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Functional analytic methods for discrete approximations of subwavelength resonator systems

Habib Ammari, Bryn Davies, Erik Orvehed Hiltunen

Abstract

We survey functional analytic methods for studying subwavelength resonator systems. In particular, rigorous discrete approximations of Helmholtz scattering problems are derived in an asymptotic subwavelength regime. This is achieved by re-framing the Helmholtz equation as a non-linear eigenvalue problem in terms of integral operators. In the subwavelength limit, resonant states are described by the eigenstates of the generalised capacitance matrix, which appears by perturbing the elements of the kernel of the limiting operator. Using this formulation, we are able to describe subwavelength resonance and related phenomena. In particular, we demonstrate large-scale effective parameters with exotic values. We also show that these systems can exhibit localised and guided waves on very small length scales. Using the concept of topologically protected edge modes, such localisation can be made robust against structural imperfections.

Functional analytic methods for discrete approximations of subwavelength resonator systems

Abstract

We survey functional analytic methods for studying subwavelength resonator systems. In particular, rigorous discrete approximations of Helmholtz scattering problems are derived in an asymptotic subwavelength regime. This is achieved by re-framing the Helmholtz equation as a non-linear eigenvalue problem in terms of integral operators. In the subwavelength limit, resonant states are described by the eigenstates of the generalised capacitance matrix, which appears by perturbing the elements of the kernel of the limiting operator. Using this formulation, we are able to describe subwavelength resonance and related phenomena. In particular, we demonstrate large-scale effective parameters with exotic values. We also show that these systems can exhibit localised and guided waves on very small length scales. Using the concept of topologically protected edge modes, such localisation can be made robust against structural imperfections.

Paper Structure

This paper contains 40 sections, 56 theorems, 199 equations, 34 figures.

Table of Contents

  1. Introduction
  2. Wave manipulation at subwavelength scales
  3. Analysis of scattering problems
  4. Functional analytic approach
  5. Capacitance coefficients
  6. Finite systems
  7. Main results of the capacitance formulation
  8. Properties of the capacitance matrix
  9. Modal decompositions
  10. Higher-order approximations
  11. Two-dimensional models
  12. Numerical approaches
  13. Periodic systems
  14. Floquet-Bloch theory
  15. Evanescent-mode resonances
  16. ...and 25 more sections

Key Result

Lemma 2.1

In the regime $\omega\rightarrow 0$, the Helmholtz problem eq:finite_scattering is equivalent to finding $\psi,\phi\in L^2(\partial D)$ such that where the operator $\mathcal{A}(\omega,\delta):L^2(\partial D)\times L^2(\partial D)\to H^1(\partial D)\times L^2(\partial D)$ is defined as where, as in eq:dtilde, $\widetilde{\delta}(x) = \delta_i$ for $x\in \partial D_i$.

Figures (34)

  • Figure 1: The functional analytic method developed here is useful for studying scattering by a system of material inclusions, which act as subwavelength resonators in an appropriate high-contrast regime. We are able to derive concise asymptotic results in terms of the capacitance matrix for the case of either finitely many resonators or a periodically repeating array of finitely many resonators.
  • Figure 2: A finite collection of $N$ resonators, with wave speeds $v_i$ for $i=1,\dots,N$, in a surrounding medium with wave speed $v$. The contrast between the $i$th resonator and the background is given by $\delta_i$, where a small value of $\delta_i$ describes a large contrast.
  • Figure 3: A system of two spherical resonators can be described using bispherical coordinates. Such a coordinate system is convenient since the boundaries of the spheres lie on level sets and the capacitance coefficients can be calculated explicitly.
  • Figure 4: The subwavelength resonant frequencies of a system of ten spherical resonators. We compare the values computed using the multipole expansion method to discretised the full boundary integral equation and the values computed using the capacitance matrix. The computations using the full multipole method took $41$ seconds while the approximations from the capacitance matrix took just $0.02$ seconds, on the same computer. Each resonator has unit radius and we use $\delta=1/5000$.
  • Figure 5: A periodic array of material inclusions. Here, three material inclusions (resonators) are sketched with periodicity in one dimension. Each interior has a different wave speed $v_1$, $v_2$, $v_3$ and the surrounding medium has a wave speed $v$. The contrast between the $i$th resonator and the background is given by $\delta_i$, where a small value of $\delta_i$ describes a large contrast.
  • ...and 29 more figures

Theorems & Definitions (94)

  • Definition 1.1: Subwavelength resonant frequency
  • Definition 1.2: Characteristic value
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Definition 2.5: Capacitance matrix
  • Definition 2.6: Generalised capacitance matrix
  • Theorem 2.7
  • Remark 2.8
  • ...and 84 more