Habib Ammari
email: habib.ammari@math.ethz.ch

Pierre Millien
email: pierre.millien@ens.fr

Matias Ruiz
email: matias.ruiz@ens.fr

Hai Zhang

Mathematical analysis of plasmonic nanoparticles: the scalar case *

Keywords: Mathematics Subject Classification (MSC2000): 35R30, 35C20 plasmonic resonance, Neumann-Poincaré operator, nanoparticle, scattering and absorption enhancements, super-resolution imaging, layer potentials

Localized surface plasmons are charge density oscillations confined to metallic nanoparticles. Excitation of localized surface plasmons by an electromagnetic field at an incident wavelength where resonance occurs results in a strong light scattering and an enhancement of the local electromagnetic fields. This paper is devoted to the mathematical modeling of plasmonic nanoparticles. Its aim is threefold: (i) to mathematically define the notion of plasmonic resonance and to analyze the shift and broadening of the plasmon resonance with changes in size and shape of the nanoparticles; (ii) to study the scattering and absorption enhancements by plasmon resonant nanoparticles and express them in terms of the polarization tensor of the nanoparticle. Optimal bounds on the enhancement factors are also derived; (iii) to show, by analyzing the imaginary part of the Green function, that one can achieve super-resolution and super-focusing using plasmonic nanoparticles. For simplicity, the Helmholtz equation is used to model electromagnetic wave propagation.

Introduction

Plasmon resonant nanoparticles have unique capabilities of enhancing the brightness of light and confining strong electromagnetic fields [START_REF] Sarid | Modern Introduction to Surface Plasmons: Theory, Mathematical Modeling, and Applications[END_REF]. A thriving interest for optical studies of plasmon resonant nanoparticles is due to their recently proposed use as labels in molecular biology [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]. New types of cancer diagnostic nanoparticles are constantly being developed. Nanoparticles are also being used in thermotherapy as nanometric heat-generators that can be activated remotely by external electromagnetic fields [START_REF] Baffou | Mapping heat origin in plasmonic structures[END_REF]. According to the quasi-static approximation for small particles, the surface plasmon resonance peak occurs when the particle's polarizability is maximized. Plasmon resonances in nanoparticles can be treated at the quasi-static limit as an eigenvalue problem for the Neumann-Poincaré integral operator, which leads to direct calculation of resonance values of permittivity and optimal design of nanoparticles that resonate at specified frequencies [START_REF] Ammari | Optimal shape design by partial spectral data[END_REF][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]. 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 [START_REF] Ammari | Surface plasmon resonance of nanoparticles and applications in imaging[END_REF], we have provided a rigorous mathematical framework for localized surface plasmon resonances. We have considered the full Maxwell equations. Using layer potential techniques, we have derived the quasi-static limits of the electromagnetic fields in the presence of nanoparticles.

We have proved that the quasi-static limits are uniformly valid with respect to the nanoparticle's bulk electron relaxation rate. We have introduced localized plasmonic resonances as the eigenvalues of the Neumann-Poincaré operator associated with the nanoparticle. We have described a general model for the permittivity and permeability of nanoparticles as functions of the frequency and rigorously justified the quasi-static approximation for surface plasmon resonances.

In this paper, 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 near-cloaking. 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]. Finally, we consider the super-resolution phenomenon in plasmonic nanoparticles. Super-resolution is 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]. 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. 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. 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 super-resolution 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 methodology developed 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], we show in this paper that one can achieve super-resolution using plasmonic nanoparticles as well.

The paper is organized as follows. In section 2 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 3 we consider the case of multiple plasmonic nanoparticles. Section 4 is devoted to the study of the scattering and absorption enhancements. We also clarify the connection between the blow up of the scattering frequencies and the plasmonic resonances. The scattering coefficients are simply the Fourier coefficients of the scattering amplitude [START_REF] Ammari | Enhancement of near-cloaking. Part II: The Helmholtz equation[END_REF][START_REF] Ammari | Enhancement of near cloaking for the full Maxwell equations[END_REF]. In section 5 we investigate the behavior of the scattering coefficients at the plasmonic resonances. In section 6 we prove that using plasmonic nanoparticles one can achieve super resolution imaging. Appendix A is devoted to the derivation of asymptotic expansions with respect to the frequency of some boundary integral operators associated with the Helmholtz equation and a single particle. These results are generalized to the case of multiple particles in Appendix B. In Appendix C we provide the technical modifications needed in order to study the shift in the plasmon resonance in the two-dimensional case. In Appendix D we prove useful sum rules for the polarization tensor.

Layer potential formulation for plasmonic resonances 2.1 Problem formulation and some basic results

We consider the scattering problem of a time-harmonic wave incident on a plasmonic nanoparticle. For simplicity, we use the Helmholtz equation instead of the full Maxwell equations. 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 (the twodimensional case is treated in Appendix C) 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 ε c < 0, ε c > 0, µ c < 0, µ 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 paper, we assume that ε m and µ m are real and strictly positive and that k c < 0 and k c > 0.

Using dimensionless quantities, we assume that the following set of conditions holds.

Condition 1. We assume that the numbers ε m , µ m , ε c , µ c are dimensionless and are of order one. We also assume that the particle has size of order one and ω is dimensionless and is of order o(1).

It is worth emphasizing that in the original dimensional variables ω refers to the ratio between the size of the particle and the wavelength. Moreover, the operating frequency varies in a small range and hence, the material parameters ε c and µ c can be assumed independent of the frequency.

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.

(2.1)

Here, ∂/∂ν denotes the normal derivative and 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.

The model problem (2.1) is referred to as the transverse magnetic case. Note that all the results of this paper 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. Let G be the Green function for the Helmholtz operator ∆ + k 2 satisfying the Sommerfeld radiation condition. In dimension three, G is given by

G(x, y, k) = - e ik|x-y| 4π|x -y| .
By using the following single-layer potential and Neumann-Poincaré integral operator

S k D [ψ](x) = ∂D G(x, y, k)ψ(y)dσ(y), x ∈ R 3 , (K k D ) * [ψ](x) = ∂D ∂G(x, y, k) ∂ν(x) ψ(y)dσ(y), x ∈ ∂D,
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, (2.2)
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 , (2.3)
where Id denotes the identity operator. 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, (2.4)
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 , (2.5)
f = F 2 + 1 µ c 1 2 Id -(K kc D ) * (S kc D ) -1 [F 1 ]. (2.6)
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 , (2.7)
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 [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].

Lemma 2.1. (i) The following Calderón identity holds:

K D S D = S D K * D ;
(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 (2.8) 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 2.2. 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 .
(2.9)

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. (i) We have (-1 2 Id + K * D )S -1 D [χ(∂D)] = 0 with χ(∂D) being the characteristic function of ∂D. (ii) Let λ 0 = 1 2
, then the corresponding eigenspace has dimension one and is spanned by the function

ϕ 0 = cS -1 D [χ(∂D)] 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 , χ(∂D)) -1 2 , 1 2 = 0 for j ≥ 1.
Here, {ϕ j } j is the set of normalized eigenfunctions of K * D .

From (2.7), it is easy to see that

A D,0 [ψ] = ∞ j=0 τ j (ψ, ϕ j ) H * ϕ j , (2.10)
where

τ j = 1 2µ m + 1 2µ c - 1 µ c - 1 µ m λ j . (2.11)
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 A, we can obtain the following result.

Lemma 2.4. 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 .
(2.12)

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 . (2.13)
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 (A.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 * , (2.14)
where A D,2 is defined by (2.12).

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 ), (2.15) ϕ j (ω) = ϕ j + ω 2 ϕ j,2 + O(ω 3 ), (2.16)
where

τ j,2 = R jj , (2.17)
ϕ j,2 = l =j R jl 1 µm -1 µc (λ j -λ l ) ϕ l .
(2.18)

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

We first introduce different notions of plasmonic resonance as follows.

Definition 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 (2.11).

(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 (2.17).

Note that quasi-static resonances are size independent and is therefore a zero-order approximation of the plasmonic resonance in terms of the particle size while the first-order corrected quasi-static plasmonic resonance depends on the size of the nanoparticle (or equivalently on ω in view of the non-dimensionalization adopted herein).

We are interested in solving the equation 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. 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 2.1. 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 paper, we always exclude 0 from the index set of resonance J.

From now on, we shall use J as our index set of resonances. For simplicity, we assume throughout that the following conditions hold. Condition 2. Each eigenvalue λ j for j ∈ J is a simple eigenvalue of the operator K * D .

Condition 3. Let λ = µ m + µ c 2(µ m -µ c ) . ( 2

.19)

We assume that λ = 0 or equivalently, µ c = -µ m .

Condition 3 implies that the set J is finite.

We 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, (2.20)
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), (2.21)
ϕ j (ω) = ϕ j + ω 2 ϕ j,2 + O(ω 2 ). (2.22)
Using the eigenfunctions ϕ j (ω), we can show that

P J (ω)[x] = j∈J x, ϕ j (ω) H * ϕ j (ω). (2.23)
Throughout this paper, 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 ]]. (2.24)
The following lemma holds.

Lemma 2.5. The norm

A D (ω) -1 P J c (ω) L(H * (∂D),H * (∂D)) is uniformly bounded in ω.
Proof. Consider the operator

A D (ω)| J c : P J c (ω)H * (∂D) → P J c (ω)H * (∂D).
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 (2.6) with respect to ω.

Lemma 2.6. 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

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 χ(∂D) + iω √ ε m µ m [d • (x -z)] + O(ω 2 )] = - e ikmd•z µ c 1 2 Id -K * D S -1 D [χ(∂D)] - ωe ikmd•z µ c 1 2 Id -K * D B D,1 [χ(∂D)] - 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 ),
where we have made use of the facts that

1 2 Id -K * D S -1 D [χ(∂D)] = 0 and B D,1 [χ(∂D)] = cS -1 D [χ(∂D)
] for some constant c; see again Appendix A.

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

Theorem 2.1. Under Conditions 1, 2, and 3 the scattered field u s = u -u i due to a single plasmonic particle has the following representation in the quasi-static regime:

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 (2.19).

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 2.6. 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 * . (2.25)
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 2.1. Assume the same conditions as in Theorem 2.1. Under the additional condition that min

j∈J |τ j (ω)| ω 3 , (2.26)
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).
Rescaling 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).
(2.27)

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, (2.28)
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 ), (2.29)
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 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 (2.15), 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 (2.17) with D replaced by B.

3 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. For ease of exposition, we consider the case of L particles with an identical shape. We assume that the following condition holds.

Condition 4. 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 + δ D, l = 1, 2, . . . , L, where D has size one and is centered at the origin. Moreover, we denote D 0 = δ D as our reference nanoparticle. Denote by

D = 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. (3.1) 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 (3.1) 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 .
3.2 First-order correction to plasmonic resonances and field behavior at plasmonic resonances in the multi-particle case

We consider the scattering in the quasi-static regime, i.e., when the incident wavelength is much greater than one. With proper dimensionless analysis, we can assume that ω 1. As a consequence, 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 l,p (ω) = 1 µ c 1 2 Id -(K kc D l ) * (S kc D l ) -1 S km D l ,Dp + 1 µ m K km D l ,Dp , 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.

Lemma 3.1. (i) Regarded as operators from H * (∂D p ) into H * (∂D l ), 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 l,p,0,1 + S l,p,0,2 + 1 µ m K l,p,0,0 + O(δ 2 ω 2 ) + O(δ 4 ).
Moreover,

1 2 Id -K * D l • S -1 D l • S l,p,0,1 = O(δ 2 ), 1 2 Id -K * D l • S -1 D l • S l,p,0,2 = O(δ 3 ), K l,p,0,0 = O(δ 2 ).
Proof. The proof of (i) follows from Lemmas 2.4 and B.3. We now prove (ii). Recall that

1 2 Id -(K kc D l ) * = 1 2 Id -K * D l + O(δ 2 ω 2 ), (S kc D l ) -1 = S -1 D l -k c S -1 D l S D l ,1 S -1 D l + O(δ 2 ω 2 ), S km D l ,Dp = S l,p,0,0 + S l,p,0,1 + S l,p,0,2 + k m S l,p,1 + k 2 m S l,p,2,0 + O(δ 4 ) + O(ω 2 δ 2 ) K km Dp,D l = K l,p,0,0 + O(ω 2 δ 2 ).
Using the identity

1 2 Id -K * D l S -1 D l [χ(D l )] = 0, we can derive that A l,p (ω) = 1 µ c 1 2 Id -K * D l (S kc D l ) -1 S km D l ,Dp + 1 µ m K l,p,0,0 + O(δ 2 ω 2 ) = 1 µ c 1 2 Id -K * D l S -1 D l S km D l ,Dp + 1 µ m K l,p,0,0 + O(δ 2 ω 2 ) = 1 µ c 1 2 Id -K * D l S -1 D l S l,p,0,0 + S l,p,0,1 + S l,p,0,2 + k m S l,p,1 + k 2 m S l,p,2,0 + O(δ 4 ) + 1 µ m K l,p,0,0 + O(δ 2 ω 2 ) = 1 µ c 1 2 Id -K * D l S -1 D l S l,p,0,1 + S l,p,0,2 + 1 µ m K l,p,0,0 + O(δ 2 ω 2 ) + O(δ 4 ).
The rest of the lemma follows from Lemmas B.3 and B.6.

Denote by

H * (∂D) = H * (∂D 1 ) × . . . × H * (∂D L ), which is equipped with the inner product (ψ, φ) H * = L l=1 (ψ l , φ l ) H * (∂D l ) .
With the help of Lemma 3.1, the following result is obvious.

Lemma 3.2. Regarded as an operator from H * (∂D) into H * (∂D), we have

A(ω) = A D,0 + A D,1 + O(ω 2 δ 2 ) + O(δ 4 ),
where

A D,0 =     A D 1 ,0 A D 2 ,0 . . . A D L ,0     , A D,1 =     0 A D,1,12 A D,1,13 . . . A D,1,21 0 A D,1,23 . . . . . . A D,1,L1 . . . A D,1,LL-1 0     with A D l ,0 = 1 2µ m + 1 2µ c Id -( 1 µ c - 1 µ m )K * D l , A D,1pq = 1 µ c 1 2 Id -K * Dp S -1 Dp S p,q,0,1 + S p,q,0,2 + 1 µ m K p,q,0,0 .
It is evident that

A D,0 [ψ] = ∞ j=0 L l=1 τ j (ψ, ϕ j,l ) H * ϕ j,l , (3.2)
where

τ j = 1 2µ m + 1 2µ c - 1 µ c - 1 µ m λ j , (3.3)
ϕ j,l = ϕ j e l (3.4)
with e l being the standard basis of R L . We take A(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. Condition 5. Each eigenvalue λ j , j ∈ J, of the operator K * D 1 is simple. Moreover, we have

ω 2 δ.
In what follows, we only use the first order perturbation theory and derive the leading order term, i.e., the perturbation due to the term A D,1 . For each l, we define an L × L matrix R l by letting

R l,pq = A D,1 [ϕ l,p ], ϕ l,q H * , = A D,1 [ϕ l e p ], ϕ l e q H * , = A D,1,pq [ϕ l ], ϕ l H * .
Lemma 3.3. The matrix R l = (R l,pq ) p,q=1,...,L has the following explicit expression:

R l,pp = 0, R l,pq = 3 4πµ c (λ j - 1 2
)

|α|=|β|=1 ∂D 0 ∂D 0 (z p -z q ) α+β |z p -z q | 5 x α y β ϕ l (x)ϕ l (y)dσ(x)dσ(y) + 1 4πµ c - 1 4πµ m (λ j - 1 2
)

∂D 0 ∂D 0 x • y |z p -z q | 3 ϕ l (x)ϕ l (y)dσ(x)dσ(y) = O(δ 3 ), p = q.
Proof. It is clear that R l,pp = 0. For p = q, we have

R l,pq = R I l,pq + R II l,pq + R III l,pq ,
where

R I l,pq = 1 µ c 1 2 Id -K * Dp S -1 Dp S p,q,0,1 [ϕ l ], ϕ l H * (∂D l ) , R II l,pq = 1 µ c 1 2 Id -K * Dp S -1 Dp S p,q,0,2 [ϕ l ], ϕ l H * (∂D l ) , R III l,pq = 1 µ m K p,q,0,0 [ϕ l ], ϕ l H * (∂D l ) .
We first consider R I l,pq . By the following identity

1 2 Id -K * Dp S D l [ϕ l ] = S D l 1 2 Id -K Dp [ϕ l ] = (λ j - 1 2 )ϕ l ,
we obtain

R I l,pq = - 1 µ c 1 2 Id -K * Dp S -1 Dp S p,q,0,1 [ϕ l ], S D l [ϕ l ] L 2 (∂D l ) , = 1 µ c (λ j - 1 2 ) S p,q,0,1 [ϕ l ], S D l [ϕ l ] L 2 (∂D l ) .
Using the explicit representation of S p,q,0,1 and the fact that (χ(∂D j ), φ l ) L 2 (∂D j ) = 0 for j = 0, we further conclude that R I l,pq = 0. Similarly, we have

R II l,pq = 1 µ c (λ j - 1 2 ) S p,q,0,2 [ϕ l ], S D l ϕ l L 2 (∂D l ) , = 1 µ c (λ j - 1 2
)

|α|=|β|=1 ∂D 0 ∂D 0 3(z p -z q ) α+β 4π|z p -z q | 5 x α y β + δ αβ x α y β 4π|z p -z q | 3 ϕ l (x)ϕ l (y)dσ(x)dσ(y) = 3 4πµ c (λ j - 1 2
)

|α|=|β|=1 ∂D 0 ∂D 0 (z p -z q ) α+β |z p -z q | 5 x α y β ϕ l (x)ϕ l (y)dσ(x)dσ(y) + 1 4πµ c (λ j - 1 2
)

|α|=1 ∂D 0 ∂D 0 1 |z p -z q | 3 x α y α ϕ l (x)ϕ l (y)dσ(x)dσ(y).
Finally, note that

K p,q,0,0 [ϕ l ] = 1 4π|z p -z q | 3 a • ν(x) = 1 4π|z p -z q | 3 3 m=1 a m ν m (x),
where a m = (y -z q ) m , ϕ l L 2 (∂Dq) , and a = (a 1 , a 2 , a 3 ) T . By identity (2.25), we have

R III l,pq = - 1 µ m K p,q,0,0 [ϕ l ], ϕ l H * (∂D l ) = - 1 4π|z p -z q | 3 µ m a • ν(x), ϕ l H * (∂D l ) = - 1 4π|z p -z q | 3 µ m 1 2 Id -K * Dp S -1 Dp (a • (x -z p )), ϕ l H * (∂D l ) = - 1 4π|z p -z q | 3 µ m (λ j - 1 2 ) a • (x -z p ), ϕ l L 2 (∂Dp) = - 1 4π|z p -z q | 3 µ m (λ j - 1 2
)

∂D 0 ∂D 0 x • yϕ l (x)ϕ l (y)dσ(x)dσ(y).
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 f presentation, we assume the following condition. Condition 6. R l has L-distinct eigenvalues.

We remark that Condition 6 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),

τ j,l (ω) = τ j + τ j,l + O(δ 4 ) + O(ω 2 δ 2 ). (3.5)
The associated perturbed eigenfunctions have the following form

ϕ j,l (ω) = L p=1 X j,l,p e p ϕ j + O(δ 4 ) + O(ω 2 δ 2 ). (3.6)
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 (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 the previous section for a single particle case.

We define

P J (ω)ϕ j,m (ω) = ϕ j,m (ω), j ∈ J, 0, j ∈ J c .
In fact,

P J (ω) = j∈J P j (ω) = j∈J 1 2πi γ j (ξ -A D (ω)) -1 dξ, (3.7)
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.

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

τ j,l (ω) = τ j + τ j,l + O(δ 4 ) + O(ω 2 δ 2 ), ϕ j,l (ω) = ϕ j,l + O(δ 4 ) + O(ω 2 δ 2 ),
where

ϕ j,l = L p=1
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

(ϕ j,p , ϕ j,q ) H * (∂D) = δ pq ,
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 3.4. In the space H * (∂D), as ω goes to zero, we have

f = ωf 0 + O(ω 2 δ 3 2 ),
where

f 0 = (f 0,1 , . . . , f 0,L ) T with f 0,l = -i √ ε m µ m e ikmd•z l 1 µ m d • ν(x) + 1 µ c 1 2 Id -K * D l S -1 D l [d • (x -z)] = O(δ 3
2 ).

Proof. We first show that

u H * (∂D 0 ) = δ 3 2 +m u H * (∂ D) , u H(∂D 0 ) = δ 1 2 +m u H(∂ D)
for any homogeneous function u such that u(δx) = δ m u(x). Indeed, we have η(u

)(x) = δ m u(x). Since η(u) H * (∂ D) = δ -3 2 u H * (∂D 0 ) (see Appendix B), we obtain u H * (∂D 0 ) = δ 3 2 η(u) H * (∂ D) = δ 3 2 +m u H * (∂ D) ,
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 2.6 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 3.1. Under Conditions 1, 2, 3, and 4, the scattered field by L plasmonic particles in the quasi-static regime has the following representation

u s = S km D [ψ],
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 2.1.

As a consequence, the following result holds.

Corollary 3.1. With the same notation as in Theorem 3.1 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 ).

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 (4.10) and (4.11), where M given by (4.7) is the polarization tensor associated with the nanoparticle D and the magnetic contrast µ c (ω)/µ m . In view of (4.15), 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 (4.17). As shown in (4.20) and (4.22), 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.

Propsition 4.1. Let x ∈ R 3 be such that |x| 1/ω. Then, we have

u s (x) = - e ikm|x| 4π|x| A ∞ x |x| + O 1 |x| 2 (4.1)
with

A ∞ x |x| = ∂D e -ikm x |x| •y ψ(y)dσ(y) (4.2)
being the scattering amplitude and ψ being defined by (2.3).

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) = Re 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 . We now consider the outward flow of energy through the sphere ∂B R of radius R and center the origin:

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).
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 .
Definition 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.

Theorem 4.1 (Optical theorem). If u i (x) = e ikmd•x , where d is a unit direction, then

Q ext =Q s + Q a = 1 k m [A ∞ (d)] , (4.3)
Q s = S 2 |A ∞ (x)| 2 dσ(x) (4.4)
with A ∞ being the scattering amplitude defined by (4.2).

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

∇u s (x) • ν(x) ∼ ik m u s (x). (4.5)
Hence, from (4.1) we get

u s (x)∇u s (x) • ν(x) -u s (x)∇u s (x) • ν(x) ∼ 2Ck m |x| 2 A ∞ x |x| 2 ,
which yields (4.4). We now compute the extinction rate. We have

∇u i (x) • ν(x) = ik m d • ν(x)e ikmd•x . (4.6)
Therefore, using 4.5 and 4.6, 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) 4π|x| d • n -ik m e ikm(|x|-d•x) 4π|x| = -ik m e ikm|x|-d•ν(x) 4π|x| (d • ν(x) -1) .
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)) 4πR (d • ν(x) -1)
.

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]) 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 ,
where

G(ν(x)) = d • ν(x) -1. Hence, ∂B R u i (x)∇u s (x) -u s (x)∇u i (x) • ν(x) ∼ -A ∞ (d) as R → ∞.
Therefore,

W = -iC A ∞ (d) -A ∞ (d) = 2C [A ∞ (d)] .
Since

|C| u i (x)∇u i (x) -u i (x)∇u i (x) = 2|C|k 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. [START_REF] Bonnetier | On the spectrum of the Poincaré variational problem for two close-to-touching inclusions in 2D[END_REF]) and let

M (λ, D) := ∂D (λId -K * D ) -1 [ν]x dσ(x) (4.7)
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 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 )) (4.8)
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 .

Assume for simplicity that ε c = ε m . We explicitly compute the scattering amplitude A ∞ in (4.1). Take u i (x) = e ikmd•x and assume again for simplicity that z = 0. Equation (4.8) yields, for |x|

1 ω , u s (x) = e ikm|x| 4π|x| ik m ik m x |x| - x |x| 2 • M (λ, D)d + O( δ 4 dist(λ, σ(K * D ))
).

Since we are in the far-field region, we can write that, up to an error of order δ 4 /dist(λ, σ(K * D )),

u s (x) = -k 2 m e ikm|x| 4π|x| x |x| • M (λ, D)d + O 1 |x| 2 .
(4.9)

In the next proposition we write the extinction and scattering cross-sections in terms of the polarization tensor.

Propsition 4.3. 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 = k m 3 [TrM (λ, D)] , (4.10)
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 16π 9 |TrM (λ, D)| 2 . (4.11)
Proof. Remark from (4.9) that the scattering amplitude A ∞ in the case of a plane wave illumination is given by

A ∞ x |x| = -k 2 m x |x| • M (λ, D)d. (4.12)
Using Theorem 4.1, we can see that for a given orientation

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

Q ext m = k m 4π S 2 d • M (λ, D)d dσ(d) .
Since M (λ, D) is symmetric, it can be written as M (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 4π S 2 x • N (λ)xdσ(x) , = k m 3 [TrN (λ)] = k m 3 [TrM (λ, D)] . (4.13)
Now, we compute the averaged scattering cross-section. Let M (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 S 2 ×S 2 |x • M (λ, D)d| 2 dσ(x) dσ(d), =k 4 m 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 16π 9 |TrM (λ, D)| 2 ,
which completes the proof.

From Theorem 4.1, we obtain that the averaged absorption cross-section is given by

Q a m = k m 3 [TrM (λ, D)] -k 4 m 16π 9 |TrM (λ, D)| 2 .
Therefore, under the condition (2.26), 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). (4.14)
For a C 1,α domain D in R d , K * D is compact and self-adjoint in H * (defined in Lemma 2.1 for d = 3 and in Lemma C.1 for d = 2). 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 2.1). 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 , (4.15)
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. 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 C). Since |λ j | < 1/2 for j ≥ 1, we obtain the result.

Lemma 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 D.

Let λ = λ + iλ . We have

(Tr(M (λ, D))) = ∞ j=1 |λ | d l=1 α (j) l,l (λ -λ j ) 2 + λ 2 . (4.16)
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 and 3 hold. Then we define the bijection ρ : N + → N + such that λ ρ(j) = -λ j and we can write

(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 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, (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 4.2 we obtain the following result.

(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.

Corollary 4.1. We have

(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 ).
(4.17)

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 [7]

M (λ, D) =      |D| λ -1 2 a-b a+b 0 0 |D| λ + 1 2 a-b a+b      . (4.18)
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 (4.15), 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 (4.19)
for j ≥ 2 and

i∈V(λ j ) α (i) 1,2 = 0 for j ≥ 1.
In view of (4.19), 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,

| (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 ).
Using Lemma 4.2 we obtain the following result. Thus, bound (4.20) 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 .

| (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 ).

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. (4.21)
The following holds [START_REF] Ammari | Mathematical and Statistical Methods for Multistatic Imaging[END_REF].

Lemma 4.3. Let D be the ellipsoid defined by (4.21). 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 (4.22) where for j = 1, 2, 3,

3 l=1 D |∇S D [ν l ]| 2 dx = (s 2 1 + s 2 2 + s 2 3 )|D|.
(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 ),
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 near-cloaking. 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 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.

The notion of scattering coefficients

From Graf's addition formula [START_REF] Ammari | Mathematical and Statistical Methods for Multistatic Imaging[END_REF] and (2.2) 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

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 (2.4).

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 (2.4) 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).

(5.1)

The coefficients W nm are called the scattering coefficients.

Lemma 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 ] .
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 C). 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 C). Finally, J m (x) = O(x m ) immediately yields the desired estimate for f (m) .

From Theorem C.1, it is easy to 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 ). (5.2)
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).
(5.3) 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 5.1, we obtain the following result.

Propsition 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. Lemma 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 (5.2) and Lemma 5.1. Changing S D by S D in Theorem

2.1 gives ( 1 2 Id -K * D ) S -1 D [|x|e iθx ], ϕ j H * = -(e iθν , ϕ j ) H * .
Using now Lemma 5.1 in (5.2) 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 (5.3) 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 5.2 and equation ( 5.3), 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 (2.19). 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)   . (5.4)
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 (4.7).

Finally, considering the above we can state the following result.

Theorem 5.1. Let A ∞ be the scattering amplitude in the far-field defined in (4.2) 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 1,1 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 (5.1).

Proof. From (4.12), 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).
We have then from (5.4) 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.

6 Super-resolution (super-focusing) by using plasmonic particles

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 [START_REF] Ammari | An Introduction to Mathematics of Emerging Biomedical Imaging[END_REF]. By modifying the homogeneous spaces with subwavelength resonators, we can introduce propagating subwavelength resonance modes to the space which encode subwavelength information in a neighborhood of the space embedded by the subwavelenghth resonators, thus 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]. 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. The results in this section can be viewed as a consequence of the results in Section 3.

Asymptotic expansion of the scattered field

In order to illustrate the superfocusing phenomenon, we set

u i (x) = G(x, x 0 , k m ) = - e ikm|x-x 0 | 4π|x -x 0 | . Lemma 6.1.
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 ).

Proof. The proof is similar to that of Lemma 2.6. 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, we consider the following domain

D δ,k = x ∈ R 3 ; min 1≤l≤L |x -z l | δ, max 1≤l≤L |x -z l | 1 k . Lemma 6.2. 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 , 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 [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, 0, k) ∂y α y α , and
∂G(x, 0, k) ∂y α = - 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

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. Theorem 6.1. Under Conditions 1, 2, 3, and 4, 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 ,

Γ(x, x 0 , k m ) = G(x, x 0 , k m ) + 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(δ 4 ) + O(ωδ 3 )

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 6.2,

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 ).

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 6.1, we obtain the following result on the imaginary part of the Green function.

Theorem 6.2. Assume the same conditions as in Theorem 6.1. 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 ), λ -λ j + 1 µ c - 1 µ m -1 τ j,l λ -λ j + 1 µ c - 1 µ m -1 τ j,l
for each l and j ∈ J, we have

Γ(x, x 0 , k m ) = G(x, x 0 , k m ) + 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, 0) + O(ωδ 3 ) + O(δ 4 )

× 1 λ -λ j + 1 µc -1 µm -1 τ j,l ,
where x, x 0 ∈ D δ,km .

Note that 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 6.2, if we have additionally that

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 Γ(x, x 0 , k m ) which is due to resonance has size one and exhibits subwavelength peak with width of order one. This breaks the diffraction limit 1/k m in the free space. We also note that the term G(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 paper, based on perturbation arguments, we studied the scattering by plasmonic nanoparticles when the frequency is close to a resonant frequency. We have shown that plasmon resonant nanoparticles provide a possible way not only of super-resolved imaging but also of scattering and absorption enhancements.

We have derived the shift and broadening of the plasmon resonance with changes in size. We have also consider the case of multiple nanoparticles under the weak interaction assumption. The localization algorithms developed in [START_REF] Ammari | Mathematical and Statistical Methods for Multistatic Imaging[END_REF][START_REF] Ammari | MUSIC-type electromagnetic imaging of a collection of small three-dimensional inclusions[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 crosssection 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 paper is under consideration and will be reported elsewhere. Another challenging problem is to optimize the super-focusing phenomenon in terms of the organization of the nanoparticles. This will be also the subject of a forthcoming publication.

A Asymptotic expansion of the integral operators: single particle

In this section, we derive asymptotic expansions with respect to k of some boundary integral operators defined on the boundary of a bounded and simply connected smooth domain D in dimension three whose size is of order one. We first consider the single layer potential

S k D [ψ](x) = ∂D G(x, y, k)ψ(y)dσ(y), x ∈ ∂D,
Proof. The proof of (i) is straightforward and we only need to prove (ii) and (iii). To prove (iii), we have

f 2 H * (∂D 0 ) = ∂D 0 ∂D 0 f (x)f (y) 4π|x -y| dσ(x)dσ(y) = δ 3 ∂ D ∂ D η(f )( x)η(f )( y) 4π| x -y| dσ( x)dσ( x) = δ 3 η(f ) 2 H * (∂ D) ,
whence (iii) follows. To prove (ii), recall that

f H(∂D 0 ) = S -1 D 0 f H * (∂D 0 ) . Let u = S -1 D 0 [f ]. Then f = S D 0 [u]. We can show that η(f ) = δS D (η(u)).
As a result, we have

η(f ) H(∂ D) = δ S D (η(u)) H(∂ D) = δ η(u) H * (∂ D) = δ -1 2 u H * (∂D 0 ) = δ -1 2 f H(∂D 0 ) ,
which proves (ii). Then,

R L(H * (∂Y ),H * (∂X)) = δ 2+m R L(H * (∂ Y ),H * (∂ X)) , R L(H * (∂Y ),H(∂X)) = δ 1+m R L(H * (∂ Y ),H(∂ X)) .
Proof. The result follows from Lemma B.1 and the following identity

R = δ 2+m η -1 • R • η.
We first consider the operators S k D j and (K k D j ) * . The following asymptotic expansions hold.

Lemma B.3. (i) Regarded as operators from H * (∂D j ) into H(∂D j ), we have

S k D j = S D j + kS D j ,1 + k 2 S D j ,2 + O(k 3 δ 3 ),
where S D j = O(1) and S D j ,m = O(δ m );

(ii) Regarded as operators from H(∂D j ) into H * (∂D j ), we have

(S k D j ) -1 = S -1 D j + kB D j ,1 + k 2 B D j ,2 + O(k 3 δ 3 ),
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

(K k D j ) * = K * D j + k 2 O(δ 2 ),
where

K * D j = O(1).
Proof. The proof immediately follows from Lemmas B.2, A.1, and A.3.

We now consider the operator S k D j ,D l . By definition,

S k D j ,D l [ψ](x) = ∂D j G(x, y, k)ψ(y)dσ(y), x ∈ ∂D l .
Using the expansion

G(x, y, k) = ∞ m=0 k m Q m (x, y),
where

Q m (x, y) = - i m |x -y| m-1 4π , we can derive that S k D j ,D l = m≥0 k m S j,l,m , where S j,l,m [ψ](x) = ∂D j Q m (x, y)ψ(y)dσ(y).
We can further write S j,l,m = n≥0 S j,l,m,n , where S j,l,m,n is defined by

S j,l,m,n [ψ](x) = ∂D j |α|+|β|=n 1 α!β! ∂ |α|+|β| ∂x α ∂y β Q m (z l , z j )(x -z l ) α (y -z j ) β ψ(y) dσ(y).
In particular, we have

S j,l,0,0 [ψ](x) = - 1 4π|z j -z l | (ψ, χ(∂D j )) H -1/2 (∂D j ),H 1/2 (∂D j ) χ(D l ), S j,l,0,1 [ψ](x) = |α|=1 (z l -z j ) α 4π|z l -z j | 3 (x -z l ) α (ψ, χ(∂D l )) H -1/2 (∂D j ),H 1/2 (∂D j ) + (y -z j ) α , ψ χ(D l ) , S j,l,0,2 [ψ](x) = |α|+|β|=2 1 α!β! ∂ 2 Q 0 (z l , z j ) ∂x α ∂y β (x -z l ) α (y -z j ) β ψ(y)dσ(y), S j,l,1 [ψ](x) = - i 4π (ψ, χ(∂D j )) H -1/2 (∂D j ),H 1/2 (∂D j ) χ(D l ), S j,l,2,0 [ψ](x) = 1 4π |z l -z j |(ψ, χ(∂D j )) H -1/2 (∂D j ),H 1/2 (∂D j ) χ(D l ).
The following estimate holds.

Lemma B.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.

Similarly, for the operator K km D j ,D l defined in the following way

K k D j ,D l [ψ](x) = ∂D j ∂G(x, y, k) ∂ν(x) ψ(y)dσ(y), x ∈ ∂D l , we have K k D j ,D l = m≥0 k m n≥0 K j,l,m,n ,
where

K j,l,m,n [ψ](x) = ∂D j |α|+|β|=n 1 α!β! ∂ n K m (z l , z j ) ∂x β ∂y α (x -z l ) β (y -z j ) α (x -y) • ν(x)ψ(y)dσ(y) with K m (x, y) = - i m (m -1)|x -y| m-3 4πm! .
In particular, we have

K j,l,0,0 [ψ](x) = 1 4π|z l -z j | 3 (x -z l ) • ν(x) ψ, χ(∂D j ) H -1/2 (∂D j ),H 1/2 (∂D j ) -ψ, (y -z j ) • ν(x) H -1/2 (∂D j ),H 1/2 (∂D j ) +(z l -z j ) • ν(x) ψ, χ(∂D j ) H -1/2 (∂D j ),H 1/2 (∂D j ) , (B.1) K j,l,1,m [ψ] = 0 for all m. (B.2)
Lemma B.5. We have K j,l,m,n L(H * (∂D j ),H * (∂D l )) O(δ n+2 ).

Proof. Note that

K j,l,m,n [ψ](x) = ∂D j |α|+|β|=n 1 α!β! ∂ n K m (z l , z j ) ∂x β ∂y α (x -z l ) β (y -z j ) α (x -z l ) • ν(x)ψ(y)dσ(y), - ∂D j |α|+|β|=n 1 α!β! ∂ n K m (z l , z j ) ∂x β ∂y α (x -z l ) β (y -z j ) α (y -z j ) • ν(x)ψ(y)dσ(y), + ∂D j |α|+|β|=n 1 α!β! ∂ n K m (z l , z j ) ∂x β ∂y α (x -z l ) β (y -z j ) α (z l -z j ) • ν(x)ψ(y)dσ(y).
After a translation of coordinates, we can apply Lemma B.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.6. (i) Regarded as an operator from H * (∂D j ) into H(∂D l ) we have,

S k D j ,D l = S j,l,0,0 + S j,l,0,1 + S j,l,0,2 + kS j,l,1 + k 2 S j,l,2,0 + O(δ 4 ) + O(k 2 δ 2 ).
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

K k D j ,D l = K j,l,0,0 + O(k 2 δ 2 ).
Moreover, K j,l,0,0 = O(δ 2 ).

C Adaptation of results to the two-dimensional case

In this section we adapt the layer potential formulation to plasmonic resonances in two dimensions. We only consider the single particle case. For the multiple particle case, a similar analysis holds.

Recall that in R 2 the single-layer potential S D :

H -1/2 (∂D) → H 1/2 (∂D) is not, in general, invertible nor injective. Hence, -(u, S D [v]) -1 2 , 1 2
does not define an inner product and the symmetrization technique described in Lemma 2.1 is no longer valid. To overcome this difficulty, a substitute of S D can be introduced as in [START_REF] Ando | Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann-Poincaré operator[END_REF] by

S D [ψ] = S D [ψ] if (ψ, χ(∂D)) -1 2 , 1 2 = 0, 1 if ψ = ϕ 0 , (C.1)
where ϕ 0 is the unique (in the case of a single particle) eigenfunction of K * D associated with eigenvalue 1/2 such that (ϕ 0 , χ(∂D)) -1 2 , 1 2 = 1. Note that, from the jump relations of the layer potentials, S D [ϕ 0 ] is constant. The operator S D : H -1/2 (∂D) → H 1/2 (∂D) is invertible. Moreover, the following Calderón

identity holds K D S D = S D K * D . With this, define (u, v) H * = -(u, S D [v]) -1 2 , 1 2 .
Thanks to the invertibility and positivity of -S D , this defines an inner product for which K * D is self-adjoint and H * is equivalent to H -1/2 . Then, if D is C 1,α , we have the following result.

Lemma C.1. Let D be a C 1,α bounded simply connected domain of R 2 and let S D be the operator defined in C.1. Then, (i) The operator K * D is compact self-adjoint in the Hilbert space H * (∂D) equipped with the inner product defined by

(u, v) H * = -(u, S D [v]) -1 2 , 1 2 (C.2) 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;

(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) H * (∂D) = H * 0 (∂D) ⊕ {µϕ 0 , µ ∈ C}, 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 . Remark C.1. Note that S -1 D [χ(∂D)] = ϕ 0 and (-1 2 Id + K * D ) = (-1 2 Id + K * D )P H * 0 , where P H * 0
is the orthogonal projection onto H * 0 (∂D). In particular, we have (-

1 2 Id + K * D ) S -1 D [χ(∂D)] = 0.
Let us now consider the single-layer potential for the Helmholtz equation in R 2

S k D [ψ](x) = ∂D G(x, y, k)ψ(y)dσ(y), x ∈ ∂D, where G(x, y, k) = - i 4 H (1)
0 (k|x -y|) and H

(1) 0 is the Hankel function of first kind and order 0. We have

- i 4
H (1) 0 (k|x -y|) = 1 2π log |x -y| + τ k + ∞ j=1 (b j log k|x -y| + c j )(k|x -y|) 2j ,
where

τ k = 1 2π (log k + γ -log 2) - i 4 , b j = (-1) j 2π 1 2 2j (j!) 2 , c j = -bj γ -log 2 - iπ 2 - j n=1 1 n ,
and γ is the Euler constant. Thus, we get

S k D = Ŝk D + ∞ j=1 k 2j log k S (1) D,j + ∞ j=1 k 2j S (2) D,j , (C.3) where Ŝk D [ψ](x) = S D [ψ](x) + τ k ∂D [ψ]dσ, S (1)
D,j [ψ](x) = ∂D b j |x -y| 2j ψ(y)dσ(y), S (2)
D,j [ψ](x) = ∂D |x -y| 2j (b j log |x -y| + c j )ψ(y)dσ(y).
Lemma C.2. The norms S

D,j L(H * (∂D),H(∂D)) and S

D,j L(H * (∂D),H(∂D)) are uniformly bounded with respect to j. Moreover, the series in (C.3) is convergent in L(H * (∂D), H(∂D)).

Proof. The proof is similar to that of Lemma A.1.

Observe that

S D -S D [ψ] = S D -S D [P H * 0 [ψ] + (ψ, ϕ 0 ) H * ϕ 0 ] = (ψ, ϕ 0 ) H * (S D [ϕ 0 ] -χ(∂D)) .
Then it follows that It is clear that Lemma 2.5 holds in the two-dimensional case. We also have the following asymptotic expansion for f in terms of ω.

Lemma C.7. In the space H * (∂D), as ω goes to zero, we have

f = ωf 1 + O(ω 2 ),
where

f 1 = -ie ikmd•z √ ε m µ m 1 µ m [d • ν(x)] + 1 µ c ( 1 2 Id -K * D ) S -1 D [d • (x -z)]
and z is the center of the domain D.

Finally, the following result holds.

Theorem C.1. Under Conditions 1, 2, and 3, the scattered field by a single plasmonic particle, u s = u -u i , has in the quasi-static limit the following representation: Then, the proof is complete. Proof. We have

f (K * D ) = ∞ i=0 a i (K * D ) i = ∞ i=0 a i ∞ j=1 λ i j (•, ϕ j ) H * ϕ j = ∞ j=1 ∞ i=0 a i λ i j (•, ϕ j ) H * ϕ j = ∞ j=1 f (λ j )(•, ϕ j ) H * ϕ j .
From Lemma D.1, we can deduce that Summing on i and using ∇ x Γ(x, x ) = -∇ x Γ(x, x ), we get .

From (D.3) it follows that

I 1 = - |D| 2 .
Since x l is harmonic, we have Replacing I 1 and I 2 by their expressions gives the desired result.

1 Introduction 2 2 8 3plasmonic nanoparticles 13 3 . 1

 12831 Layer potential formulation for plasmonic resonances 4 2.1 Problem formulation and some basic results . . . . . . . . . . . . . . . . . . . . . 4 2.2 First-order correction to plasmonic resonances and field behavior at the plasmonic resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple Layer potential formulation in the multi-particle case . . . . . . . . . . . . . . . . 13 3.2 First-order correction to plasmonic resonances and field behavior at plasmonic resonances in the multi-particle case . . . . . . . . . . . . . . . . . . . . . . . . . 15

Lemma 4 . 1 .

 41 We have α (j) l,l ≥ 0, j ≥ 1.

Theorem 4 . 2 .

 42 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,

Figure 1 :

 1 Figure 1: Optimal bound for ellipses.

Corollary 4 . 2 .

 42 For any ellipse D of semi-axes of length a and b, we have

(4. 20 )Figure 1

 201 Figure 1 shows (4.20) and the average extinction of two ellipses of semi-axis a and b, where the ratio a/b = 2 and a/b = 4, respectively. We can see from (4.16), Lemma 4.1 and the first sum rule in Lemma 4.2 that for an arbitrary shape B, | (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 2

For a rotatedCorollary 4 . 3 .

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

Lemma B. 2 .

 2 Let X and Y be bounded and simply connected smooth domains in R 3 . Assume 0 ∈ X, Y and X = δ X, Y = δ Y . Let R and R be two boundary integral operators from D (∂Y ) to D (∂X) and D (∂ Y ) to D (∂ X), respectively. Here, D denotes the Schwartz space. Assume that both operators have the same Schwartz kernel R with the following homogeneous scaling property R(δx, δy) = δ m R(x, y).

1 D 1 DS 1 )D, 1 + k 2 S ( 2 )D, 1 + 1 ( Ŝk D ) - 1 . 1

 1111221111 Ŝk D [ψ] = S D [ψ] + (ψ, ϕ 0 ) H * (S D [ϕ 0 ] -χ(∂D)) + τ k ∂D ψ 0 + (ψ, ϕ 0 ) H * ϕ 0 dσ = S D [ψ] + Υ k [ψ], where Υ k [ψ] = (ψ, ϕ 0 ) H * (S D [ϕ 0 ] -χ(∂D) + τ k ) . (C.4)Therefore, we arrive at the following result.Lemma C.3. For k small enough Ŝk D : H * (∂D) → H(∂D) is invertible. Proof. Υ k is clearly a compact operator. Since S D is invertible, the invertibility of Ŝk D is equivalent to that of Ŝk D S -1 D = Id + Υ k S -1 D .By the Fredholm alternative we only need to prove the injectivity ofId + Υ k S -1 D . Since ∀ v ∈ H 1/2 , Υ k S -1 D [v] ∈ C, for Id + Υ k S -[v] = 0, we need v = S D [αϕ 0 ] = α ∈ C. We have Id + Υ k S -D [αϕ 0 ] = α(S D [ϕ 0 ] + τ k ) = 0 iff S D [ϕ 0 ] = -τ k or α = 0.Since we can always find a small enough k such that S D [ϕ 0 ] = -τ k , we need α = 0. This yields the stated result.Lemma C.4. For k small enough, the operator S k D : H * (∂D) → H(∂D) is invertible.Proof. The operator S k D -Ŝk D is a compact operator. Because Ŝk D is invertible for k small enough, by the Fredholm alternative only the injectivity of S k D is necessary. From the uniqueness of a solution to the Helmholtz equation we get the result.We can write (C.3) asS k D = Ŝk D + G k ,whereG k = k 2 log kS (O(k 4 log k).From the two lemmas above we get the identity(S k D ) -1 = Id + ( Ŝk D ) -1 G k -It is clear that ( Ŝk D ) -1 L(H(∂D),H * (∂D))is bounded in k. Thus, for k small enough, we can formally write(S k D ) -1 = ( Ŝk D ) -1 -( Ŝk D ) -1 G k ( Ŝk D ) -1 + O(k 4 log 2 k).We have the identity( Ŝk D ) -1 = S -1 k = Id + (•, ϕ 0 ) H * (S D [ϕ 0 ] -χ(∂D) + τ k )ϕ 0 .Then,Λ -1 k = Id -(•, ϕ 0 ) H * S D [ϕ 0 ] -χ(∂D) + τ k S D [ϕ 0 ] + τ k ϕ 0 ,and therefore,( Ŝk D ) -1 = S -1 D -( S -1 D [•], ϕ 0 ) H * ϕ 0 + ( S -1 D [•], ϕ 0 ) H * S D [ϕ 0 ] + τ k ϕ 0 .Finally, we get(S k D ) -1 = L D + U k -k 2 log kL D S(1)D,1 L D -k 2 L D S(2)D,1 L D -log k(U k S(1)D,1 L D + L D S(1)D,1 U k ) +O(k 2 log -1 k) with L D = P H * 0 S -1 D and U k = ( S -1 D [•], ϕ 0 ) H * S D [ϕ 0 ] + τ k ϕ 0 . We note that U k = O(log -1 k).We now consider the expansion for the boundary integral operator (K k D ) * . We have(K k D ) * = K * D + µc (λ j -λ l )ϕ l , (C.9) and R jl = (A D,1 [ϕ j ], ϕ l ) H * .

u 1 1 2

 11 s = S km D [ψ],whereψ = j∈J ik m e ikmd•z d • ν(x), ϕ j H * ϕ j + O(ω 3 log ω) λ -λ j + O(ω 2 log ω) + O(ω)with λ being defined by(2.19).Proof. We haveψ = 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 3 (log ω)) µm λ j + O(ω 2 log ω) + O(ω). Since d • (x -z) is a harmonic function, changing S D by S D in Theorem 2.1 yields ( Id -K * D )S -1 D [d • (x -z)], ϕ j H * = -(d • ν(x), ϕ j ) H * .

Corollary C. 1 . 1 µm- 1 f

 111 Assume the same conditions as in Theorem 2.1. Then, under the additional condition minj∈J |τ j (ω)| ω 2 ,we haveψ = j∈J ik m e ikmd•z d • ν(x), ϕ j H * ϕ j + O(ω 3 log ω) λ -λ j + ω 2 log ω 1 µc -τ j,1 + O(ω).D Sum rules for the polarization tensorLet f be a holomorphic function defined in an open set U ⊂ C containing the spectrum of K * ∂D .Then, we can write f (z) = ∞ j=0 a j z j for every z ∈ U .Definition 4. Let f (K * D ) := ∞ j=0 a j (K * D ) j , where (K * D ) j := K * D • K * D • .. • K * (λ j )(•, ϕ j ) H * ϕ j .

∂Dx

  l f (K * D )[ν m ](x) dσ(x) = .1) yields the summation rules for the entries of the polarization tensor.In order to prove that∞ j=1 α (j) l,m = δ l,m |D|, we take f (λ) = 1 in (D.1) to get 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 ](x) ∂ν -D [ν l ](x) ∂ν -dσ(x). (D.2)Integrating by parts we arrive at∂D x l ∂S D [ν l ](x) ∂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 ](x) ∂ν -(x)dσ(x) = D e l (x) • ∂D ∇ x Γ(x, x )ν l (x )dσ(x ) dx.

K

  D [ν l ] ∂ν(x) - (x)dσ(x) = -D ∂D ν(x ) • ∇ x Γ(x, x )dσ(x ) dx, D Dis the double-layer potential. Hence, summing equation (D.2) for i = 1, . . . , d, we get the result.Finally, we show that|∇S D [ν l ]| 2 dx. Taking f (λ) = λ 2 in (D.1) yields l (K * D ) 2 [ν l ](x) dσ(x) D [y l ](x)K * D [ν l ](x) dσ(x)

  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 ](x) ∂ν -D [ν l ]| 2 dx.

supported by the ERC Advanced Grant Project MULTIMOD-267184.

where G(x, y, k) = -e ik|x-y| 4π|x -y| is the Green function of Helmholtz equation in R 3 , subject to the Sommerfeld radiation condition. Note that

(ik|x -y|) j-1 j! .

We get

where

(i|x -y|) j-1 j! ψ(y)dσ(y).

In particular, we have

is uniformly bounded with respect to j. Moreover, the series in (A.1) is convergent in L(H * (∂D), H(∂D)).

where C is independent of j. On the other hand, a similar estimate also holds for the operator S * D,j . It follows that S D,j L(H -1 (∂D),L 2 (∂D)) ≤ C.

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 (S k D ) -1 S k D = Id, we can derive that

We can also derive other lower-order terms B D,j .

Lemma A.2. The series in (A.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 Asymptotic expansion of the integral operators: multiple particles

In this section, we consider the three-dimensional case. 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 considered in the paper that are 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

We first state some useful results.

Lemma B.1. The following scaling properties hold:

where

Lemma C.5. The norms K

D,j L(H * (∂D),H * (∂D)) and K

D,j L(H * (∂D),H * (∂D)) are uniformly bounded for j ≥ 1. Moreover, the series in (C.5) is convergent in L(H * (∂D), H * (∂D)).

Proof. The proof is similar to that of Lemma A.1. Recall (2.5) and (2.6), we can show that the following result holds.

Lemma C.6. Regarding A D (ω) as an operator from H * (∂D) to H * (∂D), we have

where

Proof. We have

, where Υ km is defined by (C.4). Hence,


Thomas Levy

Laura Marchand

Vincent Stroobant

Luc Pilotte

Benoît Van Den Eynde

Frédéric Rodriguez

Evelyne Delfourne
email: delfourn@chimie.ups-tlse.fr

IDO1 and TDO inhibitory evaluation of analogues of the marine pyrroloiminoquinone alkaloids:wakayin and tsitsikammamines

Keywords: marine alkaloids, pyrroloiminoquinone, bispyrroloquinone, wakayin, tsitsikammamine, IDO1 and TDO

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In the recent period, immunotherapy has proven to be a promising method for cancer treatment. Among the enzymes involved in the mechanisms allowing tumors to resist or escape immune rejection, indoleamine-2,3-dioxygenase (IDO1) and tryptophane-2,3dioxygenase (TDO) play an essential role. 1,2 They are two main enzymes which catalyse the rate-determining first step of the tryptophan catabolization along the Kynurenine pathway.

Despite the continuous efforts to develop IDO1 and TDO inhibitors, only nine compounds have reached clinical trials 3 , of which, only two target both the two enzymes. This indicates that there is still an urgent need for the discovery of new molecules with this mode of action.

Exiguanine A, a marine alkaloid isolated from the sponge Neopetrosia exigua, has been reported so far, as the most potent IDO1 inhibitor issued from natural source (Ki = 41 nM). 4 In our group we are interested in a family of alkaloids, mainly found in sponges and ascidians, based on the pyrrolo[4,3,2-de]quinolone scaffold, and more particularly in a subgroup of this class of compounds including wakayin 5 and tsitsikammamines A and B 6 , which possess a unique tetracyclic bispyrroloiminoquinone ring system.

Figure 1: Alkaloids issued from marine source

As part of our research program to define new anticancer agents, we were led to study IDO1 and TDO inhibitory activities of tsitsikammamines analogues involved in the synthesis of tsitsikammamine A. 7,8 Some of the compounds described, including methoxytsitsikammamine have shown submicromolar potency in an enzymatic assay whereas a number of other were also active against IDO1 in a cellular assay. 9 Recently, the structurally related derivative Roxyl-WL which constitutes one of the most highly potent and selective inhibitor targeting IDO1 discovered so far (IC 50 = 1 nM), has been described. 10

Results and discussion

Chemistry

Compounds 1-4 were prepared by adapting the reaction sequence previously reported by Zhang et al., the methoxy-naphtoquinone being replaced by N-tosylindoledione 21. 11

Scheme 1: Synthesis of analogues 1-4

In the first step, the tosylindoledione 21 was condensed with tryptamine to give a mixture of two regioisomers 1a and 2a both with their corresponding detosylated analogues 1 and 2 in 8, 11, 46 and 22% yield respectively. It has to be noted that tosyltryptamine was isolated as a byproduct. 12 The same reaction in which tryptamine was replaced by N-methyl tryptamine led to compounds 1b and 2b which were subsequently reacted with DDQ to form derivatives 3 and 4 (28 and 75% yield respectively). The structural assignment of these compounds was achieved from a previously described empirical method. 13 It has to be noted that this last synthetic sequence constitutes a new strategy to build the bispyrroloquinonic framework.

Derivatives 5-7 were obtained by action of 21 on different -ketoamines 14 , whereas pyrazolic analogues 8-20 have been prepared on the basis of a [3+2] dipolar cycloaddition reaction between various quinones and diazo species as previously reported by our group. 15,16 Thus, compounds 8 and 9 were issued from the reaction involving indoledione 21 and the diazoaminopropane derivative 22, and compounds 10-20 resulted from the reaction of indoledione 23 with diazo-derivatives 24.

Scheme 2: Synthesis of bispyrroloquinonic and pyrrazolic analogues

Bioactivity assay

The synthesized derivatives were evaluated for their ability to inhibit tryptophane degradation and kynurenine production in HEK 293-EBNA cell lines expressing hIDO1 or hTDO cell lines expressing IDO1 and TDO. This global cellular assay is informative for drug development, as it evaluates a combination of the drugs' TDO/IDO inhibitory potency, their potential cytotoxicity, the inhibition of tryptophan transporters, and the effects of their metabolites. The assay was performed as previously described, the results are reported in Table1. 17 Among the twenty compounds tested in the present study, six: 1, 3, 5, 6, 7 and 19 exhibit inhibitory activities ranging from 23 to 63% against IDO1 at 12.5 M with only two of which 5 and 6 retain good activity when their concentration was reduced to 3.12 M. Three of these derivatives 1, 5 and 19 are also active against TDO at the same concentration. It seemed relevant to us to compare these results with those obtained for tsitsikammamine analogues which were the subject of a previous work (Figure 4). The best IDO1 inhibitors in this series and in this family so far, are compounds 5 and 6 which are also the only ones comprising an additional substituent in -position of the nitrogen of the substituted pyrrole ring.

Three derivatives 1, 5 and 19 are dual inhibitors, as they also show activity against TDO, the most active being compound 5 whereas none were active in the previous series.

Molecular docking

In order to understand the affinity observed within IDO1, we studied the binding mode of compounds by molecular docking 18 using Molegro Virtual Docker program. 19 Prior to modelling studies, a classification of know Protein Data Bank (PDB) 20 structures related to Uniprot P14902 domain [START_REF]-The UniProt consortium[END_REF] was done. The PDB structures were split and all chains structurally aligned against 2D0T 22 (chain A, formerly 2D0Ta) using UCSF Chimera/MatchMaker. 23,24 The resulting 52 chains were clustered using a combination of descriptors at protein or ligand levels, resulting in five groups. The 2DTOa (used for structural alignment) and 5EK2a 25 (used for docking as target) were included in the same group (5EK2a being the representative of a sub-cluster characterized by a binding cavity a little more opened). Two docking protocols BBJ and HQS are found in PDB structures 6E40 and 6E41. 26 These two structures are classified in the same cluster 2D0Ta. The tosyl (Ts) group allows the classical L shaped form upper the heme's plane and shows interactions (hydrogen bonds) with SER167 and TYR126.

The cycles (Ts and bispyrrololoquinonic frame) are found in the same plane as most of the IDO1 ligands complexed with heme whereas the phenyl groups share non unusual placement with compounds found in structures of IDO1 at entry of binding cavity (Figure 6 for an example). Interestingly, one of the two carbonyl groups of 5 is aligned with the hydroxylamine of BBJ and HQS.

The best pose of 6 is different from its counterpart 5, but shares the overall T shaped form, allowed by the placement of Ts group inside the cavity and normal to the heme. The Ts group exhibits the same interaction (hydrogen bond) with SER167. Some similarities (shape, planes) are found with known ligands of IDO1, particularly D09, PKJ and DYC from 4PK5 (chain A), 6KOF (chain A) and 6KW7 (chain A) respectively. 27,28 These protein structures are classified in the same cluster than 5EK2a used for the docking studies. The pattern of interactions is different from the one of 5, with more involvement of Pi (cycles, cations, sulfur) interactions. Interestingly, one of five-membered cycles of 6 is very well placed over the heme center, even if a nitrogen atom is not found in front of the Fe.

In conclusion, derivatives of marine pyrroloquinolines wakayin and tsitsikammamine have shown in a cell-based assay, interesting inhibitory activity on IDO1 and TDO, two enzymes involved in tumoral immune resistance. A molecular modeling study of compounds 5 and 6 in the IDO1active site gives insights to optimize these compounds. Particularly the impact of one of the two carbonyl groups of 5 close enough to Fe could be investigated to make electrostatic interactions as in the case of the hydroxylamine group of BBJ. In compound 6, the replacement of a carbon by a nitrogen capable of dative bonding with Fe would be a first approach before further optimizing the compound to create a second interaction involving the carboxyl group of heme.

Figure 2 :

 2 Figure 2: Roxyl-WL

Figure 3 :

 3 Figure 3: Analogues of the marine alkaloids wakayin and tsitsikammamine

9 Compounds 1

 91 and 21 which are open structures have the same inhibition capacity against IDO, the nature of the aromatic substituent at the end of the chain and the presence of a hydroxyl group does not seem to have much influence on the activity. Compounds 3 and 4 are less active than their counterpart compounds in which the second five-membered cycle is not aromatic. This is in agreement with the previous observation which showed that the fully aromatic analogue of 22b had an activity reduced by half. None of the aza-analogues evaluated in this work, except compound 19, showed activity against IDO, 19 inhibiting the enzyme to a moderate level. In general, contrary to what one would have expected, given the activity of Roxyl-WL, the replacement of a pyrrole ring by a pyrrazole ring, did not lead to better inhibitors.

Figure 4 :

 4 Figure 4: Tsitsikammamine analogues studied in a previous work and exhibiting the best inhibitory activities (the percentage obtained at 10 M is indicated in brackets), a and b refer to iso1 and iso2.

  were used: P1-GPU for screening and P2-OPT (full flexibility of binding site) for investigating the best compounds 5 and 6 (supplementary data for the overall in silico processes). The key interactions stabilizing the compounds within the IDO binding cleft are depicted in Figures5 and 6respectively (2D interactions diagrams are available in supplementary data).

Figure 5 :

 5 Figure 5: Best 5 docked pose (pink, big sticks) of 5 in 5EK2a binding site (light green, molecular surface) relatively to crystallographied ligands BBJ (purple, from 6E40, chain A) and HQS (kaki, from 6E41, chain A) after structural alignment.

Figure 6 :

 6 Figure 6: Best 6 docked pose (blue, big sticks) of 6 in 5EK2a binding site (light green, molecular surface) relatively to crystallographied ligands DO9 (red, from 6KOFa), PKJ (light blue, from 4PK5a) and DYC (brown, from 6KW7a) after structural alignment.

Table 1 :

 1 Inhibition and cell viability data for wakayin and tsitsikammamine analogues

	Compounds Viability IDO TDO Viability IDO TDO Viability IDO TDO Viability IDO TDO
		25 µM	25 µM	12,5 µM	12,5 µM	6,25 µm	6,25 µM	3,12 µM	3,12 µM
		mean mean mean mean mean mean mean mean mean mean mean mean
	1	26	39	67	56	38	31	69	28	12	75	23	5
	2	84	11	0	99	13	0	98	9	-3	98	2	-8
	3	95	28	-8	97	26	-1	98	20	2	100	12	2
	4	88	27	-3	96	21	0	96	12	3	98	8	2
	5	111	61	28	102	62	55	125	58	27	99	52	15
	6	100	63	-1	105	63	3	97	63	6	94	61	4
	7	75	24	-9	82	25	-2	81	22	1	87	16	-1
	8	67	-88 -35	85	-80 -34	88	-58 -28	87	-55 -19
	9	115	-3	3	110	-4	-6	98	-3	-7	107	0	-7
	10	91	6	7	97	9	13	91	9	14	92	10	12
	11	93	11	15	110	15	23	106	12	19	87	10	15
	12	90	-1	9	91	0	8	92	2	8	93	2	6
	13	78	18	-10	79	16	2	95	12	5	91	10	5
	14	96	23	21	123	22	18	126	19	13	91	14	10
	15	84	5	5	94	5	6	93	4	5	93	4	3
	16	63	18	12	65	11	-4	80	7	-2	87	5	-3
	17	136	20	0	121	14	-10	86	6	-6	90	3	-8
	18	84	3	1	96	2	-4	98	0	0	103	-1	-9
	19	72	36	35	77	23	21	89	16	18	93	11	3
	20	31	14	27	84	1	6	107	3	5	104	1	-7

IDO and TDO cell assay inhibition: Inhibitory activity (%) was estimated by measuring tryptophan degradation and kynurenine production in the supernatant of 24-h cultures performed in the presence of compound at different concentrations. Tryptophan and kynurenine were measured by HPLC. Cell viability (%) was estimated by an MTS/PMS assay in the same cultures. The indicated values are the result of two independent determinations.