Abstract
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.
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1 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.
2 Scattering resonances of a dielectric cavity
2.1 Model
We consider the scattering of linearly polarized light by a dielectric cavity in a time-harmonic regime. Let \(\Omega \) be a bounded domain in \({\mathbb {R}}^d\) for \(d=2,3,\) with smooth boundary \(\partial \Omega \). Assume \(\varepsilon \equiv \tau \varepsilon _c + \varepsilon _m\) inside \(\Omega \) and \(\varepsilon = \varepsilon _m\) outside \(\Omega \), and \(\mu = \mu _m\) everywhere. Here, \(\varepsilon _c,\varepsilon _m,\) and \(\tau \) are positive constants. Since we are interested in scattering resonances, we look for solutions u of the homogeneous Helmholtz equation at complex frequency \(\omega \):
It is known that the above scattering problem attains a unique solution for \(\omega \) with \(\mathfrak {I}\omega \ge 0\). Using analytic continuation, the solution also exists and is unique for all complex \(\omega \) except for a countable number of points, which are the scattering resonances (see, for instance, [19]).
Let \(\Gamma _m\) be the outgoing fundamental solution of \(\Delta + \varepsilon _m \mu _m \omega ^2\) in free space. We define the following integral operator:
Definition 2.1
Let
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 \(\Omega \) is a solution of
where I is the identity operator.
According to [12], the following spectral decomposition of the operator \(K_\Omega ^\omega \) holds:
Lemma 2.3
For \(\omega \in {\mathbb {C}}\), the operator \(K_\Omega ^\omega \) is bounded from \(L^2(\Omega )\) into \(H^2(\Omega )\). Moreover, it is a Hilbert–Schmidt operator. Therefore, its spectrum is
where \(|\lambda _j(\omega )| \rightarrow 0\) as \(j\rightarrow +\infty \) and \(\{0\} = \sigma (K_\Omega ^\omega ) \setminus \sigma _p(K_\Omega ^\omega )\) with \(\sigma _p(K_\Omega ^\omega )\) being the point spectrum.
Let \(H_j\) be the generalized eigenspace associated with \(\lambda _j(\omega )\). Then, again from [12], it follows that \(L^2(\Omega ) \) is the closure of \(\bigcup _j H_j\).
Lemma 2.4
We have
Moreover, if we assume that for any j, \(\text {dim }H_j=1\), and denote by \(e_j\) a unitary basis vector for \(H_j\), then the functions
form a normal basis for \(L^2(\Omega \times \Omega )\) and the following completeness relation holds:
Remark 2.5
Note that \(\mathfrak {I}\lambda _j(\omega ) \ne 0\) for all j and \(\omega \in {\mathbb {R}}\) because of the Rellich lemma.
Since \(K_\Omega ^\omega \) is a holomorphic family of compact operators for \(\omega \in {\mathbb {C}}\) and \(\left( I - \omega ^2 \tau \varepsilon _c \mu _m K_\Omega ^\omega \right) ^{-1}\) exists for \(\omega \in {\mathbb {R}}\), then by the Fredholm analytic theory, \(\left( I - \omega ^2 \tau \varepsilon _c \mu _m K_\Omega ^\omega \right) ^{-1}\) is a meromorphic family of operators for \(\omega \in {\mathbb {C}}\).
Definition 2.6
In view of Lemmas 2.3 and 2.4, we say that \(\omega _0\) is a scattering resonance for the open cavity problem if there exists a \(j_0\) such that
We say that the scattering resonance \(\omega _0\) is a non-exceptional scattering resonance if the following assumptions hold:
-
(i)
We have
$$\begin{aligned} 1 - \omega ^2 \tau \varepsilon _c \mu _m \lambda _{j_0} (\omega ) = R(\omega ) (\omega - \omega _0), \end{aligned}$$where \(R(\omega _0) \ne 0\) and \(\omega \mapsto R(\omega )\) is holomorphic;
-
(ii)
The generalized eigenspace \(H_{j_0}(\omega )\) is of dimension 1.
Remark 2.7
Note that the assumption \(\varepsilon > \varepsilon _m\) in \(\Omega \) is to insure that the imaginary parts of the scattering resonances converge to zero as \(\tau \) goes to infinity (see, for instance, [23]) and therefore, shifts due to the presence of small particles are measurable.
2.2 Pole pencil decomposition of the Green’s function
We denote by \( G(x,y; \omega ) \) 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 \(\omega \) is close to a non-exceptional scattering resonance. We refer to “Appendix A” for its proof.
Proposition 2.8
Assume that \(\omega _0\) is a non-exceptional scattering resonance. There exists a complex neighborhood \(V(\omega _0)\) of \(\omega _0\) such that for \(\omega \) in \(V(\omega _0) \setminus \{\omega _0\}\),
where \(\mathrm {vect}(e_{j_0})= H_{j_0}\). Moreover, \(\omega \mapsto {\widetilde{R}}(x,y;\omega )\), \(\omega \mapsto e_{j_0}(\,\cdot , \omega )\), and \(\omega \mapsto c_{j_0}(\omega )\) are all holomorphic in \(V(\omega _0)\), and \((x,y) \mapsto {\widetilde{R}}(x,y;\omega )\) is smooth.
3 Shift of the scattering resonances by internal small particles
Now let \(D \Subset \Omega \) be a small particle of the form \(D= z+\delta B\), where \(\delta \) 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 \(\mu _c\) that is different from \(\mu _m\), and consider the operator
where \(\mu = \mu _c\) in D and \(\mu = \mu _m\) outside D.
As \(\delta \rightarrow 0\), we seek an \(\omega _\delta \) in a neighborhood of \(\omega _0\) such that there exists a non-trivial solution to
subject to the outgoing radiation condition.
The following asymptotic expansion of \(\omega _\delta \) holds.
Proposition 3.1
Assume that \(\omega _0\) is a non-exceptional scattering resonance. Then, as \(\delta \rightarrow 0\), 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, \(T_D^{{\omega }}\) 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:
\(\square \)
From, for instance [9, Appendix B], the operator \(T_D^\omega \) is well defined. Therefore, we seek \(\omega _\delta \) such that there is a non-trivial \(v \in L^2(D)^d\) satisfying
or equivalently,
Hence, as the characteristic size \(\delta \) of D goes to zero, we seek \(\omega _\delta \) in a neighborhood of \(\omega _0\) such that \(1/( (\mu _m/\mu _c) - 1)\) is an eigenvalue of \(T_D^{\omega _\delta }\).
From the pole-pencil decomposition (4) of G, we have
where \(R: L^2(D)^d \rightarrow L^2(D)^d\) is an operator with smooth kernel that is holomorphic in \(\omega \in V(\omega _0).\) Let
Then, it follows that
where \((\, \cdot , \cdot \,)\) denotes the \(L^2\) real scalar product on D.
Let
where \(N_D^0:= N_D^{\omega =0}\). Then, (10) can be rewritten as:
where \({\widetilde{R}}: L^2(D)^d \rightarrow L^2(D)^d\) is an operator with smooth kernel that is holomorphic in \(\omega \in V(\omega _0).\)
Now, we make use of the orthogonal decomposition of \(L^2(D)\) and the spectral analysis of \(N_D^0\) on \(L^2(D)\) that can be found in [17, 18]. More precisely, recall that
where \( H^1_0(D)\) is the set of \(H^1\)-functions in D with trace zero on \(\partial D\), \({H}(\mathrm {div\ }0,D)\) is the space of divergence free \(L^2\)-vector fields and W is the space of gradients of harmonic \(H^1\) functions. Here, \(H^1\) is the set of functions in \(L^2\) having their weak derivatives in \(L^2\). We will use the following lemma proved in [9]:
Lemma 3.3
The operator \(N_D^0\) is a bounded self-adjoint map on \(L^2(D)\) with \( \nabla H^1_0(D)\), \({H}(\text {div } 0,D)\) and W as invariant subspaces. On \(\nabla H^1_0(\Omega )\), \(N_D^0[\phi ]=\phi \), on \({H}(\text {div } 0,D)\), \(N_D^0[\phi ]=0\) and on W:
where \(\nu \) is the outward normal on \(\partial D\) and \(K_D^*: L^2(\partial D) \rightarrow L^2(\partial D)\) is the Neumann–Poincaré operator associated with \(\partial D\). Recall that \(K_D^*\) is given for \(\varphi \in L^2(\partial D)\) by
where \(\Gamma ^{(0)}\) is the fundamental solution of the Laplacian in \({\mathbb {R}}^d\).
Moreover, \((1/2) I - N_D^0|_W: {W} \longrightarrow {W}\) 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 3.3, we have
So, since
see [7] and [9, Lemma 4.2], the term \(L^{-1} {\widetilde{R}}[v]\) 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. \(\square \)
To conclude this section, it is worth mentioning that in the case where the parameter \(\varepsilon \) 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 \(\varepsilon _D\), is different from the background parameter. Then, by extending the representation formula (9) to this case, we can show that \(\omega _\delta -\omega _0\) can be approximated by
4 Shift of the scattering resonances by external small particles
Now consider the case where the particle is outside \(\Omega \). The main difference in this case is that the modes of \(K_\Omega ^\omega \) are not defined on D, and therefore, we must first write the expansion for G outside of \(\Omega \). We start by recalling the Lippmann–Schwinger equation for \(v=G-\Gamma _m\):
Now, using Proposition 2.8 for z and \(z'\) inside \(\Omega \), we have
and we can write an expansion for \(v(x,x_0)\) for \(x\in {\mathbb {R}}^d \setminus \Omega \):
for \(x \in {\mathbb {R}}^d ,x_0\in \Omega .\) The latter equality can be written as:
for \(x \in {\mathbb {R}}^d ,x_0\in \Omega \), where \(R_1\) is regular in space and holomorphic in \(\omega \). Let
We have
Now, let \( x,x_0\in {\mathbb {R}}^d\). By using the Lippmann–Schwinger equation (13), it follows that
We can now use expansion (15) to obtain that
Therefore, we have an expansion for v outside of \(\Omega \):
Analogous to the calculations in the previous section, we have
for some operator R with smooth kernel that is holomorphic in \(\omega \) in a neighborhood \(V(\omega _0)\) of \(\omega _0\). Therefore, by exactly the same method as in the previous section, the following asymptotic expansion can be obtained.
Proposition 4.1
Assume that \(\omega _0\) is a non-exceptional scattering resonance. Then, as \(\delta \rightarrow 0\), we have
where \({g}_{j_0}\) is defined by (14) and \( M(\mu _m /\mu _c,B)\) is given by (7).
5 Shift of the scattering resonances due to resonant particles
Let \(D \Subset \Omega \). Suppose that D is such that \(\mu _c\) depends on \(\omega \), and, for a discrete set of frequencies \(\omega \), problem (8) (or equivalently the operator \(\displaystyle {\big (\frac{\mu _m+\mu _c(\omega )}{2(\mu _m - \mu _c(\omega ))} I- K_D^* \big )}\)) is singular, see [4, 10, 11]. We call such frequencies subwavelength resonances. In that case, we have the following scattering resonance problem: Find \(\omega \) such that there is a non-trivial solution v to
where \(L(\omega )\) is defined by (11). Using, for instance, the Drude model for \(\mu _c\), we have \(\mu _c(\omega ) = \mu _m (1 - \omega _p^2 / \omega ^2)\), where \(\omega _p\) is a given real constant.
It is easy to see that the singular character of (8) is linked to the non-invertibility of \(L(\omega )\) on W.
Denote by \(P_1\ :\ L^2(D) \longrightarrow L^2(D) \) the orthogonal projector on \(\nabla H^1_0(D)\) and \(P_2\ : \ L^2(D) \longrightarrow L^2(D) \) the orthogonal projector on \({H}(\mathrm {div\ }0,D)\). Using Lemma 3.3, we can write the resolvent operator \(L^{-1}(\omega )\) as follows:
where \((\lambda _j, \varphi _j)_j\) are the pairs of eigenvalues and associated orthonormal eigenfunctions of \(N_D^0\). We can then rewrite equation (17) as follows:
Now, taking the scalar product on \(L^2(D)\) with \(\nabla e_{j_0}\) and multiplying by \((\omega -\omega _0) (\lambda (\omega )-\lambda _j)\), we obtain that
Since R is holomorphic in \(\omega \), the remainder \((\omega -\omega _0) (\lambda (\omega )-\lambda _j) L^{-1}R[v]\) is negligible in a neighborhood of \(\omega _0\). Hence, we arrive at the following proposition:
Proposition 5.1
As \(\delta \rightarrow 0\), we have
Note that if \(\lambda (\omega ) - \lambda _j \simeq O(\omega - \omega _0)\) for \(\omega \) close to \(\omega _0\), then we obtain
Hence, we have a significant shift in the scattering resonances if the particle D is resonant near or at the frequency \(\omega _0\). 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].
6 Asymptotic analysis near exceptional scattering resonances
In this section, we consider the asymptotic behavior of an exceptional scattering resonance for a particular form of the Green’s function. These exceptional resonances are due to the non-Hermitian character of the operator \(T_D^\omega \), see [12, 22]. For simplicity and in view of the Jordan-type decomposition of the operator \(T_D^\omega \) established in [12], we assume that, for \(\omega \) near \(\omega _0\), \(G(x,y;\omega )\) behaves like
for two functions \(h^{(1)}\) and \(h^{(2)}\) in \(L^2(D)\). Here, the functions \(\omega \mapsto c_j(\omega )\), \(j=1,2\) and \(\omega \mapsto R(x,y;\omega ) \) are all holomorphic in a neighborhood of \(\omega _0\), and \((x,y) \mapsto {R}(x,y; \omega )\) is smooth.
In this simple case, where the exceptional scattering resonance is of second order, we characterize the splitting of the scattering resonance \(\omega _0\) due to the small particle D, which is assumed for simplicity to be non-resonant.
Following the same arguments as those in the previous sections, we neglect R in (18) and seek a non-trivial v such that
or equivalently,
By multiplying the above equations by \(\nabla h^{(1)}\) and \(\nabla h^{(2)}\), respectively, and integrating by parts over D, we obtain the following system of equations:
Therefore, the following result holds.
Proposition 6.1
Assume that the decomposition (18) holds for \(\omega \) near \(\omega _0\). Then, the perturbed scattering resonance problem (due to the particle D) can be approximately reformulated as a search for \(\omega \) near \(\omega _0\) such that the matrix
is singular.
In view of Proposition 6.1, the second-order exceptional scattering frequency is split into two scattering frequencies which can be computed approximately by finding the values of \(\omega \) for which the determinant of the matrix \({\mathcal {A}}(\omega )\) is zero. When D is a disk or a sphere, the functions \(h^{(j)}\) and the functions \(c_j(\omega )\) for \(j=1,2\), defined in (18), can be computed explicitly for \(\omega \) near a resonance \(\omega _0\) (see, for instance, [8]) and hence, an expression for \({\mathcal {A}}(\omega )\) can be obtained. In the general case, it seems difficult to obtain accurate approximations of the functions \(h^{(j)}\) and the functions \(c_j(\omega )\) for \(j=1,2\).
Assume that \((L^{-1}[\nabla {h^{(1)}}], \nabla h^{(2)}) =0.\) Then, the values of \(\omega \) near \(\omega _0\) such that the determinant of \({\mathcal {A}}(\omega )\) is zero are determined by
for \(j=1\) or 2. Since \(c_j(\omega ) (L^{-1}[\nabla {h^{(j)}}], \nabla h^{(j)}) = O(|D|)\) for \(j=1,2,\) and \(\omega \) near \(\omega _0\), it can be easily seen that the splitting corresponding to \(j=2\) of \(\omega _0\) is of order the square root of the volume of the particle. This is in contrast with (6), where the perturbation induced by the small particle is proportional to its volume.
It is worth emphasizing that the derivations presented in this section can be generalized to the case of exceptional points of arbitrary order N. In this case, it is expected that the strength of the splitting induced by a small particle on an exceptional scattering frequency of order N is proportional to the volume of the particle to the power 1/N.
7 Concluding remarks
In this paper, the leading-order term in the shifts of scattering resonances of a radiating dielectric cavity due to the presence of small particles is derived. The formula describes the dependency of the frequency shifts on the position and the polarization tensor of the particle. It is also proved that the shift is significantly enhanced if the particle is a subwavelength resonant particle which resonates near or at a scattering resonance of the cavity. A characterization of the splitting of the scattering resonances due to small particles near an exceptional scattering resonance is performed. It would be challenging to develop a general theory near such frequencies. This would be the subject of a forthcoming paper.
Change history
28 May 2021
The original online version of this article was revised: The funding note was incorrect and it has been corrected now.
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Appendices
A Proof of Proposition 2.8
Proof
Under the assumption that for any j, \(\text {dim } H_j =1\), the proof follows an idea from [12]. Denote by v the difference
One can check that \(v(\cdot ,x_0)\) is a solution of the following integral equation:
Therefore,
Under the assumption that \(\omega _0\) is a non-exceptional scattering resonance (see Definition 2.6) we can perform a pole pencil decomposition of the resolvent of \(K_\Omega ^\omega \). We start from the spectral decomposition of the compact operator \(K_\Omega ^\omega \) on \(L^2(\Omega )\). By assumption, the eigenspace associated with the eigenvalue \(\frac{1}{\omega _0^2\tau \varepsilon _c \mu _m}\) is of dimension one. We denote by \(e_{j_0}\) its basis. One can then write
where \((\; ,\;)\) denotes the \(L^2\) real scalar product on \(\Omega \), and \(\omega \mapsto {\hat{R}}(\cdot ,\omega )\in L^2(\Omega )\) is holomorphic in a complex neighborhood V of \(\omega _0\). Using
and composing with \(K_\Omega ^\omega \), we obtain that
Now we note that
Using the completeness relation given in Lemma 2.4 yields
Now, we can write
to arrive at
\(\square \)
B Proof of Lemma 3.2
Proof
The operator \(T_D\) is a singular integral operator of the Calderón–Zygmund type, see [14]. This type of singular operator often arises in electrostatic and magnetostatic theories (see the appendix of [9] for a simple review of the properties of these operators within the formalism of Green’s functions). The fact that \(T_D^\omega \) is well defined can be deduced directly from Proposition 2.8. Since G can be written as \(G(x,y)= \Gamma _m(x,y) + K(x,y)\) where K is a smooth kernel, we can see that the singularity of the derivatives of G is the same as that of the derivatives of \(\Gamma _m\), that is, \(\partial _{x_i,x_j}G(x,y) = \partial _{x_i,x_j}\Gamma _m(x,y) + K_{i,j}(x,y)\). Therefore, it is easy to see that the singular part of \(\partial _{x_i,x_j}G(x,y)\) satisfies the same cancellation property as \(\partial _{x_i,x_j}\Gamma _m(x,y)\), that is,
Hence, the fact that \(T_D\) is defined on \(L^2(D)\) follows directly from classical Calderón–Zygmund theory and the cancellation property above. \(\square \)
C Proof of estimate (12)
Here, we give some more details on how to obtain (12) from the results of [9].
Lemma C.1
As \(\delta \rightarrow 0\), we have
Proof
From [9, Proposition 3.1], one can see that if \(\varphi \) satisfies
then \(\nabla \varphi \) solves the integral equation
which is exactly
Now, replacing \(\nabla e_{j_0}\) by its average and controlling the reminder via the Cauchy–Schwarz inequality, we have:
But the average of \(\nabla \varphi \) is exactly the dipole moment, which is given by the polarization tensor applied to the average of the exciting field:
Since \(\frac{1}{|D|} \mathop {\int }\limits _D \nabla e_{j_0}(x) dx - \nabla e_{j_0}(z) = O(\delta ) \) (recall that \(e_j\) is a mode of the cavity, and is therefore independent of \(\delta \)), we can replace the average of \(\nabla e_{j_0}\) by its value at the center of D to get the result. \(\square \)
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Ammari, H., Dabrowski, A., Fitzpatrick, B. et al. Perturbations of the scattering resonances of an open cavity by small particles: Part II—the transverse electric polarization case. Z. Angew. Math. Phys. 72, 80 (2021). https://doi.org/10.1007/s00033-021-01521-x
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DOI: https://doi.org/10.1007/s00033-021-01521-x
Keywords
- Open dielectric resonator
- Shift of scattering resonances
- Exceptional scattering resonances
- Splitting of scattering resonances
- Subwavelength resonant nanoparticles
- Pole-pencil decomposition