Shape and Size Dependence of Dipolar Plasmonic Resonance of Nanoparticles

Shape and size dependence of dipolar plasmonic resonance of nanoparticles

H. Ammari and P. Millien

Research Report No. 2018-18

April 2018 Latest revision: July 2018

Seminar für Angewandte Mathematik Eidgenössische Technische Hochschule CH-8092 Zürich Switzerland

______Shape and size dependence of dipolar plasmonic resonance of nanoparticles

Habib Ammari∗ Pierre Millien†

Abstract The aim of this paper is to present a new approach, based on singular volume integral equations, in order to compute the size dependency of plasmonic resonances. The paper also provides rigorous derivations of the extinction and absorption cross sections for elliptical particles.

Mathematics Subject Classiﬁcation (MSC2000). 35R30, 35C20. Keywords. plasmonic resonance, volume integral equation, singular integrals.

1 Introduction

1.1 Position of the problem The optical properties of metallic nanoparticle have been a subject of great interest in the past decades. They have the ability to exhibit plasmonic resonances, which are strong enhancement of the scattering and absorption cross sections at certain frequencies. This capacity to interact strongly with light is a key to many major innovations in nanophotonics [37, 9, 41], in biomedical imaging [25, 31], cancer treatment therapy [10]. For a nice review of some of these applications we refer the reader to [19]. These resonances have been theoretically and experimentally studied by the physics community. It has been experimentally shown [32] (via measurements of the extinction and absorption cross sections) and numerically (simulations of the Maxwell equations, often via a coupled dipoles method, see [29, 23]) that the frequency at which a metallic nanoparticle resonates depends on

(i) the shape of the particle;

(ii) the type of metal;

(iii) the surrounding medium;

(iv) the size of the particle.

∗Department of Mathematics, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland ([email protected]). †Institut Langevin, 1 Rue Jussieu, 75005 Paris, France ([email protected]).

1 Plasmonic resonances have been the subject of some theoretical work as well in the physics community. In the case of a spherical particle, the classical Mie theory explains points (ii) to (iv). In the case where the particle is not spherical, using the quasi-static approximation and solving Laplace equation, computations of the polarizability for some simple shapes have given a lot of insights on points (i), (ii), and (iii); see, for instance, [44]. Moreover, the conservation of energy fails in the quasi-static theory, due to the absence of radiative loss. This issue has been dealt by adding a radiative correction [1]. The size dependence has been more problematic. Some corrections of the quasi-static approximation, sometimes called the modified long-wavelength approximation, or computations of a dynamic polarizability have tackled this issue [33, 42, 29, 35]. Nevertheless, they heavily rely on strong assumptions and are valid only for spheroidal shapes. In the mathematical community, plasmonic resonances are a more recent subject of interest. In the quasi-static approximation, plasmonic resonances were shown to be an eigenvalue problem linked to the Neumann Poincaréoperator [22, 6, 26]. It was then showed that Maxwell’s equation yields a similar type of eigenvalue problems, and a computation of the polarizability for small plasmonic particle was given, solving items (i)to(iii) for a general regular shape [2]. Note that these studies were all done in the case where the shape of the particle is assumed to have some regularity, and the theory breaks down when the particle has corners. Some recent progress has been made on this topic [11, 24, 39]. The size dependance has been justified in [4, 7] in the scalar case (transverse electric or transverse magnetic) and in [5] in the Maxwell setting. However, practical computations of this size dependency remains complicated. We aim here at presenting a new approach, based on a singular volume integral equation, to compute this size dependency. Our integral volume approach can be extended to the case where the shape of the particle has corners.

1.2 Main contribution In this work, using a volume integral equation, we show that the resonant frequencies at which a nanoparticle of characteristic size δ exhibits plasmonic resonances occurs can be written as a nonlinear eigenvalue problem:

Find ω such that f(ω) σ (ωδ) (1.1) ∈ T for some nonlinear function f and some operator (ωδ) (see Deﬁnition 2.2). T These types of problems are extremely diﬃcult to handle in their generality. In this work, we add some assumptions arising from experimental observations and classical electromagnetic theory to compute solutions of (1.1) in a regime that corresponds to practical situations. The perturbative analysis presented in this work is based on the following assumptions:

(i) The size δ of the particle is small compared to the wavelenght of the lights at plasmonic frequencies: ω δ 1; c ≪

(ii) The particle is constituted of metal, whose permittivity can be described by a Drude- Lorentz type model [38].

2 In this regime, we show that (Theorem 5.1):

∂ δ σ( (ωδ)) 1. ∂ω T ∼ c ≪ And using that we give the following procedure for solving (1.1):

Find ω such that f(ω ) σ( (0)); • 0 0 ∈ T Compute σ( (δω0)) by a perturbative method; • T Find ω such that f(ω ) σ( (δω0)). • 1 1 ∈ T Since, in practical situations ∂ f(ω) δ (this comes from the fact that the particle is metallic ∂ω ≫ c and can be checked numerically, see Appendix A for more details), one can see that ω1 is a good approximated solution of problem (1.1).

1.3 Additional contributions In this paper, we also show that in the case where the particle has an elliptic shape, the dipole resonance of the nanoparticle (and its dependence on the size of the particle) can be very easily computed using the L dyadic that can be found in the physics literature [47, 48]. This dyadic L is often incorrectly derived in the literature. In Appendix B we give a correct derivation of L, as well as some precisions on some common misconceptions about singular integrals found in the classical literature on electromagnetic ﬁelds. We also give formulas for the computations of some observable quantities such as the extinction and absorption cross sections for elliptical particles (see Section 6). To the best of our knowledge, this is the ﬁrst time that a formal proof is given for these type of computations.

2 Model and deﬁnition

2.1 Maxwell’s equations We consider the scattering problem of a time-harmonic wave incident on a plasmonic nanoparticle. Denote by ε0 and µ0 the electric permittivity and the magnetic permeability of the vacuum −1/2 and by c0 =(ε0µ0) the speed of light in the vacuum. The homogeneous medium is characterized by its relative electric permittivity εm and relative magnetic permeability µm, while the particle occupying a bounded and simply connected domain of center of mass z0:

3 D = z0 + δB ⋐ R with 1,α boundary is characterized by its electric permittivity ε and its magnetic permeability C c µ , both of which may depend on the frequency. We assume that ε < 0, ε > 0 and deﬁne c ℜ c ℑ c ω ω km = √εmµm, kc = √εcµc, (2.1) c0 c0 and

ε (ω)= ε χ(R3 D¯)+ ε (ω)χ(D¯), µ = µ χ(R3 D¯)+ µ χ(D), (2.2) D m \ c D m \ c

3 Es(x,ω)=?

δ εc(ω) εm z0

D = z0 + δB δω 1 Ei(x,ω) ≪

Figure 1: Schematic representation of the scattering problem. where χ denotes the characteristic function. We assume that the particle is nonmagnetic, i.e., µm = µc. Throughout this paper, we assume that εm is real and strictly positive and that k < 0 and k > 0. ℜ c ℑ c For a given plane wave solution (Ei, Hi) to the Maxwell equations

Ei = iωµ Hi in R3, ∇× m Hi = iωε Ei in R3, ∇× − m let (E, H) be the solution to the following Maxwell equations:

E = iωµ H in R3 ∂D, ∇× D \ H = iωε E in R3 ∂D, (2.3)  ∇× − D \ [ν E]=[ν H] = 0 on ∂D,  × × subject to the Silver-M¨uller radiation condition:

[ν E]=(ν E) (ν E) , [ν H]=(ν H) (ν H) . × × + − × − × × + − × −

Proposition 2.1. If εc = 0 , then problem (2.3) is well posed. Moreover, if we denote by I εm 6 3 (E, H) its unique solution,h theni (E, H) H(curl,D) and (E, H) 3 H (curl, R D). D ∈ R \D ∈ loc \

Proof. The well-posedness is addressed in [45, 17, 4].

4 We also denote by Gkm the scalar outgoing Green function for the homogeneous medium, i.e., the unique solution in the sense of distributions of

ω2 ∆+ ε µ Gkm ( , z)= δ in R3, (2.4) c2 m m · z 0 subject to the Sommerfeld radiation condition. Gkm is given by (see [36]):

eikm|x−z| Gkm (x, z)= . (2.5) 4π x z | − | 2.2 Volume integral equation for the electric field We start by defining a singular integral operator, sometimes known as the magnetization integral operator [20]. Definition 2.1. Introduce

L2(D, R2) L2(D, R2) −→ k : TD f k2 Gk(x,y)f(y)dy Gk( ,y) f(y)dy. 7−→ −∇ ∇ · · ZD ZD We then give the equation satisfied by the electric field: Proposition 2.2. The electric field inside the particle satisfies the volume integral equation (or Lippmann-Schwinger equation):

ε ε m I k E = m Ei. (2.6) ε ε −TD ε ε m − c m − c Proof. See [15, Chapter 9] or [16].

2.3 Plasmonic resonances as an eigenvalue problem Deﬁnition 2.2. We say there is a plasmonic resonance if

εc k σ D . εm εc ∈ T − 2.4 Dipole resonance Deﬁnition 2.3. The dipole moment of a particle is given by

P = p(x)dx = ε χ(x)E(x)dx = (ε ε )E(x)dx. m c − m ZD ZD ZD

5 We say that there is a dipolar plasmonic resonance if the dipole moment P satisﬁes

P (ε ε ) Ei(x)dx . | | ≫ c − m ZD

Therefore, we want to compute the values of εc and εm such that

(i)

εc k λ := σ D ; εm εc ∈ T −

(ii) One of the eigenvectors ϕλ associated with λ has non zero average: 1 ϕ = 0. D λ 6 | | ZD 3 The quasi-static approximation

In this section, we study the case when the particle has finite size δ = 0 and δk 1. This 6 m ≪ corresponds to the usual quasi-static approximation. It has already been shown in [2, 4] that the solution of Maxwell’s or Helmholtz equation converge uniformly when δk 0 to the solution m → of the quasi-static problem in the case of negative index materials. Proposition 3.1. In the quasi-static approximation, the excitation field Ei becomes constant, the electric field can be written as the gradient of a potential E = u and the scattering problem ∇ described by (2.6) becomes:

(ε (x) u) = 0, ∇· D ∇ Find u such that i −1 ( u(x) E x = O x . − · | | Equivalently, E = u is a solution of the following integral equation: ∇ ε ε m I 0 E = m Ei. (3.1) ε ε −TD ε ε m − c m − c Remark 3.1. These types of transmission / exterior problems have been extensively treated in the literature. For more details on the well posedness, the appropriate functional spaces, and the study of small conductivity inhomogeneities we refer to [36, 3]. i i Proposition 3.2. Let y = z0 + δy˜ and write u˜(˜y)= u(y), and u˜ (˜y)= u (y). Then u˜ solves:

ε ε c I 0 u˜ = c u˜i. ε ε −TB ∇ ε ε ∇ m − c m − c Proof. This is a direct consequence of Theorem B.3.

6 3.1 Spectral analysis of the static operator, link with Neumann-Poincaré operator It has been shown in [2, 4] that the plasmonic resonances are linked to the eigenvalues of the Neumann-Poincaréoperator. In this subsection, we show that the surface integral approach and the volume integral approach are coherent. The link between the volume integral operator and the Neumann Poincaréoperator is summed up in Corollary 3.1. We first recall the definition of the Neumann-Poincaréoperator Definition 3.1. The operator ∗ : L2(∂D) L2(∂D) is defined by KD → 1 (x y) ν(x) ∗ [ϕ](x) := − · ϕ(y)dσ(y) , (3.2) KD ω x y d d Z∂D | − | with ν(x) being the outward normal at x ∂D, ω the measure of the unit sphere in dimension ∈ d d, and σ the Lebesgue measure on ∂D. We note that ∗ maps L2(∂D) onto itself (see, for KD 0 instance, [3]). Recall the orthogonal decomposition

L2(D, R3)= H1(D) H(div 0,D) W, ∇ 0 ⊕ ⊕ where H(div 0,D) is the space of divergence free L2 vector ﬁelds and W is the space of gradients of harmonic H1 functions. We start with the following result from [17]: Proposition 3.3. The operator 0 is a bounded self-adjoint map on L2(D, R2) with H1(D), TD ∇ 0 H(div 0,D) and W as invariant subspaces. On H1(Ω), 0[ϕ]= ϕ, on H(div 0,D), 0[ϕ] = 0 ∇ 0 TD TD and on W: 1 ν 0[ϕ]= + ∗ [ϕ ν] on ∂D. ·TD 2 KD · Proof. The proof can be found in [20, 17].

From this, it immediately follows that the following corollary holds. Corollary 3.1. Let λ = 1. Let ϕ 0 be such that 6 D 6≡ λϕ 0[ϕ ] = 0 in D. D −TD D Then,

ϕ W, D ∈ ϕ = 0 in D, ∇· D λϕ = [ϕ ν] in D, D ∇SD D · 1 λϕ ν = + ∗ [ϕ ν] on ∂D. D · 2 KD D ·

7 Letting u = [ϕ ν], we have: D SD D · 3 ∆uD = 0 in R ∂D, \ (3.3) [∂ u ]= ϕ ν on ∂D. ( n D D · Proposition 3.4. If the boundary of D is 1,α, then 0 : W W is a compact operator. C TD W −→ Proof. The operator 0 is a bounded map from W to H 1(D, R3) [20, 34]. The 1,α regularity TD C of ∂D and the usual Sobolev embedding theorems ensure its compactness (see [13, Chapter 9]).

Proposition 3.5. The set of eigenvalues (λ ) N of 0 is discrete, and the associated eigen- n n∈ TD W functions (ϕ ) form a basis of W. We have: n

0 = λ ϕ , ϕ . TD W nh n ·i n n X

3.2 Link between σ(L ) and σ( 0), and the dipole resonances for an ellipse B TB As explained in Section 2.4, to understand the dipole resonances of the particle, we need to compute the eigenvectors of 0 that have a non zero average, i.e., that are not orthogonal in TB the L2(B) sense to every e , i 1, 2, 3 , where (e , e , e ) is an orthonormal basis of R3. i ∈ { } 1 2 3 In general, constant vector ﬁelds over B are not eigenvectors of . Nevertheless in the case TB where B is an ellipse, then constant vector ﬁelds can be eigenvectors for . This is essentially TB a corollary of Newton’s shell theorem. Theorem 3.1. If B is an ellipse centered at the origin, then the following holds: Let ϕ ∈ L2(B, R3) and let λ R 0, 1 be such that ∈ \{ } λϕ 0[ϕ] = 0, −TB  ϕ = 0.  6 ZB Then, 

(λI L ) ϕ = 0. − B 0 ZB Remark 3.2. The operator we are considering is essentially the double derivative of a classical Newtonian potential. When the domain is an ellipse, the Newtonian potential of a constant is a second order polynomial. Therefore, its second derivative is a constant. Hence, the possibility to have constant eigenvectors for occurs. This property characterizes ellipses. In fact, it is TB the weak Eshelby conjecture; see [27] for more details.

Proof. For the proof we need the following lemma from [18]: Lemma 3.1. If B is an ellipse, then, for any ϕ R3, 0 ∈ [ϕ ]= L ϕ . TB 0 B 0

8 Combining this with Proposition 3.5, and using the orthogonality between the eigenvectors of 0, one gets the result. TB

Corollary 3.2. Let E R3 0 be such that λE = L E. Then, ∈ \{ } B 0[E]= λE. T Proof. This is a direct consequence of Lemma 3.1.

3.3 Static polarizability of an ellipse In this subsection, we assume that B is an ellipse. Definition 3.2. The polarizability is the matrix linking the average incident electrical field M to the induced dipolar moment. It is defined by

1 p = Ei . M D | | ZD Theorem 3.2. The static polarizability of the particle B is given by M

ε −1 = δ3ε (ε 1) m + L . (3.4) M 0 c − ε ε B m − c Remark 3.3. The polarizability is used to compute diﬀerent observables such as the scattering and extinction cross sections of the particle (see Section 6).

Proof. We recall equation (3.1) for the electric ﬁeld inside the particle

ε ε m I 0 u = m ui. ε ε −TD ∇ ε ε ∇ m − c m − c We now remark that the operator

: L2(D, R3) L2(D, R3) P −→ f f 7−→ ZD is the projector onto the subspace of L2-spanned by constant functions. Compose the previous equation with we get: P ε c I 0 [ u]= [ ui]. P ◦ ε ε −TD ∇ P ∇ m − c We now use Proposition 3.5 to diagonalise 0: T

εc i + λi ϕn, u ϕn = [ u ]. P εm εc h ∇ i ! P ∇ Xi −

9 Since the particle is an ellipse, we know by Corollary 3.2 that E R3, the eigenvectors of L i ∈ B associated with the eigenvalues λL , are also eigenvectors for 0 for some eigenvalues λ , λ ,i T i1 i2 and λi3 . Moreover, λi1 = λL,1, λi2 = λL,2 and λi3 = λL,3 We can also note that (E1, E2, E3) span the image of the projector . By orthogonality of the eigenvectors of 0, we obtain that P T

εc εm + λi ϕn, u ϕn = + λL,i Ei, u Ei P εm εc h ∇ i ! εm εc h ∇ i Xi − Xi − Noticing that E , u = E , ( u) = E ( u), we obtain that the right-hand side of the h i ∇ i h i P ∇ i i ·P ∇ previous equation is exactly the expression of

ε c + L ( u) ε ε B P ∇ m − c in the basis (E1, E2). Therefore, we have shown that

ε ε c I 0 [ u]= c + L ( u). P ◦ ε ε −TD ∇ ε ε B P ∇ m − c m − c Thus

ε −1 u = c + L ui. ∇ ε ε B ∇ ZD m − c ZD Using Deﬁnition 2.3 of the induced dipole moment, we get the result.

3.4 Static polarizability of an arbitrary particle In the case where the particle occupies an arbitrary 1,α domain, the volume integral approach C does not yield a simple expression for the polarizability. Nevertheless, the layer potential approach gives the well known polarization tensor. The validity of the polarization tensor formula for negative index material has been shown in [4, 2]. We recall here the formula for completeness: Theorem 3.3. [4, 2] The static polarizability is given by

ε + 1 −1 α = δ3ε (ε 1) y I ∗ [ν](y)dσ(y), 0 − 2(ε 1) −KB Z∂B − where ε = εc . εm

10 4 Perturbative approach: spectral analysis of the dynamic operator

In this section, we aim at ﬁnding λ˜ such that there exists some f 0 L2(B, R3) such that 6≡ ∈ λI˜ δk [f] = 0. −TB Let λ be an eigenvalue of 0. Let V C be a neighborhood of λ such that λI 0 is n0 T ⊂ n0 −T invertible for every λ V . Let ϕ L2(B, R3) be a unitary eigenvector associated with λ . ∈ n0 ∈ n0 Lemma 4.1. For any λ V , the following decomposition holds: ∈

0 −1 ϕn0 , λI = h ·i ϕn0 + (λ), −T λ λ 0 R − n where C L2(B, R3) L2(B, R3) −→ → λ (λ) 7−→R is holomorphic in λ.

Proof. Denote by : L2(B, R3) L2(B, R3) and : L2(B, R2) L2(B, R3) the orthogonal P1 → P2 → projections on H1(B) and H(div 0,B), respectively. Using Propositions 3.3 and 3.5, we can ∇ 0 write:

λI 0 = (λ λ ) ϕ , ϕ +(λ 1) + λ . −T − n h n ·i n − P1 P2 X The result immediately follows.

Lemma 4.2. Let λ be an eigenvalue for 0. Then, if k is small enough, there exists a n0 T | | neighborhood V C of λ such that δk has exactly one eigenvalue in V . ⊂ n0 TB Proof. We start by writing:

λI δk = λI 0 + 0 δk . −T −T T −T Recall that there exists V C such that λI 0 is invertible for every λ V λ . Therefore, ⊂ −T ∈ \{ n0 } for λ V λ , ∈ \{ n0 } −1 λI δk = λI 0 I + λI 0 0 δk . −T −T −T T −T Using Lemma 4.1, we get

0 δk δk ϕn0 , [f] 0 δk λI [f]= f + h T −T iϕn0 + (λ) [f]. −T λ λn0 R T −T −

11 We can show that

0 δk 0 (δk 0). T −T −→ →

Since λ (λ) is holomorphic, the compact operator 7→ R λ (λ) 0 δk 7−→ R T −T converges uniformly to 0 with respect to λ when k goes to 0. Since the operator

L2(B, R3) L2(B, R3) k −→ : ϕ , 0 δk [f] K n0 f h T −T iϕn0 7−→ λ λ 0 − n is a rank one linear operator, the operator I + δk is invertible. KB Therefore, there exists K > 0 such that λI k = I + δk + (λ) 0 δk is invertible −T K R T −T for every λ V λ and every δk

We can now give an asymptotic formula for the perturbed eigenvalues λ˜ of δk: T Proposition 4.1. The following asymptotic formula for the perturbed eigenvalues holds:

˜ 0 δk λ λn0 ϕn0 , ϕn0 . (4.1) ∼ − T −T L2(B,R3) D E Proof. We use the same notations as in the previous lemmas. We have:

λ˜ σ δk V λ f 0 such that λI˜ δk [f] = 0 ∈ T ∪ \{ n0 } ⇔∃ 6≡ −T −1 f 0 such that I + 0 δk λI˜ 0 [f] = 0. ⇔ ∃ 6≡ T −T −T −1 Using the decomposition stated in Lemma 4.1 for λI˜ 0 , we get the following equation −T for f and λ˜: ϕ , f f + h n0 i 0 δk [ϕ ]+ 0 δk (λ˜)[f] = 0. (4.2) ˜ n0 λ λn0 T −T T −T R − We start by proving that ϕ , f = 0. Indeed, if one has ϕ , f = 0, then (4.2) becomes h n0 i 6 h n0 i I + 0 δk (λ˜) [f] = 0. T −T R If k is close enough to 0, then 0 δk (λ˜) < 1 and then I + 0 δk (λ˜) is invertible k T −T R k T −T R and we have f = 0, which is a contradiction. f hϕn0 , i 0 δk We then note that f and ˜ [ϕn0 ] are terms of order O( f ) whereas the λ−λn0 T −T | | regular part, the term 0 δk (λ˜)[f] is of order O(δk f ). We drop the regular part, and T −T R | |

12 take the scalar product against ϕn0 to obtain that

ϕ , f ϕ , f + h n0 i 0 δk [ϕ ], ϕ = 0. (4.3) n0 ˜ n0 n0 h i λ λn0 T −T − D E Finally, dividing equation (4.3) by ϕ , f = 0 yields h n0 i 6 λ˜ = λ 0 δk [ϕ ], ϕ . n0 − T −T n0 n0 D E

We also prove the following lemma, giving an approximation of the resolvent of k along TD the direction of the eigenfunction ϕn0 : Proposition 4.2. Let g L2(B, R3). If f L2(B, R3) is a solution of ∈ ∈ λI δk f = g, −TB then, for λ λ , the following holds: ∼ n0

2 3 g, ϕn0 L (B,R ) f, ϕ 2 R3 h i . n0 L (B, ) 0 δk h i ∼ λ λ + ( )[ϕ ], ϕ 2 3 − n0 T −T n0 n0 L (B,R )

Proof. The result follows directly from Lemma 4.1 and identity (4.2) with g in the right-hand side.

5 Dipolar resonance of a ﬁnite sized particle

5.1 Computation of the perturbation via change of variables

i i By the change of variables: y = z0 + δy˜, E˜(˜y)= E(y), E˜ (˜y)= E (y), and (2.6) becomes:

ε ε c I δk E˜ = c Ei. ε ε −TB ε ε m − c m − c We are now exactly in the right frame to apply the results of Section 4. We know that there is a neighborhood V C of λ(0), i 1, 2, 3 such that δkm has exactly one eigenvalue in V i ⊂ i ∈ { } TB i (Lemma 4.2) and that the perturbed eigenvalue is given by Theorem 5.1. We have

˜ (0) 0 δkm λ λi B B ϕi, ϕi , (5.1) ∼ − T −T L2(B,R3) D E where ϕ is a unitary eigenvector of 0 associated with λ(0). i TB i

13 5.2 The case of an ellipse 5.2.1 The perturbative matrix In the case where B is an ellipse, since the eigenmodes associated with a dipole resonance are constant ϕ E R3 (see Section 3.2) and therefore formula (5.1) simpliﬁes to: i ≡ i ∈ Proposition 5.1. We have

0 δkm [ϕ ], ϕ = E Mδkm E TB −TB i i i · B i D E with

δkm 0 δkm ⊤ MB := G (˜x, z˜) G (˜x, z˜) ν(˜x)ν(˜z) dσ(˜x)dσ(˜z). ∂B×∂B − ZZ Proof. From

0 δkm 0 δkm B B [ϕi], ϕi = Ei G (x,y) G (x,y) dydxEi, T −T · B ∇ B ∇ − D E Z Z h i an integration by parts yields the result.

5.2.2 The algorithmic procedure We now give a practical way to compute this perturbation in the case of an elliptical particle:

(i) Compute the resonant value associated with the static problem:

Compute the matrix L M (R); • B ∈ 3 Compute its spectrum λ , λ , λ and corresponding unitary eigenvectors E , E and • 1 2 3 1 2 E3;

δkm (ii) Compute the perturbative matrix MB and the perturbed eigenvalues

λ˜ = λ E Mδkm E . i i − i · B i 6 Computation of observables for an elliptical nanoparticle

6.1 Dipole moment beyond the quasi-static approximation

Denote by λj, j = 1, 2, 3, the three eigenvalues of LB. Denote by Ej the three eigenvectors ( R3) associated with λ such that (E , E , E ) forms an orthonormal basis of R3. Denote by ∈ j 1 2 3 Q = E , E , E3 O(3) the matrix associated with this basis. 1 2 ∈ Since E are eigenmodes for 0, we can use Lemma 4.2 to ﬁnd that j T

Ei, E E, E h ji . h ji∼ λ λ + ( 0 δk)[E ], E − j h T −T j ii

14 We can then write:

1 0 δk 0 0 λ λ1 + ( )[E1], E1  − h T −T i  1 1 3 0 0 t i P δ ε0(ε 1)Q  0 δk  Q E . (6.1) ∼ −  λ λ2 + ( )[E2], E2  D D  − h T −T i  | | Z  1   0 0 0  λ λ + ( 0 δk)[E ], E   3 3 3   − h T −T i  Remark 6.1. This expression is valid near the resonant frequencies when the corresponding mode is excited, i.e., when λ λ and ui(z ) E ui(z ) . ∼ i ∇ 0 · i ∼ |∇ 0 | Remark 6.2. The expression

1 0 0 0 λ−λ1+ (T −T δk)[E1],E1 h i 1 3 0 0 0 t dyn := δ ε0(ε 1)Q  λ−λ2+ (T −T δk)[E2],E2  Q M − h i 1 0 0 0 0  3 δk E3 E3   λ−λ + (T −T )[ ],   h i  is a dynamic version of the usual quasi-static polarization tensor.

6.2 Far-ﬁeld expansion Assume that the incident ﬁelds are plane waves given by

d d Ei(x)= Ei eikm ·x, Hi(x)= d Ei eikm ·x, 0 × 0 with d S2 and Ei R3, such that Ei d = 0. ∈ 0 ∈ 0 · Since we have an approximation of the dipole moment of the particle we can find an approximation of the electric field radiated far away from the particle. The far-field expansion written in [5] is still valid (Theorem (4.1) in the aforementioned paper), one just has to replace the dipole moment M(λ,D)Ei where M is the usual polarization tensor defined with the Neumann Poincaréoperator by the new corrected expression obtained in (6.1). The scattered far field has the form

k2 1 k2 m Ei(y)dy = m P. 4π Mdyn D 4π | | ZD 6.3 Scattering and absorption cross sections Having an approximation of the dipole moment, we can compute the extinction and scattering cross sections of the particle. Proposition 6.1. ([12, Chapter 13]) The power radiated by an oscillating dipole P can be

15 written

4 µmω 2 Pr = P , 12πc0 | | and the power removed from the incident plane wave (absorption and scattering) can be written

2 4π Ei km P 0 · 4π Pe = i 2 . km I E " | 0| # In the same spirit as in [4, 5] we can then give upper bounds for the cross sections as follows. Proposition 6.2. Near plasmonic resonant frequencies, the leading-order term of the average over the orientation of the extinction (respectively absorption) cross section of a randomly oriented nanoparticle is bounded by

Qext k [Tr ] , m ∼ mI Mdyn k4 Qa m Tr 2 , m ∼ 6π | Mdyn| where Tr denotes the trace.

Proof. We start from equation (6.1) and get that

i P = MdynE0 [1 + f(D,km, d)] with 1 ikmd·x f(D,km,d)= 1 e dx. D D − | | Z Here, f represents the correction of the average illuminating ﬁeld over the particle due to the ﬁnite ratio between the size of the particle and the wavelength. Its magnitude is of the order i of δk. If we take the average over all directions for E0 and d, then we obtain that

2 ext 1 4π km Qm = [e0 dyne0(1 + f(D,km, d)]dσ(e0)dσ(d). (4π)2 S2 k 4π I ·M ZZ m

The term f(D,km, d) is a small correction of the order of kmδ that is due to the fact that what determines the dipole moment is not the incident ﬁeld at the center of the particle, but the average of the ﬁeld over the particle. Therefore, it reduces the dipole response of the particle. Ignoring it and considering only the leading order term gives:

ext km Qm [e0 Mdyne0]dσ(e0) ∼ 4π S1 I · Z k [Tr ] . ∼ mI Mdyn A similar computation gives the leading term of the absorption cross section.

16 1.35

1.3

1.25

1.2

1.15 ) ω ( f 1.1

1.05

0.95

0.9 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 ω 1015 · Figure 2: Numerical values of the real part of f(ω) = λ(ω) = εc(ω) for a gold nanoparticle εm−εc(ω) in water.

A Justiﬁcation of the asymptotic regime

To quickly justify the model and the regime we are working in, we give some values for the physical parameters used in the model corresponding to practical situations. In practice: ω [2, 5] 1015Hz; δ [5, 100] 10−9m; ε 1.8ε 1.5 10−11F m−1 for ∈ · ∈ · m ∼ 0 ∼ · · water; µ 12 10−7H m−1; c 3 108m s−1; k 107m−1. 0 ∼ · · 0 ∼ · · m ∼ Therefore, one can see that we have δk 10−2 for very small particles, and δk 1 for bigger ≤ ∼ particles (of size 100nm). For the permittivity of the metal, one can use a Lorentz-Drude type model:

2 ωp εc(ω)= ε0 1 −1 − ω(ω + iτ )! with τ = 10−14 s; • ω = 2 1015s−1. • p · This model is enough to understand the behavior of ε but for numerical computations, it is better to use the tabulated parameters that can be found in [40]. We plot f(ω) on Figure 2 and df on Figure 3. One can see that df is of order 10−15 while δ 10−16 for size particle under dω dω c ∼ 100nm. So the procedure described in Section 1.2 is justiﬁed.

B Singular integrals, Calder´on Zygmund type operators

There is an abundant literature on singular integral operators, yet these types of principal value integrals are misunderstood and misused in some of the physics literature. We include here some properties that are well known for people who are familiar with these types of operators, but seem to be often misunderstood.

17 10−15 0.4 ·

0.2

0.2 − f ω d d 0.4 −

0.6 −

0.8 −

1 − 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 ω 1015 · df Figure 3: Numerical values of the real part of dω (ω) for a gold nanoparticle in water.

There have been numerous contributions in the twentieth century. Some notable ones are: Tricomi (1928) [46]; Kellogg (1929) [28]; Calder´on-Zygmund (1952) [14]; Seeley (1959) [43]; Gel’fand-Shilov (1964) [21], and Mikhlin (1965) [34]. In the following, we do not state the results in their most general settings and assumptions. We use some notations and hypotheses that are adapted to our problem (Green’s function method).

B.1 Principal value integral Let D Rd be a bounded domain. We are concerned with the existence and the manipulation ⊂ of integrals of the type

x y 1 f − u(y)dy, x D, (B.1) x y x y d ∈ ZD | − | | − | where u is a function defined on D and f a function defined on Sd−1. We denote by B(x,ε) the ball centered at x of radius ε. Definition B.1. The principal value of the integral (B.1) is defined by

x y 1 lim f − u(y)dy. ε→0 x y x y d ZD\B(x,ε) | − | | − | We now give suﬃcient conditions for the existence of the principal value. 0,α Theorem B.1. If u (D), α > 0 and 1 f(θ)dθ = 0, then the principal value of (B.1) ∈ C Sd− does exist. R Remark B.1. These conditions are not necessary, and singular integrals can be deﬁned for a much larger class of functions. f can be replaced by f(x,θ) and u does not need to be chosen as

18 H¨older continuous. One can choose u in some Lebesgue space u Lp(D). ∈ Example 1. Consider the Green function for the free space Laplace equation in two and three dimensions: 1 log x y if d = 2, 2π | − | G(x,y)=  1 1  if d = 3,  4π x y | − |  x−y 1  then ∂xi,xj G(x,y)= f |x−y| |x−y|d with 1 (x y )(x y ) δ 2 i − i j − j if d = 2, x y 2π ij − x y 2 f − =  | − | x y  1 (xi yi)(xj yj) | − |  δij 3 − − if d = 3. − 4π − x y 2 | − |   0,α One can check that 1 f(θ)dθ = 0. Therefore, for u C (D) one can write: Sd− ∈ R ∂x ,x G(x,y)u(y)dy = lim ∂x ,x G(x,y)u(y)dy. i j ε→0 i j ZD ZD\B(x,ε) B.2 Non spherical volume of exclusion One common misconception found in the physics literature is that the limit of the integral over the domain minus a small volume around the singularity does not depend on the shape of the volume when the maximum cord of the volume of exclusion goes to zero. The limit does depend on the shape of the volume. This issue has been dealt with by Mikhlin [34, p. 40]. We include here the formula for the limit, using the notations used in physics literature. Assume that V (x,ε) D is a small volume of exclusion such that its boundary is given, in polar coordinates ⊂ by:

x y ∂V (x,ε)= y D, x y = εβ − . ∈ | − | x y | − | Theorem B.2. Under the assumptions of Theorem B.1,

x y 1 x y 1 lim f − u(y)dy = f − u(y)dy ε→0 x y x y d x y x y d ZD\V (x,ε) | − | | − | ZD | − | | − | u(x) f(θ) log β(θ)dθ. − Sd−1 Z Example 2. Let d = 2 and let

x y 1 (x y )2 f − = 1 2 1 − 1 x y 2π − x y 2 | − | | − | be corresponding to the angular term of ∂1,1G(x,y). If V (x,ε) is an ellipse of semi-axis ε and

19 eccentricity e where x is at one of the focal point

(1 e2) ∂V (x,ε)= y D, y x = ε − , ∈ | − | 1 e x1−y1 ( − |x−y| ) then the correction term is

2 2π 2 u(x) 2 1 e u(x) 2 1 e 1 2θ1 log − dθ = (1 2 cos (t)) log − dt. 2π S1 − 1 eθ 2π − 1 e cos(t) Zθ∈ − 1 Z0 − Remark B.2. Note that the correction term does not only depend on the shape of the volume of exclusion, but also on the position of x inside it. In the previous example, if x is at the center of the ellipse instead of being one of the focal point, the polar equation, hence the correction term, is modiﬁed.

B.3 Change of variables This issue has also been dealt with by Seeley [43] and Mikhlin [34, p. 41]. The classical formula for a change of variables in an integral cannot be applied in a straightforward way, and some precautions have to be taken into account. Consider a region D and an homeomorphism ψ : D D. Consider f˜ = f ψ−1,u ˜ = u ψ−1, and denote by J(˜x) the non vanishing −→ ◦ ◦ Jacobian of ψ−1. The corrective term to the usual change of variablese formula is given by the reciprocal imagee of the unit sphere by ψ. One can establish formulae of the form:

ψ(x) ψ(y) x y 2 = ψ(x) ψ(y) 2F ψ(x), − + O ψ(x) ψ(y) 3 , | − | | − | ψ(x) ψ(y) | − | | − | and then the change of variables can be written as follows: Theorem B.3. We have

x y 1 x˜ y˜ 1 f − u(y)dy = f˜ − u˜(˜y)J(˜y)d˜y x y x y d e x˜ y˜ x˜ y˜ d ZD | − | | − | ZD | − | | − | +u ˜(˜x)J(˜x) f˜(θ˜) log F (˜x, θ˜)dθ.˜ Sd−1 Z x−z0 Remark B.3. For a dilation : x˜ = δ , the image of the unit sphere is still a sphere and therefore, F = 1 and the usual change of variables formula is valid.

B.4 Diﬀerentiation of weakly singular integrals, integration by parts We want to diﬀerentiate integrals of the type

x y 1 g − u(y)dy, x D. x y x y d−1 ∈ ZD | − | | − | The following results can be found in [43, 34]: Theorem B.4. If u is H¨older continuous and if g and its ﬁrst derivative are bounded, then:

20 (i) Diﬀerentiation formula under the integral sign:

∂ x y 1 ∂ x y 1 g − u(y)dy = g − u(y)dy ∂x x y x y d−1 ∂x x y x y d−1 i ZD | − | | − | ZD i | − | | − | + u(x) g(θ)θidθ; Sd−1 Z (ii) Integration by parts formula:

x y 1 ∂ ∂ x y 1 g − [f(y)]dy = g − f(y)dy x y x y d−1 ∂x − ∂x x y x y d−1 ZD | − | | − | i ZD i | − | | − | x y f(y) + g − d−1 ν(y) eidσ(y)+ f(x) g(θ)θidθ. x y x y · Sd−1 Z∂D | − | | − | Z Remark B.4. Once again we only give suﬃcient conditions for the validity of theses formulas, corresponding to our framework. These formulas are valid for u Lp and for more general ∈ kernels. Example 3. Let d = 3 and consider the second derivative of a Newtonian potential: 1 1 x y ∂ x y −1u(y)dy = ∂ j − j u(y)dy. xi,xj 4π | − | − xi 4π x y 3 ZD ZD | − |

We can apply Theorem B.4 with g(θ)= θj and get:

1 1 δ ∂ x y −1u(y)dy = ∂ x y −1 u(y)dy + u(x) ij . xi,xj 4π | − | 4π xi,xj | − | 3 ZD ZD B.5 The L dyadic Lemma B.1. Let x D. Denote by L (x) the matrix ∈ D G(x,y)ν⊤(y)dσ(y). ∇ Z∂D Assume that D can be written in polar coordinates as

x y D = y Rd, x y ρ − . ∈ | − |≤ x y | − | Then, 1 (LD(x))i,j = ∂xj G(x,y)ν(y) eidσ(y)= + fi,j(θ) log ρ(θ)dθ · −d Sd−1 Z∂D Zθ∈ with fi,j being deﬁned in Example 1.

1 Proof. We start by using the integration by part formula from Theorem B.4, with g(θ)= 2π θj

21 1 if d = 2 and g(θ)= 4π θj if d = 3, and f = 1. We obtain 1 ∂ G(x,y)dy = ∂ G(x,y)ν(y) e dσ(y)+ . xi,xj xj · i d ZD Z∂D

In order to compute D ∂xi,xj G(x,y)dy, we use the change of variables [43, 34] : y = x + tθ, θ Sd−1 and t [0, ρ(θ)] to arrive at ∈ ∈ R

−d d−1 ∂xi,xj G(x,y)dy = fi,j(θ)t t dtdθ Sd−1 ZD Zθ∈ Zt∈[0,ρ(θ)]

= fi,j(θ) log ρ(θ)dθ. Sd−1 Zθ∈

B.6 Second derivative of a Newtonian potential In this section we give a correct simple derivation of the formula found in [47, p. 73] and [48, 30, 8]. Proposition B.1. Let V ∗ Rd be such that ⊂ (i) 0 V ∗; ∈ (ii) ∂V ∗ is a piecewise smooth; (iii) V ∗ is radially convex with respect to the origin. Let V (x,ε)= x + εV ∗. Then,

∂x ,x G(x,y)u(y)dy = lim ∂x ,x G(x,y)u(y)dy (LV ∗ ) u(x). i j ε→0 i j − ij ZD ZD\V (x,ε) Proof. Let x D and let V (x,ε) D. Assume that V (x,ε) can be described by some polar ∈ ⊂ equation:

x y V (x,ε)= y D, x y ερ − . (B.2) ∈ | − |≤ x y | − |

Before the computation we also recall that ∂xi,xj G(x,y) can be written as

x y ∂ G(x,y)= f − x y −d, xi,xj ij x y | − | | − | as it was seen in Example 1. Then we have

∂xi,xj G(x,y)u(y)dy =∂xi ∂xj G(x,y)u(y)dy (B.3) ZD ZD 1 = u(x)+ ∂ G(x,y)u(y)dy. (B.4) d xi,xj ZD

22 Using Theorem B.2 we obtain:

∂xi,xj G(x,y)u(y)dy = lim ∂xi,xj G(x,y)u(y)dy u(x) fij(θ) log ρ(θ)dθ, ε→0 − Sd−1 ZD ZD\V (x,ε) Z and therefore,

1 ∂xi ∂xj G(x,y)u(y)dy = lim ∂xi,xj G(x,y)u(y)dy u(x) + fi,j(θ) log ρ(θ)dθ . ε→0 − −d Sd−1 ZD ZD\V (x,ε) Zθ∈ Finally, using Lemma B.1 we arrive at

∂x ∂x G(x,y)u(y)dy = lim ∂x ,x G(x,y)u(y)dy (LV ∗ ) u(x). i j ε→0 i j − ij ZD ZD\V (x,ε)

Remark B.5. There are several issues and misconceptions in the literature with this formula:

(i) The shape V ∗ cannot be completely arbitrary as often mentioned. It has to satisfy some regularity condition, since the construction of LV ∗ uses some integration on the boundary of V ∗ involving the normal vector.

(ii) The exclusion volume V (x,ε) needs to be taken small in the numerical evaluation of the integral. Only if the test function u is constant then ε does not need to be small.

(iii) The derivation of this formula often contains mistakes. One common derivation of this formula is through a splitting of the integral of the form

∂xi ∂xj G(x,y)u(y)dy = ∂xi,xj G(x,y)u(y)dy ZD ZD\V (x,ε) + ∂ G(x,y)[u(y) u(x)]dy + u(x)∂ ∂ G(x,y)dy , xi,xj − xi xj ZV (x,ε) ZV (x,ε) ! which is a wrong application of the diﬀerentiation under the sign theorem. The reason why it is wrong is that, if the limit when ε 0 is to be taken, then one has to take into → R account the dependency of the volume of integration on D V (x,ε) on the variable x and \ use Reynold’s transport theorem to compute the derivative and add some boundary integral terms. The correct splitting would be:

∂ ∂ G(x,y)u(y)dy = ∂ G(x,y)u(y)dy ∂ G(x,y)u(y)ν(y) e dσ(y) xi xj xi,xj − xj · i ZD ZD\V (x,ε) Z∂V (x,ε) + ∂ G(x,y)[u(y) u(x)]dy + ∂ G(x,y)[u(y) u(x)] ν(y) e dσ(y) xi,xj − xj − · i ZV (x,ε) Z∂V (x,ε) 1 +∂ u(x) ∂ G(x,y)dy+u(x) ∂ G(x,y)ν(y) e dσ(y)+ + ∂ G(x,y)dy . xi xj xj · i d xi,xj ZV (x,ε) Z∂V (x,ε) ZV (x,ε) !

23 In the limit ε 0, the extra terms compensate each other and the ﬁrst (wrong) splitting → gives the same (correct) result as the second one.

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