Perturbations of the scattering resonances of an open cavity by small particles: Part II—the transverse electric polarization case

This paper is concerned with the scattering resonances of open cavities. It is a follow-up of Ammari et al. (ZAMP 71:102, 2020), where the transverse magnetic polarization was assumed. In that case, using the method of matched asymptotic expansions, the leading-order term in the shifts of scattering resonances due to the presence of small particles of arbitrary shapes was derived and the effect of radiation on the perturbations of open cavity modes was characterized. The derivations were formal. In this paper, we consider the transverse electric polarization and prove a small-volume formula for the shifts in the scattering resonances of a radiating dielectric cavity perturbed by small particles. We show a strong enhancement in the frequency shift in the case of subwavelength particles with dipole resonances. We also consider exceptional scattering resonances and perform small-volume asymptotic analysis near them. A significant observation is the large-amplitude splitting of exceptional scattering resonances induced by small particles. Our method in this paper relies on pole-pencil decompositions of volume integral operators.


Introduction
In this paper, which is a follow-up of [1], we consider dielectric radiating cavities [13,15,25] and rigorously obtain asymptotic formulas for the shifts in the scattering resonances that are due to a small particle of arbitrary shape. Our formula shows that the perturbations of the scattering resonances are proportional to the polarization tensor of the small particle. Therefore, the shift of the scattering frequencies induced by the small particle is of the order of the particle's volume. When the particle is excited near its resonant frequencies, its polarization tensor blows up and consequently, as shown in this paper, an anomalous shift of the scattering resonances can be observed when the resonant particle is coupled to the cavity modes. We also consider the case where the scattering resonances are exceptional. Exceptional scattering resonances can be defined as the poles of the Green's function associated with the radiating cavity which are not simple [2,3,12,20]. They owe their existence to the non-Hermitian character of the scattering resonance problem [12,20]. Optical cavities that operate at exceptional scattering frequencies can be exploited for enhanced nanoparticle sensing [16,21]. In this paper, we prove that a small particle inside a cavity perturbs the system from its exceptional points, leading to frequency splitting. Moreover, the splitting induced by the particle is of a much larger amplitude than suggested by the particle's volume. In fact, we consider exceptional points of order two and derive a formula for the splitting of such resonances induced by a small particle. We prove that the strength of the splitting of the exceptional scattering frequencies is proportional to the square root of the volume of the particle. Our method for proving various formulas that describe the shifts in the scattering resonances due to small particles is based on pole-pencil decompositions (see, for instance, [5,7]) of the volume integral operator associated with the radiating dielectric cavity problem.
The new technique introduced in this paper cannot be easily extended to the transverse magnetic case considered in [1] due to the hyper-singular character of the associated volume-integral operator.
The paper is organized as follows. In Sect. 2, we characterize the scattering resonances of dielectric cavities in terms of the spectrum of a volume integral operator. In Sect. 3, using the method of pole-pencil decompositions, we derive the leading-order term in the shifts of scattering resonances of an open dielectric cavity due to the presence of internal particles. In Sect. 4, using a Lippmann-Schwinger representation formula for the Green's function associated with the open cavity, we generalize the formula obtained in Sect. 3 to the case of external particles. In Sect. 5, we consider the perturbation of an open dielectric cavity by subwavelength resonant particles. The formula obtained for the shifting of the frequencies shows a strong enhancement in the frequency shift in the case of subwavelength resonant particles. In Sect. 6, we perform an asymptotic analysis for the shift of exceptional scattering resonances. The paper ends with some concluding remarks.

Model
We consider the scattering of linearly polarized light by a dielectric cavity in a time-harmonic regime.
Let Ω be a bounded domain in R d for d = 2, 3, with smooth boundary ∂Ω. Assume ε ≡ τ ε c + ε m inside Ω and ε = ε m outside Ω, and μ = μ m everywhere. Here, ε c , ε m , and τ are positive constants. Since we are interested in scattering resonances, we look for solutions u of the homogeneous Helmholtz equation at complex frequency ω: u satisfies the outgoing radiation condition.
It is known that the above scattering problem attains a unique solution for ω with ω ≥ 0. Using analytic continuation, the solution also exists and is unique for all complex ω except for a countable number of points, which are the scattering resonances (see, for instance, [19]).
Let Γ m be the outgoing fundamental solution of Δ + ε m μ m ω 2 in free space. We define the following integral operator: The following Lippmann-Schwinger representation formula holds:

Proposition 2.2. u is a solution of (1) if and only if the restriction of u on Ω is a solution of
where I is the identity operator.
According to [12], the following spectral decomposition of the operator K ω Ω holds: For ω ∈ C, the operator K ω Ω is bounded from L 2 (Ω) into H 2 (Ω). Moreover, it is a Hilbert-Schmidt operator. Therefore, its spectrum is being the point spectrum. Let H j be the generalized eigenspace associated with λ j (ω). Then, again from [12], it follows that L 2 (Ω) is the closure of j H j .

Lemma 2.4. We have
Moreover, if we assume that for any j, dim H j = 1, and denote by e j a unitary basis vector for H j , then the functions form a normal basis for L 2 (Ω × Ω) and the following completeness relation holds:

Remark 2.5.
Note that λ j (ω) = 0 for all j and ω ∈ R because of the Rellich lemma.
Since K ω Ω is a holomorphic family of compact operators for ω ∈ C and Definition 2.6. In view of Lemmas 2.3 and 2.4, we say that ω 0 is a scattering resonance for the open cavity problem if there exists a j 0 such that We say that the scattering resonance ω 0 is a non-exceptional scattering resonance if the following assumptions hold: Remark 2.7. Note that the assumption ε > ε m in Ω is to insure that the imaginary parts of the scattering resonances converge to zero as τ goes to infinity (see, for instance, [23]) and therefore, shifts due to the presence of small particles are measurable.

Pole pencil decomposition of the Green's function
We denote by G(x, y; ω) the Green's function associated with problem (1), that is, the solution in the sense of distributions of satisfying the outgoing radiation condition.
We can give the following expansion for G when ω is close to a non-exceptional scattering resonance. We refer to "Appendix A" for its proof.

Shift of the scattering resonances by internal small particles
Now let D Ω be a small particle of the form D = z + δB, where δ is the characteristic size of D, z is its location, and B is a smooth bounded domain containing the origin. We suppose that D has a material parameter μ c that is different from μ m , and consider the operator where μ = μ c in D and μ = μ m outside D.
As δ → 0, we seek an ω δ in a neighborhood of ω 0 such that there exists a non-trivial solution to subject to the outgoing radiation condition.
The following asymptotic expansion of ω δ holds.
Proposition 3.1. Assume that ω 0 is a non-exceptional scattering resonance. Then, as δ → 0, we have where M is the polarization tensor given by with v (1) being such that Before proving the above result, we state the following useful lemma. We refer to "Appendix B" for its proof.

Lemma 3.2. Let
Then, T ω D is a well-defined operator from L 2 (D) into itself. Proof (of Proposition 3.1). The outgoing solution to problem (5) admits the following Lippmann-Schwinger representation formula: From, for instance [9, Appendix B], the operator T ω D is well defined. Therefore, we seek ω δ such that there is a non-trivial Hence, as the characteristic size δ of D goes to zero, we seek ω δ in a neighborhood of ω 0 such that Then, it follows that where ( ·, · ) denotes the L 2 real scalar product on D. Let where N 0 D := N ω=0 D . Then, (10) can be rewritten as: where R : L 2 (D) d → L 2 (D) d is an operator with smooth kernel that is holomorphic in ω ∈ V (ω 0 ). Now, we make use of the orthogonal decomposition of L 2 (D) and the spectral analysis of N 0 D on L 2 (D) that can be found in [17,18]. More precisely, recall that where ν is the outward normal on ∂D and K * D : L 2 (∂D) → L 2 (∂D) is the Neumann-Poincaré operator associated with ∂D. Recall that K * D is given for ϕ ∈ L 2 (∂D) by

is a compact operator and hence, its spectrum is discrete and the associated eigenfunctions form a basis of W .
We refer the reader to [5] for the properties of the Neumann-Poincaré operator K * D . Therefore, using Lemma 3.
see [7] and [9,Lemma 4.2], the term L −1 R[v] can be neglected, and the following asymptotic expansion holds: Moreover, from [9, Proposition 3.1] (see also "Appendix C"), it follows that where M is the polarization tensor given by (7); see [6]. The proof is then complete.
To conclude this section, it is worth mentioning that in the case where the parameter ε inside the small particle is different from the background one, an asymptotic formula for the shift of the scattering resonance can be derived. Say, for instance, that the parameter inside the particle, which we denote by ε D , is different from the background parameter. Then, by extending the representation formula (9) to this case, we can show that ω δ − ω 0 can be approximated by

Shift of the scattering resonances by external small particles
Now consider the case where the particle is outside Ω. The main difference in this case is that the modes of K ω Ω are not defined on D, and therefore, we must first write the expansion for G outside of Ω. We start by recalling the Lippmann-Schwinger equation for v = G − Γ m : Now, using Proposition 2.8 for z and z inside Ω, we have v(z, z ; ω) = c j0 (ω) e j0 (z; ω)e j0 (z ; ω) ω − ω 0 + R(z, z ; ω), and we can write an expansion for v(x, The latter equality can be written as: where R 1 is regular in space and holomorphic in ω. Let We have Now, let x, x 0 ∈ R d . By using the Lippmann-Schwinger equation (13), it follows that We can now use expansion (15) Therefore, we have an expansion for v outside of Ω: Analogous to the calculations in the previous section, we have for some operator R with smooth kernel that is holomorphic in ω in a neighborhood V (ω 0 ) of ω 0 . Therefore, by exactly the same method as in the previous section, the following asymptotic expansion can be obtained.

Shift of the scattering resonances due to resonant particles
Let D Ω. Suppose that D is such that μ c depends on ω, and, for a discrete set of frequencies ω, problem (8) (or equivalently the operator μ m + μ c (ω) 2(μ m − μ c (ω)) I − K * D ) is singular, see [4,10,11]. We call such frequencies subwavelength resonances. In that case, we have the following scattering resonance problem: Find ω such that there is a non-trivial solution v to where L(ω) is defined by (11). Using, for instance, the Drude model for μ c , we have μ c (ω) = μ m (1−ω 2 p /ω 2 ), where ω p is a given real constant.
Hence, we have a significant shift in the scattering resonances if the particle D is resonant near or at the frequency ω 0 . In fact, the shift in the scattering resonance is proportional to the square root of the particle's volume. This anomalous effect has been observed in [24].