Brian Abdul

Alexander, Giovanni, Guanghui Tomas I David

Jorge Gonazález

Sofia Olivares

Felipe Soto

Ignacio Atal

Francisco Vial

Tomas Yany

Michel Lemaire

Juan Pablo

Keywords: Neumann Poincaré operator, layer potentials, plasmonic nanoparticles, plasmonic resonances, asymptotic analysis

I would like to start by thanking the most important person in the achievement 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

• We say that ∂D is of type C 1,α , for 0 < α < 1, if ∂D is locally Lipschitz of order 0 < α < 1;

• ν denotes the outward normal to ∂D and ∂ ∂ν the outward normal derivative;

• ϕ ± (x) = lim t→0 + ϕ(x ± tν), x ∈ ∂D;

• Id denotes the identity operator;

• H s (∂D) denotes the usual Sobolev space of order s on ∂D;

• (•, •) -1 2 , 1 2
denotes the duality pairing between H -1 2 (∂D) and H 1 2 (∂D);

• For any functional space F (∂D) defined on ∂D, F 0 (∂D) denotes its zero mean subspace;

• L(E, F ) denotes the set of bounded linear applications from E to F and L(E) := L(E, E);

• For α = (α 1 , α 2 ) ∈ N 2 , ∂ α := ∂ α 1 1 ∂ α 2 d and α! := α 1 !α 2 !;

• χ(S), denotes the characteristic function of the set S;

• ℜz denotes the real part of z;

• ℑz denotes the imaginary part of z;

• |x| denotes the norm of x ∈ R d ;

• We denote by the Sommerfeld radiation condition for a function u in dimension d = 2, 3, the following condition:

∂u ∂|x| -ik m u ≤ C|x| -(d+1)/2
as |x| → +∞ for some constant C independent of x.

xi 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 Nations General Assembly as the "International Year of Light and Light-based Technologies". In the last decades, the field of photonics has seen a revolution 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 opportunities, 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 nanotechnologies.

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 premeabilitty has negative real part. When an external light wave is incident on the nanoparticle, the cloud of free electrons on the surface of the particule 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 [START_REF] Born | Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light[END_REF] 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 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 [START_REF] Link | Shape and size dependence of radiative, nonradiative and photothermal properties of gold nanocrystals[END_REF]. 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", consisting 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 [START_REF] Jain | Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: Applications in biomedical imaging and biomedicine[END_REF].

Nanoparticles are also being used in thermotherapy as nanometric heatgenerators 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 [START_REF] Govorov | Generating heat with metal nanoparticles[END_REF].

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 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 [START_REF] Giannini | Scaling behavior of individual nanoparticle plasmon resonances[END_REF][START_REF] Link | Shape and size dependence of radiative, nonradiative and photothermal properties of gold nanocrystals[END_REF]. It was experimentally observed, for instance, in [START_REF] Giannini | Scaling behavior of individual nanoparticle plasmon resonances[END_REF][START_REF] Scaffardi | Size dependence of refractive index of gold nanoparticles[END_REF] 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 [START_REF] Giannini | Scaling behavior of individual nanoparticle plasmon resonances[END_REF][START_REF] Palomba | Blue-shifted plasmon resonance of individual size-selected gold nanoparticles[END_REF][START_REF] Scaffardi | Size dependence of refractive index of gold nanoparticles[END_REF]. 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 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 [START_REF] Ammari | A mathematical theory of super-resolution by using a system of sub-wavelength Helmholtz resonators[END_REF], 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 [START_REF] Ammari | Mathematical analysis of plasmonic nanoparticles: the scalar case[END_REF][START_REF] Habib Ammari | Shape reconstruction of nanoparticles from their associated plasmonic resonances[END_REF][START_REF] Ammari | Heat generation with plasmonic nanoparticles[END_REF][START_REF] Ammari | Mathematical analysis of plasmonic resonances for nanoparticles: the full Maxwell equations[END_REF][START_REF] Ammari | Mathematical and numerical framework for metasurfaces using thin layers of periodically distributed plasmonic nanoparticles[END_REF][START_REF] Ammari | Reconstructing fine details of small objects by using plasmonic spectroscopic data[END_REF].

Part I

Mathematical Analysis of Plasmonic Resonances

Chapter 1

The Quasi-Static Limit

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) [START_REF] Sonnichsen | Plasmons in metal nanostructures[END_REF]. 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 [START_REF] Grieser | The plasmonic eigenvalue problem[END_REF]. 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 [START_REF] Ammari | Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory[END_REF]. Here, we extend the validity of the Maxwell-Garnett effective medium theory in order to describe the behavior of a system of arbitraryshaped 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 arbitraryshaped nanoparticles.

Preliminaries

In this section we recall important properties of the layer potentials for the Laplacian that will be of great use throughout this thesis. A more detailed analysis for the case d = 2 is given in Appendix A.

Layer potentials for the Laplace equation

The Neumann-Poincaré operator (NP) K * D associated with D is defined as follows:

K * D [ϕ](x) = 1 2π ∂D x -y, ν x ) |x -y| d ϕ(y)dσ(y), x ∈ ∂D.
It is related to the single layer potential S D by the following jump relation:

∂S D [ϕ] ∂ν ± = (± 1 2 I + K * D )[ϕ] for ϕ ∈ H -1/2 (∂D). (1.1)
It can be shown that the operator λI -K * D : H -1/2 (∂D) → H -1/2 (∂D) is invertible for any |λ| > 1/2. Furthermore, K * D is compact, its spectrum is discrete and contained in ] -1/2, 1/2] with 0 being an accumulation point; see for instance [START_REF] Ammari | Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory[END_REF][START_REF] Ando | Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann-Poincaré operator[END_REF] for more details.

In general, K * D is not symmetric for the pairing (•,

•) 1 2 ,- 1 2
. Nevertheless, using Calderon's identity

K D S D = S D K * D ,
K * D can be symmetrized with the following inner product

(u, v) H * := -(S D [v], u) 1 2 ,- 1 2 .
It can be shown that in R 3 , (•, •) H * defines a Hilbert space, equivalent to H -1/2 (∂D) [START_REF] Ammari | Mathematical and Statistical Methods for Multistatic Imaging[END_REF][START_REF] Ando | Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann-Poincaré operator[END_REF][START_REF] Kang | Spectral resolution of the Neumann-Poincaré operator on intersecting disks and analysis of plamson resonance[END_REF][START_REF] Khavinson | Poincaré's variational problem in potential theory[END_REF]. In R 2 a similar analysis can be done to symmetrize K * D . We refer the reader to Appendix A. Let (λ j , ϕ j ), j = 0, 1, 2, . . . be the eigenvalue and normalized eigenfunction pair of K * D in H * (∂D). From the spectral theorem, we know that λ j → 0 for j → ∞ and ϕ j form a base of H * (∂D). Therefore, the following representation formula holds: for any ϕ ∈ H -1/2 (∂D),

K * D [ϕ] = ∞ j=0 λ j (ϕ, ϕ j ) H * ⊗ ϕ j .
Chapter 1. The Quasi-Static Limit

From the jump formula (1.1), we can see that 1/2 is always an eigenvalue of K * D . If the D is simply connected, there is only one eigenvalue taking the value 1/2. We denote this eigenvalue by λ 0 and its corresponding eigenfunction ϕ 0 .

In R 3 , let H(∂D) be the space H 1 2 (∂D) equipped with the following equivalent inner product

(u, v) H = ((-S D ) -1 [u], v) -1 2 , 1 2 .
(1.2)

Then, S D is an isometry between H * (∂D) and H(∂D).

A smilar result can be found in R 2 , see Appendix A.

Layer potential formulation for the scattering problem

We consider the scattering problem of a time-harmonic wave u i incident on a plasmonic nanoparticle. The homogeneous medium is characterized by its electric permittivity ε m that we assume to be real and strictly positive. The particle occupying a bounded and simply connected domain D ⋐ R d of class C 1,α for some 0 < α < 1 is characterized by electric isotropic permittivity ε c which may depend on the frequency of the incoming wave ω by the Drude model as

ε c = ε c (ω) = 1 - ω 2 p ω(ω + iγ) ε 0 .
Here, ω p is called the plasmon frequency, γ the damping parameter and ε 0 is the permittivity of the free space. Assume that ℜε c < 0, ℑε c > 0, and define

ε D = ε m χ(R d \D) + ε c χ(D).
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 D ∇u = 0 in R d \∂D, u + -u -= 0 on ∂D, ε c ∂u ∂ν + -ε m ∂u ∂ν -= 0 on ∂D, u -u i = O( 1 |x| d-1 ), |x| → ∞. (1.3)
Here u corresponds to the electric potential. For some ϕ ∈ H -1 2 (∂D), the solution u can be written as

u(x) = u i + S D [ϕ].
From Lemma 1.2.1 we can see that only the transmission conditions

ε c ∂u ∂ν + -ε m ∂u ∂ν -= 0 on ∂D,
1.3. Layer potential formulation for the scattering problem 13 need to be satisfied. This translates into

ε c ∂S D [ϕ] ∂ν (x) + -ε m ∂S D [ϕ] ∂ν (x) -= (ε m -ε c ) ∂u i ∂ν on ∂D.
From the jump formula for the single layer potential S D , i.e

∂S D [ϕ] ∂ν (x) ± = ± 1 2 Id + K * D [ϕ](x), x ∈ ∂D.
we have

λ -K * D [ϕ] = ∂u i ∂ν , (1.4)
with

λ = ε m + ε c 2(ε m -ε c ) . Finally u = u i + S D (λ -K * D ) -1 [ ∂u i ∂ν ] = u i + ∞ j=1 ( ∂u i ∂ν , ϕ j ) H * λ -λ j S D [ϕ j ]. (1.5)
Recall that λ j are eigenvalues K * D and they satisfy |λ j | < 1/2. In the plasmonic case, ℜε c (ω) can take negative values. Then it holds that |ℜλ(ω)| < 1/2 and 0 ≤ ℑε c (ω) ≪ 1. So, for a certain frequency ω j , the value of λ(ω j ) can be very close to an eigenvalue λ j of the NP operator. Then, in (1.5), the mode S D [ϕ j ] will be amplified provided that ( ∂u i ∂ν , ϕ j ) H * is nonzero. As a result, the scattered field u -u i will show a resonant behavior. This phenomenon is called the plasmonic quasi-static resonance.

Contracted generalized polarization tensors

Decomposition (1.5) of u together with

u i (x) = α∈N d 1 α! ∂ α u i (0)x α and G(x, y) = +∞ |β|=0 (-1) |β| β! ∂ β x Γ(x)y β , y in a compact set, |x| → +∞,
where G(x, y) is the fundamental solution to the Laplacian, yields the farfield behavior [4, p. 77]

(u -u i )(x) = |α|,|β|≥1 1 α!β! ∂ α u i (0) ∂D y β (λI -K * D ) -1 [ ∂x α ∂ν ](y)dσ(y) ∂ β G(x, 0)(1.6)
Chapter 1. The Quasi-Static Limit as |x| → +∞. Introduce the generalized polarization tensors [4]:

M αβ (λ, D) := ∂D y β (λI -K * D ) -1 [ ∂x α ∂ν ](y) dσ(y), α, β ∈ N d .
They will be of great use in chapter 8. We call M := M αβ for |α| = |β| = 1 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

u s (x) = u i (x) -∇ y G(x, 0)M (λ, D)∇u i (0) + O δ d+1 dist(λ, σ(K * D ))
.

For a positive integer m, let P m (x) be the complex-valued polynomial

P m (x) = (x 1 + ix 2 ) m := |α|=m a m α x α + i |β|=m b m β x β .
(1.7)

Using polar coordinates x = re iθ , the above coefficients a m α and b m β can also be characterized by

We introduce the contracted generalized polarization tensors to be the following linear combinations of generalized polarization tensors using the coefficients in (1. It is clear that

M cc mn = ∂D ℜ(P n )(λI -K * D ) -1 [ ∂ℜ(P m ) ∂ν ] dσ, M cs mn = ∂D ℑ(P n )(λI -K * D ) -1 [ ∂ℜ(P m ) ∂ν ] dσ, M sc mn = ∂D ℜ(P n )(λI -K * D ) -1 [ ∂ℑ(P m ) ∂ν ] dσ, M ss mn = ∂D ℑ(P n )(λI -K * D ) -1 [ ∂ℑ(P m ) ∂ν ] dσ.
We refer to [START_REF] Ammari | Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory[END_REF] for further details As recently shown [START_REF] Ammari | Generalized polarization tensors for shape description[END_REF][START_REF] Ammari | Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory[END_REF], 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 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.

Plasmonic resonances for the anisotropic problem

In this section, we consider the scattering problem of a time-harmonic wave u i , incident on a plasmonic anisotropic nanoparticle. The homogeneous medium is characterized by its electric permittivity ε m , while the particle occupying a bounded and simply connected domain Ω ⋐ R 3 of class C 1,α 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:

∇ • ε m Idχ(R 3 \ Ω) + Aχ(Ω) ∇u = 0, |u -u i | = O(|x| -2 ), |x| → +∞, (1.9)
where χ denotes the characteristic function and u i is a harmonic function in R 3 . We are interested in finding the plasmonic resonances for problem (1.9). First, introduce the fundamental solution to the operator ∇•A∇ in dimension three

G A (x) = - 1 4π det(A)|A * x| with A * = √ A -1
. From now on, we denote G A (x, y) := G A (x -y). The single-layer potential associated with A is

S A Ω [ϕ] : H -1 2 (∂Ω) -→ H 1 2 (∂Ω) ϕ -→ S A Ω [ϕ](x) = ∂Ω G A (x, y)ϕ(y)dσ(y), x ∈ R 3 .
We can represent the unique solution to (1.9) in the following form [START_REF] Ammari | Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory[END_REF]:

u(x) = u i + S Ω [ψ], x ∈ R 3 \ Ω, S A Ω [φ], x ∈ Ω,
where (ψ, φ) ∈ H -1 2 (∂Ω) 2 is the unique solution to the following system of integral equations on ∂Ω:

     S Ω [ψ] -S A Ω [φ] = -u i , ε m ∂S Ω [ψ] ∂ν + -ν • A∇S A Ω [φ] - = -ε m ∂u i ∂ν .
(1.10) Lemma 1.4.1. The operator S A Ω : H -1 2 (∂Ω) → H

S A Ω = T A * S Ω T -1 A * r -1 ν ,
and in particular S A Ω is invertible and its inverse

(S A Ω ) -1 = r ν T A * S -1 Ω T -1 A * . Note that, for x ∈ ∂Ω, ν( x) = A -1 * ν(x) |A -1 * ν(x)|
, where ν( x) is the outward normal to ∂ Ω at x = A * x. We have

ν • A∇S A Ω ± = ν • A∇ x T A * S Ω T -1 A * r -1 ν ± = ν • AA * T A * ∇ x S Ω T -1 A * r -1 ν ± = |A -1 * ν| ν • T A * ∇ x S Ω T -1 A * r -1 ν ± = ± 1 2 Id + (r ν T A * )K * Ω (r ν T A * ) -1 . (1.11)
The result follows from a change of variables in the expression of the operator

(K A Ω ) * := (r ν T A * )K * Ω (r ν T A * ) -1 . Lemma 1.4.2. S A Ω is negative definite for the duality pairing (•, •) -1 2 , 1 2
and we can define a new inner product

(u, v) H * A = -(u, S A Ω [v]) -1 2 , 1 2 with which H * A (∂Ω), the space induced by (•, •) H * A , is equivalent to H -1 2 (∂Ω). Proof. Let ϕ ∈ H -1 2 (∂Ω). Using Lemma 1.4.1, we have ϕ = ν • A∇S A Ω [ϕ] + -ν • A∇S A Ω [ϕ] - . Thus ∂Ω ϕ(x)S A Ω [ϕ](x)dσ(x) = ∂Ω ν • A∇S A Ω [ϕ] + (x)S A Ω [ϕ](x)dσ(x) - ∂Ω ν • A∇S A Ω [ϕ] - (x)S A Ω [ϕ](x)dσ(x) = - R 3 \ Ω ∇S A Ω [ϕ](x) • A∇S A Ω [ϕ](x)dσ(x) - R 3 \ Ω S A Ω [ϕ](x)∇ • A∇S A Ω [ϕ](x)dσ(x) - Ω ∇S A Ω [ϕ](x) • A∇S A Ω [ϕ](x)dσ(x) + Ω S A Ω [ϕ](x)∇ • A∇S A Ω [ϕ](x)dσ(x) = - R 3 ∇S A Ω [ϕ](x) • A∇S A Ω [ϕ](x)dσ(x) ≤ 0,
where the equality is achieved if and only if ϕ = 0. Here we have used an integration by parts, the fact that

S A Ω [ϕ](x) = O(|x| -1 ) as |x| → ∞, ∇ • A∇S A Ω [ϕ](x) = 0 for x ∈ R 3
\∂Ω and that A is positive-definite. In the same manner, it is known that

ϕ 2 H * = - ∂Ω ϕ(x)S Ω [ϕ](x)dσ(x) = R 3 |∇S Ω [ϕ](x)| 2 dσ(x).
Since A is positive-definite we have

c ϕ 2 H * ≤ - ∂Ω ϕ(x)S A Ω [ϕ](x)dσ(x) ≤ C ϕ 2 H *
for some constants c, C > 0.

Using the fact that H * (∂Ω) is equivalent to H -1 2 (∂Ω), we get the desired result.

From (1.10) we have φ = (S A Ω ) -1 (S Ω [ψ] + u i ), whereas, by Lemma 1.4.1, the following equation holds for ψ:

Q A [ψ] = F (1.12) with Q A = 1 2 ε m Id + (S A Ω ) -1 S Ω + ε m K * Ω -(K A Ω ) * (S A Ω ) -1 S Ω , (1.13)
and

F = -ε m ∂u i ∂ν + ν • A∇S A Ω [(S A Ω ) -1 u i ] - . Propsition 1.4.1. Q A has a countable number of eigenvalues. Proof. It is clear that (K A Ω ) * : H -1 2 (∂Ω) → H -1 2 (∂Ω) is a compact operator. Hence, ε m K * Ω -(K A Ω ) * (S A Ω ) -1 S Ω is compact as well. Therefore, only the invertibility of 1 2 ε m Id + (S A Ω ) -1 S Ω needs to be proven. Since S A Ω is invertible, the invertibility of 1 2 ε m Id + (S A Ω ) -1 S Ω is equivalent to that of ε m S A Ω + S Ω . Consider now, the bilinear form, for (ϕ, ψ) ∈ (H -1 2 (∂Ω)) 2 B(ϕ, ψ) = -ε m ∂Ω ϕ(x)S A Ω [ψ](x)dσ(x) - ∂Ω ϕ(x)S Ω [ψ](x)dσ(x),
where ε m > 0. From Lemma 1.4.2, we have

B(ψ, ψ) ≥ c ψ 2 H -1 2 (∂Ω)
for some constant c > 0.

It follows then, from the Lax-Milgram theorem that ε m S A Ω + S Ω is invertible in H -1 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 Q A is frequency dependent and we should write Q A (ω). We say that ω is a plasmonic resonance if

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

ω = arg max ω Q -1 A (ω) L(H * (∂Ω)) . (1.14)
From now on, we suppose that A is an anisotropic perturbation of an isotropic parameter, i.e., A = ε c (Id + P ), with P being a symmetric matrix and P ≪ 1.

Lemma 1.4.3. Let A = ε c (Id + δR), with R being a symmetric matrix, R = O(1) and δ ≪ 1. Let Tr denote the trace of a matrix. Then, as δ → 0, we have the following asymptotic expansions:

S A Ω = 1 ε c S Ω + δS Ω,1 + o(δ) , (S A Ω ) -1 = ε c S -1 Ω + δB Ω,1 + o(δ) , (K A Ω ) * = K * Ω + δK * Ω,1 + o(δ) with S Ω,1 [ϕ](x) = - 1 2 Tr(R)S Ω [ϕ](x) - 1 2 ∂Ω R(x -y), x -y 4π|x -y| 3 ϕ(y)dσ(y), B Ω,1 = -S -1 Ω S Ω,1 S -1 Ω , K * Ω,1 = - 1 2 Tr(R)K * Ω [ϕ](x) - 3 2 ∂Ω R(x -y), x -y x -y, ν(x) 4π|x -y| 5 ϕ(y)dσ(y).
Proof. Recall that, for δ small enough,

(I + δR) -1 = Id - δ 2 R + O(δ 2 ), det(I + δR) = = 1 + δTr(R) + o(δ), (1 + δx + o(δ)) s = 1 + δsx + o(δ), s ∈ R.
The results follow then from asymptotic expansions of -1 4π det(A)|A * x| β , β = 1, 3 and the identity

(S A Ω ) -1 = ε c (Id + δS -1 Ω S Ω,1 + o(δ)) -1 S -1 Ω .
Plugging the expressions above into the expression of Q A we get the following result. Lemma 1.4.4. As δ → 0, the operator Q A has the following asymptotic expansion

Q A = Q A,0 + δQ A,1 + o(δ),
where

Q A,0 = ε m + ε c 2 Id + (ε m -ε c )K * Ω , Q A,1 = ε c ( 1 2 Id -K * Ω )B Ω,1 S Ω -K * Ω,1 .
We regard the operator Q A as a perturbation of Q A,0 . As in section 3.3, we use the standard perturbation theory to derive the perturbed eigenvalues and eigenvectors in H * (∂Ω).

Let (λ j , ϕ j ) be the eigenvalue and normalized eigenfunction pairs of K * Ω in H * (∂Ω) and τ j the eigenvalues of Q A,0 . We have

τ j = ε m + ε c 2 + (ε m -ε c )λ j .
For simplicity, we consider the case when λ j is a simple eigenvalue of the operator K * Ω . Define

P j,l = (Q A,1 [ϕ j ], ϕ l ) H * .
As δ → 0, the perturbed eigenvalue and eigenfunction have the following form:

τ j (δ) = τ j + δτ j,1 + o(δ), ϕ j (δ) = ϕ j + δϕ j,1 + o(δ), where τ j,1 = P jj , ϕ j,1 = l =j P jl ε m -ε c (λ j -λ l ) ϕ l .

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 ⋐ R 3 of class C 1,α for 0 < α < 1, filled with a composite material that consists of a matrix of constant electric permittivity ε m and a set of periodically distributed plasmonic nanoparticles with (small) period η and electric permittivity ε c . Let Y =] -1/2, 1/2[ 3 be the unit cell and denote δ = η β for β > 0. We set 20

Chapter 1. The Quasi-Static Limit the (re-scaled) periodic function

γ = ε m χ(Y \ D) + ε c χ(D),
where D = δB with B ⋐ R 3 being of class C 1,α and the volume of B, |B|, is assumed to be equal to 1. Thus, the electric permittivity of the composite is given by the periodic function

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

∇ • γ η ∇u η = 0 in Ω, (1.15)
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 = 0 in Ω
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 * = (γ * pq ) p,q=1,2,3 is defined by [START_REF] Ammari | Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory[END_REF] γ

* pq = Y γ(x)∇u p (x) • ∇u q (x)dx,
where u p , for p = 1, 2, 3, is the unique solution to the cell problem

       ∇ • γ∇u p = 0 in Y, u p -x p periodic (in each direction) with period 1, Y u p (x)dx = 0.
(1.16) Using Green's formula, we can rewrite γ * in the following form:

γ * pq = ε m ∂Y u q (x) ∂u p ∂ν (x)dσ(x).
(1.17)

The matrix γ * depends on η as a parameter and cannot be written explicitly.

The following lemmas are from [START_REF] Ammari | Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory[END_REF].

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

u p (x) = x p + C p + S D♯ (λ ε Id -K * D♯ ) -1 [ν p ](x) in Y,
1.5. A Maxwell-Garnett theory for plasmonic nanoparticles

21

where C p is a constant, ν p is the p-component of the outward unit normal to ∂D, λ ε is defined by (3.17), and

S D♯ [ϕ](x) = ∂D G ♯ (x, y)ϕ(y)dσ(y), K * D♯ [ϕ](x) = ∂D ∂G ♯ (x, y) ∂ν(x) ϕ(y)dσ(y)
with G ♯ (x, y) being the periodic Green function defined by

G ♯ (x, y) = - n∈Z 3 \{0} e i2πn•(x-y) 4π 2 |n| 2 .
Lemma 1.5.2. Let S D♯ and K * D♯ be the operators defined as in Lemma 1.5.1. Then the following trace formula holds on ∂D

(± 1 2 Id + K * D♯ )[ϕ] = ∂S D♯ [ϕ] ∂ν ± .
For the sake of simplicity, for p = 1, 2, 3, we set

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

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

γ * pq = ε m δ pq + ∂Y y q ∂S D♯ [φ p ] ∂ν (y)dσ(y) . (1.19)
In view of the periodicity of S D♯ [φ p ], the divergence theorem applied on Y \ D and Lemma 1.5.2 yields (see [START_REF] Ammari | Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory[END_REF])

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

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

γ * = ε m (Id + f P ), (1.20)
where f = |D| = δ 3 (= η 3β ) is the volume fraction of D and P = (P pq ) p,q=1,2,3 is given by

P pq = ∂B y q ψ p (y)dσ(y). (1.21)
To proceed with the computation of P we will need the following Lemma [START_REF] Ammari | Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory[END_REF].

Chapter 1. The Quasi-Static Limit Lemma 1.5.3. There exists a smooth function R(x) in the unit cell Y such that

G ♯ (x, y) = - 1 4π|x -y| + R(x -y).
Moreover, the Taylor expansion of R(x) at 0 is given by

R(x) = R(0) - 1 6 (x 2 1 + x 2 2 + x 2 3 ) + O(|x| 4 ).
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

f ≪ dist(λ ε (ω), σ(K * B )) 3/5 and (Id -δ 3 R -1 λε(ω) T 0 ) -1 = O(1), (1.22)
where R -1 λε(ω) and T 0 are to be defined and dist(λ ε (ω), σ(K * D )) is the distance between λ ε (ω) and the spectrum of K * B . Theorem 1.5.1. Assume that (1.22) holds. Then we have

γ * = ε m Id + f M (Id - f 3 M ) -1 + O f 8/3 dist(λ ε (ω), σ(K * B )) 2 , (1.23)
uniformly in ω. Here, M = M (λ ε (ω), B) is the polarization tensor (3.36) associated with B and λ ε (ω).

Proof. In view of Lemma 1.5.3 and (1.18), we can write, for x ∈ ∂D,

(λ ε (ω)Id -K * D )[φ p ](x) - ∂D ∂R(x -y) ∂ν(x) φ p (y)dσ(y) = ν p (x),
which yields, for x ∈ ∂B,

(λ ε (ω)Id -K * B )[ψ p ](x) -δ 2 ∂B ∂R(δ(x -y)) ∂ν(x) ψ p (y)dσ(y) = ν p (x).
By virtue of Lemma 1.5.3, we get

∇R(δ(x -y)) = - δ 3 (x -y) + O(δ 3 )
uniformly in x, y ∈ ∂B. Since ∂B ψ p (y)dσ(y) = 0, we now have

(R λε(ω) -δ 3 T 0 + δ 5 T 1 )[ψ p ](x) = ν p (x),
and so

(Id -δ 3 R -1 λε(ω) T 0 + δ 5 R -1 λε(ω) T 1 )[ψ p ](x) = R -1 λε(ω) [ν p ](x), (1.24)
where

R λε(ω) [ψ p ](x) = (λ ε (ω)Id -K * B )[ψ p ](x), T 0 [ψ p ](x) = ν(x) 3 • ∂B yψ p (y)dσ(y), T 1 L(H * (∂B)) = O(1
(λ ε (ω)Id -K * B ) -1 L(H * (∂B)) ≤ c dist(λ ε (ω), σ(K * B )) (1.25)
for a constant c.

It is clear that T 0 is a compact operator. From the fact that the imaginary part of R λε(ω) is nonzero, it follows that Id -δ 3 R -1 λε(ω) T 0 is invertible. Under the assumption that

(Id -δ 3 R -1 λε(ω) T 0 ) -1 = O(1), δ 5 ≪ dist(λ ε (ω), σ(K * B )),
we get from (1.24) and (1.25)

ψ p (x) = (Id -δ 3 R -1 λε(ω) T 0 + δ 5 R -1 λε(ω) T 1 ) -1 R -1 λε(ω) [ν p ](x), = (Id -δ 3 R -1 λε(ω) T 0 ) -1 R -1 λε(ω) [ν p ](x) + O δ 5 dist(λ ε (ω), σ(K * B ))
.

Therefore, we obtain the estimate for ψ p

ψ p = O 1 dist(λ ε (ω), σ(K * B ))
. Now, we multiply (1.24) by y q and integrate over ∂B. We can derive from the estimate of ψ p that

P (Id - f 3 M ) = M + O δ 5 dist(λ ε (ω), σ(K * B )) 2 ,
and therefore,

P = M (Id + f 3 M ) -1 + O δ 5 dist(λ ε (ω), σ(K * B )) 2
with P being defined by (1.21). Since f = δ 3 and

M = O δ 3 dist(λ ε (ω), σ(K * B ))
, 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

f M = O(1), which is equivalent to the scale f = O dist(λ ε (ω), σ(K * B )) , the matrix f M (Id -f 3 M ) -1
may have a negative-definite symmetric real part. 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 Chapter 1. The Quasi-Static Limit plasmonic nanoparticles, following (1.14)

ω = arg max ω Q -1 γ * (ω) L(H * (∂Ω)) .

Concluding remarks

In this chapter we have analyzed the plasmonic resonance phenomena assuming 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 K * D associated with D. We remark that this 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 [START_REF] Ammari | Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory[END_REF]. 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.

Chapter 2

The Helmholtz Equation

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 [START_REF] Giannini | Scaling behavior of individual nanoparticle plasmon resonances[END_REF][START_REF] Palomba | Blue-shifted plasmon resonance of individual size-selected gold nanoparticles[END_REF][START_REF] Scaffardi | Size dependence of refractive index of gold nanoparticles[END_REF].

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 [START_REF] Ammari | Surface plasmon resonance of nanoparticles and applications in imaging[END_REF], 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. Moreover, 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 [START_REF] Ammari | Enhancement of nearcloaking. Part II: The Helmholtz equation[END_REF] for scalar wave propagation problems and in [START_REF] Ammari | Enhancement of near cloaking for the full Maxwell equations[END_REF] 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 [START_REF] Ammari | Super-resolution in highly contrasted media from the perspective of scattering coefficients[END_REF] the experimental results reported in [START_REF] Arhab | Nanometric resolution with far-field optical profilometry[END_REF].

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 [START_REF] Ammari | Enhancement of nearcloaking. Part II: The Helmholtz equation[END_REF][START_REF] Ammari | Enhancement of near cloaking for the full Maxwell equations[END_REF]. In section 6.4 we investigate the behavior of the scattering coefficients at the plasmonic resonances.

Preliminaries

Layer potentials for the Helmholtz equation

Let G be the Green function for the Helmholtz operator ∆ + k 2 satisfying the Sommerfeld radiation condition.

The Sommerfeld radiation condition can be expressed in dimension d = 2, 3, as follows:

∂u ∂|x| -ik m u ≤ C|x| -(d+1)/2
as |x| → +∞ for some constant C independent of x.

Preliminaries

In R 3 , G is given by

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

S k D [ϕ](x) = ∂D G(x, y, k)ϕ(y)dσ(y), x ∈ R 3 , (K k D ) * [ϕ](x) = ∂D ∂G(x, y, k) ∂ν(x) ϕ(y)dσ(y), x ∈ ∂D,
Let us recall some well known properties [START_REF] Ammari | Mathematical and Statistical Methods for Multistatic Imaging[END_REF]:

(i) S k D : H -1 2 (∂D) → H 1 2 (∂D), H 1 loc (R 2 \∂D) is bounded; (ii) (∆ + k 2 )S k D [ϕ](x) = 0 for x ∈ R 2 \∂D, ϕ ∈ H -1 2 (∂D); (iii) (K k D ) * : H -1 2 (∂D) → H -1 2 (∂D) is compact; (iv) S k D [ϕ], ϕ ∈ H -1 2 (∂D), satisfies the Sommerfeld radiation condition at infinity; (v) ∂S k D [ϕ] ∂ν ± = (± 1 2 I + (K k D ) * )[ϕ].
We have that, for any ψ, φ ∈ H -1 2 (∂D),

u := u i + S km D [ψ], x ∈ R 2 \D, S kc D [φ], x ∈ D, (2.1)
with

k m = ω √ ε m µ m and k c = ω √ ε c µ c , satisfies ∆u + k 2 m u = 0 in R 2 \ D, ∆u + k 2
c u = 0 in D and u -u i satisfies the Sommerfeld radiation condition. To satisfy the boundary transmission conditions, ψ, φ ∈ H -1 2 (∂D) need to satisfy the following system of integral equations on ∂D

     S km D [ψ] -S kc D [φ] = -u i , 1 εm
1 2 I + (K km D ) * [ψ] + 1 εc 1 2 I -(K kc D ) * [φ] = - 1 ε m ∂u i ∂ν .
(2.

2)

The following result shows the existence of such a representation [START_REF] Ammari | Boundary Layer Techniques for Solving the Helmholtz Equation in the Presence of Small Inhomogeneities[END_REF].

Theorem 2.2.1. The operator

T : H -1 2 (∂D) 2 → H 1 2 (∂D) × H -1 2 (∂D) (ψ, φ) → S km D [ψ] -S kc D [φ], 1 ε m 1 2 I + (K km D ) * [ψ] + 1 ε c 1 2 I -(K kc D ) * [φ]
is invertible.

Chapter 2. The Helmholtz Equation

Layer potential formulation for the scattering problem

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 ε m and its magnetic permeability µ m , while the particle occupying a bounded and simply connected domain D ⋐ R 3 (the two-dimensional case can be treated similarly using results from Appendix B.3) of class C 1,α for some 0 < α < 1 is characterized by electric permittivity ε c and magnetic permeability µ c , both of which may depend on the frequency. Assume that ℜµ c < 0, ℑµ c > 0, ℑε c > 0, and define

k m = ω √ ε m µ m , k c = ω √ ε c µ c , and
ε D = ε m χ(R 3 \ D) + ε c χ( D), µ D = ε m χ(R 3 \ D) + ε c χ(D),
where χ denotes the characteristic function. Let u i (x) = e ikmd•x be the incident wave. Here, ω is the frequency and d is the unit incidence direction. Throughout this chapter, we assume that ε m and µ m are real and strictly positive and that ℑk c > 0.

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 ε m , µ m , ε c , µ c are dimensionless and are of order one. In addition, ℑµ c = o(1). We also assume that ω 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 ε c and µ c 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 µ D ∇u + ω 2 ε D u = 0 in R 3 \∂D, u + -u -= 0 on ∂D, 1 µ m ∂u ∂ν + - 1 µ c ∂u ∂ν - = 0 on ∂D,
u s := u -u i satisfies the Sommerfeld radiation condition.

(2.3)

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 ε D and µ D are interchanged.

Let

F 1 (x) = -u i (x) = -e ikmd•x , F 2 (x) = - 1 µ m ∂u i ∂ν (x) = - i µ m k m e ikmd•x d • ν(x)
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

u(x) = u i + S km D [ψ], x ∈ R 3 \ D, S kc D [φ], x ∈ D, (2.4)
where ψ, φ ∈ H -1 2 (∂D) satisfy the following system of integral equations on ∂D [START_REF] Ammari | Mathematical and Statistical Methods for Multistatic Imaging[END_REF]:

S km D [ψ] -S kc D [φ] = F 1 , 1 µm 1 2 Id + (K km D ) * [ψ] + 1 µc 1 2 Id -(K kc D ) * [φ] = F 2 , (2.5)
where Id denotes the identity operator. In the sequel, we set S 0 D = S D . We are interested in the scattering in the quasi-static regime, i.e., for ω ≪ 1. Note that for ω small enough, S kc D is invertible [START_REF] Ammari | Mathematical and Statistical Methods for Multistatic Imaging[END_REF]. We have φ = (S kc D ) -1 S km D [ψ] -F 1 , whereas the following equation holds for ψ

A D (ω)[ψ] = f, (2.6)
where

A D (ω) = 1 µ m 1 2 Id + (K km D ) * + 1 µ c 1 2 Id -(K kc D ) * (S kc D ) -1 S km D , (2.7)
f = F 2 + 1 µ c 1 2 Id -(K kc D ) * (S kc D ) -1 [F 1 ]. (2.8)
It is clear that

A D (0) = A D,0 = 1 µ m 1 2 Id+K * D + 1 µ c 1 2 Id-K * D = 1 2µ m + 1 2µ c Id- 1 µ c - 1 µ m K * D , (2.9)
where the notation K * D = (K 0 D ) * is used for simplicity. We are interested in finding A D (ω) -1 . We first recall some basic facts about the Neumann-Poincaré operator K * D stated in chaper 1. See also [START_REF] Ammari | Mathematical and Statistical Methods for Multistatic Imaging[END_REF][START_REF] Ando | Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann-Poincaré operator[END_REF][START_REF] Kang | Spectral resolution of the Neumann-Poincaré operator on intersecting disks and analysis of plamson resonance[END_REF][START_REF] Khavinson | Poincaré's variational problem in potential theory[END_REF]. (ii) The operator K * D is self-adjoint in the Hilbert space H -1 2 (∂D) equipped with the following inner product

(u, v) H * = -(u, S D [v]) -1 2 , 1 2 (2.10) Chapter 2. The Helmholtz Equation with (•, •) -1 2 , 1 2
being the duality pairing between H -1 2 (∂D) and H 1 2 (∂D), which is equivalent to the original one;

(iii) Let H * (∂D) be the space H -1 2 (∂D) with the new inner product. Let (λ j , ϕ j ), j = 0, 1, 2, . . . be the eigenvalue and normalized eigenfunction pair of K * D in H * (∂D), then λ j ∈ (-1 2 , 1 2 ] and λ j → 0 as j → ∞;

(iv) The following trace formula holds: for any ψ ∈ H * (∂D), (-

1 2 Id + K * D )[ψ] = ∂S D [ψ] ∂ν - ;
(v) The following representation formula holds: for any ψ ∈ H -1/2 (∂D),

K * D [ψ] = ∞ j=0 λ j (ψ, ϕ j ) H * ⊗ ϕ j .
It is clear that the following result holds.

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

(u, v) H = ((-S D ) -1 [u], v) -1 2 , 1 2 .
(2.11)

Then, S D is an isometry between H * (∂D) and H(∂D).

We now present other useful observations and basic results. The following holds.

Lemma 2.3.3. (i) We have (-

1 2 Id + K * D )S -1 D [1] = 0.
(ii) Let λ 0 = 1 2 . Then the corresponding eigenspace has dimension one and is spanned by the function ϕ 0 = cS -1 D [START_REF] Abboud | Diffraction at a curved gratings: TM and TE cases, Homogenization[END_REF] for some constant c such that ||ϕ 0 || H * = 1.

(iii) Moreover, H * (∂D) = H * 0 (∂D) ⊕ {µϕ 0 , µ ∈ C}, where H * 0 (∂D) is the zero mean subspace of H * (∂D) and ϕ j ∈ H * 0 (∂D) for j ≥ 1, i.e., (ϕ j , 1) -1 2 , 1 2 = 0 for j ≥ 1. Here, {ϕ j } j is the set of normalized eigenfunctions of K * D .

From (2.9), it is easy to see that

A D,0 [ψ] = ∞ j=0 τ j (ψ, ϕ j ) H * ϕ j , (2.12)
where

τ j = 1 2µ m + 1 2µ c - 1 µ c - 1 µ m λ j . (2.13)
From (2.9), it is easy to see that

A D,0 [ψ] = ∞ j=0 τ j (ψ, ϕ j ) H * ϕ j , (2.14)
where

τ j = 1 2µ m + 1 2µ c - 1 µ c - 1 µ m λ j .
(2.15)

We now derive the asymptotic expansion of the operator A(ω) as ω → 0. Using the asymptotic expansions in terms of k of the operators S k D , (S k D ) -1 and (K k D ) * proved in Appendix B.1, we can obtain the following result.

Lemma 2.3.4. As ω → 0, the operator A D (ω) : H * (∂D) → H * (∂D) admits the asymptotic expansion

A D (ω) = A D,0 + ω 2 A D,2 + O(ω 3 ),
where

A D,2 = (ε m -ε c )K D,2 + ε m µ m -ε c µ c µ c ( 1 2 Id -K * D )S -1 D S D,2 .
(2.16)

Proof. Recall that A D (ω) = 1 µ m 1 2 Id + (K km D ) * + 1 µ c 1 2 Id -(K kc D ) * (S kc D ) -1 S km D .
(2.17)

By a straightforward calculation, it follows that

(S kc D ) -1 S km D = Id + ω √ ε c µ c B D,1 S D + √ ε m µ m S -1 D S D,1 + ω 2 ε c µ c B D,2 S D + √ ε c µ c ε m µ m B D,1 S D,1 + ε m µ m S -1 D S D,2 + O(ω 3 ), = Id + ω √ ε m µ m - √ ε c µ c S -1 D S D,1 + ω 2 (ε m µ m -ε c µ c )S -1 D S D,2 + √ ε c µ c ( √ ε c µ c - √ ε m µ m )S -1 D S D,1 S -1 D S D,1 +O(ω 3 ),
where B D,1 and B D,2 are defined by (B.5). Using the facts that

1 2 Id -K * D S -1 D S D,1 = 0 and 1 2 Id -(K k D ) * = 1 2 Id -K * D -k 2 K D,2 + O(k 3 ),
the lemma immediately follows.

We regard A D (ω) as a perturbation to the operator A D,0 for small ω. Using standard perturbation theory [START_REF] Reed | Methods of Modern Mathematical Physics. IV Analysis of Operators[END_REF], we can derive the perturbed eigenvalues and their associated eigenfunctions. For simplicity, we consider the case when λ j is a simple eigenvalue of the operator K * D . We let

R jl = A D,2 [ϕ j ], ϕ l H * , (2.18)
where A D,2 is defined by (2.16).

As ω goes to zero, the perturbed eigenvalue and eigenfunction have the following form:

τ j (ω) = τ j + ω 2 τ j,2 + O(ω 3 ), (2.19)
ϕ j (ω) = ϕ j + ω 2 ϕ j,2 + O(ω 3 ), (2.20)
where

τ j,2 = R jj , (2.21)
ϕ j,2 = l =j R jl 1 µm -1 µc (λ j -λ l ) ϕ l . (2.22)

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 j (ω)| ≪ 1 and is locally minimal for some j.

(ii) We say that ω is a quasi-static plasmonic resonance if |τ j | ≪ 1 and is locally minimized for some j. Here, τ j is defined by (2.15).

(iii) We say that ω is a first-order corrected quasi-static plasmonic resonance if |τ j + ω 2 τ j,2 | ≪ 1 and is locally minimized for some j. Here, the correction term τ j,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 nondimensionalization adopted herein).

We are interested in solving the equation A D (ω)[φ] = f when ω is close to the resonance frequencies, i.e., when τ j (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 ϕ j (ω). We introduce the following definition. Definition 2.2. We call J ⊂ N index set of resonance if τ j 's are close to zero when j ∈ J and are bounded from below when j ∈ J c . More precisely, we choose a threshold number η 0 > 0 independent of ω such that

|τ j | ≥ η 0 > 0 for j ∈ J c .
Remark 2.3.1. Note that for j = 0, we have τ 0 = 1/µ m , 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 j for j ∈ J is a simple eigenvalue of the operator

K * D . Condition 2.3. Let λ = µ m + µ c 2(µ m -µ c ) . ( 2

.23)

We assume that λ = 0 or equivalently, µ c = -µ 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 [START_REF] Bonnet-Ben Dhia | T-coercivity for scalar interface problems between dielectrics and metamaterials[END_REF][START_REF] Costabel | A direct boundary integral equation method for transmission problems[END_REF][START_REF] Nguyen | Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients[END_REF] for detailed discussion on this case. Remark 2.3.2. Note that in the ideal case when ℑµ c = 0, we know that τ j = 0 if λ defined in (2.23) is equal to λ j . This the usual definition in the quasi-static limiting case when the wavelength is infinite. In the case ℑµ c = 0 but ℑµ c = o(1), one may neglect the imaginary part and still use the definition 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 P J (ω) such that

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

P J (ω) = j∈J P j (ω) = j∈J 1 2πi γ j (ξ -A D (ω)) -1 dξ, (2.24)
where γ j is a Jordan curve in the complex plane enclosing only the eigenvalue τ j (ω) among all the eigenvalues.

To obtain an explicit representation of P J (ω), we consider the adjoint operator A D (ω) * . By a similar perturbation argument, we can obtain its perturbed eigenvalue and eigenfunction, which have the following form τ j (ω) = τ j (ω),

(2.25)

ϕ j (ω) = ϕ j + ω 2 ϕ j,2 + o(ω 2 ). (2.26)
Using the eigenfunctions ϕ j (ω), we can show that

P J (ω)[x] = j∈J x, ϕ j (ω) H * ϕ j (ω). (2.27)
Throughout this chapter, for two Banach spaces X and Y , by L(X, Y ) we denote the set of bounded linear operators from X into Y .

We are now ready to solve the equation

A D (ω)[ψ] = f . First, it is clear that ψ = A D (ω) -1 [f ] = j∈J f, ϕ j (ω) H * τ j (ω) + A D (ω) -1 [P J c (ω)[f ]].
(2.28)

The following lemma holds.

Lemma 2.3.5. The norm A D (ω) -1 P J c (ω) L(H * (∂D),H * (∂D)) is uniformly bounded in ω for ω sufficiently small.

Proof. Consider the operator

A D (ω)| J c : P J c (ω)H * (∂D) → P J c (ω)H * (∂D).
Chapter 2. The Helmholtz Equation

For ω small enough, we can show that dist(σ

(A D (ω)| J c ), 0) ≥ η 0 2 , where σ(A D (ω)| J c ) is the discrete spectrum of A D (ω)| J c . Then, it follows that A D (ω) -1 (P J c (ω)f ) = A D (ω)| P J c -1 (P J c (ω)f ) 1 η 0 exp( C 1 η 2 0 ) P J c (ω)f ,
where the notation A B means that A ≤ CB for some constant C. On the other hand,

P J (ω)f = j∈J f, ϕ j (ω) H * ϕ j (ω) = j∈J f, ϕ j + O(ω) H * ϕ j + O(ω) = j∈J f, ϕ j H * ϕ j (ω) + O(ω).
Thus,

P J c (ω) = (Id -P J (ω)) (1 + O(ω)),
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

f 1 = -i √ ε m µ m e ikmd•z 1 µ m [d • ν(x)] + 1 µ c 1 2 Id -K * D S -1 D [d • (x -z)]
and let z be the center of the domain D. In the space H * (∂D), as ω goes to zero, we have

f = ωf 1 + O(ω 2 ),
in the sense that, for ω small enough,

f -ωf 1 H * ≤ Cω 2
for some constant C independent of ω.

Proof. A direct calculation yields Finally, we are ready to state our main result in this section.

f = F 2 + 1 µ c 1 2 Id -(K kc D ) * (S kc D ) -1 [F 1 ] = -ω i µ m √ ε m µ m e ikmd•z [d • ν(x)] + O(ω 2 ) + 1 µ c 1 2 Id -K * D (S D ) -1 + ωB D,1 + O(ω 2 ) [-e ikmd•z 1 + iω √ ε m µ m [d • (x -z)] +O(ω 2 )] = - e ikmd•z µ c 1 2 Id -K * D S -1 D [1] - ωe ikmd•z µ c 1 2 Id -K * D B D,1 [1] - ωi √ ε m µ m e ikmd•z 1 µ m [d • ν(x)] + 1 µ c 1 2 Id -K * D S -1 D [d • (x -z)] + O(ω 2 ) = -ωi √ ε m µ m e ikmd•z 1 µ m [d • ν(x)] + 1 µ c 1 2 Id -K * D S -1 D [d • (x -z)] +O(ω 2 ),
Theorem 2.3.1. Let D has size of order one. Under Conditions 2.1, 2.2, and 2.3 the scattered field u s = u -u i due to a single plasmonic particle D has the following representation:

u s = S km D [ψ],
where

ψ = j∈J ω f 1 , ϕ j (ω) H * ϕ j (ω) τ j (ω) + O(ω), = j∈J ik m e ikmd•z d • ν(x), ϕ j H * ϕ j + O(ω 2 ) λ -λ j + O(ω 2 ) + O(ω)
with λ being given by (2.23).

Proof. We have

ψ = j∈J f, ϕ j (ω) H * ϕ j (ω) τ j (ω) + A D (ω) -1 (P J c (ω)f ), = j∈J ω f 1 , ϕ j H * ϕ j + O(ω 2 ) 1 2µm + 1 2µc -1 µc -1 µm λ j + O(ω 2 )
+ O(ω).

We now compute f 1 , ϕ j H * with f 1 given in Lemma 2.3.6. We only need to show that

1 2 Id -K * D S -1 D [d • (x -z)] , ϕ j H * = (d • ν(x), ϕ j ) H * . (2.29)
Chapter 2. The Helmholtz Equation Indeed, we have

( 1 2 Id -K * D )S -1 D [d • (x -z)], ϕ j H * = -S -1 D [d • (x -z)], 1 2 Id -K D S D [ϕ j ] -1 2 , 1 2 = -S -1 D [d • (x -z)], S D 1 2 Id -K * D [ϕ j ] -1 2 , 1 2 = -d • (x -z), 1 2 Id -K * D [ϕ j ] -1 2 , 1 2 = -d • (x -z), - ∂S D [ϕ j ] ∂ν --1 2 , 1 2 = ∂D ∂[d • (x -z)] ∂ν S D [ϕ j ]dσ - D ∆[d • (x -z)]S D [ϕ j ] -∆S D [ϕ j ][d • (x -z)] dx = -d • ν(x), ϕ j H * ,
where we have used the fact that S D [ϕ j ] is harmonic in D. This proves the 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

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

ψ = j∈J ik m e ikmd•z d • ν(x), ϕ j H * ϕ j + O(ω 2 ) λ -λ j + ω 2 1 µc -1 µm -1 τ j,2 + O(ω).
More generally, under the additional condition that

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

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

µ c (ω) = µ ′ (ω) + iµ ′′ (ω).
(2.31)

Because of causality, the real and imaginary parts of µ c obey the following Kramer-Kronig relations:

µ ′′ (ω) = - 1 π p.v. +∞ -∞ 1 ω -s µ ′ (s)ds, µ ′ (ω) = 1 π p.v. +∞ -∞ 1 ω -s µ ′′ (s)ds, (2.32)
where p.v. stands for the principle value.

The magnetic permeability µ c (ω) can be described by the Drude model; see, for instance, [START_REF] Sarid | Modern Introduction to Surface Plasmons: Theory, Mathematical Modeling, and Applications[END_REF]. We have

µ c (ω) = µ 0 (1 -F ω 2 ω 2 -ω 2 0 + iτ -1 ω ), (2.33)
where τ > 0 is the nanoparticle's bulk electron relaxation rate (τ -1 is the damping coefficient), F is a filling factor, and ω 0 is a localized plasmon resonant frequency. When

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

ℜ µ m + µ c (ω) 2(µ m -µ c (ω)) = λ j
for some j, where λ j is an eigenvalue of the Neumann-Poincaré operator

K * D (= K * B ).
It is clear that such definition is independent of the nanoparticle's size. In view of (8.11), the shifted plasmonic resonance is defined by

argmin 1 2µ m + 1 2µ c (ω) - 1 µ c (ω) - 1 µ m λ j + ω 2 δ 2 τ j,2 ,
where τ j,2 is given by (2.21) with D replaced by B.

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 µ c (ω)/µ m . 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.

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.

Chapter 2. The Helmholtz Equation Propsition 2.4.1. Let u i = e ikmd•x with d being a unit vector. Let x ∈ R 3 be such that |x| ≫ 1/ω. Then, we have

u s (x) = e ikm|x| |x| A ∞ x |x| , d + O 1 |x| 2 (2.34) with A ∞ x |x| , d = - 1 4π ∂D e -ikm x |x| •y ψ(y)dσ(y) (2.35)
being the scattering amplitude and ψ being defined by (2.5).

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

u s (x) = S km D [ψ](x) = - 1 4π ∂D e ikm|x-y| |x -y| ψ(y)dσ(y).
From

|x -y| = |x| 1 - x • y |x| 2 + O( 1 |x| 2 ) , it follows that u s (x) = - e ikm|x| 4π|x| ∂D e -ikm x |x| •y ψ(y) 1 + (x • y) |x| 2 dσ(y) + o 1 |x| 2 ,
which yields the desired result.

Energy flow

The following definitions are from [START_REF] Born | Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light[END_REF]. 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 [START_REF] Born | Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light[END_REF]. We recall that for a real monochromatic field

U (x, t) = ℜ u(x)e -iωt ,
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/µ m while in the transverse magnetic case C = ω/ε m . Assume that the particle is contained in the ball B R of radius R and center the origin. We now consider the outward flow of energy through the sphere ∂B R :

W = ∂B R F (x) • ν(x)dσ(x),
where ν(x) is the outward normal at x ∈ ∂B R .

As the total field can be written as U = u s + u i , the flow can be decomposed into three parts:

W = W i + W s + W ′ ,
where

W i = -iC ∂B R u i (x)∇u i (x) -u i (x)∇u i (x) • ν(x) dσ(x), W s = -iC ∂B R [u s (x)∇u s (x) -u s (x)∇u s (x)] • ν(x) dσ(x), W ′ = -iC ∂B R u i (x)∇u s (x) -u s (x)∇u i (x) -u i (x)∇u s (x) + u s (x)∇u i (x) • ν(x) dσ(x).
It is straightforward to check that W, W i , W a , and W ′ in the above definitions are independent of the radius R as long as the particle is contained in B R . In the case where u i is a plane wave, we can see that W i = 0:

W i = -iC ∂B R u i (x)∇u i (x) -u i (x)∇u i (x) dσ(x), = -iC ∂B R e -ikmd•x ik m de ikmd•x + e ikmd•x k m de -ikmd•x • ν(x) dσ(x), = 2Ck m d • ∂B R ν(x) dσ(x), = 0.
In a non absorbing medium with non absorbing scatterer, W is equal to zero 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

W a = -W.
Therefore, we have

W a + W s = -W ′ .
Here, W ′ is called the extinction rate. It is the rate at which the energy is 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.

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

Denote by U i the quantity U i (x) = u i (x)∇u i (x) -u i (x)∇u i (x) . In the case of a plane wave illumination, U i (x) is independent of x and is given by

U i = 2k m .
Chapter 2. The Helmholtz Equation Definition 2.3. The scattering cross-section Q s , the absorption cross-section Q a and the extinction cross-section are defined by

Q s = W s U i , Q a = W a U i , Q ext = -W ′ U i .
Note that these quantities are independent of x for a plane wave illumination.

Theorem 2.4.1 (Optical theorem). If u i (x) = e ikmd•x , where d is a unit direction, then

Q ext =Q s + Q a = 4π k m ℑ [A ∞ (d, d)] , (2.36)
Q s = S 2 |A ∞ (x, d)| 2 dσ(x) (2.37)
with A ∞ being the scattering amplitude defined by (2.35).

Proof. The Sommerfeld radiation condition gives, for any x ∈ ∂B R ,

∇u s (x) • ν(x) ∼ ik m u s (x). (2.38)
Hence, from (2.34) we get

u s (x)∇u s (x) • ν(x) -u s (x)∇u s (x) • ν(x) ∼ - 2ik m |x| 2 A ∞ x |x| , d 2 ,
which yields (2.37). We now compute the extinction rate. We have

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

u i (x)∇u s (x) • ν(x) -u s (x)∇u i (x) • ν(x) ∼ ik m e ikm(|x|-d•x) |x| d • ν + ik m e ikm(|x|-d•x) |x| A ∞ x |x| , d = ik m e ikm|x|-d•ν(x) |x| (d • ν(x) + 1) A ∞ x |x| , d .
For x ∈ ∂B R , we can write

u i (x)∇u s (x) • ν(x) -u s (x)∇u i (x) • ν(x) ∼ ik m e -ikmRν(x)•(d-ν(x)) R (d • ν(x) + 1) A ∞ x |x| , d .
We now use Jones' lemma (see, for instance, [START_REF] Born | Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light[END_REF]Chapter 13.3]) to write the following asymptotic expansion as

R → ∞ 1 R ∂B R G(ν(x))e -ikmd•ν(x) dσ(x) ∼ 2πi k m G(d)e -ikmR -G(-d)e ikmR , to obtain ∂B R u i (x)∇u s (x) -u s (x)∇u i (x) • ν(x) ∼ -4πA ∞ (d, d) as R → ∞.
Therefore,

W ′ = -i4πC A ∞ (d) -A ∞ (d) = 8πCℑ [A ∞ (d)] .
Since

u i (x)∇u i (x) -u i (x)∇u i (x) = 2k m ,
we get the result.

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

M (λ, D) := ∂D (λId -K * D ) -1 [ν]x dσ(x) (2.40)
be the polarization tensor. The following asymptotic expansion holds. It can be proved by exactly the same arguments as those in [START_REF] Ammari | Surface plasmon resonance of nanoparticles and applications in imaging[END_REF].

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

u s (x) = -k 2 m ε c ε m -1 |D|G(x, z, k m )u i (z) -∇ z G(x, z, k m ) • M (λ, D)∇u i (z) +O δ 4 dist(λ, σ(K * D ))
(2.41) for x away from D. Here, dist(λ, σ(K * D )) denotes min j |λ -λ j | with λ j being the eigenvalues of K * D .

We denote the first term in the right hand side of (2.41) by u s 1 and the second term by u s 2 . It is clear that u s 1 represent monopole radiation and u s 2 the dipole radiation. We explicitly compute the scattering amplitude A ∞ in (2.34). Take u i (x) = e ikmd•x and assume again for simplicity that z = 0. Note that

u s 2 (x) = e ikm|x| 4π|x| ik m ik m x |x| - x |x| 2 • M (λ, D)d.
In the far-field region, i.e. for |x| ≫ 1 ω ,

u s 2 (x) = -k 2 m e ikm|x| 4π|x| x |x| • M (λ, D)d + O 1 |x| 2 .
On the other hand,

u s 1 (x) = k 2 m e ikm|x| 4π|x| ε c ε m -1 • |D|.
Throughout the chapter, we are interested in the case when the frequency is near the plasmonic resonant frequency, then the polarization tensor M (λ, D) blow up and hence the magnitude of the dipole part u s 2 is much Chapter 2. The Helmholtz Equation greater than that of the monopole part u s 1 . Therefore, the leading term in the scattered field (2.41) is given by the dipole part, i.e.

u s (x) ≈ -k 2 m e ikm|x| 4π|x| x |x| • M (λ, D)d . (2.42)
In the next proposition we write the extinction and scattering crosssections 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

Q ext m = 4πk m 3 ℑ [TrM (λ, D)] , (2.43)
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

Q s m = k 4 m 9π |TrM (λ, D)| 2 .
(2.44)

Proof. Remark from (2.42) that the scattering amplitude A ∞ in the case of a plane wave illumination is given by

A ∞ x |x| , d = - k 2 m 4π x |x| • M (λ, D)d. (2.45)
Using Theorem 2.4.1, we can see that for a given orientation

Q ext = -4πk m ℑ [d • M (λ, D)d] .
Therefore, if we integrate Q ext over all illuminations we find that

Q ext m = -k m ℑ S 2 d • M (λ, D)d dσ(d) .
Since ℑM (λ, D) is symmetric, it can be written as ℑM (λ, D) = P t N (λ)P where P is unitary and N is diagonal and real. Then, by the change of variables d = P t x and using spherical coordinates, it follows that

Q ext m = -k m S 2 x • N (λ)xdσ(x) ,
and therefore,

Q ext m = - 4πk m 3 [TrN (λ)] = - 4πk m 3 ℑ [TrM (λ, D)] .
(2.46)

2.4. Scattering and absorption enhancements 43 Now, we compute the averaged scattering cross-section. Let ℜM (λ, D) = P t N (λ) P where P is unitary and N is diagonal and real. We have

Q s m = k 4 m 16π 2 S 2 ×S 2 |x • M (λ, D)d| 2 dσ(x) dσ(d), = k 4 m 16π 2 S 2 ×S 2 x • N (λ) d 2 dσ( x)dσ( d) + S 2 ×S 2 x • N (λ) d 2 dσ( x) dσ( d) .
Then a straightforward computation in spherical coordinates gives

Q s m = k 4 m 9π |TrM (λ, D)| 2 ,
which completes the proof.

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

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

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 averaged extinction cross-section Q ext m of a randomly oriented nanoparticle. Recall that the entries M l,m (λ, D) of the polarization tensor M (λ, D) are given by

M l,m (λ, D) := ∂D x l (λI -K * D ) -1 [ν m ](x) dσ(x).
(2.47)

For a C 1,α domain D in R d , K * D is compact and self-adjoint in H * . Thus, we can write

(λId -K * D ) -1 [ψ] = ∞ j=0 (ψ, ϕ j ) H * ⊗ ϕ j λ -λ j ,
with (λ j , ϕ j ) being the eigenvalues and eigenvectors of K * D in H * (see Lemma 2.3.1). Hence, the entries of the polarization tensor M can be decomposed as

M l,m (λ, D) = ∞ j=1 α (j) l,m λ -λ j , (2.48)
where α

(j) l,m := (ν m , ϕ j ) H * (ϕ j , x l ) -1 2 , 1 2 . Note that (ν m , χ(∂D)) -1 2 , 1 2 = 0. So,
considering the fact that λ 0 = 1/2, we have (ν m , ϕ 0 ) H * = 0 and so, α

l,m = 0. The following lemmas are useful for us.

Lemma 2.4.1. We have

α (j) l,l ≥ 0, j ≥ 1.
Proof. For d = 3, we have

(ϕ j , x l ) -1 2 , 1 2 = 1 2 -λ j -1 1 2 Id -K * D [ϕ j ], x l -1 2 , 1 2 = -1 1/2 -λ j ∂S D [ϕ j ] ∂ν - , x l -1 2 , 1 2 = ∂D ∂x l ∂ν S D [ϕ j ]dσ - D ∆x l S D [ϕ j ] -x l ∆S D [ϕ j ] dx = (ν l , ϕ j ) H * 1/2 -λ j ,
where we used the fact that

S D [ϕ j ] is harmonic in D. The same result holds for d = 2 if we change S D by S D (see Appendix B.3). Since |λ j | < 1/2 for j ≥ 1, we obtain the result. Lemma 2.4.2. Let M l,m (λ, D) = ∞ j=1 α (j) l,m
λ -λ j be the (l, m)-entry of the polarization tensor M associated with a C 1,α domain D ⋐ R d . Then, the following properties hold:

(i) ∞ j=1 α (j) l,m = δ l,m |D|; (ii) ∞ j=1 λ i d l=1 α (j) l,l = (d -2) 2 |D|; (iii) ∞ j=1 λ 2 j d l=1 α (j) l,l = (d -4) 4 |D| + d l=1 D |∇S D [ν l ]| 2 dx.
Proof. The proof can be found in Appendix C.

Let λ = λ ′ + iλ ′′ . We have

ℑ(Tr(M (λ, D))) = ∞ j=1 |λ ′′ | d l=1 α (j) l,l (λ ′ -λ j ) 2 + λ ′′2 . (2.49) For d = 2 the spectrum σ(K * D )\{1/2} is symmetric. For d = 3
this is no longer true. Nevertheless, for our purposes, we can assume that σ(K * D )\{1/2} is symmetric by defining α (j) l,l = 0 if λ j 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 λ ρ(j) = -λ j and we can write

ℑ(Tr(M (λ, D))) = 1 2   ∞ j=1 |λ ′′ |β j (λ ′ -λ j ) 2 + λ ′′2 + ∞ j=1 |λ ′′ |β (ρ(j)) (λ ′ + λ j ) 2 + λ ′′2   = |λ ′′ | 2 ∞ j=1 (λ ′2 + λ ′′2 + λ 2 j )(β (j) + β (ρ(j)) ) + 2λ ′ λ j (β (j) -β (ρ(j)) ) (λ ′ -λ j ) 2 + λ ′′2 (λ ′ + λ j ) 2 + λ ′′2 ,
where

β j = d l=1 α (j) l,l .
From Lemma 2.4.1 it follows that

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

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

ℑ(Tr(M (λ, D))) ≤ |λ ′′ | 2 ∞ j=1 (λ ′2 + λ ′′2 + λ 2 j )(β (j) + β (ρ(j)) ) + 2λ ′ (λ j β (j) + λ ρ(j) β (ρ(j)) ) λ ′′2 (4λ ′2 + λ ′′2 ) + O( λ ′′2 4λ ′2 + λ ′′2 ).
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

C 1,α domain D ⋐ R d with λ = λ ′ + iλ ′′ such that |λ ′′ | ≪ 1 and |λ ′ | < 1/2. Then, ℑ(Tr(M (λ, D))) ≤ d|λ ′′ ||D| λ ′′2 + 4λ ′2 + 1 |λ ′′ |(λ ′′2 + 4λ ′2 ) dλ ′2 |D| + (d -4) 4 |D| + d l=1 D |∇S D [ν l ]| 2 dx + 2λ ′ (d -2) 2 |D| + O( λ ′′2 4λ ′2 + λ ′′2 ).
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 j | < 1 2 , ∞ j=1 λ 2 j d l=1 α (j) l,l < d|D| 4 .
Hence, we can find a geometry independent, but not optimal, bound.

Chapter 2. The Helmholtz Equation Corollary 2.4.1. We have

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

Bound for ellipses

If D is an ellipse whose semi-axes are on the x 1 -and x 2 -axes and of length a and b, respectively, then its polarization tensor takes the form [START_REF] Ammari | Mathematical and Statistical Methods for Multistatic Imaging[END_REF]

M (λ, D) =      |D| λ -1 2 a-b a+b 0 0 |D| λ + 1 2 a-b a+b      . (2.51)
On the other hand, it is known that in H * (∂D) [START_REF] Khavinson | Poincaré's variational problem in potential theory[END_REF] σ

(K * D )\{1/2} = ± 1 2 a -b a + b j , j = 1, 2, . . . .
Then, from (2.48), we also have

M (λ, D) =         ∞ j=1 α (j) 1,1 λ -1 2 a-b a+b j ∞ j=1 α (j) 1,2 λ -1 2 a-b a+b j ∞ j=1 α (j) 1,2 λ -1 2 a-b a+b j ∞ j=1 α (j) 2,2 λ -1 2 a-b a+b j         . Let λ 1 = 1 2 a -b a + b and V(λ j ) = {i ∈ N such that K * D [ϕ i ] = λ j ϕ i }. It is clear now that i∈V(λ 1 ) α (i) 1,1 = i∈V(-λ 1 ) α (i) 2,2 = |D|, i∈V(λ j ) α (i) 1,1 = i∈V(-λ j ) α (i) 2,2 = 0 (2.52) for j ≥ 2 and i∈V(λ j ) α (i) 1,2 = 0 for j ≥ 1.
In view of (2.52), we have

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

|ℑ(Tr(M (λ, D)))| ≤ |λ ′′ | 2 ∞ j=1 4λ ′2 β (j) + λ ′′2 (β (j) + β (j) ) λ ′′2 (4λ ′2 + λ ′′2 ) + O( λ ′′2 4λ ′2 + λ ′′2 ).
Note that for for any ellipse D of semi-axes of length a and b, ℑ(Tr(M (λ, D))) = ℑ(Tr(M (λ, D))). Then using Lemma 2.4.2 we obtain the following result.

Corollary 2.4.2. For any ellipse D of semi-axes of length a and b, we have 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,

|ℑ(Tr(M (λ, D)))| ≤ | D|4λ ′2 |λ ′′ |(λ ′′2 + 4λ ′2 ) + 2|λ ′′ || D| λ ′′2 + 4λ ′2 + O( λ ′′2 4λ ′2 + λ ′′2 ). (2.53)
|ℑ(Tr(M (λ, B)))| is a convex combination of |λ ′′ | (λ ′ -λ j ) 2 +λ ′′2 for λ j ∈ σ(K * B )\{1/2}.
Since ellipses put all the weight of this convex combination in ±λ 1 = ± 1

|ℑ(Tr(M (λ * , B)))| ≤ |ℑ(Tr(M (λ * , D)))| with λ * = ± 1 2 a-b a+b + iλ ′′ .
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 limit) is an ellipse of semi-axis a and b such that

λ ′ (ω * ) = ± 1 2 a-b a+b .
Chapter 2. The Helmholtz Equation

Bound for ellipsoids

Let D be an ellipsoid given by

x 2 1 p 2 1 + x 2 2 p 2 2 + x 2 3 p 2 3 = 1.
(2.54)

The following holds [START_REF] Ammari | Mathematical and Statistical Methods for Multistatic Imaging[END_REF].

Lemma 2.4.3. Let D be the ellipsoid defined by (2.54). Then, for x ∈ D,

S D [ν l ](x) = s l x l , l = 1, 2, 3,
where

s l = - p 1 p 2 p 3 2 ∞ 0 1 (p 2 l + s) (p 2 1 + s)(p 2 2 + s)(p 2 3 + s)
ds.

Then we have

3 l=1 D |∇S D [ν l ]| 2 dx = (s 2 1 + s 2 2 + s 2 3 )|D|.
For a rotated ellipsoid D = RD with R being a rotation matrix, we have

M (λ, D) = RM (λ, D)R T and so Tr(M (λ, D)) = Tr(M (λ, D)).
Therefore, for any ellipsoid D of semi-axes of length p 1 , p 2 and p 3 the following result holds.

Corollary 2.4.3. For any ellipsoid D of semi-axes of length p 1 , p 2 and p 3 , we have

ℑ(Tr(M (λ, D))) ≤ | D| 3λ ′2 + λ ′ -1 4 + (s 2 1 + s 2 2 + s 2 3 ) |λ ′′ |(λ ′′2 + 4λ ′2 ) + 3|λ ′′ || D| λ ′′2 + 4λ ′2 +O( λ ′′2 4λ ′2 + λ ′′2 ), (2.

55)

where for j = 1, 2, 3,

s j = - p 1 p 2 p 3 2 ∞ 0 1 (p 2 j + s) (p 2 1 + s)(p 2 2 + s)(p 2 3 + s)
ds.

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 [START_REF] Ammari | Enhancement of nearcloaking. Part II: The Helmholtz equation[END_REF]. It plays a key role in constructing cloaking structures. It was extended in [START_REF] Ammari | Enhancement of near cloaking for the full Maxwell equations[END_REF] 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

The notion of scattering coefficients

From Graf's addition formula [START_REF] Ammari | Mathematical and Statistical Methods for Multistatic Imaging[END_REF] and (2.4) the following asymptotic formula holds as |x| → ∞

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

u i (x) = m∈Z a m (u i )J m (k m |x|)e imθx , where a m (u i ) = e im( π 2 -θ d )
. By the superposition principle, we get

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

f (m) := F (m) 2 + 1 µ c 1 2 Id -(K kc D ) * (S kc D ) -1 [F (m) 1 ]
with

F (m) 1 (x) = -J m (k m |x|)e imθx , F (m) 2 (x) = - 1 µ m ∂J m (k m |x|)e imθx ∂ν .
We have

u s (x) = (u -u i )(x) = - i 4 n∈Z H (1) n (k m |x|)e inθx m∈Z W nm e im( π 2 -θ d ) ,
where

W nm = ∂D J n (k m |y|)e -inθy ψ m (y)dσ(y). (2.56)
The coefficients W nm are called the scattering coefficients.

Lemma 2.5.1. In the space H * (∂D), as ω goes to zero, we have

f (0) = O(ω 2 ), f (±1) = ωf (±1) 1 + O(ω 2 ), f (m) = O(ω m ), |m| > 1,
where

f (±1) 1 = ∓ √ ε m µ m 2 1 µ m e i±θν + 1 µ c ( 1 2 Id -K * D ) S -1 D [|x|e i±θx ] .
Chapter 2. The Helmholtz Equation

Proof. Recall that J 0 (x) = 1 + O(x 2 ). By virtue of the fact that

1 2 Id -(K kc D ) * (S kc D ) -1 [χ(∂D)] = O(ω 2 ),
we arrive at the estimate for f (0) (see Appendix B.3). Moreover,

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

1 2 Id -(K kc D ) * (S kc D ) -1 = ( 1 2 Id -K * D ) S -1 D + O(ω 2 log ω)
gives the expansion of f (±1) in terms of ω (see Appendix B.3). Finally, J m (x) = O(x m ) immediately yields the desired estimate for f (m) .

From Theorem 3.3.2, we can see that

ψ m = j∈J f (m) , ϕ j (ω) H * ϕ j (ω) τ j (ω) + A D (ω) -1 (P J c (ω)f ). (2.57)
Hence, from the definition of the scattering coefficients,

W nm = j∈J f (m) , ϕ j (ω) H * ϕ j (ω), J n (k m |x|)e -inθx -1 2 , 1 2 τ j (ω) + ∂D J n (k m |y|)e -inθy O(ω)dσ(y).
(2.58) Since

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

Propsition 2.5.1. For |n|, |m| > 0, we have

|W nm | ≤ O(ω |n|+|m| ) min j∈J |τ j (ω)| C |n|+|m| |n| |n| |m| |m|
for a positive constant C independent of ω.

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,

ψ 0 = j∈J O(ω 2 ) τ j (ω) + O(ω), ψ ±1 = j∈J ±ω √ εmµm 2 1 µm -1 µc (e ±iθν , ϕ j ) H * ϕ j + O(ω 3 log ω) τ j (ω) + O(ω).
Proof. The expression of ψ 0 follows from (2.57) and Lemma 2.5.1. Chang-

ing S D by S D in Theorem 3.3.2 gives ( 1 2 Id -K * D ) S -1 D [|x|e iθx ], ϕ j H * = -(e iθν , ϕ j ) H * . Using now Lemma 2.5.1 in (2.57) yields the expression of ψ ±1 .
Recall that in two dimensions,

τ j (ω) = 1 2µ m + 1 2µ c - 1 µ c - 1 µ m λ j + O(ω 2 log ω),
where λ j is an eigenvalue of K * D and λ 0 = 1/2. Recall also that for 0 ∈ J we need τ j → 0 and so µ m → ∞, which is a limiting case that we can ignore. In practice, P J (ω)[ϕ 0 (ω)] = 0. We also have (ϕ j , χ(∂D)) -1 2 , 1 2 = 0 for j = 0. It follows then from the above lemmas and the expression (2.58) of the scattering coefficients that

W 00 = j∈J O(ω 4 log ω) τ j (ω) + O(ω), W 0±1 = j∈J O(ω 3 log ω) τ j (ω) + O(ω), W ±10 = j∈J O(ω 3 ) τ j (ω) + O(ω 2 ).
Note that W ±1±1 has a special structure. Indeed, from Lemma 2.5.2 and equation (2.58), we have

W ±1±1 = j∈J ± ± ω √ εmµm 2 1 µm -1 µc ϕ j , J 1 (k m |x|)e ∓iθx -1 2 , 1 2 e ±iθν , ϕ j H * + O(ω 4 log ω) τ j (ω) +O(ω 2 ), = j∈J ± ± ω 2 εmµm 4 1 µm -1 µc ϕ j , |x|e ∓iθx -1 2 , 1 2 e ±iθν , ϕ j H * + O(ω 4 log ω) τ j (ω) + O(ω 2 ), = k 2 m 4   j∈J ± ± ϕ j , |x|e ∓iθx -1 2 , 1 2 e ±iθν , ϕ j H * + O(ω 2 log ω) λ -λ j + O(ω 2 log ω) + O(1)   ,
where λ is defined by (2.23). Now, assume that min j∈J |τ j (ω)| ≫ ω 2 log ω. Then,

W ±1±1 = k 2 m 4   j∈J ± ± ϕ j , |x|e ∓iθx -1 2 , 1 2 e ±iθν , ϕ j H * λ -λ j + O(1)   . (2.59)
Define the contracted polarization tensors by

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

N +,+ (λ, D) = M 1,1 (λ, D) -M 2,2 (λ, D) + i2M 1,2 (λ, D), N +,-(λ, D) = M 1,1 (λ, D) + M 2,2 (λ, D), N -,+ (λ, D) = M 1,1 (λ, D) + M 2,2 (λ, D), N -,-(λ, D) = M 1,1 (λ, D) -M 2,2 (λ, D) -i2M 1,2 (λ, D),
where M l,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 in (2.35) for the incoming plane wave u i (x) = e ikmd•x . Assume Conditions 1 and 2 and min

j∈J |τ j (ω)| ≫ ω 2 log ω.
Then, A ∞ admits the following asymptotic expansion

A ∞ x |x| = x |x| T W 1 d + O(ω 2 ),
where

W 1 = W -11 + W 1-1 -2W 11 i W 1-1 -W -11 i W 1-1 -W -11 -W -11 -W 1-1 -2W 11 .
Here, W nm are the scattering coefficients defined by (2.56).

Proof. From (2.45), we have

A ∞ x |x| = -k 2 m x |x| T M (λ, D)d.
Since K * D is compact and self-adjoint in H * , we have

N ±,± (λ, D) = ∞ j=1 ϕ j , |x|e ±iθx -1 2 , 1 2 e ±iθν , ϕ j H * λ -λ j = j∈J ϕ j , |x|e ±iθx -1 2 , 1 2 e ±iθν , ϕ j H * λ -λ j + O(1).

Concluding remarks 53

We have then from (2.59) that

- k 2 m 4 N +,+ (λ, D) = W -11 + O(ω 2 ), - k 2 m 4 N +,-(λ, D) = -W 11 + O(ω 2 ), - k 2 m 4 N -,+ (λ, D) = -W 11 + O(ω 2 ), - k 2 m 4 N -,-(λ, D) = W 1-1 + O(ω 2 ).
In view of

M 11 = 1 4 (N +,+ + N -,-+ 2N +,-) , M 22 = 1 4 (-N +,+ -N -,-+ 2N +,-) , M 12 = -i 4 (N +,+ -N -,-) ,
we get the result.

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 [START_REF] Ammari | Mathematical and Statistical Methods for Multistatic Imaging[END_REF][START_REF] Bao | A multi-frequency inverse source problem[END_REF] 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

Introduction

The optical response of plasmon resonant nanoparticles is dominated by the appearance of plasmon resonances over a wide range of wavelengths [START_REF] Link | Shape and size dependence of radiative, nonradiative and photothermal properties of gold nanocrystals[END_REF]. 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 [START_REF] Giannini | Scaling behavior of individual nanoparticle plasmon resonances[END_REF][START_REF] Link | Shape and size dependence of radiative, nonradiative and photothermal properties of gold nanocrystals[END_REF]. It was experimentally observed, for instance, in [START_REF] Giannini | Scaling behavior of individual nanoparticle plasmon resonances[END_REF][START_REF] Scaffardi | Size dependence of refractive index of gold nanoparticles[END_REF] 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 [START_REF] Ammari | Surface plasmon resonance of nanoparticles and applications in imaging[END_REF][START_REF] Grieser | The plasmonic eigenvalue problem[END_REF][START_REF] Mayergoyz | Electrostatic (plasmon) resonances in nanoparticles[END_REF][START_REF] Mayergoyz | Numerical analysis of plasmon resonances in nanoparticules[END_REF]. 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 commonly 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 K * D associated with the particle 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(K * D ). We explain 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 R 3 the spectrum of the Neumann-Poincaré operator has symmetry, that is, if λ ∈ σ(K * D ) so does -λ. For instance, such symmetry is not present in the case of a spherical 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.

Preliminaries

Here and throughout this chapter, we assume that D is bounded, simply connected, and of class C 1,α for 0 < α < 1. We note by ∇× the curl operator for a vector field in R 3 . We denote by H s (∂D) the usual Sobolev space of order s on ∂D and

H s T (∂D) = ϕ ∈ H s (∂D) 3 , ν • ϕ = 0 .
We also need the space H 1 loc (R 3 ) of functions locally in H 1 (R 3 ). We introduce the surface gradient, surface divergence and Laplace-Beltrami operator and denote them by ∇ ∂D , ∇ ∂D • and ∆ ∂D , respectively. We define the vectorial and scalar surface curl by curl

∂D ϕ = -ν × ∇ ∂D ϕ for ϕ ∈ H 1 2 (∂D) and curl ∂D ϕ = -ν • (∇ ∂D × ϕ) for ϕ ∈ H -1 2 T (∂D), respec- tively. We remind that ∇ ∂D • ∇ ∂D = ∆ ∂D , curl ∂D curl ∂D = -∆ ∂D , ∇ ∂D • curl ∂D = 0, curl ∂D ∇ ∂D = 0.
We introduce the following functional space:

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

(∆ + k 2 )G(x, y, k) = δ y ,
where δ y is the Dirac mass at y, subject to the Sommerfeld radiation condition in dimension three

lim |x|→+∞ |x| ∂G ∂|x| -ikG = 0, uniformly in x/|x|.
The Green function G is given by

G(x, y, k) = - e ik|x-y| 4π|x -y| , x = y. (3.1)
Define the following boundary integral operators and refer to [START_REF] Ammari | Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory[END_REF][START_REF] Nédélec | Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems[END_REF] for their mapping properties:

S k D [ϕ] : H -1 2 T (∂D) -→ H 1 2 T (∂D) or H 1 loc (R 3 ) 3 (3.2) ϕ -→ S k D [ϕ](x) = ∂D G(x, y, k)ϕ(y)dσ(y), x ∈ ∂D or x ∈ R 3 ; S k D [ϕ] : H -1 2 (∂D) -→ H 1 2 (∂D) or H 1 loc (R 3 ) (3.3) ϕ -→ S k D [ϕ](x) = ∂D G(x, y, k)ϕ(y)dσ(y), x ∈ ∂D or x ∈ R 3 ; K * D [ϕ] : H -1 2 (∂D) -→ H -1 2 (∂D) (3.4) ϕ -→ K * D [ϕ](x) = ∂D ∂G(x, y, 0) ∂ν(x) ϕ(y)dσ(y), x ∈ ∂D; M k D [ϕ] : H -1 T (div, ∂D) -→ H -1 2 T (div, ∂D) (3.5) ϕ -→ M k D [ϕ](x) = ∂D ν(x) × ∇ x × G(x, y, k)ϕ(y)dσ(y), x ∈ ∂D; L k D [ϕ] : H -1 2 T (div, ∂D) -→ H -1 2 T (div, ∂D) (3.6) ϕ -→ L k D [ϕ](x) = ν(x) × k 2 S k D [ϕ](x) + ∇S k D [∇ ∂D • ϕ](x) , x ∈ ∂D.
Throughout this chapter, we denote

S 0 D , S 0 D , M 0 D by S D , S D , M D , respec- tively. We also denote K D by the (•, •) -1 2 , 1 2 -adjoint of K * D , where (•, •) -1 2 , 1 2
is the duality pairing between H -1 2 (∂D) and H 1 2 (∂D). We recall now some useful results on the operator K * D . See chapter 1 and [START_REF] Ammari | Mathematical and Statistical Methods for Multistatic Imaging[END_REF][START_REF] Ando | Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann-Poincaré operator[END_REF][START_REF] Kang | Spectral resolution of the Neumann-Poincaré operator on intersecting disks and analysis of plamson resonance[END_REF][START_REF] Khavinson | Poincaré's variational problem in potential theory[END_REF]. (ii) The operator K * D is compact self-adjoint in the Hilbert space H -1 2 (∂D) equipped with the following inner product

(u, v) H * = -(u, S D [v]) -1 2 , 1 2 , (3.7)

Preliminaries

59

with which H * (∂D), the space induced by (•, •) H * , is equivalent to H -1 2 (∂D);

(iii) Let (λ j , ϕ j ), j = 0, 1, 2, . . . be the eigenvalue and normalized eigenfunction pair of K * D in H * (∂D). Then, λ j ∈ (-1 2 , 1 2 ], λ j = 1/2 for j ≥ 1, λ j → 0 as j → ∞ and ϕ j ∈ H * 0 (∂D) for j ≥ 1, where H * 0 (∂D) is the zero mean subspace of H * (∂D);

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

K * D [ψ] = ∞ j=0 λ j (ψ, ϕ j ) H * ⊗ ϕ j ;
(v) The following trace formula holds: for any ψ ∈ H * (∂D),

(± 1 2 Id + K * D )[ϕ] = ∂S D [ϕ] ∂ν ± .
(vi) Let H(∂D) be the space H 1 2 (∂D) equipped with the following equivalent inner product

(u, v) H = -(S -1 D [u], v) -1 2 , 1 2 . (3.8)
Then, S D is an isometry between H * (∂D) and H(∂D).

In (vi) in Lemma 3.2.1, we refer to [START_REF] Ammari | Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory[END_REF] for the invertibility of the singlelayer potential S D in three dimensions.

The following result holds.

Lemma 3.2.2. The following Helmholtz decomposition holds [START_REF] Buffa | On traces for H(curl, Ω) in Lipschitz domains[END_REF]: The following results on the operator M D are of great importance. We refer to [START_REF] Nédélec | Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems[END_REF] for a proof of the following compactness property of M D .

H -1 2 T (div, ∂D) = ∇ ∂D H 3 2 (∂D) ⊕ curl ∂D H 1 2 (∂D).
Lemma 3.2.3. The operator M D : H -1 2 T (div, ∂D) -→ H -1 2 T (div, ∂D) is a compact operator. Lemma 3.2.4.
The following identities hold [START_REF] Ammari | Surface plasmon resonance of nanoparticles and applications in imaging[END_REF][START_REF] Griesmaier | An asymptotic factorization method for inverse electromagnetic scattering in layered media[END_REF]:

M D [ curl ∂D ϕ] = curl ∂D K D [ϕ], ∀ϕ ∈ H 1 2 (∂D), M D [∇ ∂D ϕ] = -∇ ∂D ∆ -1 ∂D K * D [∆ ∂D ϕ] + curl ∂D R D [ϕ], ∀ϕ ∈ H 3 2 (∂D),
where

R D = -∆ -1 ∂D curl ∂D M D ∇ ∂D .
(3.9)

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 ε m and magnetic permeability µ m , while the particle occupying a bounded and simply connected domain D ⋐ R 3 of class C 1,α for 0 < α < 1 is characterized by electric permittivity ε c and magnetic permeability µ c , both of which depend on the frequency. Define

k m = ω √ ε m µ m , k c = ω √ ε c µ c , and
ε D = ε m χ(R 3 \ D) + ε c χ(D), µ D = ε m χ(R 3 \ D) + ε c χ(D),
where χ denotes the characteristic function.

For a given incident plane wave (E i , H i ), solution to the Maxwell equations in free space

∇ × E i = iωµ m H i in R 3 , ∇ × H i = -iωε m E i in R 3 ,
the scattering problem can be modeled by the following system of equations

∇ × E = iωµ D H in R 3 \∂D, ∇ × H = -iωε D E in R 3 \∂D, (3.10)
ν × E + -ν × E -= ν × H + -ν × H -= 0 on ∂D,
subject to the Silver-Müller radiation condition:

lim |x|→∞ |x| √ µ m (H -H i )(x) × x |x| - √ ε m (E -E i )(x) = 0
uniformly in x/|x|. Here and throughout the chapter, the subscripts ± indicate, 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 [START_REF] Torres | Maxwell's equations and dielectric obstacles with Lipschitz boundaries[END_REF]

E(x) = E i (x) + µ m ∇ × S km D [ψ](x) + ∇ × ∇ × S km D [φ](x), x ∈ R 3 \ D, µ c ∇ × S kc D [ψ](x) + ∇ × ∇ × S kc D [φ](x), x ∈ D, (3.11) and H(x) = - i ωµ D (∇ × E)(x) x ∈ R 3 \∂D, (3.12)
where the pair (ψ, φ) ∈ H

-1 2 T (div, ∂D) 2 is the unique solution to    µ c + µ m 2 Id + µ c M kc D -µ m M km D L kc D -L km D L kc D -L km D k 2 c 2µ c + k 2 m 2µ m Id + k 2 c µ c M kc D - k 2 m µ m M km D    ψ φ = ν × E i iων × H i ∂D .
(3.13) Let D = z + δB where B contains the origin and |B| = O(1). For any x ∈ ∂D, let x = x-z δ ∈ ∂B and define for each function f defined on ∂D, a corresponding function defined on B as follows

η(f )( x) = f (z + δ x). (3.14)
Throughout this chapter, for two Banach spaces X and Y , by L(X, Y ) we denote the set of bounded linear operators from X into Y . We will also denote by L(X) the set L(X, X).

Lemma 3.3.1. For ϕ ∈ H -1 2
T (div, ∂D), the following asymptotic expansion holds

M k D [ϕ](x) = M B [η(ϕ)](x) + ∞ j=2 δ j M k B,j [η(ϕ)](x),
where

M k B,j [η(ϕ)]( x) = ∂B -(ik) j 4πj! ν(x) × ∇ x × |x -ỹ| j-1 η(ϕ)(ỹ)dσ(ỹ). Moreover, M k B,j L H -1 2
T (div,∂B)
is uniformly bounded with respect to j. In particular, the convergence holds in L H

-1 2 T (div, ∂B) and M k D is analytic in δ.
Proof. We can see, after a change of variables, that

M k D [ϕ](x) = ∂B ν(x) × ∇ x × G(x, ỹ, δk)η(ϕ)(ỹ)dσ(ỹ).
A Taylor expansion of G(x, ỹ, δk) yields

G(x, ỹ, δk) = - ∞ j=0 (iδk|x -ỹ|) j j!4π|x -ỹ| = - 1 4π|x -ỹ| + ∞ j=1 δ j (ik) j 4πj! |x -ỹ| j-1 .
Hence,

M k D [ϕ](x) = M B [η(ϕ)](x) + ∞ j=2 δ j ∂B -(ik) j 4πj! ν(x) × ∇ x × |x -ỹ| j-1 η(ϕ)(ỹ)dσ(ỹ).
Note that from the regularity of |x-

ỹ| j-1 , j ≥ 2, M k B,j [η(ϕ)] H -1 2
T (div,∂B)
is uniformly bounded with respect to j, and therefore,

M k B,j L H -1 2
T (div,∂B)
Chapter 3. The Full Maxwell Equations is uniformly bounded with respect to j as well.

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

(L kc D -L km D )[ϕ](x) = ∞ j=1 δ j ωL B,j [η(ϕ)](x),
where

L B,j [η(ϕ)](x) = C j ν(x) × ∂B |x -ỹ| j-2 η(ϕ)(ỹ)dσ(ỹ) - ∂B |x -ỹ| j-2 (x -ỹ) j + 1 ∇ ∂B • η(ϕ)(ỹ)dσ(ỹ) ,
and

C j = i j (k j+1 c -k j+1 m ) ω4π(j -1)! . Moreover, L B,j L H -1 2
T (div,∂B)
is uniformly bounded with respect to j. In particular, the convergence holds in L H

-1 2 T (div, ∂B) and L k D is analytic in 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:

W B (δ) η(ψ) ωη(φ) =     η(ν × E i ) µ m -µ c η(iν × H i ) ε m -ε c     ∂B , (3.15)
where

W B (δ) =      λ µ Id -M B + δ 2 µ m M km B,2 -µ c M kc B,2 µ m -µ c 1 µ m -µ c (δL B,1 + δ 2 L B,2 ) 1 ε m -ε c (δL B,1 + δ 2 L B,2 ) λ ε Id -M B + δ 2 ε m M km B,2 -ε c M kc B,2 ε m -ε c      + +O(δ 3 ), (3.16)
and the material parameter contrasts λ µ and λ ε are given by

λ µ = µ c + µ m 2(µ m -µ c ) , λ ε = ε c + ε m 2(ε m -ε c ) . (3.17)
It is clear that

W B (0) = W B,0 = λ µ Id -M B 0 0 λ ε Id -M B .
Moreover,

W B (δ) = W B,0 + δW B,1 + δ 2 W B,2 + O(δ 3 ),
3.3. Layer potential formulation for the scattering problem 63 in the sense that

W B (δ) -W B,0 -δW B,1 -δ 2 W B,2 ≤ Cδ 3 for a constant C independent of δ. Here A = sup i,j A i,j H -1 2 T (div,∂B)
for any operator-valued matrix A with entries A i,j .

We are now interested in finding W -1 B (δ). For this purpose, we first consider solving the problem

(λId -M B ) [ψ] = ϕ (3.18) for (ψ, ϕ) ∈ H -1 2 T (div, ∂B) 2 and λ ∈ σ(M B ), where σ(M B ) is the spectrum of M B .
Using the Helmholtz decomposition of H

-1 2
T (div, ∂B) in Lemma 3.2.2, we can reduce (3.18) to an equivalent system of equations involving some well known operators.

Definition 3.1. For u ∈ H -1 2
T (div, ∂B), we denote by u (1) and u (2) any two functions in H 3 2 0 (∂B) and H 1 2 (∂B), respectively, such that u = ∇ ∂B u (1) + curl ∂B u (2) .

Note that u (1) is uniquely defined and u (2) is defined up to a constant function.

Lemma 3.3.3. Assume λ = 1 2 , then problem (3.18) is equivalent to (λId -M B ) ψ (1) ψ (2) = ϕ (1) ϕ (2) , (3.19)
where (ϕ (1) , ϕ (2) ) ∈ H

3 2 0 (∂B) × H 1 2 (∂B)
and

M B = -∆ -1 ∂B K * B ∆ ∂B 0 R B K B .
Here, R B is defined by (3.9) with D replaced with B.

Proof. Let (ψ (1) , ψ (2) ) ∈ H

3 2 0 (∂B) × H 1 2 (
∂B) be a solution (if there is any) to (3.19) where (ϕ (1) , ϕ (2) (2) .

) ∈ H 3 2 0 (∂B) × H 1 2 (∂B) satisfies ϕ = ∇ ∂B ϕ (1) + curl ∂B ϕ
We have

λId + ∆ -1 ∂B K * B ∆ ∂B [ψ (1) ] = ϕ (1) , (3.20) λψ (2) -R B [ψ (1) ] -K B [ψ (2) ] = ϕ (2) . (3.21)
Taking ∇ ∂B in (3.20), curl ∂B in (3.21), adding up and using the identities of Lemma 3.2.4 yields (2) .

(λId -M B ) [∇ ∂B ψ (1) + curl ∂B ψ (2) ] = ∇ ∂B ϕ (1) + curl ∂B ϕ
Chapter 3. The Full Maxwell Equations Therefore ψ = ∇ ∂B ψ (1) + curl ∂B ψ (2) , is a solution of (3.18).

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

) ∈ H 3 2 0 (∂B) × H 1 2 (∂B) such that ψ = ∇ ∂B ψ (1) + curl ∂B ψ
(λId -M B ) [∇ ∂B ψ (1) + curl ∂B ψ (2) ] = ∇ ∂B ϕ (1) + curl ∂B ϕ (2) . (3.22)
Taking ∇ ∂B • in the above equation and using the identities of Lemma 3.2.4 yields (1) ] = ∆ ∂B ϕ (1) .

∆ ∂B λId + ∆ -1 ∂B K * B ∆ ∂B [ψ
Since (ψ (1) , ϕ (1) ) ∈ (H (1) ] = ϕ (1) .

3 2 0 (∂B)) 2 we get λId + ∆ -1 ∂B K * B ∆ ∂B [ψ
Taking curl ∂B in (3.22) and using the identities of Lemma 3.2.4 yields ∆ ∂B (λψ (2) -R B [ψ (1) ] -K B [ψ (2) ]) = ∆ ∂B ϕ (2) .

Therefore, there exists a constant c such that λψ (2) -R B [ψ (1) ] -K B [ψ (2) ] = ϕ (2) + cχ(∂B).

Since K B (χ(∂B)) = 1 2 χ(∂B) we have λ ψ (2) - c λ -1/2 -R B [ψ (1) ] -K B ψ (2) - c λ -1/2 = ϕ (2) .
Hence, ψ (1) , ψ

(2) - c λ -1/2 ∈ H 3 2 0 (∂B)×H 1 2 (∂B) is a solution to (3.19)
Let us now analyze the spectral properties of M B in

H(∂B) := H 3 2 0 (∂B) × H 1 2 (∂B), (3.23)
equipped with the inner product

(u, v) H(∂B) = (∆ ∂B u (1) , ∆ ∂B v (1) ) H * + (u (2) , v (2) ) H , which is equivalent to H 3 2 0 (∂B) × H 1 2 (∂B)
. 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. Recall that K * B and K B are compact and self-adjoint in H * (∂B) and H(∂B), respectively. Since K B is the

(•, •) -1 2 , 1 2 adjoint of K * B , we have σ(K B ) = σ(K * B ), where σ(K B ) (resp. σ(K * B )) is the (discrete) spectrum of K B (resp. K * B ). Define σ 1 = σ(-K * B )\ σ(K B ) ∪ {- 1 2 } , σ 2 = σ(K B )\σ(-K * B ), (3.24)
σ 3 = σ(-K * B ) ∩ σ(K B ).
Let λ j,1 ∈ σ 1 , j = 1, 2 . . . and let ϕ j,1 be an associated normalized eigenfunc-

tion of K * B as defined in Lemma 3.2.1. Note that ϕ j,1 ∈ H -1 2 0 (∂B) for j ≥ 1. Then, ψ j,1 = ∆ -1 ∂B ϕ j,1 (λ j,1 Id -K B ) -1 R B [∆ -1 ∂B ϕ j,1 ] satisfies M B [ψ j,1 ] = λ j,1 ψ j,1 .
Let λ j,2 ∈ σ 2 and let ϕ j,2 be an associated normalized eigenfunction of K B . Then,

ψ j,2 = 0 ϕ j,2 satisfies M B [ψ j,2 ] = λ j,2 ψ j,2 .
Now, assume that Condition 3.1 holds. Let λ j,3 ∈ σ 3 , let ϕ

(1) j,3 be the associated normalized eigenfunction of K * B and let ϕ

(2) j,3 be the associated normalized eigenfunction of K B . Then,

ψ j,3 = 0 ϕ (2) j,3 satisfies M B [ψ j,3 ] = λ j,3 ψ j,3 ,
and λ j,3 has a first-order generalized eigenfunction given by

ψ j,3,g =   c∆ -1 ∂B ϕ (1) j,3 (λ j,3 Id -K B ) -1 P span{ϕ (2) j,3 } ⊥ R B [c∆ -1 ∂B ϕ (1) j,3 ]   (3.25)
for a constant c such that

P span{ϕ (2) j,3 } R B [c∆ -1 ∂B ϕ (1) j,3 ] = -ϕ (2)
j,3 . Here, span{ϕ

j,3 } is the vector space spanned by ϕ

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

M B [ψ j,3,g ] = λ j,3 ψ j,3,g + ψ j,3 .
Consequently, the following result holds.

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

Proof. It is clear that λ-M B is bijective if and only if λ / ∈ σ(-K * B )∪σ(K * B )\ {-1 2 }. It is only left to show that ψ j,1 , ψ j,2 , ψ j,3 , ψ j,3,g , j = 1, 2, . . . form a non- orthogonal basis of H(∂B). Indeed, let ψ = ψ (1) ψ (2) ∈ H(∂B). Since ψ (1) j,1 ∪ ψ (1)
j,3,g , j = 1, 2, . . . form an orthogonal basis of H * 0 (∂B), which is equivalent to H -1 2 0 (∂B), there exist α κ , κ ∈ I 1 := {(j, 1) ∪ (j, 3, g) : j = 1, 2, . . . } such that

ψ (1) = κ∈I 1 α κ ∆ -1 ∂B ψ (1) κ ,
and

κ∈I 1 |α κ | 2 ≤ ∞. It is clear that ψ (2) κ H 1 2 (∂B)
is uniformly bounded with respect to κ ∈ I 1 .

Then

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

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

ψ = κ∈I 1 ∪I 2 α κ ψ κ ,
and

κ∈I 1 ∪I 2 |α κ | 2 ≤ ∞.
To have the compactness of M B , we need the following condition.

Condition 3.2. σ 3 is finite.

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

ψ = κ α(ψ, ψ κ )ψ κ .
The following lemma follows from the Fredholm alternative.

Lemma 3.3.4. Let ψ = ψ (1) ψ (2) ∈ H(∂B).
Then,

α(ψ, ψ κ ) =                      (ψ, ψ κ ) H(∂B) (ψ κ , ψ κ ) H(∂B) , κ = (j, i), i = 1, 2, (ψ, ψ κ ′ ) H(∂B) (ψ κ , ψ κ ′ ) H(∂B) , κ = (j, 3, g), κ ′ = (j, 3), (ψ, ψ κg ) H(∂B) -α(ψ, ψ κg )(ψ κg , ψ κg ) H(∂B) (ψ κ , ψ κg ) H(∂B) , κ = (j, 3), κ g = (j, 3, g),
where

ψ κ ∈ Ker( λκ -M * B ) for κ = (j, i), i = 1, 2, 3; ψ κ ∈ Ker( λκ -M * B ) 2 for κ = (j, 3, g) and M * B is the H(∂B)-adjoint of M B .
The following remark is in order.

Chapter 3. The Full Maxwell Equations Remark 3.3.1. Note that, since ϕ j,1 and ϕ

(1) j,3 form an orthogonal basis of H * 0 (∂B), equivalent to H -1 2 0 (∂B), we also have α(ψ, ψ κ ) = (∆ ∂B ψ (1) , ϕ j,1 ) H * , κ = (j, 1),

1 c (∆ ∂B ψ (1) , ϕ (1)
j,3 ) H * , κ = (j, 3, g),
where c is defined in (3.25).

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

(λId -M B ) -1 [ψ j,i ] = ψ j,i λ -λ j,i , (λId -M B ) -1 [ψ j,3,g ] = ψ j,3,g λ -λ j,3 + ψ j,3 (λ -λ j,3 ) 2 .
Now we turn to the original equation (3.13). The following result holds. Lemma 3.3.5. The system of equations (3.13) is equivalent to (1) η(ψ) (2) ωη(φ) (1) ωη(φ) (2)

W B (δ)     η(ψ)
    =             η(ν × E i ) (1) µ m -µ c η(ν × E i ) (2) µ m -µ c η(iν × H i ) (1) ε m -ε c η(iν × H i ) (2) ε m -ε c             ∂B , (3.26)
where

W B (δ) = W B,0 + δW B,1 + δ 2 W B,2 + O(δ 3 ) with W B,0 = λ µ Id -M B O O λ ε Id -M B , W B,1 =    O 1 µ m -µ c L B,1 1 ε m -ε c L B,1 O    , W B,2 =    1 µ m -µ c M µ B,2 1 µ m -µ c L B,2 1 ε m -ε c L B,2 1 ε m -ε c M ε B,2    ,
and

M B = -∆ -1 ∂B K * B ∆ ∂B 0 R B K B , M µ B,2 = ∆ -1 ∂B ∇ ∂B • (µ m M km B,2 -µ c M kc B,2 )∇ ∂B ∆ -1 ∂B ∇ ∂B • (µ m M km B,2 -µ c M kc B,2 ) curl ∂B -∆ -1 ∂B curl ∂B (µ m M km B,2 -µ c M kc B,2 )∇ ∂B -∆ -1 ∂B curl ∂B (µ m M km B,2 -µ c M kc B,2 ) curl ∂B ,
3.3. Layer potential formulation for the scattering problem 69

M ε B,2 = ∆ -1 ∂B ∇ ∂B • (ε m M km B,2 -ε c M kc B,2 )∇ ∂B ∆ -1 ∂B ∇ ∂B • (ε m M km B,2 -ε c M kc B,2 ) curl ∂B -∆ -1 ∂B curl ∂B (ε m M km B,2 -ε c M kc B,2 )∇ ∂B -∆ -1 ∂B curl ∂B (ε m M km B,2 -ε c M kc B,2 ) curl ∂B , L B,s = ∆ -1 ∂B ∇ ∂B • L B,s ∇ ∂B ∆ -1 ∂B ∇ ∂B • L B,s curl ∂B -∆ -1 ∂B curl ∂B L B,s ∇ ∂B -∆ -1 ∂B curl ∂B L B,s curl ∂B , for s = 1, 2.
Moreover, the eigenfunctions of W B,0 in H(∂B) 2 are given by

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

Ψ 1,j,3,g = ψ j,3,g O , Ψ 2,j,3,g = O ψ j,3,g
, associated to eigenvalues λ µ -λ j,3 and λ ε -λ j,3 , respectively, all of which 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 W B (δ) as a perturbation of the operator W B,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 B,0 and by Γ g = (k, j, 3, g) : k = 1, 2; j = 1, 2, . . . the set of indices for the generalized eigenfunctions. We denote by γ g the generalized eigenfunction index corresponding to eigenfunction index γ and vice-versa. We also denote by

τ γ = λ µ -λ j,i , k = 1, λ ε -λ j,i , k = 2. (3.27) Condition 3.3. λ µ = λ 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 j,i ∈ σ 1 ∪ σ 2 (see (3.24) for the definitions). We call this subset Γ sim . Note that Conditions 3.1 and 3.3 imply that the eigenvalues of W B,0 indexed by γ ∈ Γ sim are simple. Theorem 3.3.1. As δ →, the perturbed eigenvalues and eigenfunctions indexed by γ ∈ Γ sim have the following asymptotic expansions:

τ γ (δ) = τ γ + δτ γ,1 + δ 2 τ γ,2 + O(δ 3 ), (3.28)
Ψ γ (δ) = Ψ γ + δΨ γ,1 + O(δ 2 ),
where

τ γ,1 = (W B,1 Ψ γ , Ψ γ ) H(∂B) 2 (Ψ γ , Ψ γ ) H(∂B) 2 = 0, τ γ,2 = (W B,2 Ψ γ , Ψ γ ) H(∂B) 2 -(W B,1 Ψ γ,1 , Ψ γ ) H(∂B) 2 (Ψ γ , Ψ γ ) H(∂B) 2 , (3.29)
(τ γ -W B,0 )Ψ γ,1 = -W B,1 Ψ γ .
Here,

Ψ γ ′ ∈ Ker(τ γ ′ -W * B,0 ) and W * B,0 is the H(∂B) 2 adjoint of W B,0 .
Using Lemma 3.3.4 and Remark 3.3.2 we can solve Ψ γ,1 . Indeed,

Ψ γ,1 = γ ′ ∈Γ γ ′ =γ α(-W B,1 Ψ γ , Ψ γ ′ )Ψ γ ′ τ γ -τ γ ′ + γ ′ g ∈Γg γ ′ =γ α(-W B,1 Ψ γ , Ψ γ ′ g ) Ψ γ ′ g τ γ -τ γ ′ + Ψ γ ′ (τ γ -τ γ ′ ) 2 + α(-W B,1 Ψ γ , Ψ γ )Ψ γ .
By abuse of notation,

α(x, Ψ γ ) = α(x 1 , ψ κ ) γ = (1, j, i), κ = (j, i), α(x 2 , ψ κ ) γ = (2, j, i), κ = (j, i), (3.30)
for

x = x 1 x 2 ∈ H(∂B) 2
with α being introduced in Definition 3.2.

Consider now the degenerate case γ ∈ Γ\Γ sim =: Γ deg = {γ = (k, i, j) ∈ Γ s.t λ j,i ∈ σ 3 }. It is clear that, for γ ∈ Γ deg , the algebraic multiplicity of the eigenvalue τ γ is 2 while the geometric multiplicity is 1. In this case every eigenvalue τ γ and associated eigenfunction Ψ γ will split into two branches, as δ goes to zero, represented by a convergent Puiseux series as [START_REF] Ammari | Splitting of resonant and scattering frequencies under shape deformation[END_REF]:

τ γ,h (δ) = τ γ + (-1) h δ 1/2 τ γ,1 + (-1) 2h δ 2/2 τ γ,2 + O(δ 3/2 ), h = 0, 1, (3.31)
Ψ γ,h (δ) = Ψ γ + (-1) h δ 1/2 Ψ γ,1 + (-1) 2h δ 2/2 Ψ γ,2 + O(δ 3/2 ), h = 0, 1,
where τ γ,j and Ψ γ,j can be recovered by recurrence formulas. We refer to [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF] for more details.

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

Recall that the electric and magnetic parameters, ε c and µ c , depend on the frequency of the incident field, ω, following the Drude model [START_REF] Ammari | Surface plasmon resonance of nanoparticles and applications in imaging[END_REF]. Therefore, the eigenvalues of the operator W B,0 and perturbation in the eigenvalues depend on the frequency as well, that is,

τ γ (δ, ω) = τ γ (ω) + δ 2 τ γ,2 (ω) + O(δ 3 ), γ ∈ Γ sim , τ γ,h (δ, ω) = τ γ + δ 1/2 (-1) h τ γ,1 (ω) + δ 2/2 (-1) 2h τ γ,2 (ω) + O(δ 3/2 ), γ ∈ Γ deg , h = 0, 1.
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 γ (δ)| ≪ 1
and is locally minimized for some γ ∈ Γ sim or |τ γ,h (δ)| ≪ 1 and is locally minimized for some γ ∈ Γ deg , h = 0, 1.

(ii) We say that ω is a quasi-static plasmonic resonance if |τ γ | ≪ 1 and is locally minimized for some γ ∈ Γ. Here, τ γ is defined by (3.27).

(iii) We say that ω is a first-order corrected quasi-static plasmonic resonance if |τ γ + δ 2 τ γ,2 | ≪ 1 and is locally minimized for some γ ∈ Γ sim or

|τ γ + δ 1/2 (-1) h τ γ,1 | ≪ 1 and is locally minimized for some γ ∈ Γ deg , h = 0, 1.
Here, the correction terms τ γ,2 and τ γ,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)

W B (δ)Ψ = f,
where (1) η(ψ) (2) ωη(φ) (1) ωη(φ

Ψ =     η(ψ)
) (2)     , f =             η(ν × E i ) (1) µ m -µ c η(ν × E i ) (2) µ m -µ c η(iν × H i ) (1) ε m -ε c η(iν × H i ) (2) ε m -ε c             ∂B
for ω close to the resonance frequencies, i.e., when τ γ (δ) is very small for some γ's ∈ Γ sim or τ γ,h (δ) is very small for some γ's ∈ Γ deg , h = 0, 1. In this case, the major part of the solution would be the contributions of the excited resonance modes Ψ γ (δ) and Ψ γ,h (δ).

We introduce the following definition.

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

|τ γ | ≥ η 0 > 0 for γ ∈ J c .
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 µ = 0, λ ε = 0 or equivalently,

µ c = -µ m , ε c = -ε m .
It follows that the set J is finite. Consider the space E J = span{Ψ γ (δ), Ψ γ,h (δ); γ ∈ J, h = 0, 1}. Note that, under Condition 3.4, E J is finite dimensional. Similarly, we define E J c as the spanned by Ψ γ (δ), Ψ γ,h (δ); γ ∈ J c , h = 0, 1 and eventually other vectors to complete the base. We have

H(∂B) 2 = E J ⊕ E J c .
We define P J (δ) and P J c (δ) as the (non-orthogonal) projection into the finite-dimensional space E J and infinite-dimensional space E J c , respectively. It is clear that, for any f ∈ H(∂B)

2 f = P J (δ)[f ] + P J c (δ)[f ].
Moreover, we have an explicit representation for P J (δ)

P J (δ)[f ] = γ∈J∩Γ sim α δ (f, Ψ γ (δ))Ψ γ (δ) + γ∈J∩Γ deg h=0,1 α δ (f, Ψ γ,h (δ))Ψ γ,h (δ).
(3.32) Here, as in Lemma 3.3.4,

α δ (f, Ψ γ (δ)) = (f, Ψγ (δ)) H(∂B) 2 (Ψ γ (δ), Ψγ (δ)) H(∂B) 2 , γ ∈ J ∩ Γ sim , α δ (f, Ψ γ,h (δ)) = (f, Ψγ,h (δ)) H(∂B) 2 (Ψ γ,h (δ), Ψγ,h (δ)) H(∂B) 2 , γ ∈ J ∩ Γ deg , h = 0, 1,
where

Ψ γ ∈ Ker(τ γ,h (δ) -W * B (δ)), Ψ γ,h ∈ Ker(τ γ,h (δ) -W * B (δ)) and W * B (δ) is the H(∂B) 2 -adjoint of W B (δ).
We are now ready to solve the equation W B (δ)Ψ = f . In view of Remark 3.3.2,

Ψ = W -1 B (δ)[f ] = γ∈J∩Γ sim α δ (f, Ψ γ (δ))Ψ γ (δ) τ γ (δ) + γ∈J∩Γ deg h=0,1 α δ (f, Ψ γ,h (δ))Ψ γ,h (δ) τ γ,h (δ) +W -1 B (δ)P J c (δ)[f ].
(3.33) The following lemma holds. A similar result was proved for δ = 0 in [START_REF] Ammari | Super-resolution in highly contrasted media from the perspective of scattering coefficients[END_REF].

Lemma 3.3.6. The norm W -1 B (δ)P J c (δ) L(H(∂B) 2 ,H(∂B) 2
) is uniformly bounded in ω and δ.

Proof. Consider the operator

W B (δ)| J c : P J c (δ)H(∂B) 2 → P J c (δ)H(∂B) 2 .
We can show that for every ω and δ, dist(σ

(W B (δ)| J c ), 0) ≥ η 0 2 , where σ(W B (δ)| J c ) is the discrete spectrum of W B (δ)| J c .
Here and throughout the chapter, dist denotes the distance. Then, it follows that

W -1 B (δ)P J c (δ)[f ] = W -1 B (δ)| J c P J c (δ)[f ] 1 η 0 exp( C 1 η 2 0 ) P J c (δ)[f ] 1 η 0 exp( C 1 η 2 0 ) f ,
where the notation A B means that A ≤ CB for some constant C independent 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 E s = E -E i due to a single plasmonic particle has the following representation:

E s = µ m ∇ × S km D [ψ](x) + ∇ × ∇ × S km D [φ](x) x ∈ R 3 \ D,
where

ψ = η -1 ∇ ∂B ψ (1) + curl ∂B ψ (2) , φ = 1 ω η -1 ∇ ∂B φ (1) + curl ∂B φ (2) , Ψ =      ψ (1) ψ (2) φ (1) φ (2)      = γ∈J∩Γ sim α(f, Ψ γ )Ψ γ + O(δ) τ γ (δ) + γ∈J∩Γ deg ζ 1 (f )Ψ γ + ζ 2 (f )Ψ γ,1 + O(δ 1/2 ) τ γ,0 (δ)τ γ,1 (δ) + O(1),
and

ζ 1 (f ) = (f, Ψγ,1 ) H(∂B) 2 τ γ -(f, Ψγ ) H(∂B) 2 (τ γ,1 + τ γ a 2 a 1 ) a 1 , ζ 2 (f ) = (f, Ψγ ) H(∂B) 2 a 1 , a 1 = (Ψ γ , Ψγ,1 ) H(∂B) 2 + (Ψ γ,1 , Ψγ ) H(∂B) 2 , a 2 = (Ψ γ , Ψγ,2 ) H(∂B) 2 + (Ψ γ,2 , Ψγ ) H(∂B) 2 + (Ψ γ,1 , Ψγ,1 ) H(∂B) 2 .
Proof. Recall that

Ψ = γ∈J∩Γ sim α δ (f, Ψ γ (δ))Ψ γ (δ) τ γ (δ) + γ∈J∩Γ deg h=0,1 α δ (f, Ψ γ,h (δ))Ψ γ,h (δ) τ γ,h (δ) + W -1 B (δ)P J c (δ)[f ].
By Lemma 3.3.6, we have

W -1 B (δ)P J c (δ)[f ] = O(1). If γ ∈ J ∩ Γ sim , an asymptotic expansion on δ yields α δ (f, Ψ γ (δ))Ψ γ (δ) = α(f, Ψ γ )Ψ γ + O(δ). If γ ∈ J ∩Γ deg then (Ψ γ , Ψγ ) H(∂B) 2 = 0.
Therefore, an asymptotic expansion on δ yields

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

a 1 = (Ψ γ , Ψγ,1 ) H(∂B) 2 + (Ψ γ,1 , Ψγ ) H(∂B) 2 , a 2 = (Ψ γ , Ψγ,2 ) H(∂B) 2 + (Ψ γ,2 , Ψγ ) H(∂B) 2 + (Ψ γ,1 , Ψγ,1 ) H(∂B) 2 .
Since τ γ,h (δ) = τ γ + δ 1/2 (-1) h τ γ,1 + O(δ), the result follows by adding the terms

α δ (f, Ψ γ,0 (δ))Ψ γ,0 (δ) τ γ,0 (δ) and α δ (f, Ψ γ,1 (δ))Ψ γ,1 (δ) τ γ,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

min γ∈J∩Γ sim |τ γ (δ)| ≫ δ 3 , min γ∈J∩Γ deg |τ γ (δ)| ≫ δ, (3.34)
we have

Ψ = γ∈J∩Γ sim α(f, Ψ γ )Ψ γ + O(δ) τ γ + δ 2 τ γ,2 + γ∈J∩Γ deg ζ 1 (f )Ψ γ + ζ 2 (f )Ψ γ,1 + O(δ 1/2 ) τ 2 γ -δτ 2 γ,1
+ O(1).

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

min γ∈J∩Γ sim |τ γ (δ)| ≫ δ 2 , min γ∈J∩Γ deg |τ γ (δ)| ≫ δ 1/2 , (3.35)
we have

Ψ = γ∈J∩Γ sim α(f, Ψ γ )Ψ γ + O(δ) τ γ + γ∈J∩Γ deg α(f, Ψ γ )Ψ γ τ γ + α(f, Ψ γ,g ) Ψ γ,g τ γ + Ψ γ τ 2 γ + O(1).
Proof. We have

lim δ→0 W -1 B (δ)P span{Ψ γ,0 (δ),Ψ γ,1 (δ)} [f ] = lim δ→0 α δ (f, Ψ γ,0 (δ))Ψ γ,0 (δ) τ γ,0 (δ) + α δ (f, Ψ γ,1 (δ))Ψ γ,1 (δ) τ γ,1 (δ) = W -1 B,0 (δ)P span{Ψγ ,Ψγ g } [f ] = α(f, Ψ γ )Ψ γ τ γ + α(f, Ψ γ,g ) Ψ γ,g τ γ + Ψ γ τ 2 γ ,
where γ ∈ J ∩ Γ deg , f ∈ H(∂B) 2 and )P spanE is the projection into the linear space generated by the elements in the set E.

Remark 3.3.3. Note that for γ ∈ J,

τ γ ≈ min dist λ µ , σ(K * B ) ∪ -σ(K * B ) , dist λ ε , σ(K * B ) ∪ -σ(K * B ) .
It is clear, from Remark 3.3.3, that resonances can occur when exciting the spectrum of K * B or/and that of -K * B . We substantiate in the following that only the spectrum of K * B can be excited to create the plasmonic resonances in the quasi-static regime.

Recall that

f =             η(ν × E i ) (1) µ m -µ c η(ν × E i ) (2) µ m -µ c η(iν × H i ) (1) ε m -ε c η(iν × H i ) (2) ε m -ε c             ∂B ,
and therefore,

f 1 := η(ν × E i ) (1) µ m -µ c = ∆ -1 ∂B ∇ ∂B • η(ν × E i ) µ m -µ c . Now, suppose γ = (1, j, 1) ∈ J (recall that J is the index set of resonances). Then τ γ = λ µ -λ 1,j , where λ 1,j ∈ σ 1 = σ(-K * B )\σ(K * B ). From Remark 3.3.1, α(f, Ψ γ ) = (∆ ∂B f 1 , ϕ j,1 ) H * = α(f, Ψ γ ) = 1 µ m -µ c (∇ ∂B • η(ν × E i ), ϕ j,1 ) H * , where ϕ j,1 ∈ H * 0 (∂B) is a normalized eigenfunction of K * B (∂B). A Taylor expansion of E i gives, for x ∈ ∂D, E i (x) = ∞ β∈N 3 (x -z) β ∂ β E i (z) |β|! .
Thus,

η(ν × E i )(x) = η(ν)(x) × E i (z) + O(δ),
and

∇ ∂B • η(ν × E i )(x) = -η(ν)(x) • ∇ × E i (z) + O(δ) = O(δ).
Therefore, the zeroth-order term of the expansion of

∇ ∂B • η(ν × E i ) in δ is zero. Hence, α(f, Ψ γ ) = 0.
In the same way, we have

α(f, Ψ γ ) = 0, α(f, Ψ γg ) = 0
for γ = (2, j, 1) ∈ J and γ g such that γ ∈ J.

As a result we see that the spectrum of -K * B is not excited in the zerothorder term. However, we note that σ(-K * B ) can be excited in higher-order terms.

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 [START_REF] Ammari | Surface plasmon resonance of nanoparticles and applications in imaging[END_REF], 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 [START_REF] Ammari | Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory[END_REF] M (λ, D)

= ∂D (λId -K * D ) -1 [ν](x)x dσ(x), (3.36)
where λ ∈ C\(-1/2, 1/2). In view of Lemma 3.2.1, we have

M (λ, D) = ∞ j=1 1 λ -λ j (ν, ϕ j ) H * (ϕ j , x) -1 2 , 1 2 , (3.37)
since (ν, ϕ 0 ) H * = 0;

The following result follows from [START_REF] Ammari | Surface plasmon resonance of nanoparticles and applications in imaging[END_REF].

Theorem 3.4.1. Let d σ = min dist λ µ , σ(K * D ) ∪ -σ(K * D ) , dist λ ε , σ(K * D ) ∪ -σ(K * D ) . Then, for D = z +δB ⋐ R 3 of class C 1,
α for 0 < α < 1, the following uniform far-field expansion holds

E s = - iωµ m ε m ∇ × G d (x, z, k m )M (λ µ , D)H i (z) -ω 2 µ m G d (x, z, k m )M (λ ε , D)E i (z) + O( δ 4 d σ ),
where

G d (x, z, k m ) = ε m G(x, z, k m )Id + 1 k 2 m D 2 x G(x, z, k m )
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 [START_REF] Nédélec | Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems[END_REF]. It is well-known that the solution (E, H) to the system (3.10) has the following far-field expansion as |x| → +∞:

E s (x) = - e ikm|x| 4π|x| A ∞ (x) + O 1 |x| 2 ,
and

H s (x) = - e ikm|x| 4π|x| x × A ∞ (x) + O 1 |x| 2 ,
where

A ∞ (x) = -iµ m k m x × ∂D e -ikm x•y ψ(y)dσ(y) -k 2 m x × x × ∂D e -ikm x•y φ(y)dσ(y),
3.5. Explicit computations for a spherical nanoparticle 77 and x = x |x| . We also need the optical cross-section theorem for the scattering of electromagnetic waves [START_REF] Born | Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light[END_REF], which can be stated as follows. Assume that the incident fields are plane waves given by

E i (x) = pe ikmd•x , H i (x) = d × pe ikmd•x ,
where p ∈ R 3 and d ∈ R 3 with |d| = 1 are such that p • d = 0. Then, the extinction cross-section is given by

Q ext = 4π k m ℑ p • A ∞ (d) |p| 2 ,
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:

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

A ∞ (x) = ωµ m k m e ikm(d-x)•z x×Id M (λ µ , D)(d×p)-k 2 m e ikm(d-x)•z Id-xx t M (λ ε , D)p.
(3.38) Formula (3.38) allows us to compute the extinction cross-section Q ext in terms of the polarization tensors associated with the particle D and the material parameter contrasts. Moreover, an estimate for the blow up of Q ext at the plasmonic resonances follows immediately from (3.37). Theorem 3.4.2. We have

Q ext = 4π k m |p| 2 ℑ p • ωµ m k m d × Id M (λ µ , D)(d × p) -k 2 m Id -dd t M (λ ε , D)p ,
where M (λ µ , D) and M (λ ε , D) are the polarization tensors associated with D and λ = λ µ and λ = λ ε , respectively.

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. Similarly, let v n,m (x) be defined by

Vector spherical harmonics

v n,m (x) = j n (k|x|)Y m n (x),
where j n is the spherical Bessel function of the first kind. Then the function v n,m satisfies the Helmholtz equation in R 3 . Next, define the vector spherical harmonics by

U n,m = 1 n(n + 1) ∇ S Y m n (x) and V n,m = x × U n,m
for m = -n, ..., n and n = 1, 2, .... Here, x ∈ S and ∇ S denote the surface gradient on the unit sphere S. The vector spherical harmonics form a complete orthogonal basis for L 2 T (S). Using the vectorial spherical harmonics, we can separate the solutions of Maxwell's equations into multipole solutions; see [START_REF] Nédélec | Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems[END_REF]Section 5.3]. Define the exterior transverse electric multipoles, i.e., E • x = 0, as

   E T E n,m (x) = -n(n + 1)h (1) n (k|x|)V n,m (x), H T E n,m (x) = - i ωµ ∇× -n(n + 1)h (1) n (k|x|)V n,m (x) , (3.39)
and the exterior transverse magnetic multipoles, i.e., H

• x = 0, as      E T M n,m (x) = i ωǫ ∇× -n(n + 1)h (1) n (k|x|)V n,m (x) , H T M n,m (x) = -n(n + 1)h (1) n (k|x|)V n,m (x).
(3.40)

The exterior electric and magnetic multipoles satisfy the Sommerfeld radiation condition. In the same manner, one defines the interior multipoles

( E T E n,m , H T E n,m ) and ( E T M n,m , H T M n,m ) with h (1)
n replaced by j n , i.e.,

     E T E n,m (x) = -n(n + 1)j n (k|x|)V n,m (x), H T E n,m (x) = - i ωµ ∇ × E T E n,m (x), (3.41)
and

   H T M n,m (x) = -n(n + 1)j n (k|x|)V n,m (x), E T M n,m (x) = i ωǫ ∇ × H T M n,m (x).
∇ × E T E n,m (k; x) = n(n + 1) |x| H n (k|x|)U n,m (x) + n(n + 1) |x| h (1) n (k|x|)Y m n (x)x (3.43) and ∇ × E T E n,m (k; x) = n(n + 1) |x| J n (k|x|)U n,m (x) + n(n + 1) |x| j n (k|x|)Y m n (x)x, (3.44
) where

J n (t) = j n (t) + tj ′ n (t), H n (t) = h (1) n (t) + t(h (1) n ) ′ (t).
For |x| > |y|, the following addition formula holds:

G(x, y, k)I = - ∞ n=1 ik n(n + 1) ǫ µ n m=-n E T M n,m (x) E T M n,m (y) T - ∞ n=1 ik n(n + 1) n m=-n E T E n,m (x) E T E n,m (y) T - i k ∞ n=1 n m=-n ∇v n,m (x)∇ v n,m (y) T . (3.45)
Alternatively, for |x| < |y|, we have

G(x, y, k)I = - ∞ n=1 ik n(n + 1) ǫ µ n m=-n E T M n,m (x)E T M n,m (y) T - ∞ n=1 ik n(n + 1) n m=-n E T E n,m (x)E T E n,m (y) T - i k ∞ n=1 n m=-n ∇ v n,m (x)∇v n,m (y) T . (3.46)

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

ν × ∇ × S k D [U n,m ] + |x|=r ′ = (-ikr)h (1) n (kr ′ )J n (kr)U n,m , (3.47)
ν × ∇ × S k D [V n,m ] + |x|=r ′ = ik r 2 r ′ j n (kr)H n (kr ′ )V n,m , (3.48)
ν × ∇ × ∇ × S k D [U n,m ] + |x|=r ′ = -ik r r ′ J n (kr)H n (kr ′ )V n,m , (3.49)
ν × ∇ × ∇ × S k D [V n,m ] + |x|=r ′ = ik(kr) 2 j n (kr)h (1) n (kr ′ )U n,m . (3.50)
Chapter 3. The Full Maxwell Equations

For r ′ < r, ν × ∇ × S k D [U n,m ] + |x|=r ′ = (-ikr)j n (kr ′ )H n (kr)U n,m , (3.51)
ν × ∇ × S k D [V n,m ] + |x|=r ′ = ik r 2 r ′ J n (kr ′ )h (1) n (kr)V n,m , (3.52)
ν × ∇ × ∇ × S k D [U n,m ] + |x|=r ′ = -ik r r ′ J n (kr ′ )H n (kr)V n,m , (3.53)
ν × ∇ × ∇ × S k D [V n,m ] + |x|=r ′ = ik(kr) 2 j n (kr ′ )h (1) n (kr)U n,m . (3.54)
Proof. We only consider (3.47). The other formulas can be proved in a similar way. From (3.43), (3.44), and the definitions of

E T E n,m , E T M n,m , E T E n,m and E T M n,m , we have ∇ x × G(x, y, k)U n,m (ŷ) = - ∞ n=1 ik n(n + 1) ǫ µ n m=-n ∇ × E T M n,m (x) E T M n,m (y) • U p,q (ŷ) + ∞ n=1 ik n(n + 1) n m=-n ∇ × E T E n,m (x) E T E n,m (y) • U p,q (ŷ) = - ∞ n=1 ik n(n + 1) ǫ µ n m=-n ∇ × E T M n,m (x) -i ωε 1 r J n (kr)U n,m (ŷ) • U p,q (ŷ) + ∞ n=1 ik n(n + 1) n m=-n ∇ × E T E n,m (x)(-1)j n (kr)V n,m (ŷ) • U p,q (ŷ)
for |y| = r and |x| > |y|. Therefore, we get on |x| = r

∇ × S k D [U n,m ] + = ∇ x × |y|=r G(x, y, k)U n,m (ŷ) = kr n(n + 1) 1 ωµ J n (kr)(∇ × E T M n,m (x))| |x|=r . (3.55) Since ∇ × E T M p,q = i ωε ∇ × ∇ × E T E p,q = i ωε k 2 E T E p,q ,
we obtain

x × ∇ × S k D [U n,m ] + = ikr n(n + 1) J n (kr)(x × E T E n,m (x))| |x|=r = (-ikr)h (1) n (kr)J n (kr)U n,m on |x| = r,
which completes the proof. Note that

ν × ∇ × S k D [φ] ± = (∓ 1 2 I + M k D )[φ] on ∂D,
and recall the following identity, which was proved in [START_REF] Torres | Maxwell's equations and dielectric obstacles with Lipschitz boundaries[END_REF],

ν × ∇ × ∇ × S k D [φ] = L k D [φ] on ∂D.
M k D =    1 2 -ikrh (1) n (kr)J n (kr) 0 0 1 2 + ikrj n (kr)H n (kr)    , (3.56)
and

L k D = 0 ik(kr) 2 j n (kr)h (1)
n (kr) -ikJ n (kr)H n (kr) 0 .

(3.57)

Asymptotic behavior of the spectrum of W B (r)

Now we consider the asymptotic expansions of the operator W B (r) and its spectrum when r ≪ 1.

It is well-known that, as t → 0,

j n (t) = t n (2n + 1)!! 1 - 1 2(2n + 3) t 2 + O(t 4 ) , h (1)
n (t) = -i((2n -1)!!)t -n-1 1 + 1 2(2n -1) t 2 + O(t 4 ) . (3.58)
By making use of these asymptotics of the spherical Bessel functions, we obtain that

iJ n (t)h (1) n ( t) = n + 1 2n + 1 t t n 1 t + n + 1 2(2n -1)(2n + 1) t t n t - n + 3 2(2n + 1)(2n + 3) t t n+1 t + O(t 3 ), ij n (t)H n ( t) = -n 2n + 1 t t n 1 t + -n + 2 2(2n -1)(2n + 1) t t n t + n 2(2n + 1)(2n + 3) t t n+1 t + O(t 3 ), ij n (t)h (1) n ( t) = 1 2n + 1 t t n 1 t + 1 2(2n -1)(2n + 1) t t n t - 1 2(2n + 1)(2n + 3) t t n+1 t + O(t 3 ), iJ n (t)H n ( t) = (-n)(n + 1) 2n + 1 t t n 1 t + (n + 1)(-n + 2) 2(2n -1)(2n + 1) t t n t + n(n + 3) 2(2n + 1)(2n + 3) t t n+1 t + O(t 3 ), (3.59)
for small t, t ≪ 1 with t ≈ t. So, we have

M k D =    (-1) 2(2n + 1) + (kr) 2 r n 0 0 1 2(2n + 1) + (kr) 2 s n    + O(r 4 ), (3.60)
and

L k D =   0 k 2 rp n n(n + 1) 2n + 1 1 r + k 2 rq n 0   + O(r 3 ), (3.61)
where

p n = 1 2n + 1 , q n = (n + 1)(n -2) 2(2n -1)(2n + 1) - n(n + 3) 2(2n + 1)(2n + 3) , r n = - n + 1 2(2n -1)(2n + 1) + (n + 3) 2(2n + 1)(2n + 3) , s n = - n -2 2(2n -1)(2n + 1) + n 2(2n + 1)(2n + 3) . (3.62)
Therefore, we can obtain

W B (r) = W B,0 + rW B,1 + r 2 W B,2 + O(r 3 ),
where

W B,0 =            λ µ - (-1) 2(2n + 1) 0 0 0 0 λ µ - 1 2(2n + 1) 0 0 0 0 λ ε - (-1) 2(2n + 1) 0 0 0 0 λ ε - 1 2(2n + 1)            , (3.63) W B,1 =     0 0 0 ωC µ p n 0 0 ωC µ q n 0 0 ωC ε p n 0 0 ωC ε q n 0 0 0     , (3.64)
W B,2 =     ω 2 D µ r n 0 0 0 0 ω 2 D µ s n 0 0 0 0 ω 2 D ε r n 0 0 0 0 ω 2 D ε s n     , (3.65)
and

C µ = µ c ε c -µ m ε m µ m -µ c , C ε = µ c ε c -µ m ε m ε m -ε c , (3.66)
D µ = ε c µ 2 c -ε m µ 2 m µ m -µ c , D ε = ε 2 c µ c -ε 2 m µ m ε m -ε c . (3.67)
By applying the standard perturbation theory, the asymptotics of eigenvalues of W B (r) are obtained as follows: up to an error term of order O(r 3 ),

λ µ - (-1) 2(2n + 1) + (rω) 2 C ε C µ p n q n λ µ -λ ε + p n + D µ r n + O(r 3 ), λ µ - 1 2(2n + 1) + (rω) 2 C ε C µ p n q n λ µ -λ ε -p n + D µ s n + O(r 3 ), λ ε - (-1) 2(2n + 1) + (rω) 2 C ε C µ p n q n λ ε -λ µ + p n + D ε r n + O(r 3 ), λ ε - 1 2(2n + 1) + (rω) 2 C ε C µ p n q n λ ε -λ µ -p n + D ε s n + O(r 3 ),
and the asymptotics of the associated eigenfunction are given by

[1, 0, 0, 0] T + rω C ε q n λ µ -λ ε + p n [0, 0, 0, 1] T + O(r 2 ), [0, 1, 0, 0] T + rω C ε 2n + 1 1 λ µ -λ ε -p n [0, 0, 1, 0] T + O(r 2 ), [0, 0, 1, 0] T + rω C µ q n λ ε -λ µ + p n [0, 1, 0, 0] T + O(r 2 ), [0, 0, 0, 1] T + rω C µ 2n + 1 1 λ ε -λ µ -p n [1, 0, 0, 0] T + O(r 2 ).

Extinction cross-section

In this subsection, we compute the extinction cross-section Q ext . We need the following lemma.

Lemma 3.5.2. Let D be a sphere with radius r > 0 and suppose that E i is given by

E i (x) = ∞ n=1 n l=-n α T E nl E T E n,l (x; k m ) + α T M nl E T M n,l (x; k m ),
for some coefficients α T E nl , α T M nl . Then the scattered wave can be represented as follows: for |x| > r,

E s (x) = ∞ n=1 n l=-n α T E nl S T E n E T E n,l (x; k m ) + α T M nl S T M n E T M n,l (x; k m ),
where S T E n and S T M n are given by

S T E n = µ c j n (k c r)J n (k m r) -µ m j n (k m r)J n (k c r) µ m J n (k c r)h (1)
n (k m r) -µ c j n (k c r)H(k m r) , S T M n = ε c j n (k c r)J n (k m r) -ε m j n (k m r)J n (k c r) ε m J n (k c r)h (1)
n (k m r) -ε c j n (k c r)H(k m r)
.

Chapter 3. The Full Maxwell Equations

Proof. Let E i = E T E n,l (x; k m ).
We look for a solution of the following form:

E = a E T E n,l (x; k c ), |x| < r E T E n,l (x; k m ) + b E T E n,l (x; k m ), |x| > r.
Then, from the boundary condition on ∂D, we easily see that

j n (k m r) 1 µm J n (k m r) = j n (k c r) -h (1)
n (k m r)

1 µc J n (k c r) -1 µm H n (k m r) a b . (3.68)
Therefore, the coefficient a and b can be obtained as follows:

1/a b/a = j n (k m r) h (1)
n (k m r)

1 µm J n (k m r) 1 µm H n (k m r) -1 j n (k c r) h (1)
n (k c r)

1 µc J n (k c r) 1 µc H n (k c r) 1 0 , = µ m k m r i 1 µm H n (k m r) -h (1)
n (k m r) -1 µm J n (k m r) j n (k m r) j n (k c r) 1 µc J n (k c r) , = -ik m r    H n (k m r)j n (k c r) - µ m µ c h (1) n (k m r)J n (k c r) -J n (k m r)j n (k c r) + µ m µ c j n (k m r)J n (k c r)    , (3.69)
where we have used the following Wronskian identity for the spherical Bessel function:

j n (t)H n (t) -h (1) n (t)J n (t) = t j n (t)(h (1) n ) ′ (t) -j ′ n (t)h (1) n (t) = i t .
Therefore, we immediately see that

b = µ c j n (k c r)J n (k m r) -µ m j n (k m r)J n (k c r) µ m J n (k c r)h (1)
n (k m r) -µ c j n (k c r)H(k m r) . Now suppose that E i = E T M n,l (x; k m ).
We look for a solution in the following form:

E = c E T M n,l (x; k c ), |x| < r, E T M n,l (x; k m ) + d E T M n,l (x; k m ), |x| > r.
Then, from the boundary conditions on |x| = r, we obtain

1 εc J n (k c r) 1 εc H n (k c r) j n (k c r) h (1) n (k c r) c 0 = 1 εm J n (k m r) 1 εm H n (k m r) j n (k m r) h (1)
n (k m r)

1 d .
(3.70)

By solving (3.70), we get

d = ε c j n (k c r)J n (k m r) -ε m j n (k m r)J n (k c r) ε m J n (k c r)h (1)
n (k m r) -ε c j n (k c r)H(k m r)
.

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 E s is given by

E s (x) = ∞ n=1 n l=-n β T E nl E T E n,l (x; k m ) + β T M nl E T M n,l (x; k m ) for R 3 \ D.
Then the scattering amplitude A ∞ can be represented as follows:

A ∞ (x) = ∞ n=1 n l=-n 4π(-i) n ik m n(n + 1) β T E nl V n,l (x) + µ m ε m β T M nl U n,l .
Proof. It is well-known that

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

E T E n,m (x; k m ) ∼ - e ikm|x| k m |x| e -i n+1 2 π n(n + 1)V n,l (x)
and

E T M n,m (x; k m ) ∼ - e ikm|x| k m |x| µ m ε m e -i n+1 2 π n(n + 1)U n,l (x)
.

By applying these asymptotics to the series expansion of E s , the conclusion follows.

A plane wave can be represented as a series expansion. The following lemma is proved in [START_REF] Klinkenbusch | Scattering of an arbitrary plane electromagnetic wave by a finite elliptic cone[END_REF]. Lemma 3.5.4. Let E i be a plane wave, that is, E i (x) = p e ikmd•x with d ∈ S and p • d = 0. Then we have the following series representation for a plane wave as follows:

E i (x) = ∞ n=1 n l=-n α pw,T E nl E T E n,l (x; k m ) + α pw,T M nl E T M n,l (x; k m ), where          α pw,T E nl = (-1)4πi n n(n + 1) i V n,l (d) • p , α pw,T M nl = (-1)4πi n n(n + 1) ε m µ m U n,l (d) • p .
Q ext = ∞ n=1 n l=-n (4π) 3 k 2 m |p| 2 ℑ (-1)S T E n (V n,l (d) • p) 2 + iS T M n (U n,l (d) • p) 2 .
Moreover, for small r > 0, we have

Q ext = 1 l=-1 (-1)(4πk m r) 3 k 2 m |p| 2 ℑ i 2 3 µ c -µ m 2µ m + µ c (V 1,l (d) • p) 2 + 2 3 ε c -ε m 2ε m + ε c (U 1,l (d) • p) 2 + O((k m r) 4 ).
Proof. Let us first compute the scattering amplitude A ∞ when E i is a plane wave. From

A ∞ (x) = ∞ n=1 n l=-n 4π(-i) n ik m n(n + 1) × α pw,T E nl S T E n V n,l (x) + µ m ε m α pw,T M nl S T M n U n,l = ∞ n=1 n l=-n (4π) 2 k m (-1)S T E n (V n,l (d) • p)V n,l + iS T M n (U n,l (d) • p)U n,l .
Therefore, we have

Q ext = 4π k m ℑ p • A ∞ (d) |p| 2 = ∞ n=1 n l=-n (4π) 3 k 2 m |p| 2 ℑ (-1)S T E n (V n,l (d) • p) 2 + iS T M n (U n,l (d) • p) 2 .
Now we assume that r ≪ 1. By applying (3.58), one can easily see that

S T E 1 = i 2 3 (µ c -µ m )(k m r) 3 2µ m + µ c + O(r 4 ), S T M 1 = i 2 3 (ε c -ε m )(k m r) 3 2ε m + ε c + O(r 4 ), S T E n , S T M n = O(r 4 ), for n ≥ 2.
Therefore, we obtain, up to an error term of order O(r 4 ),

Q ext = 1 l=-1 (-1)(4π) 3 k 2 m |p| 2 ℑ i 2 3 (µ c -µ m )(k m r) 3 2µ m + µ c (V 1,l (d) • p) 2 + 2 3 (ε c -ε m )(k m r) 3 2ε m + ε c (U 1,l (d) • p) 2 .
The proof is complete.

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 simple, we apply degenerate perturbation theory in order to compute the first order effect of the size.

Explicit representation of boundary integral operators

Let D s and D c be a spherical shell with radius r s and r c with r s > r c > 0. Let

(ε, µ) =      (ε m , µ m ) in D c , (ε s , µ s ) in D s \ Dc , (ε m , µ m ) in R 3 \ Ds . Let ρ = r c r s .
The solution to the transmission problem can be represented as follows

E(x) =                      µ c ∇ × S kc Ds [ψ s ](x) + ∇ × ∇ × S kc Ds [φ s ](x) +µ c ∇ × S kc Dc [ψ c ](x) + ∇ × ∇ × S kc Dc [φ c ](x) x ∈ D c , µ s ∇ × S ks Ds [ψ s ](x) + ∇ × ∇ × S ks Ds [φ s ](x) +µ s ∇ × S ks Dc [ψ c ](x) + ∇ × ∇ × S ks Dc [φ c ](x) x ∈ D s \ Dc , E i + µ m ∇ × S km Ds [ψ s ](x) + ∇ × ∇ × S km Ds [φ s ](x) +µ m ∇ × S km Dc [ψ c ](x) + ∇ × ∇ × S km Dc [φ c ](x) x ∈ R 3 \ Ds , (3.71) and H
(x) = - i ωµ D (∇ × E)(x) x ∈ R 3 \∂D, (3.72)
where the pair

(ψ s , φ s , ψ c , φ c ) ∈ H -1 2 T (div, ∂D s ) 2 × H -1 2 T (div, ∂D c ) 2 is the unique solution to W sh     ψ s φ s ψ c φ c     := W sh 11 W sh 12 W sh 21 W sh 22     ψ s φ s ψ c φ c     =     ν × E i iων × H i 0 0     with W sh 11 =    µ s + µ m 2 Id + µ s M ks Ds -µ m M km Ds L ks Ds -L km Ds L ks Ds -L km Ds k 2 s 2µ s + k 2 m 2µ m Id + k 2 s µ s M ks Ds - k 2 m µ m M km Ds    , (3.73)
W sh 12 =   µ s ν × ∇ × S ks Dc -µ m ν × ∇ × S km Dc ν × ∇ × ∇ × S ks Dc -ν × ∇ × ∇ × S km Dc ν × ∇ × ∇ × S ks Dc -ν × ∇ × ∇ × S km Dc k 2 s µ s ν × ∇ × S ks Dc - k 2 m µ m ν × ∇ × S km Dc   , (3.74)
Chapter 3. The Full Maxwell Equations

W sh 21 =   -µ c ν × ∇ × S kc Ds + µ s ν × ∇ × S ks Ds -ν × ∇ × ∇ × S kc Ds + ν × ∇ × ∇ × S ks Ds -ν × ∇ × ∇ × S kc Ds + ν × ∇ × ∇ × S ks Ds - k 2 c µ c ν × ∇ × S kc Ds + k 2 s µ s ν × ∇ × S ks Ds   , (3.75)
W sh 22 =    - µ c + µ s 2 Id -µ c M kc Dc + µ s M ks Dc -L kc Dc + L ks Dc -L kc Dc + L ks Dc - k 2 c 2µ c + k 2 s 2µ s Id - k 2 c µ c M kc Dc + k 2 s µ s M ks Dc    .
(3.76) Note that W sh 11 and W sh 22 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. By Lemma 3.5.1, we can represent ν × ∇ × S k D | |x|=r ′ and ν × ∇ × ∇ × S k D | |x|=r ′ in a matrix form as follows(using U n,m , V n,m as basis):

(i) For r ′ > r, ν × ∇ × S k D | |x|=r ′ = (-ikr)J n (kr)h (1)
n (kr ′ ) 0 0 ik r 2 r ′ j n (kr)H n (kr ′ ) , (3.77)
ν × ∇ × ∇ × S k D | |x|=r ′ = 0 ik(kr) 2 j n (kr)h (1)
n (kr ′ ) -ik r r ′ J n (kr)H n (kr ′ ) 0 ;

(3.78)

(ii) For r ′ < r, ν × ∇ × S k D | |x|=r ′ = (-ikr)j n (kr ′ )H n (kr) 0 0 ik r 2 r ′ J n (kr ′ ) (1)
n (kr) ,

ν × ∇ × ∇ × S k D | |x|=r ′ = 0 ik(kr) 2 j n (kr ′ )h (3.79)
n (kr) -ik r r ′ J n (kr ′ )H n (kr) 0 .

(3.80)

Using the above formulas, the matrix representation of the operators W sh 12 and W sh 21 can be easily obtained. We now consider scaling of W sh . First, we need some definitions. Let D s = z + r s B s where B s contains the origin and |B s | = O(1). Let B c be defined in a similar way. For any x ∈ ∂D s (or ∂D c ), let x = x-z rs ∈ ∂B s (or ∂B c with r s replaced by r c ) and define for each function f defined on ∂D s (or ∂D c ), a corresponding function defined on B as follows

η s (f )( x) = f (z + r s x), η c (f )( x) = f (z + r c x).
(3.81)

3.6. Explicit computations for a spherical shell 89 Then, in a similar way to the sphere case, let us write

W sh B (r s )     η s (ψ s ) ωη s (φ s ) η c (ψ c ) ωη c (φ c )     =      η(ν×E i ) µm-µs η(iν×H i ) εm-εs 0 0      . Using (U n,m , V n,m , U n,m , V n,m ) × (U n,m , V n,m , U n,m , V n,m
) as basis, we can represent W sh B (r s ) in a 8×8 matrix form in a subspace H n,m (∂B s )×H n,m (∂B c ). Then, by using (3.59), their asymptotic expansion can also be obtained.

Here, the resulting asymptotics of the matrix W sh B are given as follows. Write

W sh B (r s ) = W sh B,0 + r s W sh B,1 + r 2 s W sh B,2 + O(r 3 s ), (3.82)
where

W sh B,0 = Λ µ,ε Λ µ,ε + P 0,n Q 0,n R 0,n -P 0,n , (3.83)
W sh B,1 = P 1,n Q 1,n R 1,n -P 1,n , W sh B,2 = P 2,n Q 2,n R 2,n -P 2,n .
Here, the matrix P j,n , Q j,n and R j,n are given by

Λ µ,ε =     λ µ λ µ λ ε λ ε     , P 0,n =     p n -p n p n -p n     , Q 0,n = ρ 2     g n f n g n f n     , R 0,n =     f n g n f n g n     , P 1,n = ω     C µ p n C µ q n C ε p n C ε q n     , P 2,n = ω 2     D µ r n D µ s n D ε r n D ε s n     , Q 1,n = ωρ     C µ pn C µ qn C ε pn C ε qn     , Q 2,n = ω 2 ρ     D µ rn D µ sn D ε rn D ε sn     , R 1,n = -ωρ -1     C µ pn C µ qn C ε pn C ε qn     , Chapter 3. The Full Maxwell Equations R 2,n = ω 2 ρ -1     D µ sn -D µ rn D ε sn -D ε rn     .
Here, p n , q n , r n , s n are defined as (3.62) and pn , qn , rn , sn , D µ and D ε are defined as follows:

f n = ρ n n 2n + 1 , g n = ρ n-1 n + 1 2n + 1 , (3.84
)

pn = 1 2n + 1 ρ n+1 , (3.85
)

qn = (n + 1)(n -2) 2(2n -1)(2n + 1) ρ n - n(n + 3) 2(2n + 1)(2n + 3) ρ n+2 , (3.86
)

rn = - n + 1 2(2n -1)(2n + 1) ρ n + (n + 3) 2(2n + 1)(2n + 3) ρ n+2 , (3.87
)

sn = - n -2 2(2n -1)(2n + 1) ρ n+1 + n 2(2n + 1)(2n + 3) ρ n+3 , (3.88)
and

D µ = ε s µ 2 s -ε m µ 2 m µ m -µ s , D ε = ε 2 s µ s -ε 2 m µ m ε m -ε s . (3.89)
3.6.2 Asymptotic behavior of the spectrum of W sh B (r s )

Let us define λ sh n = 1 2(2n + 1)

1 + 4n(n + 1)ρ 2n+1 .

Note that ±λ sh n are eigenvalues of the Neumann-Poincaré operator on the shell.

It turns out that the eigenvalues of W sh B,0 are as follows

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

λ µ + λ sh n -→ E 0 1 := (λ sh n + p n )e 1 + f n e 5 , E 0 2 := (λ sh n -p n )e 2 + g n e 6 , λ µ -λ sh n -→ E 0 3 := (-λ sh n + p n )e 1 + f n e 5 , E 0 4 := (-λ sh n -p n )e 2 + g n e 6 , λ ε + λ sh n -→ E 0 5 := (λ sh n + p n )e 3 + f n e 7 , E 0 6 := (λ sh n -p n )e 4 + g n e 8 , λ ε -λ sh n -→ E 0 7 := (-λ sh n + p n )e 3 + f n e 7 , E 0 8 := (-λ sh n -p n )e 4 + g n e 8 ,
where {e i } 8 i=1 is standard unit basis in R 8 . 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

T 16,n = C ε (λ sh n -p n )a 1,n -b 1,n |E 0 1 ||E 0 6 | , T 18,n = C ε (-λ sh n -p n )a 1,n -b 1,n |E 0 1 ||E 0 8 | , T 25,n = C ε (λ sh n + p n )a 2,n -b 2,n |E 0 2 ||E 0 5 | , T 27,n = C ε (-λ sh n + p n )a 2,n -b 2,n |E 0 2 ||E 0 7 | , T 36,n = C ε (λ sh n -p n )a 3,n -b 3,n |E 0 3 ||E 0 6 | , T 38,n = C ε (-λ sh n -p n )a 3,n -b 3,n |E 0 3 ||E 0 8 | , T 45,n = C ε (λ sh n + p n )a 4,n -b 4,n |E 0 4 ||E 0 5 | , T 47,n = C ε (-λ sh n + p n )a 4,n -b 4,n |E 0 4 ||E 0 7 | , T 52,n = C µ C ε T 16,n , T 54,n = C µ C ε T 18,n , T 61,n = C µ C ε T 25,n , T 63,n = C µ C ε T 27,n , T 72,n = C µ C ε T 36,n , T 74,n = C µ C ε T 38,n , T 81,n = C µ C ε T 45,n , T 83,n = C µ C ε T 47,n ,
where

a 1,n = (λ sh n + p n )q n + ρf n qn , a 2,n = (λ sh n -p n )p n + ρg n pn , a 3,n = (-λ sh n + p n )q n + ρf n qn , a 4,n = (-λ sh n -p n )p n + ρg n pn , and
b 1,n = f n g n q n + ρ -1 (λ sh n + p n )g n qn , b 2,n = f n g n p n + ρ -1 (λ sh n -p n )f n pn , b 3,n = f n g n q n + ρ -1 (-λ sh n + p n )g n qn , b 4,n = f n g n p n + ρ -1 (-λ sh n -p n )f n pn .
We also define

K 1,n = D µ (λ sh n + p n )((λ sh n + p n )r n + ρf n rn ) + f n ((λ sh n + p n )ρ -1 sn -f n r n ) |E 0 1 | 2 , K 2,n = D µ g n ((-λ sh n + p n )ρ -1 rn -g n s n ) + (λ sh n -p n )((λ sh n -p n )s n + ρg n sn ) |E 0 2 | 2 , K 3,n = D µ (-λ sh n + p n )((-λ sh n + p n )r n + ρf n rn ) + f n ((-λ sh n + p n )ρ -1 sn -f n r n ) |E 0 3 | 2 , K 4,n = D µ g n ((λ sh n + p n )ρ -1 rn -g n s n ) + (-λ sh n -p n )((-λ sh n -p n )s n + ρg n sn ) |E 0 4 | 2 , K 5,n = D ε D µ K 1,n , K 6,n = D ε D µ K 2,n , K 7,n = D ε D µ K 3,n , K 8,n = D ε D µ K 4,n .
Now we are ready to state the result. The followings are asymptotics of eigenvalues of W sh B (r s )

λ µ + λ ε + (r s ω) 2 T 16,n T 61,n λ µ -λ ε + T 18,n T 81,n λ µ -λ ε + 2λ sh n + K 1,n + O(r 3 s ), λ µ + λ ε + (r s ω) 2 T 16,n T 61,n λ µ -λ ε + T 18,n T 81,n λ µ -λ ε + 2λ sh n + K 2,n + O(r 3 s ), λ µ -λ ε + (r s ω) 2 T 36,n T 63,n λ µ -λ ε -2λ sh n + T 38,n T 83,n λ µ -λ ε + K 3,n + O(r 3 s ), λ µ -λ ε + (r s ω) 2 T 36,n T 63,n λ µ -λ ε -2λ sh n + T 38,n T 83,n λ µ -λ ε + K 4,n + O(r 3 s ), λ ε + λ µ + (r s ω) 2 T 52,n T 25,n λ ε -λ µ + T 54,n T 45,n λ ε -λ µ + 2λ sh n + K 5,n + O(r 3 s ), λ ε + λ µ + (r s ω) 2 T 52,n T 25,n λ ε -λ µ + T 54,n T 45,n λ ε -λ µ + 2λ sh n + K 6,n + O(r 3 s ), λ ε -λ µ + (r s ω) 2 T 72,n T 27,n λ ε -λ µ -2λ sh n + T 74,n T 47,n λ ε -λ µ + K 7,n + O(r 3 s ), λ ε -λ µ + (r s ω) 2 T 72,n T 27,n λ ε -λ µ -2λ sh n + T 74,n T 47,n λ ε -λ µ + K 8,n + O(r 3 s ).
We also have the following asymptotic expansions of the eigenfunctions:

E 0 1 + r s ω T 16,n λ µ -λ ε E 0 6 + T 18,n λ µ -λ ε + 2λ sh n E 0 8 + O(r 2 s ), E 0 2 + r s ω T 25,n λ µ -λ ε E 0 5 + T 27,n λ µ -λ ε + 2λ sh n E 0 7 + O(r 2 s ), E 0 3 + r s ω T 36,n λ µ -λ ε -2λ sh n E 0 6 + T 38,n λ µ -λ ε E 0 8 + O(r 2 s ), E 0 4 + r s ω T 45,n λ µ -λ ε -2λ sh n E 0 5 + T 47,n λ µ -λ ε E 0 7 + O(r 2 s ), E 0 5 + r s ω T 52,n λ µ -λ ε E 0 2 + T 54,n λ µ -λ ε + 2λ sh n E 0 4 + O(r 2 s ), E 0 6 + r s ω T 61,n λ µ -λ ε E 0 1 + T 63,n λ µ -λ ε + 2λ sh n E 0 3 + O(r 2 s ), E 0 7 + r s ω T 72,n λ µ -λ ε -2λ sh n E 0 2 + T 74,n λ µ -λ ε E 0 4 + O(r 2 s ), E 0 8 + r s ω T 81,n λ µ -λ ε -2λ sh n E 0 1 + T 83,n λ µ -λ ε E 0 3 + O(r 2 s ).
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.

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.

The particle dimension and interparticle distances are considered to be infinitely 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.

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 [START_REF] Sarid | Modern Introduction to Surface Plasmons: Theory, Mathematical Modeling, and Applications[END_REF]. This translates into an efficient heat generation in the presence of electromagnetic radiation. Moreover, plasmonic nanoparticles biocompatibility makes them suitable for use in nanotherapy [START_REF] Baffou | Mapping heat origin in plasmonic structures[END_REF].

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 [START_REF] Govorov | Generating heat with metal nanoparticles[END_REF]. In this nanomedical application, the temperature increase is the most important parameter [START_REF] Marangon | Tumor stiffening, a key determinant of tumor progression, is reversed by nanomaterial-induced photothermal therapy[END_REF][START_REF] Qin | Quantitative comparison of photothermal heat generation between gold nanospheres and nanorods[END_REF]. 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 [START_REF] Govorov | Generating heat with metal nanoparticles[END_REF].

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 [START_REF] Govorov | Generating heat with metal nanoparticles[END_REF].

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.

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 u i is a transverse electric or transverse magnetic (TE or TM) polarized wave. This approximation, also called paraxial approximation [START_REF] Greivenkamp | Field Guide to Geometrical Optics[END_REF], is a good model for a laser beam which are used, in particular, in fullfield 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 ⋐ R 2 of class C 1,α for some 0 < α < 1. Furthermore, let D = z + δB, where B is centered at the origin and |B| = O(1).

We denote by ε c (x) and µ c (x), x ∈ D, the electric permittivity and magnetic permeability of the particle, respectively, both of which may depend on the frequency ω of the incident wave. Assume that ε

c (x) = ε 0 ε ′ c , µ c (x) = µ 0 µ ′ c and that ℜε ′ c < 0, ℑε ′ c > 0, ℜµ ′ c < 0, ℑµ ′ c > 0.
Here and throughout, ε 0 and µ 0 are the permittivity and permeability of vacuum.

Similarly, we denote by ε m (x) = ε 0 ε ′ m and µ m (x) = µ 0 µ ′ m , x ∈ R 2 \D the permittivity and permeability of the host medium, both of which do not depend on the frequency ω of the incident wave. Assume that ε m and µ m are real and strictly positive.

The index of refraction of the medium (with the nanoparticle) is given by

n(x) = ε ′ c µ ′ c χ(D)(x) + ε ′ m µ ′ m χ(R 2 \D)(x),
where χ denotes the indicator function.

The scattering problem for a TE incident wave u i is modeled as follows: with ν being the outward unit normal vector to ∂D.

                   ∇ • c 2 n 2 ∇u + ω 2 u = 0 in R 2 \∂D, u + -u -= 0 on ∂D, 1 ε m ∂u ∂ν + - 1 ε c ∂u ∂ν - = 0 on ∂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 [START_REF] Baffou | Heat generation in plasmonic nanostructures: Influence of morphology[END_REF]

                   ρC ∂τ ∂t -∇ • γ∇τ = ω 2π ℑ(ε)|u| 2 in (R 2 \∂D) × (0, T ), τ + -τ -= 0 on ∂D, γ m ∂τ ∂ν + -γ c ∂τ ∂ν - = 0 on ∂D, τ (x, 0) = 0, (4.2)
where

ρ = ρ c χ(D)+ρ m χ(R 2 \D) is the mass density, C = C c χ(D)+C m χ(R 2 \D) is the thermal capacity, γ = γ c χ(D) + γ m χ(R 2 \D) is the thermal conductiv- ity, T ∈ R is the final time of measurements and ε = ε c χ(D) + ε m χ(R 2 \D).
We further assume that ρ c , ρ m , C c , C m , γ c , γ m are real positive constants. Note that ℑ(ε) = 0 in R 2 \D and so, outside D, the heat equation is homogeneous.

The coupling of equations (4.1) and (4.2) describes the physics of our problem.

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 ∆ + k 2 satisfying the Sommerfeld radiation condition. In dimension two, G is given by

G(x, k) = - i 4 H (1) 0 (k|x|),
where H

(1) 0

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

S k D [ϕ](x) = ∂D G(x, y, k)ϕ(y)dσ(y), x ∈ ∂D or x ∈ R 2 ,
and

(K k D ) * [ϕ](x) = ∂D ∂G(x, y, k) ∂ν(x) ϕ(y)dσ(y), x ∈ ∂D.
Our main results in this chapter are the following.

Theorem 4.2.1. For an incident wave u i ∈ C 2 (R 2 ), the solution u to (4.1), inside a plasmonic particle occupying a domain D = z +δB, has the following asymptotic expansion as δ → 0 in L 2 (D),

u = u i (z)+ δ(x -z) + S D λ ε Id -K * D -1 [ν] •∇u i (z)+O δ 3 dist(λ ε , σ(K * D ))
, where ν is the outward normal to D, σ(K * D ) denotes the spectrum of K * D in H -1 2 (∂D) and

λ ε := ε c + ε m 2(ε c -ε 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 ),

τ (x, t) = F D (x, t, b c ) -V bc D (λ γ Id -K * D ) -1 [ ∂F D (•, •, b c ) ∂ν ](x, t) + O δ 4 log δ dist(λ ε , σ(K * D )) 2 ,

Heat generation

101
where ν is the outward normal to D and

λ γ := γ c + γ m 2(γ c -γ m ) , b c := ρ c C c γ c , F D (x, t, b c ) := ω 2πγ c ℑ(ε c ) t 0 D e - |x-y| 2 4bc(t-t ′ ) 4πb c (t -t ′ ) |u| 2 (y)dydt ′ , V bc D [f ](x, t) := t 0 ∂D e - |x-y| 2 4bc(t-t ′ ) 4πb c (t -t ′ ) f (y, t ′ )dydt ′ .
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 R 3 is straightforward and the same type of small volume approximation can be found using the techniques presented in this chapter. In fact, in R 3 , the operators involved in the first term of the temperature small volume expansion are

F D (x, t, b c ) := ω 2πγ c ℑ(ε c ) t 0 D e - |x-y| 2 4bc(t-t ′ ) 4πb c (t -t ′ ) 3 2 |E| 2 (y)dydt ′ , V bc D [f ](x, t) := t 0 ∂D e - |x-y| 2 4bc(t-t ′ ) 4πb c (t -t ′ ) 3 2 f (y, t ′ )dydt ′ .
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.

Heat generation

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

                                                             ∇ • c 2 n 2 ∇u + ω 2 u = 0 in R 2 \∂D, u + -u -= 0 on ∂D, 1 ε m ∂u ∂ν + - 1 ε c ∂u ∂ν - = 0 on ∂D,
u s := u -u i satisfies the Sommerfeld radiation condition at infinity,

ρ c C c γ c ∂τ ∂t -∆τ = ω 2πγ c ℑ(ε c )|u| 2 in D × (0, T ), ρ m C m γ m ∂τ ∂t -∆τ = 0 in (R 2 \D) × (0, T ), τ + -τ -= 0 on ∂D, γ m ∂τ ∂ν + -γ c ∂τ ∂ν - = 0 on ∂D, τ (x, 0) = 0. (4.3)
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.

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).

Recall that D = z + δB. For any x ∈ ∂D, x := x-z δ ∈ ∂B and for each function f defined on ∂D, we introduce a corresponding function defined on ∂B as follows η(f

)( x) = f (z + δ x). (4.4)
It follows that

S k D [ϕ](x) = δS δk B [η(ϕ)]( x), (K k D ) * [ϕ](x) = (K δk B ) * [η(ϕ)]( x), (4.5)
so system (2.2) becomes

       S δkm B [η(ψ)] -S δkc B [η(φ)] = - η(u i ) δ , 1 εm 1 2 Id + (K δkm B ) * [η(ψ)] + 1 εc 1 2 Id -(K δkc B ) * [η(φ)] = - 1 ε m η( ∂u i ∂ν ).
(4.6) Note that the system is defined on ∂B. For δ small enough S δkm B is invertible (see Appendix B.3). Therefore,

η(ψ) = (S δkm B ) -1 S δkc B [η(φ)] -(S δkm B ) -1 [ η(u i ) δ ].
Hence, we have the following equation for η(φ):

A I B (δ)[η(φ)] = f I ,
where

A I B (δ) = 1 εm 1 2 I + (K δkm B ) * (S δkm B ) -1 S δkc B + 1 εc 1 2 Id -(K δkc B ) * , f I = - 1 ε m η( ∂u i ∂ν ) + 1 εm 1 2 Id + (K δkm B ) * (S δkm B ) -1 [ η(u i ) δ ]. (4.7)
Chapter 4. Heat Generation with Plasmonic Nanoparticles

Straightforward calculations and the fact that S B is harmonic in B yields

δS δkc B (A I B (δ)) -1 f I = u i (z) + δ x + S B λ ε Id -K * B -1 [ν] • ∇u i (z) + O δ 2 dist(λ ε , σ(K * B ))
in L 2 (B). Using Lemma A.0.3 to scale back the estimate to D leads to the desired result.

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 [START_REF] Costabel | Boundary integral operators for the heat equation[END_REF][START_REF] Hohage | Numerical solution of a heat diffusion problem by boundary element methods using the Laplace transform[END_REF][START_REF] Lubich | Time discretizations of parabolic boundary integral equations[END_REF].

Consider equation (4.3) and define the Laplace transform of a function g(t) by

L(g)(s) = ∞ 0 e -st g(t)dt.
Taking the Laplace transform of the equations on τ in (4.3) we formally obtain the following system: A rigorous justification for the derivation of system (4.11) and the validity of the inverse transform of the solution can be found in [START_REF] Hohage | Numerical solution of a heat diffusion problem by boundary element methods using the Laplace transform[END_REF].

                             s ρ c C c γ c τ (•, s) -∆τ (•, s) = L(g u )(•, s) in D, s ρ m C m γ m τ (•, s) -∆τ (•, s) = 0 in R 2 \D, τ+ (•, s) -τ-(•, s) = 0 on ∂D, γ m ∂ τ ∂ν + -γ c ∂ τ ∂ν - = 0
Using layer potential techniques we have that, for any p, q ∈ H -1 2 (∂D), τ defined by

τ := -S βγ m D [p], x ∈ R 2 \D, -FD (•, y, β γc ) -S βγ c D [q], x ∈ D, (4.12

Heat generation 107

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

     -S βγ m D [p] + S βγ c D [q] = -FD (•, β γc ), -γ m 1 2 Id + (K βγ m D ) * [p] + γ c -1 2 Id + (K βγ c D ) * [q] = -γ c ∂ FD (•, β γc ) ∂ν .
(4.13)

Re-scaling of the equations

Recall that D = z + δB, for any x ∈ ∂D, x := x-z δ ∈ ∂B, for each function f defined on ∂D, η is such that η(f )( x) = f (z + δ x) and

S k D [ϕ](x) = δS δk B [η(ϕ)]( x), (K k D ) * [ϕ](x) = (K δk B ) * [η(ϕ)]( x).
We can also verify that

FD (x, β γc ) = δ 2 FB (x, δβ γc ), ∂ FD ∂ν (x, β γc ) = δ ∂ FB ∂ν (x.δβ 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 FB to refer to FB (•, δβ γc ).

We rescale system (4.13) to arrive at

     -S δβγ m B [η(p)] + S δβγ c B [η(q)] = -δ FB , -γ m 1 2 Id + (K δβγ m B ) * [η(p)] + γ c -1 2 Id + (K δβγ c B ) * [η(q)] = -γ c δ ∂ FB ∂ν .
For δ small enough, S

δβγ c B is invertible (see Appendix B.3). Therefore, it follows that η(p) = (S δβγ m B ) -1 S δβγ c B [η(q)] + (S δβγ m B ) -1 δ FB .
Hence, we have the following equation for η(q):

A h B (δ)[η(q)] = f h ,
where

A h B (δ) = -γ m 1 2 Id + (K δβγ m B ) * (S δβγ m B ) -1 S δβγ c B + γ c -1 2 Id + (K δβγ c B ) * , f h = -γ c δ ∂ FB ∂ν + γ m 1 2 Id + (K δβγ m B ) * (S δβγ m B
) -1 δ FB . 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 Chapter 4. Heat Generation with Plasmonic Nanoparticles an asymptotic expansion for η(q) on δ to later compute δS δβγ c B

[η(q)] on ∂B, scale back to D and take Laplace inverse.

Using the asymptotic expansions of Appendix B.3 the following asymptotic for A h B (δ) holds in L(H * (∂B))

A h B (δ) = A h 0 +O(δ 2 log δ),
where

A h 0 = - 1 2 γ c + γ m Id -γ c -γ m K * B .
In the same manner, in H * (∂B),

f h = -γ c δ ∂ FB ∂ν + γ m 1 2 Id + K * B S -1 B [δ FB ] + O δ 5 log δ dist(λ ε , σ(K * D )) 2 = -γ c δ ∂ FB ∂ν -γ m 1 2 Id -K * B S -1 B [δ FB ] + γ m S -1 B [δ FB ] + O δ 5 log δ dist(λ ε , σ(K * D )) 2 .
Here the remainder comes from the fact that FB = O

δ 2 dist(λε,σ(K * D )) 2 . Note that ∆ FB = η(L(g u )) -δ 2 β 2 γc FB in B and ∆ FB = 0 in R 2 \ D.
We can further verify that FB satisfies the assumption required in Lemma A.0.4. Thus we have

1 2 Id -K * B S -1 B [δ FB ] = -δ ∂ FB ∂ν + C u ϕ 0 + γ m S -1 B [δ FB ] + O δ 5 dist(λ ε , σ(K * D )) 2 ,
where C u is a constant such that

C u = O δ 3 dist(λε,σ(K * D )) 2 .
After replacing the above in the expression of f h we find that [START_REF] Ammari | Generalized polarization tensors, inverse conductivity problems, and dilute composite materials: a review[END_REF] where

η(q) = (A h B (δ)) -1 f h = (λ γ Id -K * B ) -1 [δ ∂ FB ∂ν ] + C u γ m (γ c -γ m )(λ γ -1 2 ) ϕ 0 + O δ 5 log δ dist(λ ε , σ(K * D )) 2 , (4.
λ γ = γ c + γ m 2(γ c -γ m ) .
Finally, in H * (∂B),

η(τ ) = -δ 2 FB -δS δβγ c B (λ γ Id-K * B ) -1 [ ∂δ FB ∂ν ]- C u γ m (γ c -γ m )(λ γ -1 2 ) δS δβγ c B [ϕ 0 ]+O δ 6 log δ dist(λ ε , σ(K * D )) 2 . (4.16)
It can be shown, from the regularity of the remainders, that the previous identity also holds in L 2 (∂B).

Using Holder's inequality we can prove that

S δβγ c B [ϕ] L ∞ (∂B) ≤ C ϕ L 2 (∂B) ,
for some constant C. Hence, we find that identity (4.16) also holds true uniformly on ∂B and C u δS

δβ γcf B [ϕ 0 ](x) = O δ 4 log δ dist(λε,σ(K * D ))
(x, s) = -FD (x, β γc ) -S βγ c D (λ γ Id -K * D ) -1 [ ∂ FD (•, β γc ) ∂ν ] + O δ 4 log δ dist(λ ε , σ(K * D )) 2 .
(4.17)

Before we take the inverse Laplace transform to (4.17) we note that (see [START_REF] Lubich | Time discretizations of parabolic boundary integral equations[END_REF])

L K(x, •, b c ) = -G(x, β γc ),
where b c := ρcCc γc and K(x, •, b c ) is the fundamental solution of the heat equation. In dimension two, K is given by

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

-FD (x, β γc ) = - D G(x, y, β γc )L(g u )(y)dy = L • 0 D K(x, y, •, t ′ , b c )g u (y)dydt ′ .
We define F D as follows

F D (x, t, b c ) := t 0 D K(x, y, t, t ′ , b c )g u (y)dydt ′ . (4.18)
Similarly, we have that for a function f

- ∂D G(x, y, β γc )L(f )(y)dy = L • 0 ∂D K(x, y, •, t ′ , b c )f (y, t ′ )dydt ′ .
We define V bc D as follows

V bc D [f ](x, t) := t 0 ∂D K(x, y, t, t ′ , b c )f (y, t ′ )dydt ′ . (4.19)
Finally, using Fubini's theorem and taking Laplace inverse we find that

τ (x, t) = F D (x, t, b c ) -V bc D (λ γ Id -K * D ) -1 [ ∂F D (•, •, b c ) ∂ν ](x, t) + O δ 4 log δ dist(λ ε , σ(K * D )) 2 ,
uniformly in (x, t) ∈ ∂D × (0, T ).

Temperature elevation at the plasmonic resonance

Suppose that the incident wave is u i (x) = e ikmd•x , 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 L 2 (D),

u ≈ e ikmd•z 1 + ik m S D λ ε Id -K * D -1 [ν] • d , 110
Chapter 4. Heat Generation with Plasmonic Nanoparticles and hence

|u| 2 ≈ 1 + 2k m ℜ iS D λ ε Id -K * D -1 [ν] • d + k m S D λ ε Id -K * D -1 [ν] • d 2 . (4.

20)

Using Lemma A.0.2, we can write

S D λ ε Id -K * D -1 [ν] • d = ∞ j=1 (ν • d, ϕ j ) H * S D [ϕ j ] λ ε -λ j ,
and therefore, for a given plasmonic frequency ω, we have

S D λ ε Id -K * D -1 [ν] • d ≈ (ν • d, ϕ j * ) H * S D [ϕ j * ] λ ε (ω) -λ j * .
Here j * is such that λ j * = ℜ(λ ε (ω)) and the eignevalue λ j * is assumed to be simple. If this was not the case, (ν • d, ϕ j * ) H * S D [ϕ j * ] should be replaced by the corresponding sum over an orthonormal basis of eigenfunctions for the eigenspace associated to λ j * .

Replacing in (4.20) we find

|u| 2 ≈ 1 + 2k m (ν • d, ϕ j * ) H * S D [ϕ j * ] |λ ε (ω) -λ j * | + k 2 m (ν • d, ϕ j * ) 2 H * S D [ϕ j * ] 2 |λ ε (ω) -λ j * | 2 .
Thus, at a plasmonic resonance ω,

F D [g u ] ≈ F D [1] + 2k m (ν • d, ϕ j * ) H * |λ ε (ω) -λ j * | F D [S D [ϕ j * ]] + k 2 m (ν • d, ϕ j * ) 2 H * |λ ε (ω) -λ j * | 2 F D [S D [ϕ j * ] 2 ] , ∂F D ∂ν ≈ 2k m (ν • d, ϕ j * ) H * |λ ε (ω) -λ j * | ∂F D [S D [ϕ j * ]] ∂ν + k 2 m (ν • d, ϕ j * ) 2 H * |λ ε (ω) -λ j * | 2 ∂F D [S D [ϕ j * ] 2 ] ∂ν .
Then, the temperature on the boundary of a nanoparticle at the plasmonic resonance can be estimated by plugging the above approximations of F D and

∂F D (x, t, b c ) ∂ν into τ (x, t) = F D (x, t, b c ) -V bc D (λ γ Id -K * D ) -1 [ ∂F D (•, •, b c ) ∂ν ](x, t) + O δ 4 log δ dist(λ ε , σ(K * D )) 2 .

Temperature elevation for two close-to-touching particles

Lemma A.0.4 implies that

∂F D (x, t, b c ) ∂ν = - 1 2 Id -K * D S -1 D [F D ](x, t) + O δ 4 log δ dist(λ ε , σ(K * D )) 2 .
Therefore, we can write the temperature on the boundary of the nanoparticle as : the complement in H * (∂D) of the eigenspace associated to the eigenvalue 1 2 of K * D . This implies that, even if λ γ is close to 1 2 , the quantity

τ (x, t) = F D (x, t, b c )+V bc D (λ γ Id-K * D ) -1 P H * \E 1 2 [ ∂F D (•, •, b c ) ∂ν ](x, t)+O δ 4 log δ dist(λ ε , σ(K * D ))
(λ γ Id -K * D ) -1 P H * \E 1 2 [ ∂F D (•, •, b c ) ∂ν ](x, t)
will remain of order O δ 2 dist(λε,σ(K * D )) 2 , provided that the second largest eigenvalue of K * D is not close to 1 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 contact of order m, a family of eigenvalues of K * D in H * \E 1 2 approaches 1 2 as (see [START_REF] Bonnetier | On the spectrum of the Poincaré variational problem for two close-to-touching inclusions in 2D[END_REF])

λ ζ n ∼ 1 2 -c n ζ 1-1 m + o(ζ 1-1 m ),
where ζ is the distance between connected subparts and c n is an increasing sequence of positive numbers. Now, λ γ ≈ 1 2 is the kind of situations encountered for metallic nanoparticles immersed in water or some biological tissue. As an example, the thermal conductivity of gold is γ c = 318 W mK and that of pure water is γ m = 0.6 W mK . This gives λ γ ≈ 0.5019.

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 refined analysis, asymptotics of the eigenfunctions of K * D should be also studied.

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 D [•] and ∂ ν F D [•] can be achieved by meshing the domain D and integrating semi-analytically inside the triangles that are close to the singularities. We used the following formula to avoid numerical differentiation:

∂F D (x, t, b c ) ∂ν = 1 2πb c D exp -|x -y| 2 4b c t y -x, ν x |x -y| 2 g u (y)dy, x ∈ ∂D. (4.22
) For all the presented simulations, we considered an incident plane wave given by u i (x) = e ikmd•x , where d = (1, 1)/ √ 2 ∈ R 2 is the illumination direction and k m = 2π/750•10 9 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 Chapter 4. Heat Generation with Plasmonic Nanoparticles of a linear combination of both components of S D (λ ǫ Id-K * D ) -1 [v] ∈ R 2 . We 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 λ ǫ . 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 [START_REF] Ammari | Surface plasmon resonance of nanoparticles and applications in imaging[END_REF]. Whenever we mention that we approach a particular eigenvalue λ j of K * D , we will adopt λ ǫ = λ j + 0.001i. With respect to the heat equation coefficients, we use realistic values of gold for nanoparticles, and water for tissues.

Single-particle simulation

We consider one elliptical nanoparticle D ⋐ R 2 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 K * D with λ ǫ , and afterwards applying it to each of the components of the normal ν to ∂D. It turns out that for some eigenfunctions of K * D , the normal of the shape is almost orthogonal, in H * (∂D), to them. Therefore, we cannot observe resonance for their associated eigenvalues. In Zeroth-order component First-order component from the first-order term of the small volume expansion formula established in Theorem 4.2.1.

i (z) + δ(x -z)∇u i (z) and S D λ ε Id -K * D -1 [ν] • ∇u i (z).
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 Theorem 4.2.2, we compute the generated heat on the surface of the nanoparticle. In The total heat, the zeroth-order and its first-order, according to formula given by Theorem 4.2.2. Notice that the firstorder 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 = (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.

Two particles simulation

We consider two elliptical nanoparticles D 1 , D 2 , D = D 1 ∪D 2 , with the same shape and orientation as the nanoparticle considered in the above example. The particle D 1 is centered at the origin and D 2 is centered at (0, 4.1 • 10 -9 ), 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 K * D with each component of the normal of D. We can observe that there are more available resonant modes. In particular we can see that when λ ǫ approaches the 36th or 37th eigenvalues, we achieve strong resonant modes. 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 K * D . 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 K * D , the stronger component is the one on the y direction, 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 K * D . In Figure 4.11 we 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. 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.

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. Combined with the methods developed in [START_REF] Ammari | Backpropagation imaging in nonlinear harmonic holography in the presence of measurement and medium noises[END_REF][START_REF] Ammari | Direct algorithms for thermal imaging of small inclusions[END_REF], our results can be used for the optical and thermal detection and localization of plasmonic nanoparticles. As reported in [START_REF] Nahas | Detection of plasmonic nanoparticles with full field-OCT: optical and photothermal detection[END_REF], 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 based on the photoacoustic signal recently analyzed in [START_REF] Triki | Mathematical modelization of the Photoacoustic effect generated by the heating of metallic nanoparticles[END_REF]. Thermoacoustic 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 relationship between the photoacoustic amplitude and the fluence [START_REF] Prost | Photoacoustic generation by a gold nanosphere: from the linear to the nonlinear thermoelastic regime[END_REF].

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 R 2 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 [START_REF] Alù | Mantle cloak: Invisibility induced by a surface[END_REF][START_REF] Montelongo | Plasmonic nanoparticle scattering for color holograms[END_REF][START_REF] Tretyakov | Metasurfaces for general transformations of electromagnetic fields[END_REF]. 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. Simulations with disks, ellipses, and rings are shown. In this connection, we refer to the recent works [START_REF] Jung | Au/SiO2 nanoring plasmon waveguides at optical communication band[END_REF][START_REF] Malinsky | Nanosphere lithography: Effect of substrate on the localized surface plasmon resonance spectrum of silver nanoparticles[END_REF][START_REF] Nordlander | The Ring: A leitmotif in plasmonics[END_REF][START_REF] Vartanyan | Metal nanosphere at an interface: revival of degeneracy of a dipole plasmon[END_REF]. 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.

Following [START_REF] Abboud | Diffraction at a curved gratings: TM and TE cases, Homogenization[END_REF], 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

       ∆u app + k 2 m u app = 0 in R 2 + , u app + δz ∂uapp ∂x 2 = 0 on ∂R 2
+ , u app -u i satisfies outgoing radiation condition at infinity, (5.2) gives the leading order approximation for u. We will refer to u app +δz ∂uapp ∂x 2 = 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 [START_REF] Chandler-Wilde | The impedance boundary value problem for the Helmholtz equation in a half-plane[END_REF].

One-dimensional periodic Green function

Consider the function G ♯ : R 2 → C satisfying

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

Lemma 5.3.1. Let x = (x 1 , x 2 ), then G ♯ (x) = 1 4π log sinh 2 (πx 2 ) + sin 2 (πx 1 ) , satisfies (5.3).
Proof. We have

∆G ♯ (x) = n∈Z δ(x + (n, 0)) = n∈Z δ(x 2 )δ(x 1 + n) = n∈Z δ(x 2 )e i2πnx 1 , (5.4)
where we have used the Poisson summation formula

n∈Z δ(x 1 + n) = n∈Z e i2πnx 1 .
On the other hand, since G ♯ is periodic in x 1 of period 1, we have

G ♯ (x) = n∈Z β n (x 2 )e i2πnx 1 , therefore ∆G ♯ (x) = n∈Z (β ′′ n (x 2 ) + (i2πn) 2 β n )e i2πnx 1 . (5.5)
Comparing (5.4) and (5.5) yields

β ′′ n (x 2 ) + (i2πn) 2 β n = δ(x 2 ).

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

β 0 = 1 2 |x 2 | + c, β n = -1 4π|n| e -2π|n||x 2 | , n = 0,
where c is a constant. Subsequently,

G ♯ (x) = 1 2 |x 2 | + c - n∈Z\{0} 1 4π|n| e -2π|n||x 2 | e i2πnx 1 = 1 2 |x 2 | + c - n∈N\{0} 1 2πn e -2πn|x 2 | cos(2πnx 1 ) = 1 4π log sinh 2 (πx 2 ) + sin 2 (πx 1 ) ,
where we have used the summation identity (see, for instance, [57, pp. 813-814])

n∈N\{0} 1 2πn e -2πn|x 2 | cos(i2πnx 1 ) = 1 2 |x 2 | - log(2) 2π - 1 4π log sinh 2 (πx 2 ) + sin 2 (πx 1 )
,

and defined c = - log (2) 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 operator, respectively, for a bounded domain B ⋐ -

1 2 , 1 2
× R which we assume to be of class C 1,α for some 0 < α < 1. Let

S B♯ : H -1 2 (∂B) -→ H 1 loc (R 2 ), H 1 2 (∂B) ϕ -→ S B,♯ [ϕ](x) = ∂B G ♯ (x, y)ϕ(y)dσ(y) for x ∈ R 2 , x ∈ ∂B and let K * B♯ : H -1 2 (∂B) -→ H -1 2 (∂B) ϕ -→ K * B,♯ [ϕ](x) = ∂B ∂G ♯ (x, y) ∂ν(x) ϕ(y)dσ(y)
for x ∈ ∂B. As in [START_REF] Khavinson | Poincaré's variational problem in potential theory[END_REF], the periodic Neumann-Poincaré operator can be symmetrized. The following lemma holds.

Lemma 5.3.2. (i) For any ϕ ∈ H -1 2 (∂B), S B♯ [ϕ] is harmonic in B and in - 1 2 , 1 2 × R\B;
which is the periodic Green's function in the upper half space with Dirichlet boundary conditions, and define

S + B♯ : H -1 2 (∂B) -→ H 1 loc (R 2 ), H 1 2 (∂B) ϕ -→ S + B,♯ [ϕ](x) = ∂B G + ♯ (x, y)ϕ(y)dσ(y) for x ∈ R 2 + , x ∈ ∂B and (K * B♯ ) + : H -1 2 (∂B) -→ H -1 2 (∂B) ϕ -→ (K * B,♯ ) + [ϕ](x) = ∂B ∂G + ♯ (x, y) ∂ν(x) ϕ(y)dσ(y) (5.10)
for x ∈ ∂B.

It can be easily proved that all the results of Lemma 5.3.2 hold true for S + B♯ and (K * B♯ ) + . Moreover, for any ϕ ∈ H -1 2 (∂B), we have

S + B,♯ [ϕ](x) = 0 for x ∈ ∂R 2 + .
Now, we can readily see that α can be represented as

α = S + B,♯ [ϕ], where ϕ ∈ H -1 2 (∂B) satisfies 1 µ m ∂S + B,♯ [ϕ] ∂ν + - 1 µ c ∂S + B,♯ [ϕ] ∂ν - = 1 µ c - 1 µ m ν 2 on ∂B.
Using the jump formula from Lemma 5.3.2, we arrive at

λ µ Id -(K * B♯ ) + [ϕ] = ν 2 ,
where

λ µ = µ c + µ m 2(µ c -µ m )
.

Therefore, using item (v) in Lemma 5.3.2 on the characterization of the spectrum of K * B♯ and the fact that the spectra of (K * B♯ ) + and K * B♯ are the same, we obtain that

α = S + B,♯ λ µ Id -(K * B♯ ) + -1 [ν 2 ]. Lemma 5.4.1. Let x = (x 1 , x 2 ).
Then, for x 2 → +∞, the following asymptotic expansion holds:

α = α ∞ + O(e -x 2 ), with α ∞ = - ∂B y 2 λ µ Id -(K * B♯ ) + -1 [ν 2 ](y)dσ(y).
Proof. The result follows from an asymptotic analysis of G + ♯ (x, y). Indeed, suppose that x 2 → +∞, we have

G + ♯ (x, y) = 1 4π log sinh 2 (π(x 2 -y 2 )) + sin 2 (π(x 1 -y 1 )) - 1 4π log sinh 2 (π(x 2 + y 2 )) + sin 2 (π(x 1 -y 1 )) = 1 4π log sinh 2 (π(x 2 -y 2 )) - 1 4π log sinh 2 (π(x 2 + y 2 )) +O log 1 + 1 sinh 2 (x 2 ) = 1 2π log e π(x 2 -y 2 ) -e -π(x 2 +y 2 ) 2 -log e π(x 2 +y 2 ) -e -π(x 2 -y 2 ) 2 + O log 1 + e -x 2 2 = -y 2 + O(e -x 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

(K * B♯ ) + : H * 0 → H * 0 is a compact self-adjoint operator
, where H * 0 is defined as in Lemma 5.3.2, we can write

α ∞ = - ∂B y 2 λ µ Id -(K * B♯ ) + -1 [ν 2 ](y)dσ(y), = - ∂B y 2 ∞ j=1 (ϕ j , ν 2 ) H * 0 ϕ j (y) λ µ -λ j dσ(y), = ∞ j=1 (ϕ j , ν 2 ) H * 0 (ϕ j , y 2 ) -1 2 , 1 2 λ µ -λ j ,
where λ 1 , λ 2 , . . . are the eigenvalues of (K * B♯ ) + and ϕ 1 , ϕ 2 , . . . is a corresponding orthornormal basis of eigenfunctions.

On the other hand, by integrating by parts we get

(ϕ j , y 2 ) -1 2 , 1 2 = 1 1 2 -λ j (ϕ j , ν 2 ) H * 0 .
This together with the fact that ℑm λ µ < 0 (by the Drude model [START_REF] Ammari | Surface plasmon resonance of nanoparticles and applications in imaging[END_REF]), yield the following lemma.

Lemma 5.4.2. We have ℑm α ∞ > 0.

Finally, we give a formula for the shape derivative [START_REF] Ammari | Layer Potential Techniques in Spectral Analysis[END_REF] 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 B η be an η-perturbation of B; i.e., let Chapter 5. Plasmonic Metasurfaces h ∈ C 1 (∂B) and ∂B η be given by

∂B η = x + ηh(x)ν(x), x ∈ ∂B .
Following [START_REF] Ammari | The generalized polarization tensors for resolved imaging. Part I: Shape reconstruction of a conductivity inclusion[END_REF] (see also [START_REF] Ammari | Mathematical and Statistical Methods for Multistatic Imaging[END_REF]), we can prove that

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

                   ∆v = 0 in R 2 + \B ∪ B, v| + -v| -= 0 on ∂B, ∂v ∂ν + - µ m µ c ∂v ∂ν - = 0 on ∂B, v -x 2 → 0 as x 2 → +∞, and                    ∆w = 0 in R 2 + \B ∪ B, µm µc w| + -w| -= 0 on ∂B, ∂w ∂ν + - ∂w ∂ν - = 0 on ∂B, w -x 2 → 0 as x 2 → +∞,
respectively. Therefore, the following lemma holds.

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

d S α ∞ (B) = ( µ m µ c -1) ∂v ∂ν - ∂w ∂ν -+ µ c µ m ∂v ∂τ - ∂w ∂τ -.
If we aim to maximize the functional J := 1 2 |α ∞ | 2 over B, then it can be easily seen that J is Fréchet differentiable and its Fréchet derivative is given by ℜe d S α ∞ (B)α ∞ (B). As in [START_REF] Ammari | Generalized polarization tensors for shape description[END_REF], in order to include cases where topology changes and multiple components are allowed, a level-set version of the optimization procedure described below can be developed.

Numerical illustrations

Setup and methods

We use the Drude model [START_REF] Ammari | Surface plasmon resonance of nanoparticles and applications in imaging[END_REF] 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 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 x 2 = 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.

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.

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 to two-dimensional arrays embedded in R 3 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.

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 [START_REF] Keren | Describing complicated objects by implicit polynomials[END_REF][START_REF] Taubin | Parametrized families of polynomials for bounded algebraic curve and surface fitting[END_REF][START_REF] Fatemi | Sampling and reconstruction of shapes with algebraic boundaries[END_REF]. 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 [START_REF] Gustafsson | Reconstructing planar domains from their moments[END_REF][START_REF] Lasserre | Algebraic-exponential data recovery from moments[END_REF]. 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.

Plasmonic resonance for algebraic domains

Algebraic domains of class

Q Let Ω be the unit disk in C. For m ∈ N and a ∈ R, define Φ m,a : C \ Ω → C by Φ m,a (ζ) = ζ + a ζ m .
Assume that Φ m,a is injective on C \ Ω. We introduce the class Q as the collection of all bounded domains D ⊂ C bounded by the curves

∂D = {Φ m,a (ζ) : |ζ| = r 0 } for some r 0 > 1, m ∈ N and a ∈ R. Note that Φ m,a is a conformal mapping from {|ζ| > r 0 } onto C \ D.
In what follows, we shall suppress the subscript m, a from Φ m,a for the ease of notation.

Conformal images of the unit disc by rational functions are also called quadrature domains. We refer to [START_REF] Gustafsson | Topics on quadrature domains[END_REF][START_REF]Quadrature Domains and Their Applications[END_REF] for details and ramifications of the theory of quadrature domains. In particular, up to the inversion z → 1/z, the complements of the domains in class Q are quadrature domains. We write for convenience ζ = e ρ+iθ . Let ρ 0 be such that r 0 = e ρ 0 . Let J be the Jacobian defined by

J = ∂ ξ Φ(e ξ ) | ξ=ρ+iθ .
In the (ρ, θ) plane, the normal derivative ∂/∂ν on ∂D is represented as

∂ ∂ν = 1 J ∂ ∂ρ .
Moreover, the boundary ∂D is parametrized by θ → Φ(e ρ 0 +iθ ) = e ρ 0 +iθ + ae -mρ 0 -imθ .

If we fix the constant a and change ρ 0 , 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)ρ 0 δ. (

Then the boundary ∂D can be represented as

θ → Φ(e ρ 0 +iθ ) = e ρ 0 (e iθ + δe -imθ ). (6.2)
Now, if we fix the constant δ and change ρ 0 , then it is clear that only the size changes and the shape stays unaffected. The parameter e ρ 0 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 ρ 0 .

Explicit computation of the Neumann-Poincaré operator

In this section, we compute the Neumann-Poincaré operator on ∂D explicitly. We need to compute K * D [J -1 cos nθ] and K * D [J -1 sin nθ] explicitly. Our 136 Chapter 6. Shape Recovery of Algebraic Domains

strategy is as follows. Let u = S D [J -1 cos nθ] and v = S D [J -1 sin nθ]. If u, v can be obtained explicitly, then K * D [J -1 cos nθ] and K * D [J -1
sin nθ] are immediately derived by using the following identity:

K * D [ϕ] = 1 2 ∂S[ϕ] ∂ν + + ∂S[ϕ] ∂ν - , (6.3)
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, ∂u ∂ν + - ∂u ∂ν - = J -1 cos nθ on ∂D, u = O(|z| -1 ) as |z| → ∞. (6.4)
Let u(ρ, θ) = (u • Φ)(e ρ+iθ ). Since Φ(ζ) is conformal on |ζ| > e ρ 0 , the above problem can be rewritten as follows:

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

We can show that, for 1 ≤ n ≤ m, u| D in equation (6.5) can be explicitly solved by using the following ansatz:

u| D (z) ∝ ℜ{z n } = ℜ ζ + a ζ m n = ℜ n k=0 n k ζ n-k a ζ m k (ζ = e ρ+iθ ) = e nρ cos nθ + n k=1 a k n k e -t mn k ρ cos t mn k θ, (6.6)
6.2. Plasmonic resonance for algebraic domains 137 where the constant t mn k is defined by

t mn k = (m + 1)k -n, 0 ≤ k ≤ n.
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| D (z) turns out to be more complicated polynomial than z n but is still a polynomial of degree n.) Let us assume 1 ≤ n ≤ m. In view of (6.6), we define

w(ρ, θ) : =              e nρ cos nθ + n k=1 a k n k e -t mn k ρ cos t mn k θ, ρ < ρ 0 , e -n(ρ-2ρ 0 ) cos nθ + n k=1 a k n k e -t mn k ρ cos t mn k θ, ρ > ρ 0 .
Note that w is harmonic in {ρ < ρ 0 } and {ρ > ρ 0 } and w = O(e -ρ ) as

ρ → ∞. Moreover,      w| + = w| - on ρ = ρ 0 , ∂w ∂ρ + - ∂w ∂ρ - = (-2)ne nρ 0 cos nθ on ρ = ρ 0 . (6.7)
Therefore, the function w is equal to u up to a multiplicative constant. More precisely, we have

u(ρ, θ) = - 1 2n e -nρ 0 w(ρ, θ). (6.8)
Now we are ready to compute K * D [J -1 cos nθ]. We can check that

1 2 ∂w ∂ρ + ρ=ρ 0 + ∂w ∂ρ - ρ=ρ 0 = n k=1 -t mn k a k n k e -t mn k ρ 0 cos t mn k θ. (6.9)
Then it follows from (6.3) and (6.8) that

K * D [J -1 cos nθ] = 1 J n k=1 δ k t mn k 2n n k cos t mn k θ (6.10)
for 1 ≤ n ≤ m. In exactly the same manner, we can show that

K * D [J -1 sin nθ] = - 1 J n k=1 δ k t mn k 2n n k sin t mn k θ. (6.11)
It is worth mentioning that we can also compute the single layer potentials for J -1 cos nθ and J -1 sin nθ:

S D [J -1 cos nθ] = - 1 2n cos nθ - 1 2n n k=1 δ k n k cos t mn k θ, (6.12)
and 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

S D [J -1 sin nθ] = - 1 2n sin nθ + 1 2n n k=1 δ k n k sin t mn k θ. ( 6
K * D [J -1 cos nθ] = δ (m + 1 -n) 2 J -1 cos(m + 1 -n)θ + O(δ 2 ), K * D [J -1 sin nθ] = -δ (m + 1 -n) 2 J -1 sin(m + 1 -n)θ + O(δ 2 ) (6.14)
for small δ and 1 ≤ n ≤ m. One can verify the decay

K * D [J -1 cos nθ], K * D [J -1 sin nθ] = O(δ 2 ) (6.15)
for small δ and n ≥ m + 1.

Let us denote by

v c n = J -1 cos nθ, v s n = J -1 sin nθ,
and let V c and V s be the subspaces defined by

V c = span{v c 1 , v c 2 , . . . , v c n , . . . , v c m , . . .} and V s = span{v s 1 , v s 2 , . . . , v s n , . . . , v s m , . . .}.
In view of (6.14) and (6.15), we can easily see that the Neunamm-Poincaré operator K * D can be approximated by a finite rank operator for small δ. To state this fact, we define a finite rank operator F c m by

F c m [v c n ] = (m + 1 -n)v c m+1-n , 1 ≤ n ≤ m, 0, n ≥ m + 1. F c m [v s n ] = 0, n ≥ 1.
Similarly, we define F s m by

F s m [v s n ] = (m + 1 -n)v s m+1-n , 1 ≤ n ≤ m, 0, n ≥ m + 1. F s m [v c n ] = 0, n ≥ 1.
Then, on the subspace V c , we have

K * D = δ 2 F c m + O(δ 2 ).
Similarly, on the subspace V s , we have

K * D = - δ 2 F s m + O(δ 2
k, ± √ 1 • m, ± 2 • (m -1), . . . , ± (k -1) • (k + 1),
and the associated eigenvectors are given by

e k , e 1 ± √ m e m , e 2 ± √ m -1 e m-1 , . . . , √ k -1 e k-1 ± √ k + 1 e k+1 ,
where e i 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 eigenvectors are given by

e 1 ± √ m e m , √ 2 e 2 ± √ m -1 e m-1 , . . . , √ k e k ± √ k + 1 e k+1 .
Using (6.16), Lemma 6.2.1 and the perturbation theory [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF], we get the following asymptotic result for K * D on V c . Theorem 6.2.1. For small δ, we have the following asymptotic expansions of eigenvalues and eigenfunctions of K * D on V c : (i) If m is odd, that is, m = 2k -1 for some k ∈ N:

Eigenvalues: up to order δ δ 2 × k, ± √ 1 • m, ± 2 • (m -1) , . . . , ± (k -1) • (k + 1)
.

Eigenfunctions: up to order δ 0

v c k , v c 1 ± √ m v c m , √ 2 v c 2 ± √ m -1 v c m-1 , . . . , √ k -1 v c k-1 ± √ k + 1 v c k+1 .
(ii) If m is even, that is, m = 2k for some k ∈ N:

Eigenvalues: up to order δ δ 2 × ± √ 1 • m, ± 2 • (m -1) , . . . , ± k • (k + 1) .
Chapter 6. Shape Recovery of Algebraic Domains

Eigenfunctions: up to order δ 0

v c 1 ± √ m v c m , √ 2 v c 2 ± √ m -1 v c m-1 , . . . , √ k v c k ± √ k + 1 v c k+1 .
Similarly, we have the following result for K * D on the subspace V s . Theorem 6.2.2. We have the following asymptotic expansion of eigenvalues and eigenfunctions of the Neumann-Poincaré operator K * D on the subspace V s for small δ:

(i) If m is odd, that is, m = 2k -1 for some k ∈ N: Eigenvalues: up to order δ - δ 2 × k, ± √ 1 • m, ± 2 • (m -1) , . . . , ± (k -1) • (k + 1)
.

Eigenfunctions: up to order δ 0

v s k , v s 1 ± √ mv s m , √ 2v s 2 ± √ m -1v s m-1 , . . . , √ k -1 v s k-1 ± √ k + 1 v s k+1 .
(ii) If m is even, that is, m = 2k for some k ∈ N:

Eigenvalues: up to order δ - δ 2 × ± √ 1 • m, ± 2 • (m -1) , . . . , ± k • (k + 1)
.

Eigenfunctions: up to order δ 0

v s 1 ± √ m v s m , √ 2 v s 2 ± √ m -1 v s m-1 , . . . , √ k v s k ± √ k + 1 v s k+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 K * D has the following 2k eigenvalues:

δ 2 × ± √ 1 • m, ± 2 • (m -1) , . . . , ± (k -1) • (k + 1), ± √ k • k .

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,

λ := ε m + ε c 2(ε m -ε c ) ,
for a domain D with permittivity ε c and backgroud with permittivity ε m . 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.

Since m is odd, the shape of D has even symmetry with respect to both x 1 -axis and x 2 -axis. Thanks to this symmetry, M (λ, D) has the following simple form [START_REF] Ammari | Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory[END_REF]:

M (λ, D) = m 11 1 0 0 1 ,
where m 11 is given by

m 11 = x 1 , (λI -K * D ) -1 [ν 1 ] 1 2 ,- 1 2
.

Let λ j and ϕ j , j ∈ N, be the eigenvalues and the (normalized) eigenfunctions of K * D , respectively. Then, from the spectral decomposition of K * D , we have (see chapter 1)

m 11 = j 1 λ -λ j (x 1 , ϕ j ) 1 2 ,-1 2 (-S D [ν 1 ], ϕ j ) 1 2 ,-1 2 (-S D [ϕ j ], ϕ j ) 1 2 ,-1 2 = j ( 1 2 -λ j ) λ -λ j |(x 1 , ϕ j ) 1 2 ,-1 2 | 2 (-S D [ϕ j ], ϕ j ) 1 2 ,- 1 2
.

By Theorems 6.2.1 and 6.2.2, one can see that only the following two eigenvalues and two eigenfunctions contribute to m 11 up to order δ:

eigenvalues λ ± := ± 1 2 δ √ m, eigenfunctions ϕ ± := v c 1 ± √ m v c m .
In fact, for other eigenfunctions, we have (x 1 , ϕ j ) 1 2 ,-

(x 1 , ϕ ± ) 1 2 ,-1 2 = 2π 0 (e ρ 0 cos θ + ae -mρ 0 cos mθ)(cos θ ± √ m cos mθ) dθ, = πe ρ 0 (1 + 2λ ± ). Now, we compute (S D [ϕ ± ], ϕ ± ) 1 2 ,- 1 2
. Note that, from (6.12), we have

S D [v c n ] = - 1 2n cos nθ - 1 2 δ cos(m + 1 -n)θ + O(δ 2 ).
Consequently,

(-S D [ϕ ± ], ϕ ± ) 1 2 ,-1 2 = (-S D [v c 1 ± √ m v c m ], v c 1 ± √ m v c m ) 1 2 ,-1 2 = 2π 0 (cos θ ± √ m cos mθ) × ( 1 2 cos θ + δ 2 cos mθ ± 1 2 √ m cos mθ ± √ m δ 2 cos θ) dθ + O(δ 2 ) = π(1 + 2λ ± ) + O(δ 2 ).
Finally, we are ready to obtain an approximation formula for m 11 .

Theorem 6.2.3. We have

m 11 = π 2 e 2ρ 0 1 λ -λ + + 1 λ -λ - + O(δ 2 ), (6.18)
as δ → 0.

Second-order contracted generalized polarization tensors

Let M cc mn , M ss mn , M sc mn , and M cs mn be the contracted generalized polarization tensors. One can easily see that M sc 22 = M cs 22 = 0 and M 12 = M 21 = 0. We only need to consider M cc 22 and M ss 22 . It turns out that only the following two eigenvalues and two eigenfunctions contribute to M cc 22 (up to the order δ):

eigenvalues λ ′ ± := ± 1 2 δ 2 • (m -1), eigenfunctions ϕ ′ ± := √ 2 v c 2 ± √ m -1 v c m-1 .
Let H := ℜ (x 1 + ix 2 ) 2 . Then we have H| ∂D = e 2ρ 0 (cos 2θ + 2δ cos mθ + δ 2 cos 2mθ).

Therefore,

(H, ϕ ′ ± ) 1 2 ,-1 2 = 2π 0 e 2ρ 0 (cos 2θ + 2δ cos mθ + δ 2 cos 2mθ)( √ 2 cos 2θ ± √ m -1 cos(m -1)θ) dθ = √ 2πe 2ρ 0 . Now we compute (-S D [ϕ ′ ± ], ϕ ′ ± ) 1 2 ,-1 2 . Since S D [v c n ] = - 1 2n cos nθ - 1 2 δ cos(m + 1 -n)θ + O(δ 2 ),
we obtain

(-S D [ϕ ′ ± ], ϕ ′ ± ) 1 2 ,-1 2 = (-S D [ √ 2 v c 2 ± √ m -1 v c m-1 ], √ 2 v c 2 ± √ m -1 v c m-1 ) 1 2 ,-1 2 = 2π 0 ( √ 2 cos 2θ ± √ m -1 cos(m -1)θ) × ( 1 2 √ 2 cos 2θ + δ √ 2 cos(m -1)θ ± 1 2 √ m -1 cos(m -1)θ ± √ m -1 δ 2 cos 2θ) dθ + O(δ 2 ) = π(1 + 2λ ′ ± ) + O(δ 2 ).
Finally, we find

M cc 22 = j ( 1 2 -λ j ) (λ -λ j ) |(H, ϕ j ) 1 2 ,-1 2 | 2 (-S D [ϕ j ], ϕ j ) 1 2 ,-1 2 , = πe 4ρ 0 ( 1 2 -λ ′ + ) ( 1 2 + λ ′ + )(λ -λ ′ + ) + ( 1 2 + λ ′ -) ( 1 2 -λ ′ -)(λ -λ ′ -) + O(δ 2

Classification of algebraic domains in the class Q

The identification of the parameters ρ 0 and m is now straightforward using the results of the previous subsection. Suppose that we can obtain the values of λ ± , λ ′ ± approximately from m 11 and M cc 22 . Then, by formula (6.18), we can easily find the parameter ρ 0 , which determines the size of D. In order to reconstruct the parameters m and δ we turn to the definitions of λ + and λ ′ + :

λ + = 1 2 δ √ m, λ ′ + = 1 2 δ 2 • (m -1).
It is worth emphasizing that the eigenvalues λ + and λ ′ + are first-order approximations of the exact eigenvalues of Neumann-Poincaré operator K * D for small δ.

Solving the above equations for m and δ yields the exact formulas:

m = λ 2 + λ 2 + -(λ ′ + ) 2 /2 , δ = 2 λ 2 + -(λ ′ + ) 2 /2.

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 R 2 . As an application of the spectral decomposition of the Neumann-Poincaré operator, we derive the (1, 1)-entry, m 11 , of the first-order polarization tensor associated with the two disks. See chapter 1 for a brief introduction on polarization tensors.

The bipolar coordinates and the boundary integral operators

Let B 1 and B 2 be two disks with conductivity σ embedded in the background with conductivity 1. The conductivity is such that 0 < σ = 1 < ∞. Let σ B 1 ∪B 2 denote the conductivity distribution, i.e.,

σ B 1 ∪B 2 = σχ(B 1 ) + σχ(B 2 ) + χ(R 2 \ (B 1 ∪ B 2 ), (6.20)
where χ is the characteristic function. Let ǫ be the distance between two disks, that is,

ǫ := dist(B 1 , B 2 ).
We set Cartesian coordinates (x 1 , x 2 ) such that x 1 -axis is parallel to the line joining the centers of the two disks.

(Definition) Each point x = (x 1 , x 2 ) in the Cartesian coordinate system corresponds to (ξ, θ) ∈ R × (-π, π] in the bipolar coordinate system through the equations

x 1 = α sinh ξ cosh ξ -cos θ and x 2 = α sin θ cosh ξ -cos θ (6.21)
with a positive number α. In fact, the bipolar coordinates can be defined using a conformal mapping. Define a conformal map Ψ by

z = x 1 + ix 2 = Ψ(ζ) = α ζ + 1 ζ -1 .
If we write ζ = e ξ-iθ , 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: (Separation of variables) The bipolar coordinate system admits separation of variables for any harmonic function f as follows:

f ξ (x, y) = x -α cosh c sinh c 2 + y 2 - α sinh c 2 (6.
f (ξ, θ) = a 0 + b 0 ξ + c 0 θ + ∞ n=1
(a n e nξ + b n e -nξ ) cos nθ+ c n e nξ + d n e -nξ ) sin nθ , (

where a n , b n , c n and d n are constants. For ξ > 0, we have

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

x 1 = sgn(ξ)α 1 + 2 ∞ n=1 e -n|ξ| cos nθ , (6.27)
and

x 2 = 2α ∞ n=1
e -n|ξ| sin nθ.

Let K * be the Neumann-Poincaré operator given by

K * :=    K * B 1 ∂ ∂ν (1) S B 2 ∂ ∂ν (2) S B 1 K * B 2    ,
and define the operator S by

S = S B 1 S B 2 S B 1 S B 2 .
Here, ν (i) is the outward normal on ∂B i , i = 1, 2. Then, from [START_REF] Ammari | Spectral analysis of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance[END_REF], K * is self-adjoint with the inner product

(ϕ, ψ) H * := -(S[ψ], ϕ) H 1/2 (∂B 1 )×H 1/2 (∂B 2 ),H -1/2 (∂B 1 )×H -1/2 (∂B 2 ) , for ϕ, ψ ∈ H -1/2 0 (∂B 1 ) × H -1/2 0 (∂B 2 ).

Neumann Poincaré-operator for two separated disks and its spectral decomposition

First we introduce some notations. Set

α = ǫ(r + ǫ 4
) and ξ 0 = sinh -1 α r , for j = 1, 2, (6.28)

where r is the radius of the two disks and ǫ their separation distance. Note that ∂B j = {ξ = (-1) j ξ 0 }, for j = 1, 2. ǫ , we use the following lemma [START_REF] Ammari | Spectral analysis of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance[END_REF]. Lemma 6.3.1. Assume that there exists u a nontrivial solution to the following equation:

             ∆u = 0 in B 1 ∪ B 2 ∪ R 2 \ (B 1 ∪ B 2 ), u| + = u| - on ∂B j , j = 1, 2, ∂u ∂ν + = σ 0 ∂u ∂ν - on ∂B j , j = 1, 2, u(x) → 0 as |x| → ∞, (6.30)
where

σ 0 = - 1 + 2λ 0 1 -2λ 0 < 0. If we set ψ j := ∂u ∂ν + ∂B j - ∂u ∂ν - ∂B j , for j = 1, 2, then ψ = ψ 1 ψ 2
is an eigenvector of K * ǫ corresponding to the eigenvalue λ 0 .

One can see that the following function u n is a solution to (6.30):

u ± n (ξ, θ) = (const.) +                ∓ 1 2|n|
(e |n|ξ 0 ∓ e -|n|ξ 0 )e |n|ξ+inθ , for ξ < -ξ 0 ,

1 2|n|
e -|n|ξ 0 (e |n|ξ ∓ e -|n|ξ )e inθ , for -ξ 0 < ξ < ξ 0 ,

1 2|n|
(e |n|ξ 0 ∓ e -|n|ξ 0 )e -|n|ξ+inθ , for ξ > ξ 0 .

(6.31) From (6.31) and Lemma 6.3.1, we obtain eigenvalues and eigenvectors to K * ǫ λ ± ǫ,n = ± 1 2 e -2|n|ξ 0 and Φ ± ǫ,n (θ) = e inθ h(-ξ 0 , θ) ∓h(ξ 0 , θ) .

Note that the above eigenvectors are not normalized. We compute

(-S[Φ ± ǫ,n ], Φ ± ǫ,n ) 1 2 ,- 1 2
. From (6.31), one can see that

S[Φ ± ǫ,n ] = (const.) + ∓ 1 2|n| (1 ∓ e -2|n|ξ 0 )e inθ 1 2|n| (1 ∓ e -2|n|ξ 0 )e inθ .
It follows that

(-S[Φ ± ǫ,n ], Φ ± ǫ,n ) 1 2 ,-1 2 = 2π |n| (1 ∓ e -2|n|ξ 0 ).
Therefore, we arrive at the following result.

Theorem 6.3.1. We have the following spectral decomposition of K * ǫ :

K * ǫ = n =0 1 2 e -2|n|ξ 0 Ψ + ǫ,n ⊗ Ψ + ǫ,n + n =0 - 1 2 e -2|n|ξ 0 Ψ - ǫ,n ⊗ Ψ - ǫ,n ,
where Ψ ± ǫ,n are the normalized eigenvectors defined by

Ψ ± ǫ,n (θ) := |n|e inθ 2π(1 ∓ e -2|n|ξ 0 )
h(-ξ 0 , θ) ∓h(ξ 0 , θ) . (6.32)

Note that

(S B 1 [Ψ ± ǫ,n,1 ] + S B 2 [Ψ ± ǫ,n,2 ])(ξ, θ) = (const.) + |n| 2π(1 ∓ e -2|n|ξ 0 ) (6.33) ×                ∓ 1 2|n|
(e |n|ξ 0 ∓ e -|n|ξ 0 )e |n|ξ+inθ , for ξ < -ξ 0 ,

1 2|n|
e -|n|ξ 0 (e |n|ξ ∓ e -|n|ξ )e inθ , for -ξ 0 < ξ < ξ 0 ,

1 2|n|
(e |n|ξ 0 ∓ e -|n|ξ 0 )e -|n|ξ+inθ , for ξ > ξ 0 .

(6.34)

The Polarization tensor

Let us compute the (1, 1)-entry m ǫ 11 of the first-order polarization tensor for two separated disks. Note that

m ǫ 11 = ϕ, (λI -K * ǫ ) -1 [ψ] 1 2 ,- 1 2
,

where φ = x 1 | ∂B 1 x 1 | ∂B 2 , ψ = ν 1 | ∂B 1 ν 1 | ∂B 2 .
The spectral decomposition of K * ǫ implies

m ǫ 11 = n =0 φ, Ψ + ǫ,n ) 1 2 ,-1 2 (Ψ + ǫ,n , ψ) H * λ -λ + ǫ,n + n =0 φ, Ψ - ǫ,n ) 1 2 ,-1 2 (Ψ - ǫ,n , ψ) H * λ -λ - ǫ,n = n =0 1 2 -λ + ǫ,n | φ, Ψ + ǫ,n ) 1 2 ,-1 2 | 2 λ -λ + ǫ,n + n =0 1 2 -λ - ǫ,n | φ, Ψ - ǫ,n ) 1 2 ,-1 2 | 2 λ -λ - ǫ,n
.

From (6.27), we derive the expansion

x 1 = sgn(ξ)α ∞ m=-∞ e -|m||ξ|+imθ . (6.35)
Therefore,

φ, Ψ + ǫ,n ) 1 2 ,-1 2 = 2 2π 0 -α ∞ m=-∞ e -|m|ξ 0 +imθ |n|h(-ξ 0 , θ)e -inθ 2π(1 -e -2|n|ξ 0 ) 1 h(-ξ 0 , θ) dθ = -2 √ 2πα |n|e -|n|ξ 0 1 -e -2|n|ξ 0 , and φ, Ψ - ǫ,n ) 1 2 ,-1 2 = 0.
As a consequence, we arrive at the following result.

Chapter 6. Shape Recovery of Algebraic Domains Propsition 6.3.1. We have

m ǫ 11 = n =0 4πα 2 |n|e -2|n|ξ 0 λ -λ + ǫ,n = 8πα 2 ∞ n=1 ne -2nξ 0 λ -1 2 e -2nξ 0
, where α is given by (6.28).

Reconstruction of the separation distance

Suppose that the first eigenvalue

λ + ǫ,1 = 1 2 e -2ξ 0
is measured. Then we immediately find the value of e ξ 0 . From (6.28), we have

r cosh ξ 0 = ǫ 2 + r.
By solving the above quadratic equation, we can determine the distance ǫ between the two disks.

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 [START_REF] Ammari | Surface plasmon resonance of nanoparticles and applications in imaging[END_REF].

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 11 | and |M cc 22 | with respect to the wavelength. This allow us, in the case of class Q of algebraic domains, to recover λ + and λ ′ + and, in the case of two separated disks, to recover λ + ǫ,1 . Figures 6.2

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 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.

Chapter 6. Shape Recovery of Algebraic Domains

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 [START_REF] Bao | Near-field imaging of infinite rough surfaces[END_REF][START_REF] Bao | Near-field imaging of the surface displacement on an infinite ground plane[END_REF]. 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 [START_REF] Ammari | A mathematical theory of super-resolution by using a system of sub-wavelength Helmholtz resonators[END_REF]. 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.

Multiple plasmonic nanoparticles

Layer potential formulation in the multi-particle case

We consider the scattering of an incident time harmonic wave u i 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 D l = z l + δB, l = 1, 2, . . . , L, where B has size one and is centered at the origin. Moreover, we denote D 0 = δB as our reference

= L l=1 D l , ε D = ε m χ(R 3 \ D) + ε c χ( D), µ D = µ m χ(R 3 \ D) + µ c χ(D).
The scattering problem can be modeled by the following Helmholtz equation:

                   ∇ • 1 µ D ∇u + ω 2 ε D u = 0 in R 3 \∂D, u + -u -= 0 on ∂D, 1 µ m ∂u ∂ν + - 1 µ c ∂u ∂ν - = 0 on ∂D, u s := u -u i satisfies the Sommerfeld radiation condition. (7.1) Let u i (x) = e ikmd•x , F l,1 (x) = -u i (x) ∂D l = -e ikmd•x ∂D l , F l,2 (x) = - ∂u i ∂ν (x) ∂D l = -ik m e ikmd•x d • ν(x) ∂D l ,
and define the operator K k Dp,D l by

K k Dp,D l [ψ](x) = ∂Dp ∂G(x, y, k) ∂ν(x) ψ(y)dσ(y), x ∈ ∂D l .
Analogously, we define

S k Dp,D l [ψ](x) = ∂Dp G(x, y, k)ψ(y)dσ(y), x ∈ ∂D l .
The solution u of (7.1) can be represented as follows:

u(x) =              u i + L l=1 S km D l [ψ l ], x ∈ R 3 \ D, L l=1 S kc D l [φ l ], x ∈ D,
where φ l , ψ l ∈ H -1 2 (∂D l ) satisfy the following system of integral equations

                   S km D l [ψ l ] -S kc D l [φ l ] + p =l S km Dp,D l [ψ p ] = F l,1 , 1 µ m 1 2 Id + (K km D l ) * [ψ l ] + 1 µ c 1 2 Id -(K kc D l ) * [φ l ] + 1 µ m p =l K km Dp,D l [ψ p ] = F l,2 , and      F l,1 = -u i on ∂D l , F l,2 = - 1 µ m ∂u i ∂ν on ∂D l .

Feld behavior at plasmonic resonances in the multiparticle 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, S kc D is invertible. Note that

φ l = (S kc D l ) -1 S km D l [ψ l ] + p =l S km Dp,D l [ψ p ] -F l,1 .
We obtain the following equation for ψ l 's,

A D (w)[ψ] = f,
where

A D (w) =      A D 1 (ω) A D 2 (ω) . . . A D L (ω)      +      0 A 1,2 (ω) • • • A 1,L (ω) A 2,1 (ω) 0 • • • A 2,L (ω) . . . • • • 0 . . . A L,1 (ω) • • • A L,L-1 (ω) 0      , ψ =      ψ 1 ψ 2 . . . ψ L      , f =      f 1 f 2 . . . f L      , and
A D l (ω) = 1 µ m 1 2 Id + (K km D l ) * + 1 µ c 1 2 Id -(K kc D l ) * (S kc D l ) -1 S km D l , A l,p (ω) = 1 µ c 1 2 Id -(K kc D l ) * (S kc D l ) -1 S km Dp,D l + 1 µ m K km Dp,D l , f l = F l,2 + 1 µ c 1 2 Id -(K kc D l ) * (S kc D l ) -1 [F l,1 ].
The following asymptotic expansions hold (see chapter 1 for the definition of H * (∂D) and chapter 2 and Appendix B for the definition of the operators). we have

A D j (ω) = A D j ,0 + O(δ 2 ω 2 ),
(ii) Regarded as operators from H * (∂D l ) into H * (∂D j ), we have

A l,p (ω) = 1 µ c 1 2 Id-K * D l S -1 D l S p,l,0,1 +S p,l,0,2 + 1 µ m K p,l,0,0 +O(δ 2 ω 2 )+O(δ 4 ).
where

ψ = j∈J L l=1 f, ϕ j,l (ω) H * ϕ j,l (ω) τ j,l (ω) + A D (ω) -1 (P J c (ω)f ) = j∈J L l=1 (d • ν(x), ϕ j ) H * (∂D 0 ) Z X j,l ϕ j,l + O(ω 2 δ 3 2 ) λ -λ j + 1 µc -1 µm -1 τ j,l + O(δ 4 ) + O(δ 2 ω 2 ) + O(ωδ 3
2 ).

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

min j∈J |τ j,l (ω)| ≫ ω q δ p ,
for some integer p and q, and τ j,l (ω) = τ j,l,p,q + o(ω q δ p ),

we have

ψ = j∈J L l=1 (d • ν(x), ϕ j ) H * (∂D 0 ) Z X j,l ϕ j,l + O(ω 2 δ 3 2 ) τ j,l,p,q + O(ωδ 3
2 ).

Super-resolution (super-focusing) by using plasmonic particles

In [START_REF] Ammari | A mathematical theory of super-resolution by using a system of sub-wavelength Helmholtz resonators[END_REF][START_REF] Ammari | Super-resolution in high contrast media[END_REF], a rigorous mathematical theory is developed to explain the superresolution phenomenon in microstructures with high contrast material around the source point. Such microstructures act like arrays of subwavelength sensors. 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.

Asymptotic expansion of the scattered field

In order to illustrate the super-resolution phenomenon, we set

u i (x) = G(x, x 0 , k m ) = - e ikm|x-x 0 | 4π|x -x 0 | . Lemma 7.3.1.
In the space H * (∂D), as ω goes to zero, we have

f = f 0 + O(ωδ 3
2 ) + O(δ 5
2 ),

where f 0 = (f 0,1 , . . . , f 0,L ) T with

f 0,l = - 1 4π|z l -x 0 | 3 1 µ m (z l -x 0 ) • ν(x) + 1 µ c ( 1 2 Id -K * D l )S -1 D l [(z l -x 0 ) • (x -z l )] = O(δ 3
2 ).

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

f l = F l,2 + 1 µ c 1 2 Id -(K kc D l ) * (S kc D l ) -1 [F l,1 ].
We can show that

F l,2 = - 1 µ m ∂u i ∂ν = - 1 4πµ m |z l -x 0 | 3 (z l -x 0 )•ν(x)+O(δ 5 2 )+O(ωδ 3 2 ) in H * (∂D l ).
Besides,

u i (x)| ∂D l = - e ikm|z l -x 0 | 4π|z l -x 0 | χ(∂D l )+ 1 4π|z l -x 0 | 3 (z l -x 0 )•(x-z l )+O(δ 5 2 )+O(ωδ 3 2 ) in H(∂D l ). Using the identity ( 1 2 Id -K * D l )S -1 D l [χ(∂D l )] = 0, we obtain that 1 µ c 1 2 Id-(K kc D l ) * (S kc D l ) -1 [F l,1 ] = - 1 4π|z l -x 0 | 3 µ c ( 1 2 Id-K * D l )S -1 D l [(z l -x 0 )•(x-z l )].
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

D δ,k,C = x ∈ R 3 ; min 1≤l≤L |x -z l | ≥ Cδ, max 1≤l≤L |x -z l | ≤ 1 Ck . Lemma 7.3.2. Let ψ l ∈ H * (∂D l ) and let v(x) = S k D l [ψ l ](x). Then we have for x ∈ D δ,k,C , v(x) = G(x, z l , k) 1 |x -z l | -ik x -z l |x -z l | • ∂D 0 yψ l (y)dσ(y) + O(δ 5
2 ) ψ l H * (∂D l ) +G(x, z l , k) ∂D 0 ψ l (y)dσ(y).
Moreover, the following estimates hold

v(x) = O(δ 3
2 ) if ∂D 0 ψ l (y)dσ(y) = 0, v(x) = O(δ 1 2 ) if ∂D 0 ψ l (y)dσ(y) = 0.
Proof. We only consider the case when l = 0. The other case follows similarly or by coordinate translation. We have

v(x) = S k D 0 [ψ](x) = ∂D 0 G(x, y, k)ψ(y)dσ(y) = - ∂D 0 e ik|x-y| 4π|x -y| ψ(y)dσ(y). Since G(x, y, k) = G(x, 0, k) + |α=1| ∂G(x, 0, k) ∂y α y α + m≥2 |α=m| ∂ m G(x,
= - e ik|x| 4π|x| 1 |x| -ik x |x| = G(x, 0, k) 1 |x| -ik x α |x| ,
we obtain the required identity for the case l = 0. The estimate follows from the fact that

y α H(∂D 0 ) = O(δ 2|α|+1 2
).

This completes the proof of the lemma.

Denote by

S j,l (x, k) = G(x, z l , k) x -z l |x -z l | 2 • ∂D 0 yϕ j (y)dσ(y), S l (x, k) = G(x, z l , k) ∂D 0 ϕ 0 (y)dσ(y), H j,l (x 0 ) = - 1 4π|z l -x 0 | 3 (z l -x 0 ) • ν(x), ϕ j H * (∂D 0 ) .
It is clear that the following size estimates hold H j,p (x 0 ) X j,l,p X j,l,q S j,q (x, k m ) + O(δ 4 ) + O(ωδ 3 )

S j,l (x, k) = O(δ 3 2 ), S l (x, k) = O(δ 1 2 ), H j,l (x 0 ) = O(δ 3 2 ) forj = 0, H O,l (x 0 ) = 0.
λ -λ j + 1 µc -1 µm -1 τ j,l + O(δ 4 ) + O(δ 2 ω 2 ) + O(δ 3 ). Proof. With u i (x) = G(x, x 0 , k m ), we have ψ = j∈J 1≤l≤L a j,l ϕ j,l + 1≤l≤L a 0,l ϕ 0,l + O(δ 3
2 ), where

a j,l = (f, ϕ j,l ) H * (∂D) = (f 0 , ϕ j,l ) H * (∂D) + O(ωδ 3
2 ) + O(δ 5
2 ), = ( 1 µ c - 1 µ m ) X j,l,p H j,p (x 0 ) + O(ωδ 3
2 ) + O(δ 5
2 ),

a 0,l = (f, ϕ 0,l ) H * (∂D) = O(δ 5
2 ).

By Lemma 7.3.2, S km D [ϕ j,l ](x) = 1≤p≤L S km D [X j,l,p ϕ j e p ](x) = 1≤p≤L X j,l,p S km Dp [ϕ j ](x) = 1≤p≤L X j,l,p S j,p (x, k m ) + O(δ 5
2 ) + O(ωδ

2 ).

7.3. Super-resolution (super-focusing) by using plasmonic particles 169 On the other hand, for j = 0, we have

S km D [ϕ 0,l ](x) = O(δ 1 2 ), τ 0,l (ω) = τ 0 + O(δ 4 ) + O(δ 2 ω 2 ) = O(1).
Therefore, we can deduce that

u s = S km D [ψ](x) = j∈J 1≤l≤L a j,l S km D [ϕ j,l ] + 1≤l≤L a 0,l S km D [ϕ 0,l ] + O(δ 3 ), = j∈J L l=1 1 τ j,l (ω) ( 1 µ c - 1 µ m )H j,p (x 0 ) X j,l,p X j,l,q S j,q (x, k m ) + O(ωδ 3 ) + O(δ 4 ) +O(δ 3 ), = j∈J L l=1 H j,p (x 0 ) X j,l,p X j,l,q S j,q (x, k m ) + O(ωδ 3 ) + O(δ 4 ) λ -λ j + 1 µc -1 µm -1 τ j,l + O(δ 4 ) + O(δ 2 ω 2 ) + O(δ 3 ).

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

λ -λ j + 1 µ c - 1 µ m -1 τ j,l ≫ O(δ 4 ) + O(δ 2 ω 2 ), ℜ λ -λ j + 1 µ c - 1 µ m -1 τ j,l ℑ λ -λ j + 1 µ c - 1 µ m -1 τ j,l
for each l and j ∈ J, we have

ℑΓ(x, x 0 , k m ) = ℑG(x, x 0 , k m ) + O(δ 3 ) + j∈J L l=1 ℜ H j,p (x 0 ) X j,l,p X j,l,q S j,q (x, 0) + O(ωδ 3 ) + O(δ 4 ) ×ℑ 1 λ -λ j + 1 µc -1 µm -1 τ j,l ,
where x, x 0 ∈ D δ,km,C .

Note that ℜ H j,p (x 0 ) X j,l,p X j,l,q S j,q (x, 0) = O(δ 3 ). Under the conditions in Theorem 7.3.2, if we have additionally that

ℑ 1 λ -λ j + 1 µc -1 µm -1 τ j,l = O( 1 δ 3 )
for some plasmonic frequency ω, then the term in the expansion of ℑΓ(x, x 0 , k m ) which is due to resonance has size one and exhibits subwavelength peak with

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 [START_REF] Anker | Biosensing with plasmonic nanosensors[END_REF], 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 superresolution 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 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 [START_REF] Ammari | Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory[END_REF] 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.

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 1 ⊂ R 2 and D 2 ⊂ R 2 of class C 1,α for some 0 < α < 1, respectively. We denote the permittivity of the ordinary particle D 1 (or the plasmonic particle D 2 ) by ε 1 (or ε 2 ), respectively. The permittivity of the background medium is denoted by ε m . In other words, the permittivity distribution ε is given by ε

:= ε 1 χ(D 1 ) + ε 2 χ(D 2 ) + ε m χ(R 2 \(D 1 ∪ D 2 )).
The permittivity ε 2 of the plasmonic particle depends on the operating frequency and is modeled by the Drude model as

ε 2 = ε 2 (ω) = 1 - ω 2 p ω(ω + iγ) .
We assume the following condition on the size of the particles D 1 and D 2 .

Condition 8.1. The plasmonic particle D 2 has size of order one and is centered at a position that we denote by z; the ordinary particle D 1 has size of order δ ≪ 1 and is centered at the origin. Specifically, we write D 1 = δB, where the domain B has size of order one.

The total electric potential u satisfies the following equation:

                           ∇ • (ε∇u) = 0 in R 2 \(∂D 1 ∪ ∂D 2 ), u| + = u| - on ∂D 1 ∪ ∂D 2 , ε m ∂u ∂ν + = ε 1 ∂u ∂ν - on ∂D 1 , ε m ∂u ∂ν + = ε 2 ∂u ∂ν - on ∂D 2 , (u -u i )(x) = O(|x| -1 ), as |x| → ∞, (8.1)
where u i (x) = d • x is the incident potential with a constant vector d ∈ R 2 .

The Green function in the presence of a small particle

Let G D 1 (•, y) be the Green function at the source point y of a medium consisting of the particle D 1 , which is embedded in the free space. For every y / ∈ D 1 , G D 1 (•, y) satisfies the following equation:

                   ∇ • ε 1 χ(D 1 ) + ε m χ(R 2 \D 1 ) ∇u = δ y in R 2 \∂D 1 , u| + = u| - on ∂D 1 , ε m ∂u ∂ν + = ε 1 ∂u ∂ν - on ∂D 1 , u(x) = O(|x| -1 ), as |x| → ∞. (8.2)
We look for a solution of the form:

G D 1 (x, y) := G(x, y) + S D 1 [ψ](x), x ∈ R 2 \D 1 . (8.3)
Note that G D 1 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 potential, the density ψ must satisfy the following equation on ∂D 1 :

ε m 1 2 Id + K * D 1 [ψ] + ε 1 1 2 Id -K * D 1 [ψ] = (ε 1 -ε m ) ∂ ∂ν G(•, y). (8.4)
So we obtain

ψ = λ D 1 Id -K * D 1 -1 ∂ ∂ν G(•, y) , λ D 1 = ε 1 + ε m 2(ε 1 -ε m )
.

Therefore, from (8.3) and the uniqueness of a solution to (8.2), we have the following representation for the Green's function G D 1 :

G D 1 (x, y) = G(x, y)+S D 1 λ D 1 Id -K * D 1 -1 ∂ ∂ν G(•, y) (x) for x, y ∈ R 2 \D 1 .
(8.5)

Representation of the total potential

Here we derive a layer potential representation of the total potential u, which is the solution to (8.1). Let u D 1 be the total field resulting from the incident field u i and the ordinary particle D 1 (without the plasmonic particle D 2 ). Note that u D 1 is given by

u D 1 (x) = u i (x) + S D 1 λ D 1 Id -K * D 1 -1 [ ∂u i ∂ν 1 ](x), for x ∈ R 2 \D 1 .
To consider the total potential u, we also need to represent the field generated by the plasmonic particle D 2 . For this, we introduce a new layer potential S D 2 ,D 1 as follows:

S D 2 ,D 1 [ϕ](x) = ∂D 2 G D 1 (x, y)ϕ(y)dσ(y).
The total potential u can be represented in the following form:

u(x) = u D 1 (x) + S D 2 ,D 1 [ψ](x), x ∈ R 2 \D 2 . (8.6)
We need to find a boundary integral equation for the density ψ. It follows from (8.5) that, for any ϕ,

S D 2 ,D 1 [ϕ](x) = S D 2 [ϕ](x) + S 1 D 2 ,D 1 [ϕ](x),
where S 1 D 2 ,D 1 is given by

S 1 D 2 ,D 1 [ϕ](x) := ∂D 2 S D 1 λ D 1 Id -K * D 1 -1 [ ∂ ∂ν 1 G(•, y)](x)ϕ(y)dσ(y).
The expression of S 1 D 2 ,D 1 [ϕ] can be further developed using the following spectral expansion of the free-space Green function G(x, y) [START_REF] Ando | Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann-Poincaré operator[END_REF]:

G(x, y) = - ∞ j=1 S D 2 [ϕ j ](x)S D 2 [ϕ j ](y) + S D 2 [ϕ 0 ](x), for x ∈ R 2 \D 2 and y ∈ D 2 ,
where ϕ j , j = 1, 2, ... are eigenfunctions of K * D 2 on H * (∂D 2 ) and ϕ 0 is an eigenfunction associated to the eigenvalue 1/2. Then, for any ϕ ∈ H * (∂D 2 ), we get

∂D 2 G(x, y)ϕ(y)dσ(y) = ∞ j=1 S D 2 [ϕ j ](x)(ϕ, ϕ j ) H * (∂D 2 ) + S D 2 [ϕ 0 ](x) ∂D 2 ϕ(y)dσ(y) = ∞ j=1 S D 2 [ϕ j ](x)(ϕ, ϕ j ) H * (∂D 2 ) .
Therefore, for any ϕ ∈ H * (∂D 2 ), we have,

S 1 D 2 ,D 1 [ϕ](x) = ∂D 2 S D 1 λ D 1 Id -K * D 1 -1 [ ∂ ∂ν 1 G(•, y)](x)ϕ(y)dσ(y) = S D 1 λ D 1 Id -K * D 1 -1 ∂ ∂ν 1 S D 2 ∞ j=0 (ϕ, ϕ j ) H * ϕ j (x) = S D 1 λ D 1 Id -K * D 1 -1 ∂S D 2 [ϕ] ∂ν 1 (x),
where we have used the notation ∂ ∂ν i to indicate the outward normal derivative on ∂D i .

Combining the boundary conditions in (8.1), the representation formula (8.6) and the jump formula (1.1) yields the following equation for ψ

(A D 2 ,0 + A D 2 ,1 ) [ψ] = ∂u D 1 ∂ν 2 ,
where

A D 2 ,0 = λ D 2 Id -K * D 2 , λ D 2 = ε 2 + ε m 2(ε 2 -ε m ) , (8.7)
A D 2 ,1 = ∂S 1 D 2 ,D 1 ∂ν 2 = ∂ ∂ν 2 S D 1 λ D 1 Id -K * D 1 -1 ∂S D 2 ∂ν 1 . ( 8
:= (λ D 1 Id -K * D 1 ) -1 ∂S D 2 [ϕ] ∂ν 1 . Since S D 2 [ϕ] is harmonic in D 1 , the Green's identity gives ∂D 1 ∂ ∂ν 1 S D 2 [ϕ] = 0. Then it can be proved that ∂D 1 ϕ = 0. So we get S D 1 [ ϕ](x) = ∂D 1 (log |x -y| -log |x|) ϕ(y)dσ(y) + log |x| ∂D 1 ϕ(y)dσ(y) = ∂D 1 (log |x -y| -log |x|) ϕ(y)dσ(y).
Therefore, since |y -x| ≥ C ′ and |y| ≤ Cδ for (y, x) ∈ (∂D 1 , ∂D 2 ), we obtain

A D 2 ,1 [ϕ] H * (∂D 2 ) = ∂ ∂ν 2 S D 1 [ ϕ] H * (∂D 2 ) ≤ Cδ ϕ H * (∂D 1 ) . Now it suffices to prove that ϕ H * (∂D 1 ) ≤ Cδ. (8.9)
Recall that D 1 = δB. Let f δ (y) = f (δy). Then the function f δ belongs to H * (∂B) for f ∈ H * (∂D 1 ). Since it is known that K * Ω is scale-invariant for any Ω, we have K *

D 1 [f ] = K * B [f δ ]. Therefore, ϕ = λ D 1 Id -K * D 1 -1 [f ]d(δσ(y)) = (λ D 1 Id -K * B ) -1 [f δ ]d(δσ(y)). Again, since |y -x| ≥ C ′ for (y, x) ∈ (∂D 1 , ∂D 2 ) and |∂D 1 | = O(δ), we arrive at ϕ H * (∂D 1 ) = (λ D 1 Id -K * B ) -1 ∂S D 2 [ϕ] ∂ν 1 δ H * (∂B) ≤ C ∂ ∂ν 1 S D 2 [ϕ] H * (∂D 1 ) ≤ Cδ.
The proof is completed.

From Proposition 8.2.1, we can view A D 2 ,1 as a perturbation of A D 2 ,0 . Using standard perturbation theory [START_REF] Reed | Methods of Modern Mathematical Physics. IV Analysis of Operators[END_REF], we can derive the perturbed eigenvalues and associated eigenfunctions.

Let λ j and ϕ j be the eigenvalues and eigenfunctions of K * D 2 on H * (∂D 2 ). For simplicity, we consider the case when λ j is a simple eigenvalue of the operator K * D 2 . Let us define

R jl = A D 2 ,1 [ϕ l ], ϕ j H * (∂D 2 ) , (8.10)
where A D 2 ,1 is given by (8.8). Note that R jl = O(δ 2 ).

Chapter 8. Sensing Beyond the Resolution Limit

The perturbed eigenvalues have the following form:

τ j (δ) = λ D 2 -λ j + P j , where P j are given by

P j = R jj + l =j R jl R lj λ j -λ l + (l 1 ,l 2 ) =j R jl 2 R l 2 l 1 R l 1 j (λ j -λ l 1 )(λ j -λ l 2 ) + (l 1 ,l 2 ,l 3 ) =j R jl 3 R l 3 l 2 R l 2 l 1 R l 1 j (λ j -λ l 1 )(λ j -λ l 2 )(λ j -λ l 3 ) + • • • . (8.11)
Also, the perturbed eigenfunctions have the following form:

ϕ j (δ) = ϕ j + O(δ 2 ). (8.12)

Here the remainder term is with respect to the norm • H * (∂D 2 ) .

Remark 8.2.1. Note that P j depends not only on the geometry and material properties of D 1 , but also on D 2 's properties, in particular its position z. As a corollary, we have the following asymptotic expansion of the scattered field u -u i . Theorem 8.2.2. We have the following far field expantion: for l, m = 1, 2.

(u -u i )(x) = ∇u i (z) • M (λ D 1 , λ D 2 ,
We remark that the scattered field in the above expression depends on the frequency (since λ D 2 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 D 1 through the perturbative terms P j 's. We shall use this fact in the next section to solve the associated inverse problem of reconstructing D 1 by using those frequencies.

Representation of the shift P j using CGPTs

Here we show that the term P j in the plasmonic resonances can be expressed in terms of the contracted generalized polarization tensors (CGPTs), see chapter 1. The CGPTs carry information on the geometry and material properties of D 1 . See [START_REF] Ammari | Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory[END_REF] for a detailed reference. We shall reconstruct the ordinary particle D 1 from the measurement of the shift P j . Here, (r y , θ y ) denote the polar coordinates of y and {ϕ j } j is an orthonormal basis of eigenfunctions of K * D 2 on H * .

Proof. To simplify the notation, let us denote

F l = S D 1 λ D 1 Id -K * D 1 -1 ∂S D 2 [ϕ l ] ∂ν 1 .
Then, from the Green's identity and the jump formula (1.1), we obtain

R jl = F l , ϕ j H * = - ∂F l ∂ν 2 , S D 2 [ϕ j ] 1 2 ,-1 2 = -F l , ∂S D 2 [ϕ j ] ∂ν 2 - 1 2 ,-1 2 = -F l , (- 1 2 + K * D 2 )[ϕ j ] 1 2 ,- 1 2
.

Since ϕ j is an eigenfunction of K * D 2 with an eigenvalue λ j , we have

R jl = 1 2 -λ j F l , ϕ j 1 2 ,- 1 2
.

Let (r x , θ x ) be the polar coordinates of x. It is known from [START_REF] Ammari | Mathematical and Statistical Methods for Multistatic Imaging[END_REF] that, for |x| < |y|, G(x, y) = x cos(nθ x ) ∂ν 1 a l n,c + ∂r n

x sin(nθ x ) ∂ν 1 a l n,s .

On the contrary, if y ∈ ∂D 1 and x ∈ ∂D 2 , then |x| > |y|. We have from (8.16) that, for any f , for some constant C > 1 independent of δ. Moreover, it can be shown that (see [START_REF] Ammari | Generalized polarization tensors, inverse conductivity problems, and dilute composite materials: a review[END_REF]) ∞ n=1 (a j 0,c , a j 0,s )M 0,n (λ D 1 , D 1 )(a l n,c , a l n,s ) t = 0.

S D 1 [f ](x) =
Then the conclusion immediately follows.

Corollary 8.2.1. We have

P j (z) - l =j R jl (z)R lj (z) λ j -λ l - (l 1 ,l 2 ) =j
R jl 2 R l 2 l 1 R l 1 j (λ j -λ l 1 )(λ j -λ l 2 )

. . . In the LHS, the summation should be truncated so that all the terms which contain R jl k • • • R l k j = O(δ 2(k+1) ) with 2(k + 1) ≤ M + N + 1 are ignored.

The inverse problem

In this section, we consider the inverse problem associated with the forward system (8.1). We assume that the plasmonic particle D 2 is known, i.e., we know its electric permittivity ε 2 = ε 2 (ω), its shape D 2 and position z. The ordinary particle D 1 is unknown. For simplicity, we assume that its permittivity ε 1 is known. For each of many different positions z of the plasmonic particle D 2 , we measure the resonant frequency and use these resonant frequencies to reconstruct the shape of the ordinary particle D 1 .

As illustrated by Theorem 8.2.2, the resonance in the scattered field occurs when λ D 2 (ω) -λ j + P j is minimized and (ν l , ϕ j ) H * (ϕ j , x m ) -1 2 , 1 2 = 0. So by varying the frequency ω, we can measure the value of λ j -P j . Moreover, in the absence of the ordinary particle, the resonance occurs when λ D 2 (ω) -λ j is minimized and (ν l , ϕ j ) H * (ϕ j , x m ) -1 2 , 1 2 = 0. Since we assume that the plasmonic particle D 2 is known, we can get the value of λ j a priori. Therefore, by comparing λ j -P j and λ j , we can measure the shift P j of the eigenvalue.

Finding P j for many different positions of D 2 will yield a linear system of equations that will allow the recovery of the CGPTs associated with D 1 . From the recovered CGPTs, we will reconstruct the ordinary particle D 1 . Here, we only consider the shape reconstruction problem. Nevertheless, by using the CGPTs associated with D 1 , it is possible to reconstruct the permittivity ε 1 of D 1 in the case it is not a priori given [START_REF] Ammari | Mathematical and Statistical Methods for Multistatic Imaging[END_REF].

From now on, we denote M m,n = M m,n (λ D 1 , D 1 ).

CGPTs recovery algorithm

We propose a recurrent algorithm to recover the GPTs of order less or equal to k up to an order δ 2k-1 , using measurements of P j at different positions of D 2 . For simplicity, we only consider the shift of a single eigenvalue λ j 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 , m + n ≤ K for a given K ∈ N from the measurements of the shift P j .

Suppose we measure precisely P j for three different positions z 1 , z 2 , z 3 of the plasmonic particle D 2 . First we reconstruct M 1,1 approximately. Since M t 1,1 = M 1,1 , the matrix M 1,1 is symmetric. We look for a symmetric matrix M

(2) 1,1 satisfying P j (z 1 ) = 1 2 -λ j a j 1 (z 1 )M

(2)

1,1 (a j 1 ) t (z 1 )

P j (z 2 ) = 1 2 -λ j a j 1 (z 2 )M (2)
1,1 (a j 1 ) t (z 2 )

P j (z 3 ) = 1 2 -λ j a j 1 (z 3 )M (2)
1,1 (a j 1 ) t (z 3 ).

The above equations can be seen as a linear system of equations for three independent components (M

1,1 ) 11 , (M

1,1 ) 12 and (M

1,1 ) 22 . We emphasize that a j m (z i ) can be a priori given because the particle D 2 is known. Since, from Corollary 8.2.1 and the fact that R jl = O(δ 2 ), we have

P j (z k ) = 1 2
-λ j a j 1 (z k )M 1,1 (a j 1 ) t (z k ) + O(δ 3 ), k = 1, 2, 3,

we see that M 1,1 is well approximated by M Next we reconstruct and update the higher order GPTs M n,m in a recursive way. Towards this, we need more measurement data of the shift P j . Let k ≥ 3. Due to the symmetry of harmonic combinations of the non contracted GPTs (see [START_REF] Ammari | Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory[END_REF]), we have M m,n = M t n,m . One can see that, by using this symmetry property, the set of GPTs M m,n satisfying m + n ≤ k contains e k independent variables where e k is given by

e k = k(k -1) + k/2, if k is even, k(k -1) + (k -1)/2, if k is odd.
Therefore, we need e k measurement data for P j to reconstruct the GPTs M m,n for m + n ≤ k.

Suppose we have e k -2 more measurement data P j at different positions z 4 , z 5 , ..., z e k . Let {M a j m (z)M (k-1) m,n (a l n ) t (z).

Note that M

(k) m,n are defined recursively. In (8.18), the summation should be truncated as in Corollary 8.2.1.

Then M Hence, from Corollary 8.2.1 and the fact that R jl = O(δ 2 ), we obtain

P (k-1) j (z i ) -   P j (z i ) - l =j R jl (z i )R lj (z i ) λ j -λ l -• • •   = O(δ 2k-1 ).
Therefore, in view of Corollary 8.2.1 and the linear system (8.17), we obtain (8.19). In conclusion, M

m,n is indeed precise up to an order δ 2k-1 .

Remark 8.3.1. In practice, P j might be subject to noise and could not be measured precisely. In this case only the low order CGPTs could be recovered.

Shape recovery from CGPTs

To recover the shape of D 1 from its CGPTs, we search to minimize the following shape functional ( [START_REF] Ammari | Mathematical and Statistical Methods for Multistatic Imaging[END_REF]) 1) mn (λ D 1 , B) -N (1) mn (λ D 1 , D

J (l) c [B] := 1 2 n+m≤k N (
where N (1) m,n (λ, D) = (M cc m,n -M ss m,n ) + i(M cs m,n -M sc m,n ).

To minimize J (l) [B] we need to compute the shape derivative, d S J (l) c , of J (l) c .

For ǫ small, let B ǫ be an ǫ-deformation of B, i.e., there is a scalar function h ∈ C 1 (∂B), such that ∂B ǫ := {x + ǫh(x)ν(x) : x ∈ ∂B}.

Then, according to [START_REF] Ammari | Generalized polarization tensors for shape description[END_REF][START_REF] Ammari | Mathematical and Statistical Methods for Multistatic Imaging[END_REF][START_REF] Ammari | The generalized polarization tensors for resolved imaging. Part I: Shape reconstruction of a conductivity inclusion[END_REF], the perturbation of a harmonic sum of GPTs due to the shape deformation is given as follows:

N (1) m,n (λ D 1 , B ǫ ) -N

Numerical Illustrations

In this section, we support our theoretical results by numerical examples. In the sequel, we assume that D 2 is an ellipse with semi-axes a = 1 and b = 2, as shown in . Thus, for a fixed position of D 2 , we can measure two shifts of the plasmonic resonance: P 1 and P 2 .

We consider the case of D 1 being a triangular-shaped and a rectangularshaped particle with known contrast λ D 1 = 1, as shown in Figure 8.3.

Figure 8.4 shows the shift in the plasmonic resonance around λ 1 , for random positions of D 2 around a triangular-shaped particle D 1 . From these measurements, P 1 can be precisely estimated from the resonance peaks and the equation P j = λ j -λ r , where λ r is the value at which we achieve the 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 D 2 ) ranges from -1/2 to 1/2 and Im(λ D 2 ) = 10 -4 . In a more 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 [START_REF] Ammari | Minnaert resonances for acoustic waves in bubbly media[END_REF][START_REF] Minnaert | On musical air-bubbles and the sounds of running water[END_REF] in bubbly fluids. These new developments will be reported in forthcoming works. From left to right, we show both, the original shape and the recovered one after 0 iterations, after 30 iterations and after 100 iterations.

(i) The operator K * D is compact self-adjoint in the Hilbert space H * (∂D) and H * (∂D) is equivalent to H -1 2 (∂D); Similarly, the Hilbert space H(∂D) is equivalent to H 1 2 (∂D).

(ii) Let (λ j , ϕ j ), j = 0, 1, 2, . . . , be the eigenvalue and normalized eigenfunction pair of K * D with λ 0 = 1 2 . Then, λ j ∈ (-1 2 , 1 2 ] and λ j → 0 as j → ∞;

(iii) The following representation formula holds: for any ϕ ∈ H -1/2 (∂D),

K * D [ϕ] = ∞ j=0 λ j (ϕ, ϕ j ) H * ⊗ ϕ j .
The following lemmas are needed in the proof of Theorem 4.2.1 and Theorem 4.2.2.

Lemma A.0.3. Let D = z + δB and η be the function such that, for every ϕ ∈ H * (∂D), η(ϕ)(x) = ϕ(z + δ x), for almost all x ∈ ∂B. Then ϕ H * (∂D) = δ η(ϕ) H * (∂B) .

Similarly, if for every ϕ ∈ L 2 (D), η(ϕ)(x) = ϕ(z + δ x), for almost all x ∈ B, then

ϕ L 2 (D) = δ η(ϕ) L 2 (B) .
Proof. We only prove the scaling in H * (∂D). From the proof of Theorem A.0.1, we have where ϕ 0 is given in Definition A.1. Here, by an abuse of notation, we still denote by g the trace of g on ∂D.

Proof. Let ϕ ∈ H * (∂D). Then .

Here we have used Holder's inequality, a standard Sobolev embedding, the trace theorem and the fact that S D : H -1 2 (∂D) → H By abuse of notation we still denote T f := v to make explicit the dependency on f . It follows that Summing on i and using ∇ x Γ(x, x ′ ) = -∇ x ′ Γ(x, x ′ ), we get Replacing I 1 and I 2 by their expressions gives the desired result.

T f 2 H * = - D f S D [T f ]dx ≤ C f L 2 (D) S D [T f ] L 2 (D) ≤ C f L 2 (D) S D [T f ] H 1 (D) ≤ C f L 2 (D) S D [T f ]

Résumé

Abstract

This thesis deals with the mathematical study of the interactions between light and certain types of nanoparticles. At the nanometer scale, metal particles such as gold or silver undergo a resonance phenomenon when their free electrons interact with an electromagnetic field. This interaction results in an enhancement of the near and far electric field, enabling them to improve the brightness and the directivity of the light, confining electromagnetic fields in advantageous directions. This phenomenon, called "plasmonic resonances for nanoparticles", opens a door to a wide range of applications, from new medical imaging techniques to efficient solar panels. Using layer potentials techniques and perturbation theory, we propose a study of the scattering of electromagnetic waves by one and several plasmonic nanoparticles in the quasi-static, Helmholtz and Maxwell framework. We then study some applications such as heat generation, metasurfaces and super-resolution.

•

  D ⋐ R d denotes that D is an open, bounded and simply connected subset of R d ; • ∂D denotes the boundary of the open set D ∈ R d ;

Figure 3 :

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

Contents 1 . 1

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

For d = 3 G 1 . 1 2 1 2 (∂D); 4 .

 31114 Consider a domain D with boundary ∂D of type C 1,α for 0 < α < 1. Let ν denote the outward normal to ∂D. Define the single layer potentialS D [ϕ](x) = ∂D G(x, y)ϕ(y)dσ(y), x ∈ ∂D, x ∈ R d \∂D,where G(x, y) is the green function for the Laplacian.For d = 2 G(x, y) = 1 2π log |x -y|. For ϕ ∈ H -1 2 (∂D), S D [ϕ] ∈ H 1 (R d \∂D) is an Harmonic function on R d \∂D; 2. S D [ϕ](x) + = S D [ϕ](x) -; 3. For d = 3, S D [ϕ](x) = O( 1 |x| 2 ) as |x| → ∞ and S D [•] : H -1 2 (∂D) → H (∂D)is invertible, negative definite and self-adjoint for the duality paring H -1 2 (∂D), H Same for d = 2 under condition that ∂D ϕdσ = 0.

  x α = r m cos mθ, and |β|=m b m β x β = r m sin mθ.

1 2 ( 1 2

 21 ∂Ω) is invertible. Moreover, we have the jump formulaν • A∇S A Ω ± = ± Id + (K A Ω ) * Chapter 1. The Quasi-Static Limit with (K A Ω ) * [ϕ](x) = ∂Ω -x -y, ν(x) 4π det(A)|A * (x -y)| 3 ϕ(y)dσ(y). Proof. Let T A * ∈ L(H s (∂ Ω), H s (∂Ω)) be such that T A * [ϕ](x) = ϕ(A * x) for ϕ ∈ H s (∂ Ω) and Ω = A * Ω. Let r ν ∈ L(H s (∂Ω), H s (∂Ω)) be such that r ν [ϕ](x) = |A -1 * ν(x)|ϕ(x). It follows by the change of variables y = A * y that dσ( y) = det √ A * |A -1 * ν(y)|dσ(y). Thus,

Lemma 2 . 3 . 1 .

 231 (i) The following Calderón identity holds:K D S D = S D K * D ;

2. 3 . 35 where we have made use of the facts that 1 2

 3352 Layer potential formulation for the scattering problem Id -K * D S -1 D [1] = 0 and B D,1 [χ(∂D)] = cS -1 D [χ(∂D)] for some constant c; see again Appendix B.1.

Figure 2 .

 2 Figure 2.1 shows (2.53) and the average extinction of two ellipses of semiaxis a and b, where the ratio a/b = 2 and a/b = 4, respectively.

Figure 2 . 1 :

 21 Figure 2.1: Optimal bound for ellipses.

  have for any ellipse D and any shape B such that |B| = | D|,

Contents 3 . 1

 31 Introduction . . . . . . . . . . . . . . . . . . . . . . 3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . 3.3 Layer potential formulation for the scattering problem . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 First-order correction to plasmonic resonances and field behavior at the plasmonic resonances . . . . . 3.4 The extinction cross-section at the quasi-static limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Explicit computations for a spherical nanoparticle 3.5.1 Vector spherical harmonics . . . . . . . . . . . . . 3.5.2 Explicit representations of boundary integral operators . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Asymptotic behavior of the spectrum of W B (r) . . 3.5.4 Extinction cross-section . . . . . . . . . . . . . . . 3.6 Explicit computations for a spherical shell . . . . 3.6.1 Explicit representation of boundary integral operators . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Asymptotic behavior of the spectrum of W sh B (r s ) . 3.7 Concluding remarks . . . . . . . . . . . . . . . . . .

Lemma 3 . 2 . 1 .

 321 (i) The following Calderón identity holds:K D S D = S D K * D ;

Remark 3 . 2 . 1 . 2 0 1 20 2 0 2 0

 3212122 The Laplace-Beltrami operator ∆ ∂D : H 3 (∂D) → H -(∂D) is invertible. Here H 3 (∂D) and H -1 (∂D) are the zero mean subspaces of H 3 2 (∂D) and H -1 2 (∂D) respectively.

Condition 3 . 1 .

 31 The eigenvalues of K * B are simple.

( 2 )

 2 j,3 } ⊥ is the orthogonal space to Chapter 3. The Full Maxwell Equations span{ϕ (2) j,3 } in H(∂B) (Lemma 3.2.1 ), and

  Let x = x |x| . For m = -n, ..., n and n = 1, 2, ..., set Y m n to be the spherical harmonics defined on the unit sphere S = {x ∈ R 3 , |x| = 1}. For a wave Chapter 3. The Full Maxwell Equations number k > 0, the function v n,m (k; x) = h (1) n (k|x|)Y m n (x) satisfies the Helmholtz equation ∆v + k 2 v = 0 in R 3 \ {0} together with the Sommerfeld radiation condition lim |x|→∞ |x| ∂v n,m ∂|x| (k; x) -ikv n,m (k; x) = 0.

(3. 42 ) 3 . 5 .

 4235 Explicit computations for a spherical nanoparticle 79 Note that one has

3. 5 .

 5 Explicit computations for a spherical nanoparticle 81 For m = -n, . . . , n and n = 1, 2, 3, . . ., let H n,m (∂D) be the subspace of H(∂D) defined by H n,m (∂D) = span{U n,m , V n,m }. Let us represent the operators M k D and L k D explicitly on the subspace H m,n (∂D). Using U n,m , V n,m as basis vectors, we obtain the following matrix representations for M k D and L k D on the subspace H n,m (∂D):

  Now we are ready to compute the extinction cross-section Q ext . Theorem 3.5.1. Assume that E i (x) = p e ikmd•x with d ∈ S and p • d = 0. Let D be a sphere with radius r. Then the extinction cross-section is given Chapter 3. The Full Maxwell Equations by

u

  s := u -u i satisfies the Sommerfeld radiation condition at infinity, (4.1) where ∂ ∂ν denotes the outward normal derivative and c = 1 √ ε 0 µ 0 is the speed of light in vacuum. We use the notation ∂ ∂ν ± tν(x)) • ν(x),

  on ∂D, τ (•, s) satisfies the Sommerfeld radiation condition at infinity, (4.11) where τ (•, s) and L(g u )(•, s) are the Laplace transforms of τ and g u := ω 2πγc ℑ(ε c )|u| 2 , respectively, and s ∈ C\(-∞, 0].

  ) satisfies the differential equations in (4.11) together with the Sommerfeld radiation condition. Here β γm := i s ρmCm γm , β γc := i s ρcCc γc and FD (•, β γc ) := D G(•, y, β γc )L(g u )(y)dy.

(4. 14 )

 14 Proof of Theorem 4.2.2

  Figure 4.1 we can see values of the inner product between the eigenfunctions of K * D and the components ν x and ν y of ν.

Figure 4 .Figure 4 . 1 :

 441 Figure 4.1: Inner product in H * (∂D) between the eigenfunctions of K * D and the components ν x and ν y of the normal ν to ∂D.

Figure 4 .

 4 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 K *D , respectively. In Figure4.3 we decompose the inner field into the zerothorder and the first-order terms respectively given by u i (z) + δ(x -z)∇u i (z) and S D λ ε Id -K *

Figure 4 .

 4 4 shows the components of thevector S D (λ ǫ Id -K * D ) -1 [v]. From Figure4.3, we can see that when we excite the nanoparticle at its resonant mode, the largest contribution to the electromagnetic field comes

Figure 4 . 2 :

 42 Figure 4.2: Absolute value of the electromagnetic field inside the nanoparticle at the first resonant modes, being those when λ ǫ approaches the second, third and sixth eigenvalue of K * D .

Figure 4 . 3 :

 43 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 :

 44 Figure 4.4: Absolute value of the vectorial components of the first-order term for the first resonant mode.

Figure 4 . 1 2D

 41 5 we plot the generated heat in three dimensions and present a two dimensional plot obtained by parameterizing the boundary. In Figure 4.6 we decompose the heat in its first-and second-order terms given by Chapter 4. Heat Generation with Plasmonic Nanoparticles formula 4.2.2, being F D (x, t, b c ) and -V bc D (λ γ Id -K * D ) -1 [ ∂F D (•, •, b c ) ∂ν](x, t) respectively. In Figure4.7, 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 T =

Figure 4 . 5 : 1 -

 451 Figure 4.5: At the left-hand side, we can see a threedimensional 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.

Figure 4 . 6 :

 46 Figure 4.6: Two-dimensional plots of the zeroth-and firstorder 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.

Figure 4 . 7 :

 47 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 firstorder term is plotted in a log-log scale.

Figure 4 . 8 :

 48 Figure 4.8: inner product in H * ∂D between the eigenvalues of K * D and each component of the normal of ∂D, ν x and ν y .

Chapter 4 .DDDFigure 4 . 9 :Figure 4 . 10 :

 449410 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 λ ǫ approaches the respective eigenvalues of K * D . The x-component The y-component

4. 5 . 1 -

 51 Concluding remarks117 3D plot of heat on ∂(D 1 ∪ D 2 ) at time T = 1.Heat on ∂D 1 at time T =

Figure 4 . 11 :

 411 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 referential 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 ∈ R 2 corresponds to the center of each nanoparticle. On the top we can see nanoparticle D 2 and on the bottom nanoparticle D 1 .

Chapter 4 . 2 - 1 - 1 -

 4211 Heat Generation with Plasmonic Nanoparticles Heat zeroth component on ∂D Heat first component on ∂D 2 at time T = Heat first component on ∂D 1 at time T =

Figure 4 . 12 :

 412 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 D 2 , on the bottom we show the values for nanoparticle D 1 .

Figure 5 . 2 :Figure 5 . 3 :Figure 5 . 4 :

 525354 Figure 5.2: |α ∞ | as a function of the wavelength for disks of different radii, ranging from 0.1 to 0.4.

Figure 5 . 5 :Figure 5 . 6 :

 5556 Figure 5.5:Delocalized resonances for three wellseparated disks.

Figure 5 . 7 :

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

Figure 5 . 8 : 6 Shape Recovery of Algebraic Domains Contents 6 . 1

 58661 Figure 5.8: Wide resonance for a ring.

.13) 6 . 2 . 3

 623 Asymptotic behavior of the Neumann-Poincaré operator K * D

1 2= 1 2 ,-1 2 and (x 1 , 1 2 ,-1 2 .

 112112 O(δ). In what follows we calculate (x 1 , ϕ ± ) S D [ϕ ± ]) First, since dσ = Jdθ and x 1 | ∂D = ℜ{Φ(e ρ 0 +iθ )} = e ρ 0 cos θ + ae -mρ 0 cos mθ, we have

sin c 2 - α sin c 2 .(

 22 22) and f θ (x, y) = x 2 + y -α cos c Basis vectors) Orthonormal basis vectors {ê ξ , êθ } are defined as follows: êξ := ∂x/∂ξ |∂x/∂ξ| and êθ := ∂x/∂θ |∂x/∂θ| . (Normal-and tangential derivatives and line element) In the bipolar coordinates, the scaling factor h is h(ξ, θ) := cosh ξ -cos θ α . The gradient of any scalar function g is ∇g = h(ξ, θ) ∂g ∂ξ êξ + ∂g ∂θ êθ . (6.23) Moreover, the normal and tangential derivatives of a function u in bipolar coordinates = ∇u • v ξ=c = -sgn(c)h(c, element dσ on the boundary {ξ = ξ 0 } is dσ = 1 h(ξ 0 , θ) dθ.

(6. 29 )

 29 Chapter 6. Shape Recovery of Algebraic Domains Let us denote the Neumann-Poincaré operator for two disks separated by a distance ǫ by K * ǫ . To find out the spectral decomposition of the Neumann-Poincaré operator K *

  , 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.

3 Figure 6 . 1 :

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

Figure 6 . 2 :

 62 Figure 6.2: From top to bottom and left to right: initial shape, reconstructed shape, |m 11 | and |M cc 22 | with respect to the wavelength for m = 3 and δ = 0.066667.

Figure 6 . 3 :Figure 6 . 4 :

 6364 Figure 6.3: From top to bottom and left to right: initial shape, reconstructed shape, |m 11 | and |M cc 22 | with respect to the wavelength for m = 5 and δ = 0.03333.

Figure 6 . 5 :Figure 6 . 6 :

 6566 Figure 6.5: From top to bottom and left to right: initial shape, reconstructed shape, |m 11 | and |M cc 22 | with respect to the wavelength for m = 4, δ = 0.05.

2 Figure 6 . 7 :

 267 Figure 6.7: From top left to bottom: initial shape, reconstructed shape and |m 11 | with respect to the wavelength for ǫ = 0.2.

1 Figure 6 . 8 : 5 Figure 6 . 9 :

 168569 Figure 6.8: From top left to bottom: initial shape, reconstructed shape and |m 11 | with respect to the wavelength for ǫ = 1.

7. 2 .

 2 Multiple plasmonic nanoparticles 159 nanoparticle. Denote by D

Lemma 7 . 2 . 1 .

 721 (i) Regarded as operators from H * (∂D p ) into H * (∂D l ),

Theorem 7 . 3 . 1 .

 731 Under Conditions 2.1, 2.2, 2.3, 7.1, and 7.3, the Green function Γ(x, x 0 , k m ) in the presence of L plasmonic particles has the following representation in the quasi-static regime: for x ∈ D δ,km,C , Γ(x, x 0 , k m ) = G(x, x 0 , k m ) + j∈J L l=1

Figure 8 . 1 :

 81 Figure 8.1: Scattering of an incident wave u i by a system of a plasmonic (D 2 ) -non plasmonic (D 1 ) particles.

Theorem 8 . 2 . 1 .

 821 If D 2 is in the intermediate regime, the scattered field u s D 2 = u -u D 1 by the plasmonic particle D 2 has the following representation:u s D 2 = S D 2 ,D 1 [ψ], where ψ satisfies ψ = ∞ j=1 ∇u i (z) • ν, ϕ j H * (∂D 2 ) ϕ j + O(δ 2 )λ D 2 -λ j + P j with λ D 2 being given by (8.7).

1 2 , 1 2 + O(δ 2 )λ D 2 -

 1222 D 1 , D 2 )∇G(x, z) + O(δ 2 ) + O δ 3 dist(λ D 2 , σ(K * D 2 )) , as |x| → ∞. Here, M (λ D 1 , λ D 2 , D 1 , D 2 ) is the polarization tensor satisfying M (λ D 1 , λ D 2 , D 1 , D 2 ) l,m = ∞ j=1 (ν l , ϕ j ) H * (∂D 2 ) (ϕ j , x m ) -λ j + P j ,(8.13)

Propsition 8 . 2 . 2 .

 822 If D 2 is in the intermediate regime, then the perturbative terms R jl can be represented using CGPTs M m,n (λ D 1 , D 1 ) associated with D 1 as follows:M m,n (λ D 1 , D 1 )(a l n ) t + O(δ M +N +1 ),(8.14)where the superscript t denotes the transpose and a j m = (a j m,c , a j m,s ) witha j m,c = -1 2πm ∂D 2 cos(mθ y )r m y ϕ j (y)dσ(y),

  sin(nθ x ).(8.15) By interchanging x and y and the fact that G(x, y) = G(y, x), we have, for |x| > |y|, sin(nθ y ).(8.16) If x ∈ ∂D 1 and y ∈ ∂D 2 , then |x| < |y|. So, applying(8.15) gives∂S D 2 [ϕ l ] ∂ν 1 (x) = ∂ ∂ν 1 ∂D 2 G(x, y)ϕ l dσ(y)

∂D 1 G 2 - 1 - 1 ∂S D 2 [ϕ l ] ∂ν 1 , ϕ j 1 2

 1211211 (x, y)[f ](y)dσ(y) cos(mθ y )[f ](y)dσ(y) sin(mθ y )[f ](y)dσ(y).Therefore, from the definition of M m,n , we getR jl = 1 λ j S D 1 λ D 1 Id -K * D , a j m,s )M m,n (λ D 1 , D 1 )(a l n,c , a l n,s ) t .For any λ ∈ C and D = δB, it is easy to check that M m,n (λ, D) = δ m+n M m,n (λ, B). Since D 2 is in the intermediate regime, a l n,c and a l n,s satisfy|a j m,c |, |a j m,s | ≤ 1 m C -m , |a l n,c |, |a l n,s | ≤ 1 n C -n ,

  M m,n (λ D 1 , D 1 )(a l n ) t + O(δ M +N +1 ).

1 =

 1 O(δ 3 ).

  } m+n≤k be the set of matrices satisfying [M(k) n,m ] t = M (k)m,n and the following linear system: e k )M(k) m,n (a j n ) t (z e k ), ) := P j (z i ) -λ j -λ l -. . . , i = 1, 2, ..., e k ,

  (k)m,n becomes a good approximation of the CGPT M m,n for m + n ≤ k. Moreover, the accuracy improves as the iteration goes on. Indeed, we can see thatM m,n -M (k) m,n = O(δ 2k-1 ), m + n ≤ k. (8.19)In fact,(8.19) can be verified by induction. We already know that this is true when k= 2. Let us assume M m,n -M (k-1) m,n = O(δ 2k-3 ), m + n ≤ k -1. Then, from Proposition 8.2.2, we have R jl (z) -R (k-1) jl (z) = O(δ 2k-3 ).

( 1 )(x 1 +

 11 m,n (λ D 1 , D 1 ) = ǫ(k λ D 1 -1) dσ(x) + O(ǫ 2 ),wherek λ D 1 = (2λ D 1 + 1)/(2λ D 1 -1),(8.21)Chapter 8. Sensing Beyond the Resolution Limit and u and v are respectively the solutions to the problems:               ∆u = 0 in B ∪ (R 2 \B) , u| + -u| -ix 2 ) m )(x) = O(|x| -1 ) as |x| → ∞ , in B ∪ (R 2 \B) , k λ D 1 v| + -v| -(x 1 + ix 2 ) n )(x) = O(|x| -1 ) as |x| → ∞ . (8.23)Here, ∂/∂T is the tangential derivative. Letw m,n (x) = (k λ D 1 -, x ∈ ∂B .The shape derivative of J (l) c at B in the direction of h is given byd S J (l) c [B], h = m+n≤k δ N w m,n , h L 2 (∂B) ,whereδ N = N (1) m,n (λ D 1 , B) -N (1)m,n (λ D 1 , D 1 ) . Next, using a gradient descent algorithm we can minimize, at least locally, the functional J (l) c .

Figure 8 . 2 . 1 2

 821 In this case the resonances in the far-field can only occur at λ 1 =

Chapter 8 .Figure 8 . 5 :

 885 Figure 8.5: Positions of D 2 for which we measure P 1 . (left) Triangular-shaped particle D 1 , (right) rectangular-shaped particle D 1 .

Figure 8 . 6 :Figure 8 . 7 :

 8687 Figure 8.6: Shape recovery of a triangular-shaped particle D 1 . From left to right, we show both, the original shape and the recovered one after 0 iterations, after 8 iterations and after 30 iterations.

ϕ 2 H 2 , 1 2 , 1 2 = 0 1 2 , 1 2 = 0 2 ∂B 2 =

 2212012022 * (∂D) = -∂D ψS D [ψ]dσ + ∂D ϕdσwhere ψ = ϕ-∂D ϕdσ ϕ 0 . Note that (ψ, 1) -and so, (η(ψ), 1) -as well.By a rescaling argument we find thatϕ 2 H * (∂D) = -δ 2 ∂B ∂B 1 2π log |δ(x -ỹ)|η(ψ)(x)η(ψ)(ỹ)dσ(x)dσ(ỹ) + δ δ 2 η(ϕ) 2 H * (∂B) . Lemma A.0.4. Let g ∈ H 1 (D) be such that ∆g = f with f ∈ L 2 (D). Then, in H * (∂D), For some T f ∈ H * (∂D) and T f H * ≤ C f L 2 (D) for a constant C. Moreover, if g ∈ H 1 loc (R 2 ), ∆g = 0 in R 2 \ D, lim |x|→∞ g(x) = 0, then T f = c f ϕ 0 + S -1 D [g],withc f = D f (x)dx -∂D S -1 D [g](y)dσ(y),

2 = 1 2 , 1 2 =

 212 -g, -∂ S D [ϕ] ∂ν --∂D ∂g ∂ν S D [ϕ]dσ -D f S D [ϕ] -∆ S D [ϕ] g dx = -∂g ∂ν , ϕ H * -D f S D [ϕ]dx.We have used the fact that S D is harmonic in D.Consider the linear applicationT f [ϕ] := -D f S D [ϕ]dx. We have |T f [ϕ]| ≤ C f L 2 (D) S D [ϕ] L 2 (D) ≤ C f S D [ϕ] H 1 (D) ≤ C f S D [ϕ]

1 2 (

 2 ∂D) is continuous. By the Riez representation theorem, there exists v ∈ H * (∂D) such that T f [ϕ] = (v, ϕ) H * , ∀ϕ ∈ H * (∂D).

H 1 2 2 f

 12 (∂D) ≤ C f L 2 (D) T f H * .We now show that in H * 0 (∂D),T f = S -1 D [g].Indeed, let ϕ ∈ H * 0 (∂D), thenS -1 D [g], ϕ H * = -S -1 D [g], S D [ϕ] S D [ϕ] -∆ S D [ϕ] g dx = -D f S D [ϕ]dx.Here we have used the assumption on g, the fact thatS D [ϕ] is harmonic in D and R 2 \ D and that for ϕ ∈ H * 0 (∂D) we have S D [ϕ](x) = O( 1 |x| ) and ∂ S D [ϕ] ∂ν (x) = O( 1 |x| ) for |x| → ∞. Therefore, T f = (T f -S -1 D [g], ϕ 0 ) H * ϕ 0 + S -1 D [g].Finally, re-scaling the definition of ϕ 0 given in Definition A.1 we obtain that(T f -S -1 D [g], ϕ 0 ) H * = D f (x)dx -= δ l,m |D|, we take f (λ) = 1 in (C.m (x) dσ(x) = δ l,m |D|. * D [ν l ](x) dσ(x), ∂D x l K * D [ν l ](x) dσ(x) = ∂D x l 1 2 ν l (x) + ∂S D [ν l ] ∂ν -(x) dσ(x), = |D| 2 + ∂D x l ∂S D [ν l ] ∂ν -(x)dσ(x). (C.2)Integrating by parts we arrive at∂D x l ∂S D [ν l ] ∂ν -(x)dσ(x) = D e l (x) • ∇S D [ν l ](x)dx + D x l ∆S D [ν l ](x)dx.Since the single-layer potential is harmonic on D, ∂D x l ∂S D [ν l ] ∂ν -(x)dσ(x) = D e l (x) • ∂D ∇ x Γ(x, x ′ )ν l (x ′ )dσ(x ′ ) dx.

K 2 . 2 .

 22 D [ν l ] ∂ν -(x)dσ(x) = -D ∂D ν(x ′ ) • ∇ x ′ Γ(x, x ′ )dσ(x ′ ) dx, = -D D D [1](x)dx, = -|D|, (C.3) where D D is the double-layer potential. Hence, summing equation (C.2) for i = 1, . . . , d, we get the result. Finally, we show that D [ν l ]| 2 dx. Appendix C. Sum Rules for the Polarization Tensor 209 Taking f(λ) = λ 2 in (C.1) yields l (K * D ) 2 [ν l ](x) dσ(x) D [y l ](x)K * D [ν l ](x) dσ(x)From (C.3) it follows thatI 1 = -|D| Since x l is harmonic, we have x l = D D [y l ](x)| --S D [ν l ](x)on ∂D, and thus, + S D [ν l ](x)) ∂S D [ν l ] ∂ν -D [ν l ]| 2 dx.

  Contents 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Layer potentials for the Helmholtz equation . . . . Problem formulation and some basic results . . . . 2.3.2 First-order correction to plasmonic resonances and field behavior at the plasmonic resonances . . . . . 2.4 Scattering and absorption enhancements . . . . .

2.3 Layer potential formulation for the scattering problem . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 2.4.1 Far-field expansion . . . . . . . . . . . . . . . . . . 2.4.2 Energy flow . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Extinction, absorption, and scattering cross-sections and the optical theorem . . . . . . . . . . . . . . . 2.4.4 The quasi-static limit . . . . . . . . . . . . . . . . 2.4.5 An upper bound for the averaged extinction crosssection . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Link with the scattering coefficients . . . . . . . . 2.5.1 The notion of scattering coefficients . . . . . . . . 2.5.2 The leading-order term in the expansion of the scattering amplitude . . . . . . . . . . . . . . . . . 2.6 Concluding remarks . . . . . . . . . . . . . . . . . .

  Let us now consider the eigenvalues and the associated eigenvectors of the matrix M D,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 M D,m has the following eigenvalues:

	6.2. Plasmonic resonance for algebraic domains			139
	Here, O(δ 2 ) is with respect to the operator norm.
	Since the operators F c m and F s m are of finite rank, they have matrix rep-resentations. Using {v c n } m m can be represented as the following n=1 as basis, F c matrix M D,m :
			0	0	. . .	1	
	M D,m :=	    	0 . . . m	. . . m -1	2 . . . 0 . . . . . . 0 . . . 0	    	(6.17)
	Clearly, F s m has the same matrix representation M D,m using {v s n } m n=1 as basis.
					).			(6.16)

  Here we introduce the concept of intermediate regime and derive the asymptotic expansion of the scattered field u -u i for small δ. Definition 8.1 says that the plasmonic particle D 2 is located not too close to D 1 nor far from D 1 . Throughout this chapter, we assume the plasmonic particle D 2 is in the intermediate regime. We have the following result. Propsition 8.2.1. If D 2 is in the intermediate regime, then A D 2 ,1 H * = O(δ 2 ) as δ → 0. Proof. Fix ϕ ∈ H * (∂D 2 ) and let ϕ

	8.2. The forward problem
	.8)
	8.2.3 Intermediate regime and asymptotic expansion of the
	scattered field

Definition 8.1 (Intermediate regime). We say that D 2 is in the intermediate regime with respect to the origin if there exist positive constants C 1 and C 2 such that C 1 < C 2 and

C 1 ≤ dist(0, D 2 ) ≤ C 2 .

  Cette thèse porte sur l'étude mathématique des interactions entre la lumière et certains types de nanoparticules. A l'échelle du nanomètre, des particules métalliques comme l'or ou l'argent subissent un phénomène de résonance lorsque leurs électrons libres interagissent avec un champ électromagnétique. Cette interaction produit une augmentation du champs électrique proche et lointain, leur permettant d'améliorer la luminosité et la directivité de la lumière, confinant des champs électromagnétiques dans des directions avantageuses. Ce phénomène, appelé "résonances plasmoniques pour des nanoparticules" ouvre une porte sur une large gamme d'applications, des nouvelles techniques d'imagerie médicale à des panneaux solaires efficaces. En utilisant des techniques issues des potentiels de couches et de la théorie de la perturbation, nous proposons une étude de la dispersion d'ondes électromagnétiques par une et plusieurs nanoparticules plasmoniques, dans le cadre quasi-statique, Helmholtz et Maxwell. Nous étudions ensuite certaines applications tel que la génération de chaleur, les métasurfaces et l'imagerie super-résolue.

	Mots Clés
	operateur de Neumann-Poincaré, potentiels de couche,
	nanoparticules plasmoniques, resonance de plasmon,
	analyse asymptotique

Acknowledgements

Part II

Applications

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 δS δkc B [η(φ)] and scale back to D. We divide the proof into three steps.

Step 1. We first compute an asymptotic for A I B (δ) and f I . Let H * (∂B) be defined by (A.3) with D replaced by B. In L(H * (∂B)), we have the following asymptotic expansion as δ → 0 (see Appendix B.3)

Let ϕ 0 be an eigenfunction of K * B associated to the eigenvalue 1/2 (see Appendix A) and let U δkm be defined by (B.12) with k replaced with δk m . Then it follows that

Therefore, in L(H * (∂B)),

and from the definition of U δkm we get

(4.8) In the same manner, in the space H * (∂B),

We can further develop f I . Indeed, for every x ∈ ∂B, a Taylor expansion yields

The regularity of u i ensures that the previous formulas hold in H * (∂B).

The fact that x • ∇u i (z) is harmonic in B and Lemma A.0.4 imply that

in H * (∂B). Thus, in H * (∂B),

Chapter 4. Heat Generation with Plasmonic Nanoparticles

From the definition of U δkm we get

(4.9)

Step 2. We compute (A I B (δ)) -1 f I . We begin by computing an asymptotic expansion of (A I B (δ)) -1 . The operator A I 0 :=

Hence, the operator defined by (which appears in the expansion of A I B (δ))

is invertible of inverse

Therefore, we can write

Since K * B is a compact self-adjoint operator in H * (∂B) it follows that

for a constant c. Therefore, for δ small enough, we obtain

.

Heat generation 105

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

Using the same arguments as those in the proof of Lemma A.0.4, we have

and consequently,

Therefore,

Step 3. Finally, we compute η(u) = δS δkc B (A I B (δ)) -1 f I . From Appendix B.3, the following holds when S δkc B is viewed as an operator from the space H * (∂B) to H(∂B):

In particular, we have

It can be verified that the same expansion holds when viewed as an operator from H * (∂B) into L 2 (B).

Note that the following identity holds

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 u i is transverse magnetic (TM) or transverse electric (TE) polarized.

Consider a particle occupying a bounded domain D ⋐ R 2 of class C 1,α for some 0 < α < 1 and with size of order δ ≪ 1. The particle is characterized by electric permittivity ε c and magnetic permeability µ c , both of which may depend on the frequency of the incident wave. Assume that ℑm ε c > 0, ℜe µ c < 0, ℑm µ c > 0 and define

where ε m and µ m are the permittivity and permeability of free space respectively and ω is the frequency. Throughout this chapter, we assume that ε m and µ m are real and positive and k m is of order 1.

We consider the configuration shown in Figure 5.1, where a particle D is repeated periodically in the x 1 -axis with period δ, and is at a distance of order δ from the boundary

We denote by D this collection of periodically arranged particles and Ω := R 2 + \ D. Let u i (x) = e ikmd•x be the incident wave. Here, d is the unit incidence direction. The scattering problem is modeled as follows

u -u i satisfies an outgoing radiation condition at infinity,

where

and ∂/∂ν denotes the outward normal derivative on ∂D.

Chapter 5. Plasmonic Metasurfaces (ii) The following trace formula holds: for any ϕ ∈ H -1 2 (∂B), (-

(iii) The following Calderón identity holds:

(iv) The operator K * B♯ : H

0 (∂B) is compact self-adjoint equipped with the following inner product

(5.6)

being the duality pairing between H (v) Let (λ j , ϕ j ), j = 1, 2, . . . be the eigenvalue and normalized eigenfunction pair of K * B♯ in H * 0 (∂B), then λ j ∈ (-1 2 , 1 2 ) and λ j → 0 as j → ∞.

Proof. First, note that a Taylor expansion of sinh 2 (πx 2 ) + sin 2 (πx 1 ) yields

where R is a smooth function such that

We can decompose the operators S B♯ and K * B♯ on H * 0 (∂B) accordingly. We have

where S B and K * B are the single layer potential and Neumann-Poincaré operator (see [START_REF] Ammari | Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory[END_REF]), respectively, and G B , F B are smoothing operators. Using this fact, the proof of the Lemma follows the same arguments as those given in [START_REF] Ammari | Mathematical and Statistical Methods for Multistatic Imaging[END_REF][START_REF] Ammari | Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory[END_REF].

Boundary layer corrector and effective impedance

In order to compute z, we introduce the following asymptotic expansion [START_REF] Abboud | Diffraction at a curved gratings: TM and TE cases, Homogenization[END_REF][START_REF] Allaire | Boundary layer tails in periodic homogenization[END_REF]:

where the leading-order term u (0) is solution to

+ , u (0) -u i satisfies an outgoing radiation condition at infinity. BL have to be exponentially decaying in the x 2 -direction. Note that according to [START_REF] Abboud | Diffraction at a curved gratings: TM and TE cases, Homogenization[END_REF][START_REF] Allaire | Boundary layer tails in periodic homogenization[END_REF], u (0) BL 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 term u (0) in the asymptotic expansion of u, while u

(1)

BL is a higher-order correction term and does not contribute to the first-order equivalent boundary condition in (5.2).

We next construct the corrector u

BL . We first introduce a function α and a complex constant α ∞ such that they satisfy the rescaled problem:

(5.8)

Here, ν = (ν 1 , ν 2 ) and B = D/δ is repeated periodically in the x 1 -axis with period 1 and B is the collection of these periodically arranged particles.

Then u

BL is defined by

The corrector u (1) can be found to be the solution to

on ∂R 2 + , u (1) satisfies an outgoing radiation condition at infinity.

By writing

u app = u (0) + δu (1) , (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 u app to (5.2) with z = -α ∞ approximates pointwisely (for x 2 > 0) the exact solution u to (5.1) as δ → 0, up to a 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

Chapter 7

Super-Resolution with Plasmonic Nanoparticles

Moreover,

Proof. The proof of (i) follows from Lemmas 2.3.4 and B.2.3. We now prove (ii). Recall that

Using the identity

The rest of the lemma follows from Lemmas B.2.3 and B.2.6.

Denote by H * (∂D) = H * (∂D 1 ) × . . . × H * (∂D L ), which is equipped with the inner product

With the help of Lemma 7.2.1, the following result is obvious. Lemma 7.2.2. Regarded as an operator from H * (∂D) into H * (∂D), we have

where

It is evident that

where

with e l being the standard basis of R L . We take A D (ω) as a perturbation to the operator A D,0 for small ω and small δ. Using a standard perturbation argument, we can derive the perturbed eigenvalues and eigenfunctions. For simplicity, we assume that the following conditions hold.

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 A D,1 . For each l, we define an L × L matrix R l by letting

Lemma 7.2.3. The matrix R l = (R l,pq ) p,q=1,...,L has the following explicit expression:

)

)

Proof. It is clear that R l,pp = 0. For p = q, we have

We first consider R I l,pq . By the following identity

we obtain

Using the explicit representation of S q,p,0,1 and the fact that (χ(∂D j ), φ l ) L 2 (∂D j ) = 0 for j = 0, we further conclude that

Similarly, we have

)

)

Finally, note that

where a m = (y -z q ) m , ϕ l L 2 (∂Dq) , and a = (a 1 , a 2 , a 3 ) T .

Chapter 7. Super-Resolution with Plasmonic Nanoparticles By identity (2.29), we have

)

This completes the proof of the lemma.

We now have an explicit formula for the matrix R l . It is clear that R l is symmetric, but not self-adjoint. For ease of presentation, we assume the following condition.

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 [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF] for a complete description of the perturbation theory.

Let τ j,l and X j,l = (X j,l,1 , • • • , X j,l,L ) T , l = 1, 2, . . . , L, be the eigenvalues and normalized eigenvectors of the matrix R j . Here, T denotes the transpose. We remark that each X j,l may be complex valued and may not be orthogonal to other eigenvectors.

Under perturbation, each τ j is splitted into the following L eigenvalues of A(ω),

The associated perturbed eigenfunctions have the following form

We are interested in solving the equation A D (ω)[ψ] = f when ω is close to the resonance frequencies, i.e., when τ j,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 ϕ j,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

In fact,

where γ j is a Jordan curve in the complex plane enclosing only the eigenvalues τ j,l (ω) for l = 1, 2, . . . , L among all the eigenvalues.

Multiple plasmonic nanoparticles 165

To obtain an explicit representation of P J (ω), we consider the adjoint operator A D (ω) * . By a similar perturbation argument, we can obtain its perturbed eigenvalue and eigenfunctions. Note that the adjoint matrix RT j = Rj has eigenvalues τ j,l and corresponding eigenfunctions X j,l . Then the eigenvalues and eigenfunctions of A D (ω) * have the following form

where

X j,l,p e p ϕ j with X j,l,p being a multiple of X j,l,p .

We normalize ϕ j,l in a way such that the following holds

which is also equivalent to the following condition X j,p T X j,q = δ pq .

Then, we can show that the following result holds.

Lemma 7.2.4. In the space H * (∂D), as ω goes to zero, we have

where f 0 = (f 0,1 , . . . , f 0,L ) T with

2 ).

Proof. We first show that

for any homogeneous function u such that u(δx) = δ m u(x). Indeed, we have

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.

Denote by Z = (Z 1 , . . . , Z L ), where Z j = ik m e ikmd•z 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

Chapter 7. Super-Resolution with Plasmonic Nanoparticles width of order one. This breaks the diffraction limit 1/k m in the free space. We also note that the term ℑG(x, x 0 , k m ) has size O(ω). Thus, we can conclude that super-resolution (super-focusing) can indeed be achieved by using a system of plasmonic particles.

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 [START_REF] Ammari | Super-resolution in high contrast media[END_REF]. Indeed, by considering a periodic arrangement of nanoparticles we could construct a high contrast media, thus allowing super-resolution. To recover geometrical properties of D 1 from measurements of P 1 , we recover the CGPTs using the algorithm described in 8.3.1 and then minimize functional (8.20) to reconstruct an approximation of D 1 .

To recover the first CGPTs of order 5 or less we make 22 measurements around D 1 as shown in Figure 8.5, and measure the shift from λ 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. 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 D 1 , given by the first order polarization recovered with Algorithm 8.3.1, i.e M (5) 11 .

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 Appendix A

Layer Potentials for the Laplacian in two Dimensions

In R 2 the single-layer potential S D :

does not define an inner product and the symmetrization technique described in 1 is no longer valid.

Here and throughout, (•,

denotes the duality pairing between H -1/2 (∂D)

and H 1/2 (∂D).

To overcome this difficulty, we will introduce a substitute of S D , in the same way as in [START_REF] Ando | Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann-Poincaré operator[END_REF].

We first need the following lemma.

Proof. It is known that

is invertible [START_REF] Ammari | Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory[END_REF]Theorem 2.26].

We can see that

Thus, by the range theorem we have

Definition A.1. We call ϕ 0 the unique element of C such that ∂D ϕ 0 dσ = 1.

Note that for every ϕ ∈ H -1/2 (∂D) we have the decomposition

where we can see that (ψ, 1) -1 2 , 1 2 = 0. This kind of decomposition, ϕ = ψ + αϕ 0 , with (ψ, 1) -1 2 , 1 2 = 0 is unique.

Note that we can decompose H -1/2 as a direct sum of elements with zero-mean and multiples of ϕ 0 , H -1/2 (∂D) = H -1/2 0 (∂D) ⊕ {µϕ 0 , µ ∈ C}. This allows us to define the following operator. Definition A.2. Let S D be the linear operator that satisfies Proof. The invertibility is a direct consequence of Lemma A.0.1. Indeed, since S D is Fredholm of zero index, so is S D . Therefore, we only need the injectivity. Suppose that, ∃ ϕ = 0 such that S D [ϕ] = 0. This mean that,

The self-adjointness comes directly form that of S D . Noticing that ϕ 0 is an eigenfunction of eigenvalue 1/2 of K * D we get the Calderón identity from a similar one satisfied by S D : Definition A.3. We define the space H * (∂D) as the Hilbert space resulting from endowing H -1/2 (∂D) with the inner product

Similarly, we let H to be the Hilbert space resulting from endowing H 1/2 with the inner product

If D is C 1,α , we have the following result.

Lemma A.0.2. Let D be a C 1,α bounded domain of R 2 and let S D be the operator introduced in Definition A.2. Then 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 R 3

We consider a domain D ⋐ R 3 whose size is of order one.

Recall the definition of the single layer potential

is the Green function of Helmholtz equation in R 3 , subject to the Sommerfeld radiation condition. Note that

We get

where

In particular, we have

) is uniformly bounded with respect to j. Moreover, the series in (B.1) is convergent in L(H * (∂D), H(∂D)).

Proof. It is clear that

where C is independent of j. On the other hand, a similar estimate also holds for the operator S * D,j . It follows that

Thus, we can conclude that S D,j L(H -1

is uniformly bounded by using interpolation theory. By the equivalence of norms in the H -1 2 (∂D) and H 1 2 (∂D), the lemma follows immediately.

Note that S D is invertible in dimension three, so is S k D for small k. By formally writing

and using the identity

We can also derive other lower-order terms B D,j .

Lemma B.1.2. The series in (B.4) converges in L(H(∂D), H * (∂D)) for sufficiently small k.

Proof. The proof can be deduced from the identity

We now consider the expansion for the boundary integral operator (K k D ) * . We have

where

In particular, we have

B.2 Asymptotic expansion in R 3 : 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 D j = z j + δ D, 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 are B.2. Asymptotic expansion in R 3 : multiple particles 199 defined on small particles in terms of their size. For this purpose, we denote by D 0 = δ D. For each function f defined on ∂D 0 , we define a corresponding function on D by η(f )( x) = f (δ x).

In this section, we denote by χ(∂D j ) the constant function equal one over the border of D j .

We first state some useful results.

Lemma B.2.1. The following scaling properties hold:

Proof. The proof of (i) is straightforward and we only need to prove (ii) and (iii). To prove (iii), we have

whence (iii) follows. To prove (ii), recall that

As a result, we have

which proves (ii). Then,

Proof. The result follows from Lemma B.2.1 and the following identity

We first consider the operators S k D j and (K k D j ) * . The following asymptotic expansions hold. (ii) Regarded as operators from H(∂D j ) into H * (∂D j ), we have

where S -1 D j = O(1) and B D j ,m = O(δ m );

(iii) Regarded as operators from H * (∂D j ) into H * (∂D j ), we have

where Using the expansion

where

We can further write S j,l,m = n≥0 S j,l,m,n , B.2. Asymptotic expansion in R 3 : multiple particles 201 where S j,l,m,n is defined by

In particular, we have

The following estimate holds.

Lemma B.2.4. We have S j,l,m,n L(H * (∂D),H(∂D)) O(δ n+1 ).

Proof. After a translation of coordinates, the stated estimate immediately follows from Lemma B.2.2.

Similarly, for the operator K km D j ,D l defined in the following way

where

In particular, we have

Lemma B.2.5. We have K j,l,m,n L(H * (∂D j ),H * (∂D l )) O(δ n+2 ).

Proof. Note that

After a translation of coordinates, we can apply Lemma B.2.2 to each one of the three terms above to conclude that K j,l,m,n = O(δ n+3 ) + O(δ n+2 ). This completes the proof of the lemma.

To summarize, we have proven the following results.

Lemma B.2.6. (i) Regarded as an operator from H * (∂D j ) into H(∂D l ) we have,

Moreover, S j,l,m,n = O(δ n+1 ).

(ii) Regarded as an operator from H * (∂D j ) into H * (∂D l ), we have

Moreover, K j,l,0,0 = O(δ 2 ).

B.3 Asymptotic expansions in R 2

Let us now consider the single-layer potential for the Helmholtz equation in R 2 given by

0 (k|x -y|) and H

(1) 0 is the Hankel function of first kind and order 0. We have, for k ≪ 1,

where

and γ e is the Euler constant. Thus, we get

D,j , (B.10)

where are uniformly bounded with respect to j. Moreover, the series in (B.10) is convergent in L(H * (∂D), H(∂D)) for k < 1.

Observe that

Then it follows that

where

Therefore, we arrive at the following result. Proof. We can write (B.10) as

where G k = k 2 log kS Hence, we have

Here,

Then,

and therefore,

It is clear that ( Ŝk D ) -1 L(H(∂D),H * (∂D)) is bounded for k small. Since ||G k || L(H(∂D),H * (∂D)) goes to zero as k goes to zero, for k small enough, we can write

which yields the desired result.

We now consider the expansion for the boundary integral operator (K k D ) * . We have Proof. We have

From Lemma C.0.1, we can deduce that