for on V . KD∗ c Theorem 6.2.1. For small δ, we have the following asymptotic expansions of eigenvalues and eigenfunctions of on V : KD∗ c (i) If m is odd, that is, m = 2k 1 for some k N: − ∈ Eigenvalues: up to order δ δ k, √1 m, 2 (m 1) ,..., (k 1) (k + 1) . 2 × ± · ± · − ± − · n p p o Eigenfunctions: up to order δ0

c c c c c c c vk, v1 √mvm, √2 v2 √m 1 vm 1, ..., √k 1 vk 1 √k + 1 vk+1. ± ± − − − − ± (ii) If m is even, that is, m = 2k for some k N: ∈ Eigenvalues: up to order δ δ √1 m, 2 (m 1) ,..., k (k + 1) . 2 × ± · ± · − ± · n p p o 140 Chapter 6. Shape Recovery of Algebraic Domains

Eigenfunctions: up to order δ0

c c c c √ c c v1 √mvm, √2 v2 √m 1 vm 1, ..., kvk √k + 1 vk+1. ± ± − − ± Similarly, we have the following result for on the subspace V . KD∗ s Theorem 6.2.2. We have the following asymptotic expansion of eigenvalues and eigenfunctions of the Neumann-Poincaré operator on the subspace KD∗ Vs for small δ: (i) If m is odd, that is, m = 2k 1 for some k N: − ∈ Eigenvalues: up to order δ

δ k, √1 m, 2 (m 1) ,..., (k 1) (k + 1) . −2 × ± · ± · − ± − · p p Eigenfunctions: up to order δ0

s s s s s s s vk, v1 √mvm, √2v2 √m 1vm 1, ..., √k 1 vk 1 √k + 1 vk+1. ± ± − − − − ± (ii) If m is even, that is, m = 2k for some k N: ∈ Eigenvalues: up to order δ

δ √1 m, 2 (m 1) ,..., k (k + 1) . −2 × ± · ± · − ± · p p Eigenfunctions: up to order δ0

s s s s √ s s v1 √mvm, √2 v2 √m 1 vm 1, ..., kvk √k + 1 vk+1. ± ± − − ± Corollary 6.2.1. Suppose that m is odd, that is, m = 2k 1 for some k N. − ∈ In other words, D is a star-shaped domain with 2k petals. Then, up to order δ, the Neumann-Poincaré operator has the following 2k eigenvalues: KD∗ δ √1 m, 2 (m 1) ,..., (k 1) (k + 1), √k k . 2 × ± · ± · − ± − · ± · p p 6.2.4 Generalized polarization tensors and their spectral representations First-order polarization tensor Let us compute the ﬁrst-order polarization tensor associated with D and λ. Recall the deﬁnition of λ, ε + ε λ := m c , 2(ε ε ) m − c for a domain D with permittivity εc and backgroud with permittivity εm. See chapter 1 for a brief introduction on generalized polarization tensors. For simplicity, we consider only the case when m is odd, that is, m = 2k 1 for some k N. The case where m is even can be treated analogously. − ∈ Numerical results are presented in section 6.4 for both cases. 6.2. Plasmonic resonance for algebraic domains 141

Since m is odd, the shape of D has even symmetry with respect to both x1-axis and x2-axis. Thanks to this symmetry, M(λ,D) has the following simple form [18]: 1 0 M(λ,D)= m , 11 0 1 where m11 is given by

1 m11 = x1, (λI D∗ )− [ν1] 1 , 1 . −K 2 − 2 Let λ and ϕ , j N, be the eigenvalues and the (normalized) eigenfunctions j j ∈ of , respectively. Then, from the spectral decomposition of , we have KD∗ KD∗ (see chapter 1)

(x1,ϕj) 1 1 ( D[ν1],ϕj) 1 1 1 2 , 2 2 , 2 m11 = − −S − λ λj ( D[ϕj],ϕj) 1 , 1 Xj − −S 2 − 2 1 2 ( λj) (x1,ϕj) 1 , 1 = 2 − | 2 − 2 | . λ λj ( D[ϕj],ϕj) 1 , 1 Xj − −S 2 − 2 By Theorems 6.2.1 and 6.2.2, one can see that only the following two eigenvalues and two eigenfunctions contribute to m11 up to order δ: 1 eigenvalues λ := δ√m, ± ±2 c c eigenfunctions ϕ := v1 √mvm. ± ±

In fact, for other eigenfunctions, we have (x1,ϕj) 1 , 1 = O(δ). In what 2 − 2 follows we calculate (x1,ϕ ) 1 , 1 and (x1, D[ϕ ]) 1 , 1 . ± 2 2 S ± 2 2 First, since dσ = Jdθ and − −

ρ0+iθ ρ0 mρ0 x = Φ(e ) = e cos θ + ae− cos mθ, 1|∂D ℜ{ } we have

2π ρ0 mρ0 (x1,ϕ ) 1 , 1 = (e cos θ + ae− cos mθ)(cos θ √m cos mθ) dθ, ± 2 2 ± − Z0 = πeρ0 (1 + 2λ ). ±

Now, we compute ( D[ϕ ],ϕ ) 1 , 1 . Note that, from (6.12), we have S ± ± 2 − 2 1 1 [vc ]= cos nθ δ cos(m + 1 n)θ + O(δ2). SD n −2n − 2 − Consequently,

c c c c ( D[ϕ ],ϕ ) 1 , 1 =( D[v1 √mvm],v1 √mvm) 1 , 1 −S ± ± 2 − 2 −S ± ± 2 − 2 2π = (cos θ √m cos mθ) ± Z0 1 δ 1 δ ( cos θ + cos mθ cos mθ √m cos θ) dθ + O(δ2) × 2 2 ± 2√m ± 2 = π(1 + 2λ )+ O(δ2). ± 142 Chapter 6. Shape Recovery of Algebraic Domains

Finally, we are ready to obtain an approximation formula for m11. Theorem 6.2.3. We have

π 2ρ0 1 1 2 m11 = e + + O(δ ), (6.18) 2 λ λ+ λ λ − − − as δ 0. → Second-order contracted generalized polarization tensors cc ss sc cs Let Mmn,Mmn,Mmn, and Mmn be the contracted generalized polarization sc cs tensors. One can easily see that M22 = M22 = 0 and M12 = M21 = 0. We cc ss only need to consider M22 and M22 . It turns out that only the following two cc eigenvalues and two eigenfunctions contribute to M22(up to the order δ): 1 eigenvalues λ′ := δ 2 (m 1), ± ±2 · − p c c eigenfunctions ϕ′ := √2 v2 √m 1 vm 1. ± ± − − Let H := (x + ix )2 . Then we have ℜ 1 2 H = e 2ρ0 (cos 2θ + 2δ cos mθ + δ2 cos 2mθ). |∂D Therefore,

2π 2ρ0 2 √ (H,ϕ′ ) 1 , 1 = e (cos 2θ + 2δ cos mθ + δ cos 2mθ)( 2 cos 2θ √m 1 cos(m 1)θ) dθ ± 2 2 ± − − − Z0 = √2πe2ρ0 .

Now we compute ( D[ϕ′ ],ϕ′ ) 1 , 1 . Since −S ± ± 2 − 2 1 1 [vc ]= cos nθ δ cos(m + 1 n)θ + O(δ2), SD n −2n − 2 − we obtain

√ c c √ c c ( D[ϕ′ ],ϕ′ ) 1 , 1 =( D[ 2 v2 √m 1 vm 1], 2 v2 √m 1 vm 1) 1 , 1 −S ± ± 2 − 2 −S ± − − ± − − 2 − 2 2π = (√2 cos 2θ √m 1 cos(m 1)θ) ± − − Z0 1 δ ( cos 2θ + cos(m 1)θ × 2√2 √2 − 1 δ cos(m 1)θ √m 1 cos 2θ) dθ + O(δ2) ± 2√m 1 − ± − 2 − 2 = π(1 + 2λ′ )+ O(δ ). ± Finally, we ﬁnd

1 2 ( λj) (H,ϕj) 1 , 1 cc 2 | 2 − 2 | M22 = − , (λ λj) ( D[ϕj],ϕj) 1 , 1 Xj − −S 2 − 2 ( 1 λ ) ( 1 + λ ) 4ρ0 2 +′ 2 ′ 2 = πe 1 − + 1 − + O(δ ). (6.19) ( 2 + λ+′ )(λ λ+′ ) ( 2 λ′ )(λ λ′ ) − − − − − 6.3. Plasmonic resonances for two separated disks 143

ss Similarly, one can show that M22 has similar asymptotic expansion.

6.2.5 Classification of algebraic domains in the class Q The identification of the parameters ρ0 and m is now straightforward using the results of the previous subsection. Suppose that we can obtain the values cc of λ ,λ′ approximately from m11 and M22. Then, by formula (6.18), we ± ± can easily find the parameter ρ0, which determines the size of D. In order to reconstruct the parameters m and δ we turn to the definitions of λ+ and λ+′ : 1 1 λ = δ√m, λ′ = δ 2 (m 1). + 2 + 2 · − p It is worth emphasizing that the eigenvalues λ+ and λ+′ are first-order approximations of the exact eigenvalues of Neumann-Poincaré operator for KD∗ small δ. Solving the above equations for m and δ yields the exact formulas:

2 λ+ 2 2 m = 2 2 , δ = 2 λ+ (λ+′ ) /2. λ (λ′ ) /2 − + − + q 6.3 Plasmonic resonances for two separated disks

In this section, we consider the spectrum of the Neumann-Poincaré operator when two conductors are located closely to each other in R2. As an application of the spectral decomposition of the Neumann-Poincaré operator, we derive the (1, 1)-entry, m11, of the ﬁrst-order polarization tensor associated with the two disks. See chapter 1 for a brief introduction on polarization tensors.

6.3.1 The bipolar coordinates and the boundary integral operators

Let B1 and B2 be two disks with conductivity σ embedded in the background with conductivity 1. The conductivity is such that 0 < σ = 1 < . Let 6 ∞ σB1 B2 denote the conductivity distribution, i.e., ∪ R2 σB1 B2 = σχ(B1)+ σχ(B2)+ χ( (B1 B2), (6.20) ∪ \ ∪ where χ is the characteristic function. Let ǫ be the distance between two disks, that is, ǫ := dist(B1,B2).

We set Cartesian coordinates (x1,x2) such that x1-axis is parallel to the line joining the centers of the two disks.

(Definition) Each point x =(x1,x2) in the Cartesian coordinate system corresponds to (ξ,θ) R ( π, π] in the bipolar coordinate system through ∈ × − the equations sinh ξ sin θ x = α and x = α (6.21) 1 cosh ξ cos θ 2 cosh ξ cos θ − − 144 Chapter 6. Shape Recovery of Algebraic Domains with a positive number α. In fact, the bipolar coordinates can be defined using a conformal mapping. Define a conformal map Ψ by ζ + 1 z = x + ix = Ψ(ζ)= α . 1 2 ζ 1 − ξ iθ If we write ζ = e − , then we can recover (6.21).

(The coordinate curve) From the deﬁnition, we can derive that the coordinate curves ξ = c and θ = c are, respectively, the zero-level set of the { } { } following two functions:

cosh c 2 α 2 f (x,y)= x α + y2 (6.22) ξ − sinh c − sinh c and cos c 2 α 2 f (x,y)= x2 + y α . θ − sin c − sin c

(Basis vectors) Orthonormal basis vectors eˆ , eˆ are deﬁned as follows: { ξ θ} ∂x/∂ξ ∂x/∂θ eˆ := and eˆ := . ξ ∂x/∂ξ θ ∂x/∂θ | | | |

(Normal- and tangential derivatives and line element) In the bipolar coordinates, the scaling factor h is cosh ξ cos θ h(ξ,θ) := − . α The gradient of any scalar function g is

∂g ∂g g = h(ξ,θ) eˆ + eˆ . (6.23) ∇ ∂ξ ξ ∂θ θ Moreover, the normal and tangential derivatives of a function u in bipolar coordinates are ∂u ∂u = u vξ=c = sgn(c)h(c,θ) , ∂ν ξ=c ∇ · − ∂ξ ξ=c  (6.24)  ∂u ∂u  = sgn(c)h(c,θ) , ∂T ξ=c − ∂θ ξ=c

 and the line element dσ on the boundary ξ = ξ is { 0} 1 dσ = dθ. h(ξ0,θ) 6.3. Plasmonic resonances for two separated disks 145

(Separation of variables) The bipolar coordinate system admits separation of variables for any harmonic function f as follows:

∞ nξ nξ nξ nξ f(ξ,θ)= a0 + b0ξ + c0θ + (ane + bne− ) cos nθ+ cne + dne− )sin nθ , n=1 X (6.25) where an, bn, cn and dn are constants. For ξ > 0, we have

ζ ζ sinh ξ i sin θ e + e− ∞ nξ − = = 1 + 2 e− (cos nθ i sin nθ), (6.26) cosh ξ cos θ eζ e ζ − − n=1 − − X with ζ =(ξ + iθ)/2. Using (6.21), we have the following harmonic expansions for the two linear functions x1 and x2:

∞ n ξ x1 = sgn(ξ)α 1 + 2 e− | | cos nθ , (6.27) " n=1 # X and ∞ n ξ x2 = 2α e− | | sin nθ. n=1 X Let K∗ be the Neumann-Poincaré operator given by ∂ ∗ B K KB1 ∂ν(1) S 2 ∗ :=  ∂  , (2) B1 B∗ 2  ∂ν S K    and deﬁne the operator S by

S = SB1 SB2 . SB1 SB2 (i) Here, ν is the outward normal on ∂Bi, i = 1, 2. Then, from [7], K∗ is self-adjoint with the inner product

∗ S 1/2 1/2 −1/2 −1/2 (ϕ,ψ) := ( [ψ],ϕ)H (∂B1) H (∂B2),H (∂B1) H (∂B2), H − × × 1/2 1/2 for ϕ,ψ H− (∂B ) H− (∂B ). ∈ 0 1 × 0 2 6.3.2 Neumann Poincaré-operator for two separated disks and its spectral decomposition First we introduce some notations. Set

ǫ 1 α α = ǫ(r + ) and ξ = sinh− , for j = 1, 2, (6.28) 4 0 r r where r is the radius of the two disks and ǫ their separation distance. Note that ∂B = ξ =( 1)jξ , for j = 1, 2. (6.29) j { − 0} 146 Chapter 6. Shape Recovery of Algebraic Domains

Let us denote the Neumann-Poincaré operator for two disks separated by K a distance ǫ by ǫ∗. To ﬁnd out the spectral decomposition of the Neumann- K Poincaré operator ǫ∗, we use the following lemma [7]. Lemma 6.3.1. Assume that there exists u a nontrivial solution to the following equation:

∆u = 0 in B B R2 (B B ), 1 ∪ 2 ∪ \ 1 ∪ 2 u + = u on ∂Bj,j = 1, 2, | |− ∂u ∂u (6.30)  = σ0 on ∂Bj,j = 1, 2, ∂ν + ∂ν x − x u( ) 0 as ,  → | | → ∞  1 + 2λ where σ = 0 < 0. If we set 0 −1 2λ − 0 + ∂u ∂u − ψj := , for j = 1, 2, ∂ν ∂Bj − ∂ν ∂Bj

ψ then ψ = 1 is an eigenvector of K corresponding to the eigenvalue λ . ψ ǫ∗ 0 2 One can see that the following function un is a solution to (6.30):

1 n ξ0 n ξ0 n ξ+inθ (e| | e−| | )e| | , for ξ < ξ , ∓2 n ∓ − 0  1 | |  n ξ0 n ξ n ξ inθ un±(ξ,θ)=(const.)+  e−| | (e| | e−| | )e , for ξ0 <ξ<ξ0, 2 n ∓ −  | | 1 n ξ0 n ξ0 n ξ+inθ (e| | e−| | )e−| | , for ξ>ξ0. 2 n ∓  | |  (6.31)  K From (6.31) and Lemma 6.3.1, we obtain eigenvalues and eigenvectors to ǫ∗

1 2 n ξ0 inθ h( ξ0,θ) λǫ,n± = e− | | and Φǫ,n± (θ)= e − . ±2 h(ξ0,θ) ∓ Note that the above eigenvectors are not normalized. S We compute ( [Φǫ,n± ], Φǫ,n± ) 1 , 1 . From (6.31), one can see that − 2 − 2

1 2 n ξ0 + + 1 2 n ξ0 K∗ = e− | | Ψ Ψ + e− | | Ψ− Ψ− , ǫ 2 ǫ,n ⊗ ǫ,n −2 ǫ,n ⊗ ǫ,n n=0 n=0 X6 X6 6.3. Plasmonic resonances for two separated disks 147

where Ψǫ,n± are the normalized eigenvectors deﬁned by

inθ n e h( ξ0,θ) Ψǫ,n± (θ) := | | − . (6.32) 2π(1 e 2 n ξ0 ) h(ξ0,θ) p ∓ − | | ∓ Note that p

ǫ Let us compute the (1, 1)-entry m11 of the ﬁrst-order polarization tensor for two separated disks. Note that

ǫ I K 1 m11 = ϕ, (λ ǫ∗)− [ψ] 1 , 1 , − 2 − 2 where x ν φ = 1|∂B1 , ψ = 1|∂B1 . x ν 1|∂B2 1|∂B2 K The spectral decomposition of ǫ∗ implies

+ + φ, Ψǫ,n) 1 , 1 (Ψǫ,n,ψ) ∗ φ, Ψǫ,n− ) 1 , 1 (Ψǫ,n− ,ψ) ∗ ǫ h 2 − 2 H h 2 − 2 H m11 = + + λ λ λ λ− n=0 ǫ,n n=0 ǫ,n X6 − X6 − 1 + + 2 1 2 2 λǫ,n φ, Ψǫ,n) 1 , 1 2 λǫ,n− φ, Ψǫ,n− ) 1 , 1 − |h 2 − 2 | − |h 2 − 2 | = + + . λ λ λ λ− n=0 ǫ,n n=0 ǫ,n X6 − X6 − From (6.27), we derive the expansion

∞ m ξ +imθ x1 = sgn(ξ)α e−| || | . (6.35) m= X−∞ Therefore,

φ, Ψǫ,n− ) 1 , 1 = 0. h 2 − 2 As a consequence, we arrive at the following result. 148 Chapter 6. Shape Recovery of Algebraic Domains

Propsition 6.3.1. We have

2 2 n ξ0 2nξ0 ǫ 4πα n e− | | 2 ∞ ne− m11 = | | + = 8πα , λ λ λ 1 e 2nξ0 n=0 ǫ,n n=1 2 − X6 − X − where α is given by (6.28).

6.3.4 Reconstruction of the separation distance Suppose that the ﬁrst eigenvalue

+ 1 2ξ0 λ = e− ǫ,1 2 is measured. Then we immediately ﬁnd the value of eξ0 . From (6.28), we have ǫ r cosh ξ = + r. 0 2 By solving the above quadratic equation, we can determine the distance ǫ between the two disks.

6.4 Numerical illustrations

In this section we illustrate our main findings in this chapter with several numerical examples. We use the material parameters of gold nanoparticles and suppose that we can measure their first- and second-order polarization tensors for a range of wavelengths in the visible regime. Figure 6.1 shows the variations of the real and imaginary parts of λ, defined by (2.23), as function of the wavelength using Drude’s model for σ = σ(ω), which is depending on the operating frequency ω [9]. As shown in Figure 6.1, the imaginary part of λ is very small. Therefore, when the real part of λ hits an eigenvalue that contributes to the first-order polarization tensor (and therefore to the plasmonic resonances), we should see a peak in the graph of m and M cc with respect to the wavelength. | 11| | 22| This allow us, in the case of class of algebraic domains, to recover λ+ and Q + λ+′ and, in the case of two separated disks, to recover λǫ,1. Figures 6.2, 6.3, and 6.4 present examples of algebraic domains and their reconstructions, where a circle of radius one has been transformed for different values of m and δ. Figures 6.5 and 6.6 present examples of algebraic domains and their reconstructions, where a circle of radius one has been transformed for m = 4 and m = 6 and δ = 0.02. Figures 6.7, 6.8, and 6.9 show examples of two circles of radius one separated by a distance ǫ, and their reconstructions.

6.5 Concluding remarks

In this chapter we have proved for a class of algebraic domains that the associated plasmonic resonances can be used to classify them. It would be 6.5. Concluding remarks 149

0.3

0.2

0.1

-0.1

) 6

-0.2 real( -0.3

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10 -3 -3 #

-3.5

-4

-4.5

) -5 6

im( -5.5

-6

-6.5

-7

-7.5 1.5 2 2.5 3 3.5 -7 wavelength #10

Figure 6.1: Real and imaginary parts of λ as function of the wavelength.

very interesting to prove a similar result for all quadrature domains or all algebraic domains. We have also reconstructed the separation distance between two nanoparticles of circular shape from measurements of their ﬁrst collective plasmonic resonances. Another challenging problem would be to generalize this result to more components and arbitrary shaped particles. 150 Chapter 6. Shape Recovery of Algebraic Domains

m = 3 / = 0.066667 1.5

0.5

-0.5

-1

-1.5 -1.5 -1 -0.5 0 0.5 1 1.5

m = 2.9403 = 0.067972 estimate /estimate 1.5

0.5

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-1

-1.5 -1.5 -1 -0.5 0 0.5 1 1.5

m = 3 / = 0.066667 350

300

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| 11

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0 1.5 2 2.5 3 3.5 -7 wavelength #10

m = 3 / = 0.066667 1400

1200

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22 cc

|M 600

400

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0 1.5 2 2.5 3 3.5 -7 wavelength #10

Figure 6.2: From top to bottom and left to right: initial cc shape, reconstructed shape, m11 and M22 with respect to the wavelength for m| = 3| and δ| = 0|.066667. 6.5. Concluding remarks 151

m = 5 / = 0.033333 1.5

0.5

-0.5

-1

-1.5 -1.5 -1 -0.5 0 0.5 1 1.5

m = 4.872 = 0.034006 estimate /estimate 1.5

0.5

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-1.5 -1.5 -1 -0.5 0 0.5 1 1.5

m = 5 / = 0.033333 350

300

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0 1.5 2 2.5 3 3.5 -7 wavelength #10

m = 5 / = 0.033333 700

600

500

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22 cc

|M 300

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Figure 6.3: From top to bottom and left to right: initial cc shape, reconstructed shape, m11 and M22 with respect to the wavelength for m| = 5| and|δ = 0|.03333. 152 Chapter 6. Shape Recovery of Algebraic Domains

m = 7 / = 0.021978 1.5

0.5

-0.5

-1

-1.5 -1.5 -1 -0.5 0 0.5 1 1.5

m = 6.8191 = 0.022247 estimate /estimate 1.5

0.5

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-1.5 -1.5 -1 -0.5 0 0.5 1 1.5

m = 7 / = 0.021978 350

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m = 7 / = 0.021978 700

600

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22 cc

|M 300

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Figure 6.4: From top to bottom and left to right: initial cc shape, reconstructed shape, m11 and M22 with respect to the wavelength for|m =| 7, δ =| 0.021978| . 6.5. Concluding remarks 153

m = 4 / = 0.05 1.5

0.5

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-1

-1.5 -1.5 -1 -0.5 0 0.5 1 1.5

m = 3.7767 = 0.05212 estimate /estimate 1.5

0.5

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-1.5 -1.5 -1 -0.5 0 0.5 1 1.5

m = 4 / = 0.05 350

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m = 4 / = 0.05 700

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Figure 6.5: From top to bottom and left to right: initial cc shape, reconstructed shape, m11 and M22 with respect to the wavelength for| m| = 4, |δ = 0|.05. 154 Chapter 6. Shape Recovery of Algebraic Domains

m = 6 / = 0.02381 1.5

0.5

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m = 5.8155 = 0.024208 estimate /estimate 1.5

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m = 6 / = 0.02381 350

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m = 6 / = 0.02381 700

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Figure 6.6: From top to bottom and left to right: initial cc shape, reconstructed shape, m11 and M22 with respect to the wavelength for| m =| 6, δ |= 0.02381| . 6.5. Concluding remarks 155

= 0.1998 0 = 0.2 0estimate 1.5 1.5

1 1

0.5 0.5

0 0

-0.5 -0.5

-1 -1

-1.5 -1.5 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4

0 = 0.2 600

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Figure 6.7: From top left to bottom: initial shape, reconstructed shape and m11 with respect to the wavelength for | | ǫ = 0.2.

= 0.99796 0 = 1 0estimate 1.5 1.5

1 1

0.5 0.5

0 0

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0 = 1 1200

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Figure 6.8: From top left to bottom: initial shape, reconstructed shape and m11 with respect to the wavelength for | | ǫ = 1. 156 Chapter 6. Shape Recovery of Algebraic Domains

= 2.4938 0 = 2.5 0estimate 1.5 1.5

1 1

0.5 0.5

0 0

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-1.5 -1.5 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4

0 = 2.5 1400

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Figure 6.9: From top left to bottom right: initial shape, reconstructed shape and m11 with respect to the wavelength| for |ǫ = 2.5. 157

Chapter 7

Super-Resolution with Plasmonic Nanoparticles

Contents 7.1 Introduction ...... 158 7.2 Multiple plasmonic nanoparticles ...... 158 7.2.1 Layer potential formulation in the multi-particle case ...... 158 7.2.2 Feld behavior at plasmonic resonances in the multi- particle case ...... 160 7.3 Super-resolution (super-focusing) by using plasmonic particles ...... 166 7.3.1 Asymptotic expansion of the scattered ﬁeld . . . . 166 7.3.2 Asymptotic expansion of the imaginary part of the Green function ...... 169 7.4 Concluding remarks ...... 170 158 Chapter 7. Super-Resolution with Plasmonic Nanoparticles

7.1 Introduction

Super-resolution is a set of techniques meant to cross the barrier of diffraction limits by reducing the focal spot size. This resolution limit applies only to light that has propagated for a distance substantially larger than its wavelength [39,40]. It is known that the resolution limit (or the diffraction limit) in a general inhomogeneous space is determined by the imaginary part of the Green function in the associated space [4]. An idea to break the resolution limit is to insert subwavelength resonators in the homogeneous space. This way, we can introduce propagating subwavelength resonance modes which, when excited at the right frequency, encode subwavelength informations. This yield a Green’s function whose imaginary part exhibits subwavelength peaks and therefore break the resolution limit (or diffraction limit) in the homogeneous space. The principle has been mathematically demonstrated in [30]. Super-focusing is the counterpart of super-resolution. It is a concept for waves to be confined to a length scale significantly smaller than the diffraction limit of the focused waves. As for the resolution problem, the focusing capacity is also determined by the imaginary part of the Green function in the associated space. The super-focusing phenomenon is being intensively investigated in the field of nanophotonics as a possible technique to focus electromagnetic radiation in a region of order of a few nanometers beyond the diffraction limit of light and thereby causing an extraordinary enhancement of the electromagnetic fields. Here, using the fact that plasmonic particles are ideal subwavelength resonators, we consider the possibility of super-resolution (super-focusing) by using a system of identical plasmonic particles. First, a precise analisys of field behavior of multiple plasmonic particles is in order.

7.2 Multiple plasmonic nanoparticles 7.2.1 Layer potential formulation in the multi-particle case We consider the scattering of an incident time harmonic wave ui by multiple weakly coupled plasmonic nanoparticles in three dimensions. Our motivation is to demonstrate the principle of super-resolution in resonant media; see Section 7.3. The analysis done in this section follows the same lines as those in chapter 2. The scattering from multiple weakly coupled, non-resonant small particles can be analyzed in the same way. However, no super-resolution can be achieved in this case. For ease of exposition, we consider the case of L particles with an identical shape. We assume that Condition 2.1 holds. Moreover, in contrast to Section 2.3 where the size of the particle is assumed to be of order one, we assume the following condition in this section. Condition 7.1. All the identical particles have size of order δ which is a small parameter and the distances between neighboring ones are of order one.

We write Dl = zl + δB, l = 1, 2,...,L, where B has size one and is centered at the origin. Moreover, we denote D0 = δB as our reference 7.2. Multiple plasmonic nanoparticles 159 nanoparticle. Denote by

L D = D , ε = ε χ(R3 D¯)+ ε χ(D¯), µ = µ χ(R3 D¯)+ µ χ(D). l D m \ c D m \ c l[=1 The scattering problem can be modeled by the following Helmholtz equation:

1 2 3 u + ω εDu = 0 in R ∂D, ∇· µD ∇ \   u+ u = 0 on ∂D,  − −  (7.1)  1 ∂u 1 ∂u  = 0 on ∂D, µm ∂ν + − µc ∂ν −  s i  u := u u satisﬁes the Sommerfeld radiation condition.  −  Let

i ikmd x u (x) = e · , i ikmd x Fl,1(x) = u (x) = e · , − ∂Dl − ∂Dl i ∂u ik md x Fl,2(x) = (x) = ikme · d ν(x) , − ∂ν − · ∂Dl ∂Dl

and deﬁne the operator k by KDp,Dl ∂G(x,y,k) k [ψ](x)= ψ(y)dσ(y), x ∂D . KDp,Dl ∂ν(x) ∈ l Z∂Dp Analogously, we deﬁne

k [ψ](x)= G(x,y,k)ψ(y)dσ(y), x ∂D . SDp,Dl ∈ l Z∂Dp The solution u of (7.1) can be represented as follows:

L i km 3 u + [ψl], x R D,¯ SDl ∈ \  l=1 u(x)=  X  L  kc [φl], x D, SDl ∈  Xl=1  1  where φ ,ψ H− 2 (∂D ) satisfy the following system of integral equations l l ∈ l

km kc km [ψl] [φl]+ [ψp]= Fl,1, SDl −SDl SDp,Dl p=l  X6  1 1 km 1 1 kc  Id +( D )∗ [ψl]+ Id ( D )∗ [φl]  µm 2 K l µc 2 − K l   1 + km [ψ ]= F , Dp,D p l,2  µm K l  p=l  X6   160 Chapter 7. Super-Resolution with Plasmonic Nanoparticles and F = ui on ∂D , l,1 − l  1 ∂ui  Fl,2 = on ∂Dl.  −µm ∂ν  7.2.2 Feld behavior at plasmonic resonances in the multi- particle case We consider the scattering in the quasi-static regime, i.e., when the incident wavelength is much greater than one. With proper dimensionless analysis, we can assume that ω 1. As a consequence, kc is invertible. Note that ≪ SD

kc 1 km km φl =( )− [ψl]+ [ψp] Fl,1 . SDl SDl SDp,Dl − p=l X6 We obtain the following equation for ψl’s,

(w)[ψ]= f, AD where (ω) 0 (ω) (ω) AD1 A1,2 ··· A1,L D2 (ω) 2,1(ω) 0 2,L(ω) D(w)=  A . +A . ··· A .  , A .. . 0 .    ···   D (ω)  L,1(ω) L,L 1(ω) 0   A L  A ··· A −      ψ1 f1 ψ2 f2 ψ =  .  , f =  .  , . .     ψ  f   L  L and    

1 1 km 1 1 kc kc 1 km Dl (ω) = Id +( D )∗ + Id ( D )∗ ( D )− D , A µm 2 K l µc 2 − K l S l S l 1 1 1 (ω) = Id ( kc ) ( kc ) 1 km + km , l,p D ∗ D − Dp,D Dp,D A µc 2 − K l S l S l µm K l

1 1 kc kc 1 fl = Fl,2 + Id ( D )∗ ( D )− [Fl,1]. µc 2 − K l S l The following asymptotic expansions hold (see chapter 1 for the deﬁnition of (∂D) and chapter 2 and Appendix B for the deﬁnition of the operators). H∗ Lemma 7.2.1. (i) Regarded as operators from (∂D ) into (∂D ), H∗ p H∗ l we have (ω)= + O(δ2ω2), ADj ADj ,0 (ii) Regarded as operators from (∂D ) into (∂D ), we have H∗ l H∗ j

1 1 1 1 2 2 4 l,p(ω)= Id D∗ l D− p,l,0,1+ p,l,0,2 + p,l,0,0+O(δ ω )+O(δ ). A µc 2 −K S l S S µm K 7.2. Multiple plasmonic nanoparticles 161

Moreover,

1 1 2 Id ∗ − p,l,0,1 = O(δ ), 2 −KDl ◦SDl ◦S 1 1 3 Id ∗ − p,l,0,2 = O(δ ), 2 −KDl ◦SDl ◦S = O(δ2). Kp,l,0,0 Proof. The proof of (i) follows from Lemmas 2.3.4 and B.2.3. We now prove (ii). Recall that

1 kc 1 2 2 Id ( )∗ = Id ∗ + O(δ ω ), 2 − KDl 2 −KDl

kc 1 1 1 1 2 2 ( )− = − kc − D ,1 − + O(δ ω ), SDl SDl − SDl S l SDl

km 2 4 2 2 = p,l,0,0 + p,l,0,1 + p,l,0,2 + km p,l,1 + k p,l,2,0 + O(δ )+ O(ω δ ) SDp,Dl S S S S mS

km 2 2 = p,l,0,0 + O(ω δ ). KDp,Dl K Using the identity 1 1 Id ∗ − [χ(Dl)] = 0, 2 −KDl SDl we can derive that 1 1 1 (ω) = Id ( kc ) 1 km + + O(δ2ω2) l,p D∗ l D − Dp,D p,l,0,0 A µc 2 −K S l S l µm K 1 1 1 = Id 1 km + + O(δ2ω2) D∗ l D− Dp,D p,l,0,0 µc 2 −K S l S l µm K 1 1 1 2 4 = Id D∗ l D− p,l,0,0 + p,l,0,1 + p,l,0,2 + km p,l,1 + km p,l,2,0 + O(δ ) µc 2 −K S l S S S S S 1 2 2 + p,l,0,0 + O(δ ω ) µm K 1 1 1 1 2 2 4 = Id D∗ l D− p,l,0,1 + p,l,0,2 + p,l,0,0 + O(δ ω )+ O(δ ). µc 2 −K S l S S µm K The rest of the lemma follows from Lemmas B.2.3 and B.2.6.

Denote by (∂D)= (∂D ) ... (∂D ), which is equipped with H∗ H∗ 1 × ×H∗ L the inner product L (ψ,φ) ∗ = (ψl,φl) ∗(∂D ). H H l Xl=1 With the help of Lemma 7.2.1, the following result is obvious. Lemma 7.2.2. Regarded as an operator from (∂D) into (∂D), we have H∗ H∗ 2 2 4 (ω) = D,0 + D,1 + O(ω δ )+ O(δ ), AD A A where 0 ... AD1,0 AD,1,12 AD,1,13 D2,0 D,1,21 0 D,1,23 ... D,0 =  A  , D,1 =  A A  A ... A ...  D ,0   D,1,L1 ... D,1,LL 1 0   A L   A A −      162 Chapter 7. Super-Resolution with Plasmonic Nanoparticles with 1 1 1 1 Dl,0 = + Id ( ) D∗ l , A 2µm 2µc − µc − µm K 1 1 1 = Id 1 + + . D,1,pq D∗ p D−p q,p,0,1 q,p,0,2 q,p,0,0 A µc 2 −K S S S µm K It is evident that

L ∞ D,0[ψ]= τj(ψ,ϕj,l) ∗ ϕj,l, (7.2) A H Xj=0 Xl=1 where 1 1 1 1 τj = + λj, (7.3) 2µm 2µc − µc − µm ϕj,l = ϕjel (7.4)

L with el being the standard basis of R . We take (ω) as a perturbation to the operator for small ω and AD AD,0 small δ. Using a standard perturbation argument, we can derive the perturbed eigenvalues and eigenfunctions. For simplicity, we assume that the following conditions hold.

Condition 7.2. Each eigenvalue λj, j J, of the operator ∗ is simple. ∈ KD0 Moreover, we have ω2 δ. ≪ In what follows, we only use the ﬁrst order perturbation theory and derive the leading order term, i.e., the perturbation due to the term . For each AD,1 l, we deﬁne an L L matrix R by letting × l

Rl,pq = D,1[ϕl,p],ϕl,q ∗ , A H = D,1[ϕlep],ϕleq , A ∗ H = D,1,pq[ϕl],ϕl ∗ . A H Lemma 7.2.3. The matrix Rl = (Rl,pq)p,q=1,...,L has the following explicit expression:

Rl,pp = 0, α+β 3 1 (zp zq) α β Rl,pq = (λj ) − 5 x y ϕl(x)ϕl(y)dσ(x)dσ(y) 4πµc − 2 zp zq α = β =1 Z∂D0 Z∂D0 | |X| | | − | 1 1 1 x y + (λ ) · ϕ (x)ϕ (y)dσ(x)dσ(y) 4πµ − 4πµ j − 2 z z 3 l l c m Z∂D0 Z∂D0 | p − q| = O(δ3), p = q. 6 Proof. It is clear that R = 0. For p = q, we have l,pp 6 I II III Rl,pq = Rl,pq + Rl,pq + Rl,pq, 7.2. Multiple plasmonic nanoparticles 163 where

I 1 1 1 R = Id ∗ − q,p,0,1[ϕl],ϕl , l,pq Dp Dp ∗ µc 2 −K S S (∂Dl) H II 1 1 1 R = Id ∗ − q,p,0,2[ϕl],ϕl , l,pq Dp Dp ∗ µc 2 −K S S (∂Dl) H III 1 Rl,pq = q,p,0,0[ϕl],ϕl ∗(∂D ). µm K H l I We ﬁrst consider Rl,pq. By the following identity 1 1 1 Id ∗ [ϕ ]= Id [ϕ ]=(λ )ϕ , 2 −KDp SDl l SDl 2 −KDp l j − 2 l we obtain

I 1 1 1 R = Id ∗ − q,p,0,1[ϕl], D [ϕl] , l,pq Dp Dp l 2 −µc 2 −K S S S L (∂Dl) 1 1 = (λj ) q,p,0,1[ϕl], Dl [ϕl] L2(∂D ). µc − 2 S S l Using the explicit representation of and the fact that (χ(∂D ),φ ) 2 = Sq,p,0,1 j l L (∂Dj ) 0 for j = 0, we further conclude that 6 I Rl,pq = 0.

Similarly, we have

II 1 1 Rl,pq = (λj ) q,p,0,2[ϕl], Dl [ϕl] L2(∂D ), µc − 2 S S l 1 1 = (λj ) µc − 2 · α+β α β 3(zp zq) α β δαβx y − 5 x y + 3 ϕl(x)ϕl(y)dσ(x)dσ(y), 4π zp zq 4π zp zq α = β =1 Z∂D0 Z∂D0 | |X| | | − | | − | α+β 3 1 (zp zq) α β = (λj ) − 5 x y ϕl(x)ϕl(y)dσ(x)dσ(y) 4πµc − 2 zp zq α = β =1 Z∂D0 Z∂D0 | |X| | | − | 1 1 1 α α + (λj ) 3 x y ϕl(x)ϕl(y)dσ(x)dσ(y). 4πµc − 2 zp zq α =1 Z∂D0 Z∂D0 |X| | − | Finally, note that

3 1 1 [ϕ ]= a ν(x)= a ν (x), Kq,p,0,0 l 4π z z 3 · 4π z z 3 m m p q p q m=1 | − | | − | X T where am = (y zq)m,ϕl 2 , and a =(a1,a2,a3) . − L (∂Dq) 164 Chapter 7. Super-Resolution with Plasmonic Nanoparticles

By identity (2.29), we have

III 1 Rl,pq = q,p,0,0[ϕl],ϕl ∗(∂D ) −µm K H l 1 = a ν(x),ϕl ∗ −4π z z 3µ · (∂Dl) | p − q| m H 1 1 1 = Id ∗ − (a (x z )),ϕ 3 Dp Dp p l −4π zp zq µm 2 −K S · − ∗(∂D ) | − | H l 1 1 = (λj ) a (x zp),ϕl 2 −4π z z 3µ − 2 · − L (∂Dp) | p − q| m 1 1 = (λ ) x yϕ (x)ϕ (y)dσ(x)dσ(y). −4π z z 3µ j − 2 · l l | p − q| m Z∂D0 Z∂D0 This completes the proof of the lemma.

We now have an explicit formula for the matrix Rl. It is clear that Rl is symmetric, but not self-adjoint. For ease of presentation, we assume the following condition.

Condition 7.3. Rl has L-distinct eigenvalues. We remark that Condition 7.3 is not essential for our analysis. Without this condition, the perturbation argument is still applicable, but the results may be quite complicated. We refer to [62] for a complete description of the perturbation theory. Let τ and X =(X , ,X )T , l = 1, 2,...,L, be the eigenvalues j,l j,l j,l,1 ··· j,l,L and normalized eigenvectors of the matrix Rj. Here, T denotes the transpose. We remark that each Xj,l may be complex valued and may not be orthogonal to other eigenvectors. Under perturbation, each τj is splitted into the following L eigenvalues of (ω), A 4 2 2 τj,l(ω)= τj + τj,l + O(δ )+ O(ω δ ). (7.5) The associated perturbed eigenfunctions have the following form

L 4 2 2 ϕj,l(ω)= Xj,l,pepϕj + O(δ )+ O(ω δ ). (7.6) p=1 X We are interested in solving the equation (ω)[ψ]= f when ω is close AD to the resonance frequencies, i.e., when τj,l(ω) are very small for some j’s. In this case, the major part of the solution would be based on the excited resonance modes ϕj,l(ω). For this purpose, we introduce the index set of resonance J as we did in chapter 2 for a single particle case. We deﬁne ϕj,m(ω), j J, P (ω)ϕj,m(ω)= ∈ J 0, j J c. ∈ In fact, 1 1 P (ω)= P (ω)= (ξ (ω))− dξ, (7.7) J j 2πi −AD j J j J γj X∈ X∈ Z where γj is a Jordan curve in the complex plane enclosing only the eigenvalues τj,l(ω) for l = 1, 2,...,L among all the eigenvalues. 7.2. Multiple plasmonic nanoparticles 165

To obtain an explicit representation of PJ (ω), we consider the adjoint operator D(ω)∗. By a similar perturbation argument, we can obtain its per- A ¯T ¯ turbed eigenvalue and eigenfunctions. Note that the adjoint matrix Rj = Rj has eigenvalues τj,l and corresponding eigenfunctions Xj,l. Then the eigenvalues and eigenfunctions of (ω) have the following form AD ∗ 4 2 2 τj,l(ω) = τj + τj,l + O(δ )+ O(ω δ ), 4 2 2 ϕj,l(ω) = ϕj,l + O(δ )+ O(ω δ ), e where e e L ϕj,l = Xj,l,pepϕj p=1 X e e with Xj,l,p being a multiple of Xj,l,p. We normalize ϕj,l in a way such that the following holds e (ϕj,p, ϕj,q) ∗(∂D) = δpq, e H which is also equivalent to the followinge condition T Xj,p Xj,q = δpq.

Then, we can show that the followinge result holds. Lemma 7.2.4. In the space (∂D), as ω goes to zero, we have H∗ 2 3 f = ωf0 + O(ω δ 2 ),

T where f0 =(f0,1,...,f0,L) with

1 1 1 3 ikmd zl 1 f0,l = i√εmµme · d ν(x)+ Id ∗ − [d (x z)] = O(δ 2 ). − µ · µ 2 −KDl SDl · − m c Proof. We ﬁrst show that

3 +m 1 +m ∗ 2 ∗ 2 u (∂D0) = δ u (∂B), u (∂D0) = δ u (∂B) k kH k kH k kH k kH for any homogeneous function u such that u(δx)= δmu(x). Indeed, we have m 3 ∗ 2 ∗ η(u)(x) = δ u(x). Since η(u) (∂B) = δ− u (∂D0) (see Appendix k kH k kH B.2), we obtain

3 3 +m ∗ 2 ∗ 2 ∗ u (∂D0) = δ η(u) (∂B) = δ u (∂B), k kH k kH k kH which proves our ﬁrst claim. The second claim follows in a similar way. Using this result, by a similar argument as in the proof of Lemma 2.3.6 we arrive at the desired asymptotic result.

ikmd z Denote by Z = (Z1,...,ZL), where Zj = ikme · j . We are ready to present our main result in this section. Theorem 7.2.1. Under Conditions 2.1, 2.2, 2.3, 7.1, and 7.3, the scattered ﬁeld by L plasmonic particles has the following representation

us = km [ψ], SD 166 Chapter 7. Super-Resolution with Plasmonic Nanoparticles where

L f, ϕj,l(ω) ∗ ϕj,l(ω) 1 ψ = H + D(ω)− (PJc (ω)f) τj,l(ω) A j J l=1 X∈ X e L 2 3 ∗ 2 (d ν(x),ϕj) (∂D0)ZXj,l ϕj,l + O(ω δ ) 3 H 2 = · 1 + O(ωδ ). 1 1 − 4 2 2 j J l=1 λ λj + µ µ τj,l + O(δ )+ O(δ ω ) X∈ X − c − m e Proof. The proof is similar to that of Theorem 3.3.2.

As a consequence, the following result holds. Corollary 7.2.1. With the same notation as in Theorem 7.2.1 and under the additional condition that

q p min τj,l(ω) ω δ , j J | | ≫ ∈ for some integer p and q, and

q p τj,l(ω)= τj,l,p,q + o(ω δ ), we have

L 2 3 ∗ 2 (d ν(x),ϕj) (∂D0)ZXj,l ϕj,l + O(ω δ ) 3 ψ = · H + O(ωδ 2 ). τj,l,p,q j J l=1 X∈ X e 7.3 Super-resolution (super-focusing) by using plasmonic particles

In [30,31], a rigorous mathematical theory is developed to explain the super- resolution phenomenon in microstructures with high contrast material around the source point. Such microstructures act like arrays of subwavelength sen- sors. A key ingredient is the calculation of the resonances and the Green function in the microstructure. By following the same methodology, we show in this section that one can achieve super-resolution using plasmonic nanoparticles as well.

7.3.1 Asymptotic expansion of the scattered ﬁeld In order to illustrate the super-resolution phenomenon, we set

eikm x x0 ui(x)= G(x,x ,k )= | − | . 0 m −4π x x | − 0| Lemma 7.3.1. In the space (∂D), as ω goes to zero, we have H∗ 3 5 f = f0 + O(ωδ 2 )+ O(δ 2 ),

T where f0 =(f0,1,...,f0,L) with

1 1 1 1 1 3 f0,l = (zl x0) ν(x)+ ( Id ∗ ) − [(zl x0) (x zl)] = O(δ 2 ). −4π z x 3 µ − · µ 2 −KDl SDl − · − | l − 0| m c 7.3. Super-resolution (super-focusing) by using plasmonic particles 167

Proof. The proof is similar to that of Lemma 2.3.6. Recall that

1 1 kc kc 1 fl = Fl,2 + Id ( D )∗ ( D )− [Fl,1]. µc 2 − K l S l We can show that

i 1 ∂u 1 5 3 F = = (z x ) ν(x)+O(δ 2 )+O(ωδ 2 ) in ∗(∂D ). l,2 −µ ∂ν −4πµ z x 3 l− 0 · H l m m| l − 0| Besides,

ikm zl x0 i e | − | 1 5 3 u (x) = χ(∂D )+ (z x ) (x z )+O(δ 2 )+O(ωδ 2 ) in (∂D ). |∂Dl −4π z x l 4π z x 3 l− 0 · − l H l | l − 0| | l − 0| 1 1 Using the identity ( Id ∗ ) − [χ(∂Dl)] = 0, we obtain that 2 −KDl SDl

1 1 kc kc 1 1 1 1 Id ( )∗ ( )− [Fl,1]= ( Id ∗ ) − [(zl x0) (x zl)]. µ 2 − KDl SDl −4π z x 3µ 2 −KDl SDl − · − c | l − 0| c This completes the proof of the lemma.

We now derive an asymptotic expansion of the scattered field in an intermediate regime which is neither too close to the plasmonic particles nor too far away. More precisely, let C be a fixed sufficient large positive number, we consider the following domain

3 1 Dδ,k,C = x R ; min x zl Cδ, max x zl . ∈ 1 l L | − |≥ 1 l L | − |≤ Ck ≤ ≤ ≤ ≤ k Lemma 7.3.2. Let ψl ∗(∂Dl) and let v(x)= [ψl](x). Then we have ∈H SDl for x D , ∈ δ,k,C

1 x zl 5 v(x) = G(x, zl,k) ik − yψl(y)dσ(y)+ O(δ 2 ) ψl ∗(∂D ) x z − x z · k kH l | − l| | − l| Z∂D0 +G(x, zl,k) ψl(y)dσ(y). Z∂D0 Moreover, the following estimates hold

3 v(x) = O(δ 2 ) if ψl(y)dσ(y) = 0, Z∂D0 1 v(x) = O(δ 2 ) if ψ (y)dσ(y) = 0. l 6 Z∂D0 Proof. We only consider the case when l = 0. The other case follows similarly or by coordinate translation. We have

eik x y v(x)= k [ψ](x)= G(x,y,k)ψ(y)dσ(y)= | − | ψ(y)dσ(y). SD0 − 4π x y Z∂D0 Z∂D0 | − | Since ∂G(x, 0,k) ∂mG(x, 0,k) G(x,y,k)= G(x, 0,k)+ yα + yα, ∂yα ∂yα α=1 m 2 α=m |X| X≥ | X | 168 Chapter 7. Super-Resolution with Plasmonic Nanoparticles and

∂G(x, 0,k) eik x 1 x 1 xα = | | ik = G(x, 0,k) ik , ∂yα −4π x x − x x − x | | | | | | | | | | we obtain the required identity for the case l = 0. The estimate follows from the fact that 2|α|+1 α 2 y (∂D0) = O(δ ). k kH This completes the proof of the lemma.

Denote by x z S (x,k) = G(x, z ,k) − l yϕ (y)dσ(y), j,l l x z 2 · j | − l| Z∂D0 Sl(x,k) = G(x, zl,k) ϕ0(y)dσ(y), Z∂D0 1 Hj,l(x0) = (zl x0) ν(x),ϕj ∗ . −4π z x 3 − · (∂D0) | l − 0| H It is clear that the following size estimates hold

3 1 3 S (x,k)= O(δ 2 ), S (x,k)= O(δ 2 ), H (x )= O(δ 2 ) forj = 0, H (x ) = 0. j,l l j,l 0 6 O,l 0 Theorem 7.3.1. Under Conditions 2.1, 2.2, 2.3, 7.1, and 7.3, the Green function Γ(x,x0,km) in the presence of L plasmonic particles has the following representation in the quasi-static regime: for x D , ∈ δ,km,C

Γ(x,x0,km) = G(x,x0,km) L 4 3 Hj,p(x0)Xj,l,pXj,l,qSj,q(x,km)+ O(δ )+ O(ωδ ) 3 + 1 + O(δ ). 1 1 − 4 2 2 j J l=1 λ λj + µ µ τj,l + O(δ )+ O(δ ω ) X∈ X − e c − m i Proof. With u (x)= G(x,x0,km), we have

3 ψ = aj,lϕj,l + a0,lϕ0,l + O(δ 2 ), j J 1 l L 1 l L X∈ ≤X≤ ≤X≤ where

3 5 aj,l = (f, ϕj,l) ∗(∂D) =(f0, ϕj,l) ∗(∂D) + O(ωδ 2 )+ O(δ 2 ), H H 1 1 3 5 = ( )Xj,l,pHj,p(x0)+ O(ωδ 2 )+ O(δ 2 ), µce− µm e 5 a0,l = (f, ϕ0,l) ∗(∂De ) = O(δ 2 ). H

By Lemma 7.3.2,e

km km km [ϕj,l](x) = [Xj,l,pϕjep](x)= Xj,l,p [ϕj](x) SD SD SDp 1 p L 1 p L ≤X≤ ≤X≤ 5 3 = Xj,l,pSj,p(x,km)+ O(δ 2 )+ O(ωδ 2 ). 1 p L ≤X≤ 7.3. Super-resolution (super-focusing) by using plasmonic particles 169

On the other hand, for j = 0, we have

1 km 2 D [ϕ0,l](x) = O(δ ), S 4 2 2 τ0,l(ω) = τ0 + O(δ )+ O(δ ω )= O(1).

Therefore, we can deduce that us = km [ψ](x)= a km [ϕ ]+ a km [ϕ ]+ O(δ3), SD j,lSD j,l 0,lSD 0,l j J 1 l L 1 l L X∈ ≤X≤ ≤X≤ L 1 1 1 3 4 = ( )Hj,p(x0)Xj,l,pXj,l,qSj,q(x,km)+ O(ωδ )+ O(δ ) τj,l(ω) µc − µm j J l=1 X∈ X +O(δ3), e L 3 4 Hj,p(x0)Xj,l,pXj,l,qSj,q(x,km)+ O(ωδ )+ O(δ ) 3 = 1 + O(δ ). 1 1 − 4 2 2 j J l=1 λ λj + µ µ τj,l + O(δ )+ O(δ ω ) X∈ X − e c − m

7.3.2 Asymptotic expansion of the imaginary part of the Green function As a consequence of Theorem 7.3.1, we obtain the following result on the imaginary part of the Green function. Theorem 7.3.2. Assume the same conditions as in Theorem 7.3.1. Under the additional assumption that

1 1 1 4 2 2 λ λj + − τj,l O(δ )+ O(δ ω ), − µc − µm ≫ 1 1 1 1 1 1 λ λ + − τ . λ λ + − τ ℜ − j µ − µ j,l ℑ − j µ − µ j,l c m c m for each l and j J, we have ∈ Γ(x,x ,k ) = G(x,x ,k )+ O(δ3)+ ℑ 0 m ℑ 0 m L H (x )X X S (x, 0) + O(ωδ3)+ O(δ4) ℜ j,p 0 j,l,p j,l,q j,q j J l=1 X∈ X 1 e , 1 1 1 ×ℑ λ λj + − τj,l ! − µc − µm where x,x D . 0 ∈ δ,km,C Note that H (x )X X S (x, 0) = O(δ3). Under the condi- ℜ j,p 0 j,l,p j,l,q j,q tions in Theorem 7.3.2, if we have additionally that e 1 1 = O( ) 1 1 1 3 ℑ λ λj + − τj,l ! δ − µc − µm for some plasmonic frequency ω, then the term in the expansion of Γ(x,x ,k ) ℑ 0 m which is due to resonance has size one and exhibits subwavelength peak with 170 Chapter 7. Super-Resolution with Plasmonic Nanoparticles

width of order one. This breaks the diﬀraction limit 1/km in the free space. We also note that the term G(x,x ,k ) has size O(ω). Thus, we can con- ℑ 0 m clude that super-resolution (super-focusing) can indeed be achieved by using a system of plasmonic particles.

7.4 Concluding remarks

In this chapter, by analyzing the imaginary part of the Green function of a medium populated by plasmonic resonators, we have shown that one can achieve super-resolution and super-focusing using plasmonic nanoparticles. We have assumed a weak interaction between nanoparticles. Results on strong interaction between plasmonic nanoparticles could be acheived using ideas of chapter 1 and [31]. Indeed, by considering a periodic arrangement of nanoparticles we could construct a high contrast media, thus allowing super-resolution. 171

Chapter 8

Sensing Beyond the Resolution Limit

Contents 8.1 Introduction ...... 172 8.2 The forward problem ...... 173 8.2.1 The Green function in the presence of a small particle174 8.2.2 Representation of the total potential ...... 175 8.2.3 Intermediate regime and asymptotic expansion of the scattered ﬁeld ...... 176

8.2.4 Representation of the shift j using CGPTs . . . . 179 P 8.3 The inverse problem ...... 181 8.3.1 CGPTs recovery algorithm ...... 181 8.3.2 Shape recovery from CGPTs ...... 183 8.4 Numerical Illustrations ...... 184 8.5 Concluding remarks ...... 187 172 Chapter 8. Sensing Beyond the Resolution Limit

8.1 Introduction

The inverse problem of reconstructing fine details of small objects by using far-field measurements is severally ill-posed. There are two main reasons for this. The first is the diffraction limit. When illuminated by an incident wave with wavelength Ω, the scattered field excited from the object which carries information on the scale smaller than Ω are confined near the object itself and only those with information on the scale greater than Ω can propagate into the far-field and be measured. As a result, from the far-field measurement one can only retrieve information about the object on the scale less than Ω. Especially in the case when the object is small (size smaller than Ω), one can only obtain very few information. The second is the low signal to noise ratio. We know that small objects scatter "weakly". This results very weak measurement in the far-field. In the presence of measurement noise, one has low signal to noise ratio and hence poor reconstruction. In this chapter, we propose a new methodology to overcome the ill-posedness of this inverse problem. Our method is motivated by plasmonic bio-sensing. The key is to use a plasmonic particle to interact with the object to propagate its near field information into far-field in term of shifts of plasmonic resonance frequencies. The plasmon resonance frequency is one of the most important characterization of a plasmonic particle. It depends not only on the electromagnetic properties of the particle and its size and shape, but also the electromagnetic properties of the environment. It is the last property which enables the sensing application of plasmonic particles. Motivated by [34], we establish in this chapter a rigorous quantitative analysis for the sensing application. We show that plasmonic resonance can be used to reconstruct fine details of small objects. We also remark that plasmonic resonance can also be used to identify the shape of the plasmonic particle itself, see chapter 6. The methodology we propose is closely related to super-resolution in imaging. Super-resolution is about the separation of point sources. In super- resolution technology near field microscopy, the basis idea is to obtain the near field of sources which contains high resolution information. This is made possible by propagating the near field information into the far field through certain near field interaction mechanism, see chapter 7. In this chapter, we are interested in reconstructing the fine details of small objects in comparison to their positions and separability which are the focus of super-resolution. The idea is similar. The near field information of the object is obtained from the near field interaction of the object and the plasmonic nanoparticle. In this chapter we consider a system composed of a known plasmonic particle and the unknown object whose geometry and electromagnetic properties are the quantities of interest. Under the illumination of incident waves with frequencies in certain range, we observe the color of the system or measure the frequencies where the peaks in the scattering field occur. These are the resonant frequencies or spectroscopic data of the system. By varying the relative position of the two particles, we obtain different resonant frequencies due to the varying interactions between the two particles. We assume that the unknown particle is small compared to the plasmonic particle. In the intermediate regime when the distance of the two particles is comparable to the size of the plasmonic particle, we show that the presence of the small unknown particle can be viewed as a small perturbation to the homogenous environment of the plasmonic particle. As a result, it induces a 8.2. The forward problem 173 small shift to the plasmonic resonance frequencies of the plasmonic particle, which can be read from the observed spectroscopic data. By using rigorous asymptotic analysis, we obtain analytical formula for the shift which shows that the shift is determined by the generalized polarization tensors [18] of the unknown object. Therefore, from the far-field measurement of the shift of resonant frequencies, we can reconstruct the fine information of the object by using its generalized polarization tensors. In this chapter, for the sake of simplicity, we consider the quasi-static approximation for the interaction between the electromagnetic field and the system of the two particles. Thus, we shall use the conductivity equation instead of the Helmholtz equation and the Maxwell equations. In addition, we only consider the intermediate interaction regime, the strong interaction regime when the object is close to the plasmonic particle is also very interesting and will be reported in future works. This chapter is organized in the following way. In Section 8.2, we consider the forward scattering problem of the incident field interacting with a system composed of a normal particle and a plasmonic particle. We derive the asymptotic of the scattered field in the case of intermediate regime. In Section 8.3, we consider the inverse problem of reconstructing the geometry of the normal particle. This is done by first constructing the generalized polarization tensors of the particles through the resonance shift it induced to the plasmonic particle. In Section 8.4, we provide numerical examples to justify our theoretical results.

8.2 The forward problem

We consider a system composed of a small ordinary particle and a plasmonic particle embedded in a homogeneous medium; see Figure 8.1. The ordinary particle and the plasmonic particle occupy a bounded and simply connected domain D R2 and D R2 of class 1,α for some 0 <α< 1, respectively. 1 ⊂ 2 ⊂ C We denote the permittivity of the ordinary particle D1 (or the plasmonic particle D2) by ε1 (or ε2), respectively. The permittivity of the background medium is denoted by εm. In other words, the permittivity distribution ε is given by ε := ε χ(D )+ ε χ(D )+ ε χ(R2 (D D )). 1 1 2 2 m \ 1 ∪ 2 The permittivity ε2 of the plasmonic particle depends on the operating frequency and is modeled by the Drude model as

ω2 ε = ε (ω) = 1 p . 2 2 − ω(ω + iγ)

We assume the following condition on the size of the particles D1 and D2.

Condition 8.1. The plasmonic particle D2 has size of order one and is centered at a position that we denote by z; the ordinary particle D1 has size of order δ 1 and is centered at the origin. Speciﬁcally, we write D = δB, ≪ 1 where the domain B has size of order one. 174 Chapter 8. Sensing Beyond the Resolution Limit

The total electric potential u satisﬁes the following equation:

(ε u) = 0 in R2 (∂D ∂D ), ∇· ∇ \ 1 ∪ 2  u + = u on ∂D1 ∂D2,  | |− ∪   ∂u ∂u εm = ε1 on ∂D1, (8.1)  ∂ν + ∂ν  −  ∂u ∂u εm = ε2 on ∂D2, ∂ν + ∂ν  −  (u u i)(x)= O( x 1), as x ,  − | |− | | → ∞   where ui(x)=d x is the incident potential with a constant vector d R2. · ∈

Figure 8.1: Scattering of an incident wave ui by a system of a plasmonic (D2) - non plasmonic (D1) particles.

8.2.1 The Green function in the presence of a small particle Let G ( ,y) be the Green function at the source point y of a medium con- D1 · sisting of the particle D1, which is embedded in the free space. For every y / D , G ( ,y) satisﬁes the following equation: ∈ 1 D1 · ε χ(D )+ ε χ(R2 D ) u = δ in R2 ∂D , ∇· 1 1 m \ 1 ∇ y \ 1 u + = u on ∂D1,  | |−   ∂u ∂u (8.2) ε = ε on ∂D ,  m ∂ν 1 ∂ν 1 + −  1 u(x)= O( x − ), as x .  | | | | → ∞  We look for a solution of the form:

G (x,y) := G(x,y)+ [ψ](x), x R2 D . (8.3) D1 SD1 ∈ \ 1

Note that GD1 satisﬁes the second and fourth conditions in (8.2). From the third condition in (8.2) and the jump formula (1.1) for the single layer 8.2. The forward problem 175

potential, the density ψ must satisfy the following equation on ∂D1: 1 1 ∂ εm Id + ∗ [ψ]+ ε1 Id ∗ [ψ]=(ε1 εm) G( ,y). (8.4) 2 KD1 2 −KD1 − ∂ν · So we obtain

1 ∂ ψ = λ Id ∗ − G( ,y) , D1 −KD1 ∂ν · ε + ε h i λ = 1 m . D1 2(ε ε ) 1 − m Therefore, from (8.3) and the uniqueness of a solution to (8.2), we have the following representation for the Green’s function GD1 :

1 ∂ 2 G (x,y)= G(x,y)+ λ Id ∗ − G( ,y) (x) for x,y R D . D1 SD1 D1 −KD1 ∂ν · ∈ \ 1 h i (8.5)

8.2.2 Representation of the total potential Here we derive a layer potential representation of the total potential u, which is the solution to (8.1). i Let uD1 be the total ﬁeld resulting from the incident ﬁeld u and the ordinary particle D1 (without the plasmonic particle D2). Note that uD1 is given by

i i 1 ∂u 2 − R uD1 (x)= u (x)+ D1 λD1 Id D∗ 1 [ ](x), for x D1. S −K ∂ν1 ∈ \ To consider the total potential u, we also need to represent the ﬁeld generated by the plasmonic particle D2. For this, we introduce a new layer potential as follows: SD2,D1

[ϕ](x)= G (x,y)ϕ(y)dσ(y). SD2,D1 D1 Z∂D2 The total potential u can be represented in the following form:

u(x)= u (x)+ [ψ](x), x R2 D . (8.6) D1 SD2,D1 ∈ \ 2 We need to ﬁnd a boundary integral equation for the density ψ. It follows from (8.5) that, for any ϕ,

[ϕ](x)= [ϕ](x)+ 1 [ϕ](x), SD2,D1 SD2 SD2,D1 where 1 is given by SD2,D1

1 1 ∂ [ϕ](x) := λ Id ∗ − [ G( ,y)](x)ϕ(y)dσ(y). SD2,D1 SD1 D1 −KD1 ∂ν · Z∂D2 1 176 Chapter 8. Sensing Beyond the Resolution Limit

The expression of 1 [ϕ] can be further developed using the following SD2,D1 spectral expansion of the free-space Green function G(x,y) [32]:

∞ G(x,y)= [ϕ ](x) [ϕ ](y)+ [ϕ ](x), for x R2 D and y D , − SD2 j SD2 j SD2 0 ∈ \ 2 ∈ 2 Xj=1 where ϕj,j = 1, 2,... are eigenfunctions of ∗ on ∗(∂D2) and ϕ0 is an KD2 H eigenfunction associated to the eigenvalue 1/2. Then, for any ϕ (∂D ), ∈H∗ 2 we get

∞ ∗ G(x,y)ϕ(y)dσ(y)= D2 [ϕj](x)(ϕ,ϕj) (∂D2) + D2 [ϕ0](x) ϕ(y)dσ(y) ∂D2 S H S ∂D2 Z Xj=1 Z ∞ ∗ = D2 [ϕj](x)(ϕ,ϕj) (∂D2). S H Xj=1 Therefore, for any ϕ (∂D ), we have, ∈H∗ 2

1 1 ∂ [ϕ](x) = λ Id ∗ − [ G( ,y)](x)ϕ(y)dσ(y) SD2,D1 SD1 D1 −KD1 ∂ν · Z∂D2 1 1 ∂ ∞ − ∗ = D1 λD1 Id D∗ D2 (ϕ,ϕj) ϕj (x) S −K 1 ∂ν S H 1 j=0 h X i 1 ∂ D [ϕ] − 2 = D1 λD1 Id D∗ 1 S (x), S −K ∂ν1 where we have used the notation ∂ to indicate the outward normal deriva- ∂νi tive on ∂Di. Combining the boundary conditions in (8.1), the representation formula (8.6) and the jump formula (1.1) yields the following equation for ψ

∂uD1 ( D2,0 + D2,1)[ψ]= , A A ∂ν2 where

= λ Id ∗ , AD2,0 D2 −KD2 ε2 + εm λD2 = , (8.7) 2(ε2 εm) 1 − ∂ D ,D ∂ 1 ∂ D S 2 1 − 2 D2,1 = = D1 λD1 Id D∗ 1 S . (8.8) A ∂ν2 ∂ν2 S −K ∂ν1 8.2.3 Intermediate regime and asymptotic expansion of the scattered field Here we introduce the concept of intermediate regime and derive the asymptotic expansion of the scattered field u ui for small δ. − Definition 8.1 (Intermediate regime). We say that D2 is in the intermediate regime with respect to the origin if there exist positive constants C1 and C2 such that C1

C dist(0,D ) C . 1 ≤ 2 ≤ 2 8.2. The forward problem 177

Deﬁnition 8.1 says that the plasmonic particle D2 is located not too close to D1 nor far from D1. Throughout this chapter, we assume the plasmonic particle D2 is in the intermediate regime. We have the following result.

Propsition 8.2.1. If D2 is in the intermediate regime, then D2,1 = kA kH∗ O(δ2) as δ 0. → Proof. Fix ϕ (∂D ) and let ∈H∗ 2

1 ∂ D2 [ϕ] ϕ := (λD1 Id D∗ 1 )− S . −K ∂ν1 h i ∂ Since D [ϕ] is harmonice in D1, the Green’s identity gives D [ϕ]= S 2 ∂D1 ∂ν1 S 2 0. Then it can be proved that ϕ = 0. So we get ∂D1 R R [ϕ](x)= (log x y logex )ϕ(y)dσ(y) + log x ϕ(y)dσ(y) SD1 | − |− | | | | Z∂D1 Z∂D1 e = (log x y log x )ϕe(y)dσ(y). e | − |− | | Z∂D1 Therefore, since y x C and y Cδe for (y,x) (∂D ,∂D ), we obtain | − |≥ ′ | |≤ ∈ 1 2 ∂ D ,1[ϕ] ∗ = D [ϕ] ∗ Cδ ϕ ∗ . 2 (∂D2) 1 (∂D2) (∂D1) kA kH ∂ν2 S H ≤ k kH

Now it suﬃces to prove that e e

∗ ϕ (∂D1) Cδ. (8.9) k kH ≤ Recall that D1 = δB. Let fδ(ye) = f(δy). Then the function fδ belongs to (∂B) for f (∂D ). Since it is known that is scale-invariant for H∗ ∈ H∗ 1 KΩ∗ any Ω, we have ∗ [f]= ∗ [fδ]. Therefore, KD1 KB 1 1 ϕ = λ Id ∗ − [f]d(δσ(y))=(λ Id ∗ )− [f ]d(δσ(y)). D1 −KD1 D1 −KB δ Again, since y x C for (y,x) (∂D ,∂D ) and ∂D = O(δ), we arrive e | − |≥ ′ ∈ 1 2 | 1| at

1 ∂ D2 [ϕ] ∗ ∗ ϕ (∂D1) = (λD1 Id B∗ )− S (∂B) k kH k −K ∂ν1 δ kH ∂ h i ∗ e C D2 [ϕ] (∂D1) Cδ. ≤ k∂ν1 S kH ≤ The proof is completed. From Proposition 8.2.1, we can view as a perturbation of . AD2,1 AD2,0 Using standard perturbation theory [85], we can derive the perturbed eigenvalues and associated eigenfunctions. Let λj and ϕj be the eigenvalues and eigenfunctions of ∗ on ∗(∂D2). KD2 H For simplicity, we consider the case when λj is a simple eigenvalue of the operator ∗ . Let us deﬁne KD2

Rjl = D ,1[ϕl],ϕj ∗ , (8.10) 2 (∂D2) A H where is given by (8.8). Note that R = O(δ2). AD2,1 jl 178 Chapter 8. Sensing Beyond the Resolution Limit

The perturbed eigenvalues have the following form:

τ (δ)= λ λ + , j D2 − j Pj where are given by Pj

RjlRlj Rjl2 Rl2l1 Rl1j j = Rjj + + P λj λl (λj λl1 )(λj λl2 ) l=j (l1,l2)=j X6 − X6 − − R R R R + jl3 l3l2 l2l1 l1j + . (8.11) (λj λl1 )(λj λl2 )(λj λl3 ) ··· (l1,l2,l3)=j X 6 − − − Also, the perturbed eigenfunctions have the following form:

2 ϕj(δ)= ϕj + O(δ ). (8.12)

∗ Here the remainder term is with respect to the norm (∂D2). k · kH Remark 8.2.1. Note that depends not only on the geometry and material Pj properties of D1, but also on D2’s properties, in particular its position z.

Theorem 8.2.1. If D2 is in the intermediate regime, the scattered ﬁeld s u = u uD by the plasmonic particle D2 has the following representa- D2 − 1 tion: us = [ψ], D2 SD2,D1 where ψ satisﬁes

i 2 u (z) ν,ϕj ∗ ϕj + O(δ ) ∞ (∂D2) ψ = ∇ · H λD2 λj + j Xj=1 − P with λD2 being given by (8.7). As a corollary, we have the following asymptotic expansion of the scattered ﬁeld u ui. − Theorem 8.2.2. We have the following far ﬁeld expantion:

3 i i 2 δ (u u )(x)= u (z) M(λD1 ,λD2 ,D1,D2) G(x, z)+ O(δ )+ O , − ∇ · ∇ dist(λD ,σ( ∗ )) 2 KD2 ! as x . Here, M(λ ,λ ,D ,D ) is the polarization tensor satisfying | | → ∞ D1 D2 1 2

2 ∗ 1 1 ∞ (νl,ϕj) (∂D2)(ϕj,xm) , + O(δ ) H − 2 2 M(λD1 ,λD2 ,D1,D2)l,m = , (8.13) λD2 λj + j Xj=1 − P for l, m = 1, 2. We remark that the scattered ﬁeld in the above expression depends on the frequency (since λD2 does so) and exhibit local peaks at certain frequencies when one of the denominators is close to zero and is minimized while the associated nominator is not zero. These frequencies are called the resonant frequencies of the system. It is clear that these resonant frequencies also depend on the geometry and the electric permittivity of D1 through the 8.2. The forward problem 179 perturbative terms ’s. We shall use this fact in the next section to solve the Pj associated inverse problem of reconstructing D1 by using those frequencies.

8.2.4 Representation of the shift using CGPTs Pj Here we show that the term in the plasmonic resonances can be expressed Pj in terms of the contracted generalized polarization tensors (CGPTs), see chapter 1. The CGPTs carry information on the geometry and material properties of D1. See [18] for a detailed reference. We shall reconstruct the ordinary particle D from the measurement of the shift . 1 Pj Propsition 8.2.2. If D2 is in the intermediate regime, then the perturbative terms Rjl can be represented using CGPTs Mm,n(λD1 ,D1) associated with D1 as follows:

M N 1 R = λ aj M (λ ,D )(al )t + O(δM+N+1), (8.14) jl 2 − j m m,n D1 1 n m=1 n=1 X X j j j where the superscript t denotes the transpose and am =(am,c,am,s) with

1 cos(mθ ) aj = y ϕ (y)dσ(y), m,c −2πm rm j Z∂D2 y 1 sin(mθ ) aj = y ϕ (y)dσ(y). m,s −2πm rm j Z∂D2 y Here, (r ,θ ) denote the polar coordinates of y and ϕ is an orthonormal y y { j}j basis of eigenfunctions of ∗ on ∗. KD2 H Proof. To simplify the notation, let us denote

1 ∂ D [ϕl] − 2 Fl = D1 λD1 Id D∗ 1 S . S −K ∂ν1 Then, from the Green’s identity and the jump formula (1.1), we obtain

∂Fl Rjl = Fl,ϕj ∗ = , D2 [ϕj] 1 , 1 H − ∂ν2 S 2 − 2

∂ D2 [ϕj] 1 = F , 1 1 = F , ( + )[ϕ ] 1 1 . l S , l D∗ 2 j , − ∂ν 2 2 − −2 K 2 2 2 − − −

Since ϕj is an eigenfunction of ∗ with an eigenvalue λj, we have KD2 1 Rjl = λj Fl,ϕj 1 , 1 . 2 − 2 − 2 Let (rx,θx) be the polar coordinates of x. It is known from [12] that, for x < y , | | | | ∞ ( 1) cos(nθ ) ( 1) sin(nθ ) G(x,y)= − y rn cos(nθ )+ − y rn sin(nθ ). (8.15) 2πn rn x x 2πn rn x x n=0 y y X 180 Chapter 8. Sensing Beyond the Resolution Limit

By interchanging x and y and the fact that G(x,y) = G(y,x), we have, for x > y , | | | | ∞ ( 1) cos(nθ ) ( 1) sin(nθ ) G(x,y)= − x rn cos(nθ )+ − x rn sin(nθ ). (8.16) 2πn rn y y 2πn rn y y n=0 x x X If x ∂D and y ∂D , then x < y . So, applying (8.15) gives ∈ 1 ∈ 2 | | | | ∂ [ϕ ] ∂ SD2 l (x) = G(x,y)ϕ dσ(y) ∂ν ∂ν l 1 1 Z∂D2 ∞ ∂rn cos(nθ ) ∂rn sin(nθ ) = x x al + x x al . ∂ν n,c ∂ν n,s n=1 1 1 X On the contrary, if y ∂D and x ∂D , then x > y . We have from ∈ 1 ∈ 2 | | | | (8.16) that, for any f,

[f](x) = G(x,y)[f](y)dσ(y) SD1 Z∂D1 ∞ 1 cos(mθ ) = x rm cos(mθ )[f](y)dσ(y) −2πm rm y y m=0 x ∂D1 X Z ∞ 1 sin(mθ ) + x rm sin(mθ )[f](y)dσ(y). −2πm rm y y m=0 x ∂D1 X Z Therefore, from the deﬁnition of Mm,n, we get

1 1 ∂ D2 [ϕl] R = λ λ Id − ,ϕ 1 1 jl j D1 D1 D∗ 1 S j , 2 − S −K ∂ν 2 2 1 − 1 ∞ = λ (aj ,aj )M (λ ,D )(al ,al )t. 2 − j m,c m,s m,n D1 1 n,c n,s m=0,n=1 X m+n For any λ C and D = δB, it is easy to check that Mm,n(λ,D)= δ Mm,n(λ,B). ∈ l l Since D2 is in the intermediate regime, an,c and an,s satisfy

j j 1 m l l 1 n a , a C− , a , a C− , | m,c| | m,s|≤ m | n,c| | n,s|≤ n for some constant C > 1 independent of δ. Moreover, it can be shown that (see [15])

∞ j j l l t (a0,c,a0,s)M0,n(λD1 ,D1)(an,c,an,s) = 0. n=1 X Then the conclusion immediately follows.

Corollary 8.2.1. We have

Rjl(z)Rlj(z) Rjl2 Rl2l1 Rl1j j(z) ... P − λj λl − (λj λl1 )(λj λl2 ) l=j (l1,l2)=j X6 − X6 − − M N 1 = λ aj M (λ ,D )(al )t + O(δM+N+1). 2 − j m m,n D1 1 n m=1 n=1 X X 8.3. The inverse problem 181

In the LHS, the summation should be truncated so that all the terms which contain R R = O(δ2(k+1)) with 2(k + 1) M + N + 1 are ignored. jlk ··· lkj ≤ 8.3 The inverse problem

In this section, we consider the inverse problem associated with the forward system (8.1). We assume that the plasmonic particle D2 is known, i.e., we know its electric permittivity ε2 = ε2(ω), its shape D2 and position z. The ordinary particle D1 is unknown. For simplicity, we assume that its permittivity ε1 is known. For each of many different positions z of the plasmonic particle D2, we measure the resonant frequency and use these resonant frequencies to reconstruct the shape of the ordinary particle D1. As illustrated by Theorem 8.2.2, the resonance in the scattered field oc- ∗ curs when λD2 (ω) λj + j is minimized and (νl,ϕj) (ϕj,xm) 1 , 1 = 0. So − P H 2 2 6 by varying the frequency ω, we can measure the value of λ .− Moreover, in j −Pj the absence of the ordinary particle, the resonance occurs when λ (ω) λ D2 − j is minimized and (νl,ϕj) ∗ (ϕj,xm) 1 , 1 = 0. Since we assume that the plas- H − 2 2 6 monic particle D2 is known, we can get the value of λj a priori. Therefore, by comparing λ and λ , we can measure the shift of the eigenvalue. j −Pj j Pj Finding for many different positions of D will yield a linear system Pj 2 of equations that will allow the recovery of the CGPTs associated with D1. From the recovered CGPTs, we will reconstruct the ordinary particle D1. Here, we only consider the shape reconstruction problem. Nevertheless, by using the CGPTs associated with D1, it is possible to reconstruct the permittivity ε1 of D1 in the case it is not a priori given [12].

From now on, we denote Mm,n = Mm,n(λD1 ,D1).

8.3.1 CGPTs recovery algorithm We propose a recurrent algorithm to recover the GPTs of order less or equal 2k 1 to k up to an order δ − , using measurements of Pj at different positions of D2. For simplicity, we only consider the shift of a single eigenvalue λj with a fixed j. To gain robustness and efficiency, the shift in other resonant frequencies could also be considered. We now explain our method for reconstructing GPTs M , m + n K m,n ≤ for a given K N from the measurements of the shift . ∈ Pj Suppose we measure precisely for three different positions z , z , z of Pj 1 2 3 the plasmonic particle D2. First we reconstruct M1,1 approximately. Since t M1,1 = M1,1, the matrix M1,1 is symmetric. We look for a symmetric matrix (2) M1,1 satisfying

1 (z ) = λ aj (z )M (2)(aj )t(z ) Pj 1 2 − j 1 1 1,1 1 1 1 (z ) = λ aj (z )M (2)(aj )t(z ) Pj 2 2 − j 1 2 1,1 1 2 1 (z ) = λ aj (z )M (2)(aj )t(z ). Pj 3 2 − j 1 3 1,1 1 3 The above equations can be seen as a linear system of equations for three (2) (2) (2) independent components (M1,1 )11, (M1,1 )12 and (M1,1 )22. We emphasize 182 Chapter 8. Sensing Beyond the Resolution Limit

j that am(zi) can be a priori given because the particle D2 is known. Since, 2 from Corollary 8.2.1 and the fact that Rjl = O(δ ), we have

1 (z )= λ aj (z )M (aj )t(z )+ O(δ3), k = 1, 2, 3, Pj k 2 − j 1 k 1,1 1 k we see that M is well approximated by M (2). Speciﬁcally, we have M 1,1 1,1 1,1 − (2) 3 M1,1 = O(δ ). Next we reconstruct and update the higher order GPTs Mn,m in a re- cursive way. Towards this, we need more measurement data of the shift . Pj Let k 3. Due to the symmetry of harmonic combinations of the non con- ≥ t tracted GPTs (see [18]), we have Mm,n = Mn,m. One can see that, by using this symmetry property, the set of GPTs M satisfying m + n k contains m,n ≤ ek independent variables where ek is given by

k(k 1) + k/2, if k is even, e = − k k(k 1) + (k 1)/2, if k is odd. − − Therefore, we need e measurement data for to reconstruct the GPTs k Pj M for m + n k. m,n ≤ Suppose we have ek 2 more measurement data j at diﬀerent positions (k)− P (k) t z4, z5, ..., zek . Let Mm,n m+n k be the set of matrices satisfying [Mn,m] = (k) { } ≤ Mm,n and the following linear system:

(k 1) 1 j (k) j t − (z )= λ a (z )M (a ) (z ) Pj 1 2 − j m 1 m,n n 1 m+n k X≤ e(k 1) 1 j (k) j t − (z )= λ a (z )M (a ) (z ) Pj 2 2 − j m 2 m,n n 2 m+n k X≤ e . . . = .

(k 1) 1 j (k) j t − (z )= λ a (z )M (a ) (z ), (8.17) Pj ek 2 − j m ek m,n n ek m+n k X≤ e where

(k 1) (k 1) (k 1) Rjl − (zi)Rlj − (zi) j − (zi) := j(zi) ..., i = 1, 2,...,ek, P P − λj λl − l=j X6 − e (8.18) and

(k 1) 1 j (k 1) l t R − (z) := λ a (z)M − (a ) (z). jl 2 − j m m,n n m+n k 1 X≤ − (k) Note that Mm,n are deﬁned recursively. In (8.18), the summation should be truncated as in Corollary 8.2.1. (k) Then Mm,n becomes a good approximation of the CGPT Mm,n for m + n k. Moreover, the accuracy improves as the iteration goes on. Indeed, ≤ we can see that

(k) 2k 1 M M = O(δ − ), m + n k. (8.19) m,n − m,n ≤ 8.3. The inverse problem 183

In fact, (8.19) can be veriﬁed by induction. We already know that this is (k 1) 2k 3 true when k = 2. Let us assume M Mm,n− = O(δ ), m + n k 1. m,n − − ≤ − Then, from Proposition 8.2.2, we have

(k 1) 2k 3 R (z) R − (z) = O(δ − ). jl − jl 2 Hence, from Corollary 8.2.1 and the fact that Rjl = O(δ ), we obtain

(k 1) Rjl(zi)Rlj(zi) 2k 1 j − (zi) j(zi) = O(δ − ). P − P − λj λl −···  l=j X6 − e   Therefore, in view of Corollary 8.2.1 and the linear system (8.17), we obtain (k) 2k 1 (8.19). In conclusion, Mm,n is indeed precise up to an order δ − . Remark 8.3.1. In practice, might be subject to noise and could not be Pj measured precisely. In this case only the low order CGPTs could be recovered.

8.3.2 Shape recovery from CGPTs

To recover the shape of D1 from its CGPTs, we search to minimize the following shape functional ( [12])

1 2 (l)[B] := N (1) (λ ,B) N (1) (λ ,D ) , (8.20) Jc 2 mn D1 − mn D1 1 n+m k X≤ where

N (1) (λ,D)=(M cc M ss )+ i(M cs M sc ). m,n m,n − m,n m,n − m,n (l) (l) To minimize [B] we need to compute the shape derivative, dS c , of (l) J J c . J For ǫ small, let Bǫ be an ǫ-deformation of B, i.e., there is a scalar function h 1(∂B), such that ∈C ∂B := x + ǫh(x)ν(x) : x ∂B . ǫ { ∈ } Then, according to [11,12,17], the perturbation of a harmonic sum of GPTs due to the shape deformation is given as follows:

N (1) (λ ,B ) N (1) (λ ,D ) m,n D1 ǫ − m,n D1 1 ∂u ∂v 1 ∂u ∂v = ǫ(k 1) h(x) + (x) dσ(x)+ O(ǫ2), λD1 − ∂B "∂ν ∂ν kλD ∂T ∂T # Z − − 1 − − where kλ = (2λD + 1)/(2λD 1), (8.21) D1 1 1 − 184 Chapter 8. Sensing Beyond the Resolution Limit and u and v are respectively the solutions to the problems:

∆u = 0 in B (R2 B) , ∪ \  u + u = 0 on ∂B , | − |−  (8.22)  ∂u ∂u  kλ = 0 on ∂B ,  ∂ν + − D1 ∂ν − m 1  (u (x1 + ix2) )(x)= O( x − ) as x ,  − | | | | → ∞  and  ∆v = 0 in B (R2 B) , ∪ \

 kλD v + v = 0 on ∂B , 1 | − |−  (8.23)  ∂v ∂v  = 0 on ∂B ,  ∂ν + − ∂ν − (v (x + ix )n)(x)= O( x 1) as x .  1 2 −  − | | | | → ∞  Here, ∂/∂T is the tangential derivative. Let

∂u ∂v 1 ∂u ∂v w (x)=(k 1) + (x), x ∂B . m,n λD1 − "∂ν ∂ν kλD ∂T ∂T # ∈ − − 1 − −

(l ) The shape derivative of c at B in the direction of h is given by J (l) d [B],h = δ w ,h 2 , h SJc i N h m,n iL (∂B) m+n k X≤ where δ = N (1) (λ ,B) N (1) (λ ,D ) . N m,n D1 − m,n D1 1 Next, using a gradient descent algorithm we can minimize, at least locally, (l) the functional c . J 8.4 Numerical Illustrations

In this section, we support our theoretical results by numerical examples. In the sequel, we assume that D2 is an ellipse with semi-axes a = 1 and b = 2, as shown in Figure 8.2. In this case the resonances in the far-ﬁeld can only 1 a b 1 1 a b 1 occur at λ = − = and λ = − = . Thus, for a ﬁxed position 1 2 a+b − 6 2 − 2 a+b 6 of D , we can measure two shifts of the plasmonic resonance: and . 2 P1 P2 We consider the case of D1 being a triangular-shaped and a rectangular- shaped particle with known contrast λD1 = 1, as shown in Figure 8.3. Figure 8.4 shows the shift in the plasmonic resonance around λ1, for random positions of D2 around a triangular-shaped particle D1. From these measurements, can be precisely estimated from the resonance peaks and P1 the equation = λ λ , where λ is the value at which we achieve the Pj j − r r maximum of the resonant peak. It is worth mentioning that, for the sake of simplicity and clarity, we plot the graph not by varying the frequency but the parameter λ directly. We assume Re(λ ) ranges from 1/2 to 1/2 and Im(λ ) = 10 4. In a more D2 − D2 − 8.4. Numerical Illustrations 185

-1

-2

-3 -3 -2 -1 0 1 2 3

Figure 8.2: Plasmonic particle D2.

realistic setting, corrections in the peaks of resonances should be included, by considering the Drude model for λD2 . But they are essentially equivalent.

To recover geometrical properties of D from measurements of , we 1 P1 recover the CGPTs using the algorithm described in 8.3.1 and then minimize functional (8.20) to reconstruct an approximation of D1. To recover the ﬁrst CGPTs of order 5 or less we make 22 measurements around D as shown in Figure 8.5, and measure the shift from λ = 1 . 1 1 − 6 In the following we show a comparison between the recovered CGPTs of order less or equal to 4 and their theoretical value, for each iteration.

Triangle-shaped D1:

Theoretical values: 0.2426 0 0 0.0215 M11 = M12 = − 0 0.2426 0.0215 0 − 0.043 0 0 0 M = M = 22 0 0.043 13 0 0 Recovered:

(2) 0.2444 0.0007 (3) 0.2438 0 M = − M = 11 0.0007 0.2408 11 0 0.2414 − (4) 0.2429 0.0001 (5) 0.2426 0 M = − M = 11 0.0001 0.2430 11 0 0.2426 −

(3) 0.0008 0.2414 (4) 0 0.2413 M = − M = − 12 0.0212 0.0087 12 0.0213 0 − − − (5) 0 0.2415 M = − 12 0.0215 0 − 186 Chapter 8. Sensing Beyond the Resolution Limit

0.8

0.6

0.4

0.2

-0.2

-0.4

-0.6

-0.8 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0.8

0.6

0.4

0.2

-0.2

-0.4

-0.6

-0.8 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Figure 8.3: Non plasmonic particles D1. Triangular- shaped (left) and rectangular-shaped (right).

0.0180 0.2204 0.0368 0.0010 M (4) = M (5) = 22 0.2204 0.0389 22 0.0010 0.0497 (4) 0.0093 0.1126 (5) 0.0032 0.0005 M = − M = − 13 0.1123 0.0019 13 0.0005 0.0032 − − − −

Rectangular-shaped D1:

Theoretical values: 0.2682 0.0000 0 0 M = M = 11 0 0.2682 12 0 0 0.0544 0 0.0054 0 M = M = 22 0 0.0402 13 0 0.0054 − Recovered: 0.2703 0.0001 0.2696 0 M (2) = M (3) = 11 0.0001 0.2661 11 0 0.2662 0.2682 0 0.2682 0 M (4) = M (5) = 11 0 0.2681 11 0 0.2681 8.5. Concluding remarks 187

104 6.5

10 6 8 5.5 6

5 4

4.5

)| 2

1 ,D

2 4 0

(D 11

|M 3.5 -2

-4 3

-6 2.5 -8 2 -10

1.5 -6 -4 -2 0 2 4 6 -0.167 -0.1669 -0.1668 -0.1667 -0.1666 -0.1665 real( ) 2

Figure 8.4: (right) Modulus of the entry (1,1) of the ﬁrst order polarization tensor given in Theorem 8.2.2, for diﬀerent positions of D2 around a triangular-shaped particle D2 (left).

(3) 0.0038 0.0001 (4) 0 0 M = − M = 12 0 0.0112 12 0 0 − 0 0 M (5) = 12 0 0

(4) 0.0530 0.0007 (5) 0.0537 0.0006 M = − M = 22 0.0007 0.0425 22 0.0006 0.0416 − (4) 0.0064 0.0003 (5) 0.0060 0.0003 M = M = − 13 0.0004 0.0063 13 0.0003 0.0059 − − − The results of minimizing the functional (8.20) with a gradient descent approach and using the recovered CGPTs of order less or equal to 5 are shown in Figures 8.6 and 8.7. We take as initial point the equivalent ellipse to D1, given by the ﬁrst order polarization recovered with Algorithm 8.3.1, (5) i.e M11 .

8.5 Concluding remarks

In this chapter, using the quasi-static model, we have shown that the ﬁne details of a small object can be reconstructed from the shift of resonant frequencies it induces to a plasmonic particle in the intermediate regime. This provides a solution for the ill-posed inverse problem of reconstructing small 188 Chapter 8. Sensing Beyond the Resolution Limit

-2

-4

-6

-8

-10

-15 -10 -5 0 5 10 15

-2

-4

-6

-8

-10

-10 -5 0 5 10

Figure 8.5: Positions of D2 for which we measure 1. (left) P Triangular-shaped particle D1, (right) rectangular-shaped particle D1.

objects from far-ﬁeld measurements and also laid a mathematical foundation for plasmonic bio-sensing. The idea can be extended in several directions: (i) to investigate the strong interaction regime when the small object is close to the plasmonic particle; (ii) to study the case when the size of object is comparable to the size of plasmonic particle; (iii) to analyze the case with multiple small objects and multiple plasmonic particles; (iv) to consider the more practical model of Maxwell equations, and (v) to investigate other types of subwavelength resonances such as Minnaert resonance [10, 75] in bubbly ﬂuids. These new developments will be reported in forthcoming works. 8.5. Concluding remarks 189

0 0 0

Figure 8.6: Shape recovery of a triangular-shaped particle D1. From left to right, we show both, the original shape and the recovered one after 0 iterations, after 8 iterations and after 30 iterations.

0 0 0

Figure 8.7: Shape recovery of a rectangular-shaped particle D1. From left to right, we show both, the original shape and the recovered one after 0 iterations, after 30 iterations and after 100 iterations.

191

Appendix A

Layer Potentials for the Laplacian in two Dimensions

In R2 the single-layer potential : H 1/2(∂D) H1/2(∂D) is not, in SD − → general, invertible. Hence, (u, D[v]) 1 , 1 does not define an inner product − S 2 2 and the symmetrization technique described− in 1 is no longer valid. 1/2 Here and throughout, ( , ) 1 , 1 denotes the duality pairing between H− (∂D) · · − 2 2 and H1/2(∂D). To overcome this difficulty, we will introduce a substitute of , in the SD same way as in [32]. We first need the following lemma. Lemma A.0.1. Let = ϕ H 1/2(∂D); α C, [ϕ]= α . We have C { ∈ − ∃ ∈ SD } dim( ) = 1. C Proof. It is known that

1/2 1/2 : H− (∂D) C H (∂D) C AD × → × (ϕ,a) D[ϕ]+ a, ϕdσ , → S ∂D Z is invertible [18, Theorem 2.26]. 1 C We can see that = Π1 D− (0, ), where Π1[(ϕ,a)] = ϕ. The invertibility C A 1 of implies that Ker(Π − (0, )) = 0 . Thus, by the range theorem we AD 1AD · { } have

1 1 1 1= dim(Im(Π − (0, ))) + dim(Ker(Π − (0, ))) = dim(Im(Π − (0, ))) = dim( ). 1AD · 1AD · 1AD · C

Deﬁnition A.1. We call ϕ the unique element of such that ϕ dσ = 1. 0 C ∂D 0 Note that for every ϕ H 1/2(∂D) we have the decompositionR ∈ −

ϕ = ϕ ϕdσ ϕ0 + ϕdσ ϕ0 := ψ + ϕdσ ϕ0, − ∂D ∂D ∂D Z Z Z where we can see that (ψ, 1) 1 , 1 = 0. This kind of decomposition, ϕ = − 2 2 ψ + αϕ0, with (ψ, 1) 1 , 1 = 0 is unique. − 2 2 1/2 Note that we can decompose H− as a direct sum of elements with 1/2 zero-mean and multiples of ϕ , H 1/2(∂D) = H− (∂D) µϕ ,µ C . 0 − 0 ⊕ { 0 ∈ } This allows us to deﬁne the following operator. 192 Appendix A. Layer Potentials for the Laplacian in two Dimensions

Definition A.2. Let be the linear operator that satisfies SD 1/2 1/2 D : H− (e∂D) H (∂D) S → D[ϕ] if (ϕ, 1) 1 , 1 = 0, e ϕ S − 2 2 → 1 if ϕ = ϕ. ( − 0 Remark A.0.1. When is invertible, is similar enough to keep the SD SD invertibility. When is not invertible, then = ker( ) and the operator SD C SD becomes an invertible alternative to e that images the kernel to the SD SD C space µχ(∂D),µ C . { ∈ } e Remark A.0.2. : H 1/2(∂D) H1(D) follows the same definition. SD − → Theorem A.0.1. D is invertible, self-adjoint and negative for ( , ) 1 , 1 and eS · · − 2 2 satisfies the following Calderón identity: D ∗ = D D. e S KD K S Proof. The invertibility is a direct consequencee of Lemmae A.0.1. Indeed, since is Fredholm of zero index, so is . Therefore, we only SD SD need the injectivity. Suppose that, ϕ = 0 such that [ϕ] = 0. This mean ∃ 6 SD that, α = 0 C such that ϕ = αϕ . Therefore, [eϕ]= α [ϕ ]= α = ∃ 6 ∈ 0 SD SD 0 − 0, which is a contradiction. Hence ϕ = 0. e The self-adjointness comes directly form that ofe . Noticinge that ϕ is SD 0 an eigenfunction of eigenvalue 1/2 of we get the Calderón identity from KD∗ a similar one satisfied by : = ; see [12, Lemma 2.12]. SD SDKD∗ KDSD It is known that ∂D ψ D[ψ]dσ < 0 if (ψ, 1) 1 , 1 = 0 and ψ = 0, see S − 2 2 6 [12, Lemma 2.10]. Therefore,R writing ϕ = ψ + ∂D ϕdσ ϕ0, with ψ = ϕ ϕdσ ϕ , and noticing that ϕ [ψ]dσR = [ϕ ]ψdσ = − ∂D 0 ∂D 0SD ∂D SD 0 ∂D ψdσR = 0, we have R R − e e R 2 ϕ D[ϕ]dσ = ψ D[ψ]dσ + ϕdσ D[ϕ0] ∂D S ∂D S ∂D S Z Z Z 2 e = ψ eD[ψ]dσ ϕdσ e< 0, ∂D S − ∂D Z Z if ϕ = 0. 6

Deﬁnition A.3. We deﬁne the space ∗(∂D) as the Hilbert space resulting 1/2 H from endowing H− (∂D) with the inner product

(u,v) ∗ := (u, D[v]) 1 , 1 . (A.1) H − S − 2 2 Similarly, we let to be the Hilbert spacee resulting from endowing H1/2 with H the inner product 1 (u,v) = ( D− [u],v) 1 , 1 . (A.2) H − S − 2 2 If D is 1,α, we have the following result. C e Lemma A.0.2. Let D be a 1,α bounded domain of R2 and let be the C SD operator introduced in Deﬁnition A.2. Then e Appendix A. Layer Potentials for the Laplacian in two Dimensions 193

(i) The operator D∗ is compact self-adjoint in the Hilbert space ∗(∂D) K 1 H and ∗(∂D) is equivalent to H− 2 (∂D); Similarly, the Hilbert space H 1 (∂D) is equivalent to H 2 (∂D). H (ii) Let (λj,ϕj), j = 0, 1, 2,..., be the eigenvalue and normalized eigenfunction pair of with λ = 1 . Then, λ ( 1 , 1 ] and λ 0 as KD∗ 0 2 j ∈ − 2 2 j → j ; → ∞ (iii) The following representation formula holds: for any ϕ H 1/2(∂D), ∈ −

∞ D∗ [ϕ]= λj(ϕ,ϕj) ∗ ϕj. K H ⊗ Xj=0 The following lemmas are needed in the proof of Theorem 4.2.1 and The- orem 4.2.2. Lemma A.0.3. Let D = z + δB and η be the function such that, for every ϕ (∂D), η(ϕ)(˜x)= ϕ(z + δx˜), for almost all x˜ ∂B. Then ∈H∗ ∈

ϕ ∗(∂D) = δ η(ϕ) ∗(∂B). k kH k kH Similarly, if for every ϕ L2(D), η(ϕ)(˜x)= ϕ(z +δx˜), for almost all x˜ B, ∈ ∈ then

ϕ 2 = δ η(ϕ) 2 . k kL (D) k kL (B) Proof. We only prove the scaling in (∂D). From the proof of Theorem H∗ A.0.1, we have

2 2 ϕ ∗(∂D) = ψ D[ψ]dσ + ϕdσ , k kH − ∂D S ∂D Z Z where ψ = ϕ ∂D ϕdσ ϕ0. Note that (ψ, 1) 1 , 1 = 0 and so, (η(ψ), 1) 1 , 1 = − − 2 2 − 2 2 0 as well. R By a rescaling argument we ﬁnd that

2 2 2 1 2 ϕ ∗ = δ log δ(˜x y˜) η(ψ)(˜x)η(ψ)(˜y)dσ(˜x)dσ(˜y)+ δ η(ϕ)dσ k k (∂D) − 2π | − | H Z∂B Z∂B Z∂B 1 2 2 = δ2 log(δ) η(ψ)dσ + δ2 η(ψ) [η(ψ)]dσ + η(ϕ)dσ −2π − SD Z∂B Z∂B Z∂B 2 2 = δ η(ϕ) ∗(∂B). k kH

Lemma A.0.4. Let g H1(D) be such that ∆g = f with f L2(D). Then, ∈ ∈ in (∂D), H∗

1 1 ∂g ( I ∗ ) − [g]= + . 2 −KD SD −∂ν Tf

For some f ∗(∂D) and fe ∗ C f L2(D) for a constant C. T ∈H 1 R2 kT kH ≤ kRk2 ¯ Moreover, if g Hloc( ), ∆g = 0 in D, lim x g(x) = 0, then ∈ \ | |→∞ 1 = c ϕ + − [g], Tf f 0 SD e 194 Appendix A. Layer Potentials for the Laplacian in two Dimensions with

1 c = f(x)dx − [g](y)dσ(y), f − SD ZD Z∂D where ϕ0 is given in Deﬁnition A.1. Here,e by an abuse of notation, we still denote by g the trace of g on ∂D.

Proof. Let ϕ (∂D). Then ∈H∗

1 1 1 1 ( I D∗ ) D− [g],ϕ = D− [g], I D D[ϕ] 2 −K S ∗ − S 2 −K S 1 , 1 H − 2 2 e e 1 1 e = D− [g], D I D∗ [ϕ] − S S 2 −K 1 , 1 − 2 2 1 = g,e I e [ϕ] D∗ 1 1 − 2 −K 2 , 2 − ∂ [ϕ] = g, D S 1 1 − − ∂ν 2 , 2 −− ∂g e = [ϕ]dσ f [ϕ] ∆ [ϕ] g dx ∂ν SD − SD − SD Z∂D ZD ∂g = ,ϕe f D[ϕe]dx. e ∗ − ∂ν − D S H Z e We have used the fact that is harmonic in D. SD Consider the linear application [ϕ] := f [ϕ]dx. We have Tf − D SD e R f [ϕ] C f L2(D) D[ϕ] L2(D) Cf D[ϕ] H1(De) Cf D[ϕ] 1 Cf ϕ − 1 . |T |≤ k k kS k ≤ kS k ≤ kS kH 2 (∂D) ≤ k kH 2 (∂D) Here we have used Holder’se inequality, ae standard Soboleve embedding, the 1 1 trace theorem and the fact that : H− 2 (∂D) H 2 (∂D) is continuous. SD → By the Riez representation theorem, there exists v (∂D) such that ∈ H∗ f [ϕ]=(v,ϕ) ∗ , ϕ ∗(∂D). e T H ∀ ∈H By abuse of notation we still denote := v to make explicit the depen- Tf dency on f. It follows that

2 f ∗ = f D[ f ]dx C f L2(D) D[ f ] L2(D) kT kH − S T ≤ k k kS T k ZD e C f L2(D) eD[ f ] H1(D) ≤ k k kS T k C f L2(D) D[ f ] 1 ≤ k k kSe T kH 2 (∂D) C f L2(D) f ∗ . ≤ k k kTe kH 1 We now show that in (∂D), = − [g]. H0∗ Tf SD e Appendix A. Layer Potentials for the Laplacian in two Dimensions 195

Indeed, let ϕ (∂D), then ∈H0∗ 1 1 − [g],ϕ = − [g], D[ϕ] D ∗ D 1 1 S − S S 2 , 2 H − e = g,ϕe e 1 1 − 2 , 2 − ∂ [ϕ] ∂ [ϕ] = g, SD SD − ∂ν + − ∂ν 1 , 1 − − 2 2 e e ∂g ∂g ∂g ∂ D[ϕ] = D[ϕ] dσ D[ϕ]dσ + D[ϕ]dσ g S dσ ∂D ∂ν S − ∂D ∂ν S ∂B∞ ∂ν S − ∂B∞ ∂ν Z Z Z Z e fe D[ϕ] ∆ D[ϕ] ge dx e − R2 S − S Z = f e[ϕ]dx. e − SD ZD e Here we have used the assumption on g, the fact that [ϕ] is harmonic SD in D and R2 D¯ and that for ϕ (∂D) we have [ϕ](x) = O( 1 ) and \ ∈ H0∗ SD x e | | ∂ D[ϕ] 1 S (x)= O( x ) for x . e ∂ν | | | | → ∞ eTherefore, 1 1 f =( f − [g],ϕ0) ∗ ϕ0 + − [g]. T T − SD H SD

Finally, re-scaling the deﬁnitione of ϕ0 given in Deﬁnitione A.1 we obtain that

1 1 ( f − [g],ϕ0) ∗ = f(x)dx − [g](y)dσ(y). T − SD H − SD ZD Z∂D e e

197

Appendix B

Asymptotic Expansions

In this section, we derive asymptotic expansions for the Helmholtz integral operators with respect to k, of some boundary integral operators deﬁned on the boundary of a bounded and simply connected smooth domain D.

B.1 Asymptotic expansions in R3

We consider a domain D ⋐ R3 whose size is of order one.

Recall the deﬁnition of the single layer potential

k [ψ](x)= G(x,y,k)ψ(y)dσ(y), x ∂D, SD ∈ Z∂D where eik x y G(x,y,k)= | − | −4π x y | − | is the Green function of Helmholtz equation in R3, subject to the Sommerfeld radiation condition. Note that

∞ (ik x y )j 1 ik ∞ (ik x y )j 1 G(x,y,k)= | − | = | − | − . − j!4π x y −4π x y − 4π j! Xj=0 | − | | − | Xj=1 We get ∞ k = + kj , (B.1) SD SD SD,j Xj=1 where i (i x y )j 1 [ψ](x)= | − | − ψ(y)dσ(y). SD,j −4π j! Z∂D In particular, we have i [ψ](x) = ψ(y)dσ(y), (B.2) SD,1 −4π Z∂D 1 [ψ](x) = x y ψ(y)dσ(y). (B.3) SD,2 −4π | − | Z∂D

Lemma B.1.1. D,j (( ∗(∂D), (∂D)) is uniformly bounded with respect to kS kL H H j. Moreover, the series in (B.1) is convergent in ( (∂D), (∂D)). L H∗ H Proof. It is clear that

D,j (L2(∂D),H1(∂D)) C, kS kL ≤ 198 Appendix B. Asymptotic Expansions where C is independent of j. On the other hand, a similar estimate also holds for the operator . It follows that SD,j∗

D,j (H−1(∂D),L2(∂D)) C. kS kL ≤

Thus, we can conclude that D,j 1 1 is uniformly bounded kS k (H− 2 (∂D),H 2 (∂D)) L 1 by using interpolation theory. By the equivalence of norms in the H− 2 (∂D) 1 and H 2 (∂D), the lemma follows immediately.

Note that is invertible in dimension three, so is k for small k. By SD SD formally writing

k 1 1 2 ( )− = − + k + k + ..., (B.4) SD SD BD,1 BD,2 and using the identity ( k ) 1 k = Id, we can derive that SD − SD 1 1 1 1 1 1 1 = − − , = − − + − − − . BD,1 −SD SD,1SD BD,2 −SD SD,2SD SD SD,1SD SD,1SD (B.5) We can also derive other lower-order terms . BD,j Lemma B.1.2. The series in (B.4) converges in ( (∂D), (∂D)) for L H H∗ suﬃciently small k.

Proof. The proof can be deduced from the identity

k 1 1 ∞ j 1 1 ( )− =(Id + − k )− − . SD SD SD,j SD Xj=1

We now consider the expansion for the boundary integral operator ( k ) . KD ∗ We have k 2 ( )∗ = ∗ + k + k + ..., (B.6) KD KD KD,1 KD,2 where

j 1 j i ∂(i x y ) − i (j 1) j 3 [ψ](x)= | − | ψ(y)dσ(y)= − x y − (x y) ν(x)ψ(y)dσ(y). KD,j −4π j!∂ν(x) − 4πj! | − | − · Z∂D Z∂D In particular, we have

1 (x y) ν(x) = 0, [ψ](x)= − · ψ(y)dσ(y). (B.7) KD,1 KD,2 4π x y Z∂D | − |

Lemma B.1.3. The norm D,j ( ∗(∂D), ∗(∂D)) is uniformly bounded for kK kL H H j 1. Moreover, the series in (B.6) is convergent in ( (∂D), (∂D)). ≥ L H∗ H∗ B.2 Asymptotic expansion in R3: multiple particles

In this section, we consider the multiple particle case in dimension three. We assume that the particles have size of order δ which is a small number and the distance between them is of order one. We write Dj = zj + δD, j = 1, 2,...,M, where D has size one and is centered at the origin. Our goal is to derive estimates for various boundary integral operators that aree e B.2. Asymptotic expansion in R3: multiple particles 199 defined on small particles in terms of their size. For this purpose, we denote by D0 = δD. For each function f defined on ∂D0, we define a corresponding function on D by e η(f)(x)= f(δx). e In this section, we denote by χ(∂Dj) the constant function equal one over e e the border of Dj. We first state some useful results. Lemma B.2.1. The following scaling properties hold:

1 (i) η(f) e = δ f 2 ; k kL2(∂D) − k kL (∂D0) 1 e 2 (ii) η(f) (∂D) = δ− f (∂D0); k kH k kH 3 e 2 ∗ (iii) η(f) ∗(∂D) = δ− f (∂D0). k kH k kH Proof. The proof of (i) is straightforward and we only need to prove (ii) and (iii). To prove (iii), we have

2 f(x)f(y) f ∗ = dσ(x)dσ(y) (∂D0) k kH 4π x y Z∂D0 Z∂D0 | − | η(f)(x)η(f)(y) = δ3 dσ(x)dσ(x) e e ∂D ∂D 4π x y 3 Z Z 2 | − | = δ η(f) e , e e ∗(∂D) e e k kH e e whence (iii) follows. To prove (ii), recall that

1 ∗ f (∂D0) = D− f (∂D0). k kH kS 0 kH 1 Let u = − [f]. Then f = D [u]. We can show that SD0 S 0 η(f)= δ e (η(u)). SD As a result, we have

1 1 e e e e 2 ∗ 2 η(f) (∂D) = δ D(η(u)) (∂D) = δ η(u) ∗(∂D) = δ− u (∂D0) = δ− f (∂D0), k kH kS kH k kH k kH k kH which proves (ii).

Lemma B.2.2. Let X and Y be bounded and simply connected smooth domains in R3. Assume 0 X,Y and X = δX, Y = δY . Let and be two ∈ R R boundary integral operators from (∂Y ) to (∂X) and (∂Y ) to (∂X), D′ D′ D′ D′ respectively. Here, denotes the Schwartze space. Assumee that bothe opera- D′ tors have the same Schwartz kernel R with the following homogeneouse scalinge property R(δx,δy)= δmR(x,y). Then,

2+m ∗ ∗ e e ( (∂Y ), (∂X)) = δ ( ∗(∂Y ), ∗(∂X)), kRkL H H kRkL H H 1+m ∗ e e ( (∂Y ), (∂X)) = δ ( ∗(∂Y ), (∂X)). kRkL H H kRke L H H e 200 Appendix B. Asymptotic Expansions

Proof. The result follows from Lemma B.2.1 and the following identity

2+m 1 = δ η− η. R ◦ R ◦ e k k We ﬁrst consider the operators and ( )∗. The following asymptotic SDj KDj expansions hold. Lemma B.2.3. (i) Regarded as operators from (∂D ) into (∂D ), we H∗ j H j have k = + k + k2 + O(k3δ3), SDj SDj SDj ,1 SDj ,2 where = O(1) and = O(δm); SDj SDj ,m (ii) Regarded as operators from (∂D ) into (∂D ), we have H j H∗ j k 1 1 2 3 3 ( )− = − + k D ,1 + k D ,2 + O(k δ ), SDj SDj B j B j 1 m where − = O(1) and D ,m = O(δ ); SDj B j (iii) Regarded as operators from (∂D ) into (∂D ), we have H∗ j H∗ j k 2 2 ( )∗ = ∗ + k O(δ ), KDj KDj

where ∗ = O(1). KDj Proof. The proof immediately follows from Lemmas B.2.2, B.1.1, and B.1.3.

We now consider the operator k . By deﬁnition, SDj ,Dl

k [ψ](x)= G(x,y,k)ψ(y)dσ(y), x ∂D . SDj ,Dl ∈ l Z∂Dj Using the expansion

∞ m G(x,y,k)= k Qm(x,y), m=0 X where im x y m 1 Q (x,y)= | − | − , m − 4π we can derive that k = km , SDj ,Dl Sj,l,m m 0 X≥ where [ψ](x)= Q (x,y)ψ(y)dσ(y). Sj,l,m m Z∂Dj We can further write = , Sj,l,m Sj,l,m,n n 0 X≥ B.2. Asymptotic expansion in R3: multiple particles 201 where is deﬁned by Sj,l,m,n 1 ∂ α + β [ψ](x)= | | | | Q (z , z )(x z )α(y z )βψ(y) dσ(y). Sj,l,m,n α!β! ∂xα∂yβ m l j − l − j Z∂Dj α + β =n | | X| | In particular, we have 1 [ψ](x) = (ψ,χ(∂D )) −1/2 1/2 χ(D ), j,l,0,0 j H (∂Dj ),H (∂Dj ) l S −4π zj zl | − | α (zl zj) α [ψ](x) = − (x z ) (ψ,χ(∂D )) −1/2 1/2 + j,l,0,1 3 l l H (∂Dj ),H (∂Dj ) S 4π zl zj − α =1 |X| | − | (y z )α,ψ χ(D ) , − j l 1 ∂2Q (z , z) [ψ](x) = 0 l j (x z )α(y z )βψ(y)dσ(y), Sj,l,0,2 α!β! ∂xα∂yβ − l − j α + β =2 | |X| | i j,l,1[ψ](x) = (ψ,χ(∂Dj)) −1/2 1/2 χ(Dl), S −4π H (∂Dj ),H (∂Dj ) 1 j,l,2,0[ψ](x) = zl zj (ψ,χ(∂Dj)) −1/2 1/2 χ(Dl). S 4π | − | H (∂Dj ),H (∂Dj ) The following estimate holds. n+1 Lemma B.2.4. We have j,l,m,n ( ∗(∂D), (∂D)) . O(δ ). kS kL H H Proof. After a translation of coordinates, the stated estimate immediately follows from Lemma B.2.2.

Similarly, for the operator km deﬁned in the following way KDj ,Dl ∂G(x,y,k) k [ψ](x)= ψ(y)dσ(y), x ∂D , KDj ,Dl ∂ν(x) ∈ l Z∂Dj we have k = km , KDj ,Dl Kj,l,m,n m 0 n 0 X≥ X≥ where 1 ∂nK (z , z ) [ψ](x)= m l j (x z )β(y z )α(x y) ν(x)ψ(y)dσ(y) Kj,l,m,n α!β! ∂xβ∂yα − l − j − · Z∂Dj α + β =n | | X| | with im(m 1) x y m 3 K (x,y)= − | − | − . m − 4πm! In particular, we have 1 j,l,0,0[ψ](x) = (x zl) ν(x) ψ,χ(∂Dj) −1/2 1/2 K 4π z z 3 − · H (∂Dj ),H (∂Dj ) | l − j| h ψ, (y zj) ν(x) −1/2 1/2 − − · H (∂Dj ),H (∂Dj ) +(zl zj) ν(x) ψ,χ(∂Dj) −1/2 1/2 , (B.8) − · H (∂Dj ),H (∂Dj ) j,l,1,m[ψ] = 0 for all m. i (B.9) K 202 Appendix B. Asymptotic Expansions

n+2 Lemma B.2.5. We have j,l,m,n ( ∗(∂D ), ∗(∂D )) . O(δ ). kK kL H j H l Proof. Note that

1 ∂nK (z , z ) [ψ](x) = m l j (x z )β(y z )α(x z ) ν(x)ψ(y)dσ(y), Kj,l,m,n α!β! ∂xβ∂yα − l − j − l · Z∂Dj α + β =n | | X| | 1 ∂nK (z , z ) m l j (x z )β(y z )α(y z ) ν(x)ψ(y)dσ(y), − α!β! ∂xβ∂yα − l − j − j · Z∂Dj α + β =n | | X| | 1 ∂nK (z , z ) + m l j (x z )β(y z )α(z z ) ν(x)ψ(y)dσ(y). α!β! ∂xβ∂yα − l − j l − j · Z∂Dj α + β =n | | X| | After a translation of coordinates, we can apply Lemma B.2.2 to each one of the three terms above to conclude that = O(δn+3)+ O(δn+2). This Kj,l,m,n completes the proof of the lemma.

To summarize, we have proven the following results. Lemma B.2.6. (i) Regarded as an operator from (∂D ) into (∂D ) H∗ j H l we have,

k = + + +k +k2 +O(δ4)+O(k2δ2). SDj ,Dl Sj,l,0,0 Sj,l,0,1 Sj,l,0,2 Sj,l,1 Sj,l,2,0 Moreover, = O(δn+1). Sj,l,m,n (ii) Regarded as an operator from (∂D ) into (∂D ), we have H∗ j H∗ l k = + O(k2δ2). KDj ,Dl Kj,l,0,0 Moreover, = O(δ2). Kj,l,0,0 B.3 Asymptotic expansions in R2

Let us now consider the single-layer potential for the Helmholtz equation in R2 given by

k [ϕ](x)= G(x,y,k)ϕ(y)dσ(y), x ∂D, SD ∈ Z∂D i where G(x,y,k)= H(1)(k x y ) and H(1) is the Hankel function of ﬁrst −4 0 | − | 0 kind and order 0. We have, for k 1, ≪ i 1 ∞ H(1)(k x y )= log x y + τ + (b log k x y + c )(k x y )2j, −4 0 | − | 2π | − | k j | − | j | − | Xj=1 where

j 1 i ( 1)j 1 iπ 1 τ = (log k+γ log 2) , b = − , c = bj γ log 2 , k 2π e− −4 j 2π 22j(j!)2 j − e − − 2 − n n=1 ! X B.3. Asymptotic expansions in R2 203

and γe is the Euler constant. Thus, we get

∞ ∞ k = ˆk + k2j log k (1) + k2j (2) , (B.10) SD SD SD,j SD,j j=1 j=1 X X where

ˆk [ϕ](x) = [ϕ](x)+ τ ϕdσ, SD SD k Z∂D (1) [ϕ](x) = b x y 2jϕ(y)dσ(y), SD,j j| − | Z∂D (2) [ϕ](x) = x y 2j(b log x y + c )ϕ(y)dσ(y). SD,j | − | j | − | j Z∂D (1) (2) Lemma B.3.1. The norms D,j ( ∗(∂D), (∂D)) and D,j ( ∗(∂D), (∂D)) kS kL H H kS kL H H are uniformly bounded with respect to j. Moreover, the series in (B.10) is convergent in ( (∂D), (∂D)) for k < 1. L H∗ H Observe that

D D [ϕ]= D D [ ∗ [ϕ]+(ϕ,ϕ0) ∗ ϕ0]=(ϕ,ϕ0) ∗ ( D[ϕ0] + 1) . S − S S − S PH0 H H S e e Then it follows that

ˆk D[ϕ]= D[ϕ]+(ϕ,ϕ0) ∗ ( D[ϕ0] + 1)+τk ∗ [ϕ]+(ϕ,ϕ0) ∗ ϕ0dσ = D[ϕ]+Υk[ϕ], S S H S PH0 H S Z∂D where e e Υk[ϕ]=(ϕ,ϕ0) ∗ ( D[ϕ0]+1+ τk) . (B.11) H S Therefore, we arrive at the following result. Lemma B.3.2. For k small enough, ˆk : (∂D) (∂D) is invertible. SD H∗ →H

Proof. Υk is clearly a compact operator. Since D is invertible, the invert- ˆk ˆk 1 S 1 ibility of D is equivalent to that of D D− = I +Υk D− . By the Fredholm S S S S 1 alternative, we only need to prove the injectivitye of I +Υ − . kSD 1/2 1 C e e1 Since v H (∂D), Υk D− [v] , for I +Υk D− [v] = 0, we need to ∀ ∈ S ∈ S e show that v = D[αϕ0]= α C. S −e ∈ e We have e 1 I +Υ − [αϕ ]= α( [ϕ ]+ τ ) = 0 iﬀ [ϕ ]= τ or α = 0. kSD SD 0 SD 0 k SD 0 − k Since we cane alwayse ﬁnd a small enough k such that [ϕ ] = τ , we need SD 0 6 − k α = 0. This yields the stated result.

Lemma B.3.3. For k small enough, the operator k : (∂D) (∂D) SD H∗ → H is invertible.

Proof. The operator k ˆk : (∂D) (∂D) is a compact operator. SD − SD H∗ → H Because ˆk is invertible for k small enough, by the Fredholm alternative only SD the injectivity of k is necessary. From the uniqueness of a solution to the SD Helmholtz equation we get the result. 204 Appendix B. Asymptotic Expansions

Lemma B.3.4. The following asymptotic expansion holds for k small enough:

k 1 1 2 1 (1) 1 2 ( D)− = ∗ − + k k log k ∗ − ∗ − + O(k ) S PH0 SD U − PH0 SD SD,1PH0 SD with e 1 e e ( D− [ ],ϕ0) ∗ k = S · H ϕ0. (B.12) U − D[ϕ0]+ τk Se Note that = O(1/ log k). Uk Proof. We can write (B.10) as

k = ˆk + , SD SD Gk where = k2 log k (1) + O(k2). From Lemma B.3.2 and Lemma B.3.3 we Gk SD,1 get the identity

1 k 1 k 1 − k 1 ( )− = I +( ˆ )− ( ˆ )− . SD SD Gk SD Hence, we have 1 k 1 1 k − 1 ( ˆ )− = − ˆ − . SD SD SD SD Λ−1 e k e Here, | {z }

Λk = I ( ,ϕ0) ∗ ( D[ϕ0]+1+ τk)ϕ0 − · H S = ∗ ( ,ϕ0) ∗ ( D[ϕ0]+ τk)ϕ0. PH0 − · H S Then,

1 1 Λ− = ∗ ( ,ϕ0) ∗ ϕ0, k PH0 − · H [ϕ ]+ τ SD 0 k and therefore, 1 k 1 1 ( D− [ ],ϕ0) ∗ ( ˆ ) = ∗ S · H ϕ . D − 0 D− 0 S PH S − D[ϕ0]+ τk Se ˆk 1 It is clear that ( D)− ( (∂D), ∗e(∂D)) is bounded for k small. Since k ( (∂D), ∗(∂D)) k S kL H H ||G ||L H H goes to zero as k goes to zero, for k small enough, we can write

k 1 k 1 k 1 k 1 4 2 ( )− =( ˆ )− ( ˆ )− ( ˆ )− + O k (log k) , SD SD − SD Gk SD which yields the desired result.

We now consider the expansion for the boundary integral operator ( k ) . KD ∗ We have

k ∞ 2j (1) ∞ 2j (2) ( )∗ = ∗ + k log k + k , (B.13) KD KD KD,j KD,j j=1 j=1 X X B.3. Asymptotic expansions in R2 205 where ∂ x y 2j (1) [ϕ](x) = b | − | ϕ(y)dσ(y), KD,j j ∂ν(x) Z∂D 2j ∂ x y (bj log x y + cj) (2) [ϕ](x) = | − | | − | ϕ(y)dσ(y). KD,j ∂ν(x) Z∂D (1) (2) Lemma B.3.5. The norms D,j ( ∗(∂D), ∗(∂D)) and D,j ( ∗(∂D), ∗(∂D)) kK kL H H kK kL H H are uniformly bounded for j 1. Moreover, the series in (B.13) is convergent ≥ in ( (∂D), (∂D)). L H∗ H∗

207

Appendix C

Sum Rules for the Polarization Tensor

Let f be a holomorphic function deﬁned in an open set U C containing ⊂ ∞ the spectrum of . Then, we can write f(z)= a zj for every z U. KD∗ j ∈ Xj=0 Deﬁnition C.1. Let ∞ j f( ∗ ) := a ( ∗ ) , KD j KD Xj=0 j where ( ) := ∗ ∗ .. ∗ . KD∗ KD ◦KD ◦ ◦KD j times Lemma C.0.1.|We have{z }

∞ f( D∗ )= f(λj)( ,ϕj) ∗ ϕj. K · H Xj=1

Proof. We have

∞ i ∞ ∞ i f( D∗ ) = ai( D∗ ) = ai λj( ,ϕj) ∗ ϕj K K · H Xi=0 Xi=0 Xj=1

∞ ∞ i = aiλj ( ,ϕj) ∗ ϕj ! · H Xj=1 Xi=0 ∞ = f(λj)( ,ϕj) ∗ ϕj. · H Xj=1

From Lemma C.0.1, we can deduce that

∞ (j) xlf( D∗ )[νm](x) dσ(x)= f(λj)αl,m. (C.1) ∂D K Z Xj=1 Equation (C.1) yields the summation rules for the entries of the polarization tensor. 208 Appendix C. Sum Rules for the Polarization Tensor

∞ In order to prove that α(j) = δ D , we take f(λ) = 1 in (C.1) to l,m l,m| | j=1 get X ∞ (j) αl,m = xlνm(x) dσ(x)= δl,m D . ∂D | | Xj=1 Z Next, we prove that

d ∞ (d 2) λ α(j) = − D . j l,l 2 | | Xj=1 Xl=1 Taking f(λ)= λ in (C.1), we obtain

d d ∞ (j) λj αl,l = xl D∗ [νl](x) dσ(x), ∂D K Xj=1 Xl=1 Xl=1 Z 1 ∂ D[νl] xl D∗ [νl](x) dσ(x)= xl νl(x)+ S (x) dσ(x), ∂D K ∂D 2 ∂ν Z Z − D ∂ D[νl] =| | + xl S (x) dσ(x). (C.2) 2 ∂D ∂ν Z −

Integrating by parts we arrive at

∂ D[νl] xl S (x)dσ(x)= el(x) D[νl](x)dx + xl∆ D[νl](x)dx. ∂D ∂ν D · ∇S D S Z − Z Z

Since the single-layer potential is harmonic on D,

∂ D[νl] xl S (x)dσ(x)= el(x) xΓ(x,x′)νl(x′)dσ(x′) dx. ∂D ∂ν D · ∂D ∇ Z − Z Z

Summing on i and using Γ(x,x )= ′ Γ(x,x ), we get ∇x ′ −∇x ′ d ∂ D[νl] xl S (x)dσ(x)= ν(x′) x′ Γ(x,x′)dσ(x′) dx, ∂D ∂ν − D ∂D ·∇ l=1 Z − Z Z X = [1](x)dx, − DD ZD = D , (C.3) −| | where is the double-layer potential. Hence, summing equation (C.2) for DD i = 1,...,d, we get the result. Finally, we show that

d d ∞ 2 (j) d 4 2 λj αl,l = − D + D[νl] dx. 4 | | D |∇S | Xj=1 Xl=1 Xl=1 Z Appendix C. Sum Rules for the Polarization Tensor 209

Taking f(λ)= λ2 in (C.1) yields

d d ∞ 2 (j) 2 λj αl,l = xl( D∗ ) [νl](x) dσ(x) ∂D K Xj=1 Xl=1 Xl=1 Z d

= D[yl](x) D∗ [νl](x) dσ(x) ∂D K K Xl=1 Z d d νl ∂ D[νl] = D[yl] dσ + D[yl] S dσ ∂D K 2 ∂D K ∂ν |− Xl=1 Z Xl=1 Z d d (d 2) yl ∂ D[νl] ∂ D[νl] = − D S dσ + D[yl] S dσ . 4 | |− ∂D 2 ∂ν ∂D D ∂ν l=1 Z − l=1 Z − − X X I1 I 2 From (C.3) it follows that | {z } | {z }

D I = | |. 1 − 2

Since xl is harmonic, we have xl = D[yl](x) D[νl](x) on ∂D, and thus, D |− −S d ∂ D[νl] I2 = (xl + D[νl](x)) S (x)dσ(x), ∂D S ∂ν Xl=1 Z − d ∂ D[νl] = D + D[νl] S dσ, −| | ∂D S ∂ν Xl=1 Z − d 2 = D + D[νl] dx. −| | D |∇S | Xl=1 Z

Replacing I1 and I2 by their expressions gives the desired result.

211

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[95] J. Yang, H. Giessen, and P. Lalanne, Simple analytical expression for the peak-frequency shifts of plasmonic resonances for sensing, Nano Lett., 15 (2015), 3439–3444. Résumé Abstract Cette thèse porte sur l’étude mathématique des interac- This thesis deals with the mathematical study of the inter- tions entre la lumière et certains types de nanoparticules. actions between light and certain types of nanoparticles. A l’échelle du nanomètre, des particules métalliques comme At the nanometer scale, metal particles such as gold or l’or ou l’argent subissent un phénomène de résonance silver undergo a resonance phenomenon when their free lorsque leurs électrons libres interagissent avec un champ electrons interact with an electromagnetic field. This in- électromagnétique. Cette interaction produit une aug- teraction results in an enhancement of the near and far mentation du champs électrique proche et lointain, leur electric field, enabling them to improve the brightness permettant d’améliorer la luminosité et la directivité de la and the directivity of the light, confining electromagnetic lumière, confinant des champs électromagnétiques dans fields in advantageous directions. This phenomenon, called des directions avantageuses. Ce phénomène, appelé "plasmonic resonances for nanoparticles", opens a door "résonances plasmoniques pour des nanoparticules" ou- to a wide range of applications, from new medical imag- vre une porte sur une large gamme d’applications, des ing techniques to efficient solar panels. Using layer po- nouvelles techniques d’imagerie médicale à des panneaux tentials techniques and perturbation theory, we propose solaires efficaces. En utilisant des techniques issues des a study of the scattering of electromagnetic waves by one potentiels de couches et de la théorie de la perturba- and several plasmonic nanoparticles in the quasi-static, tion, nous proposons une étude de la dispersion d’ondes Helmholtz and Maxwell framework. We then study some électromagnétiques par une et plusieurs nanoparticules applications such as heat generation, metasurfaces and plasmoniques, dans le cadre quasi-statique, Helmholtz super-resolution. et Maxwell. Nous étudions ensuite certaines applications tel que la génération de chaleur, les métasurfaces et l’imagerie super-résolue. Mots Clés Keywords operateur de Neumann-Poincaré, potentiels de couche, Neumann Poincaré operator, layer potentials, plasmonic nanoparticules plasmoniques, resonance de plasmon, nanoparticles, plasmonic resonances, asymptotic analy- analyse asymptotique sis