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Surface plasmons in metamaterial cavities: Scattering by obstacles with negative wave speed

Yan-Long Fang, Jeffrey Galkowski

TL;DR

This work analyzes resonances for metamaterial cavities with negative index, focusing on surface plasmons that localize near the boundary. The authors develop a semiclassical exterior–interior framework, recasting the problem via Dirichlet-to-Neumann maps and factorizing near the boundary to obtain precise microlocal estimates. They establish criteria for the absence or presence of resonances near the real axis, prove plasmonicity for resonances with small imaginary parts, and derive a Weyl-type law for the plasmon count. The results bridge spectral theory and boundary microlocal analysis, providing sharp asymptotics and localization descriptions that advance understanding of plasmonic phenomena in non-trapping metamaterial cavities.

Abstract

We study scattering by metamaterials with negative indices of refraction, which are known to support \emph{surface plasmons} -- long-lived states that are highly localized at the boundary of the cavity. This type of states has found uses in a variety of modern technologies. In this article, we study surface plasmons in the setting of non-trapping cavities; i.e. when all billiard trajectories outside the cavity escape to infinity. We characterize the indices of refraction which support surface plasmons, show that the corresponding resonances lie super-polynomially close to the real axis, describe the localization properties of the corresponding resonant states, and give an asymptotic formula for their number.

Surface plasmons in metamaterial cavities: Scattering by obstacles with negative wave speed

TL;DR

This work analyzes resonances for metamaterial cavities with negative index, focusing on surface plasmons that localize near the boundary. The authors develop a semiclassical exterior–interior framework, recasting the problem via Dirichlet-to-Neumann maps and factorizing near the boundary to obtain precise microlocal estimates. They establish criteria for the absence or presence of resonances near the real axis, prove plasmonicity for resonances with small imaginary parts, and derive a Weyl-type law for the plasmon count. The results bridge spectral theory and boundary microlocal analysis, providing sharp asymptotics and localization descriptions that advance understanding of plasmonic phenomena in non-trapping metamaterial cavities.

Abstract

We study scattering by metamaterials with negative indices of refraction, which are known to support \emph{surface plasmons} -- long-lived states that are highly localized at the boundary of the cavity. This type of states has found uses in a variety of modern technologies. In this article, we study surface plasmons in the setting of non-trapping cavities; i.e. when all billiard trajectories outside the cavity escape to infinity. We characterize the indices of refraction which support surface plasmons, show that the corresponding resonances lie super-polynomially close to the real axis, describe the localization properties of the corresponding resonant states, and give an asymptotic formula for their number.

Paper Structure

This paper contains 27 sections, 51 theorems, 366 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Inhomogeneous metrics
  3. Outline and ideas from the proof
  4. Structure of the paper
  5. Preliminaries
  6. Semiclassical rescaling and pseudodifferential operators
  7. The operator in Fermi Normal Coordinates
  8. Semiclassical defect measures
  9. An estimate for the Neumann Trace
  10. Reformulation of negative index of refraction problem as an exterior problem
  11. Review of estimates for the Dirichlet resolvent in $\Omega_{{_{\mathcal{O}}}}$
  12. Review of estimates for the Dirichlet resolvent in $\Omega_{{_{\mathcal{I}}}}$
  13. Reformulation of the negative index of refraction problem
  14. Microlocal description of the Dirichlet-to-Neumann map
  15. Semiclassical Lee-Uhlmann constructions
  16. ...and 12 more sections

Key Result

Theorem 1.1

Suppose that $\Omega_{{_{\mathcal{I}}}}$ is non-trapping and $n\in C^\infty(\overline{\Omega}_{{_{\mathcal{I}}}};(0,\infty))$ satisfies $n|_{\partial\Omega}<1$. Then for all $M>0$ there is $C>0$ such that

Figures (4)

  • Figure 1: Examples of trapping and non-trapping domains.
  • Figure 2: The figure shows the resonances for \ref{['e:nProblem']} with $n|_{\partial\Omega}<1$ as x's. The resonance free region is determined by Theorem \ref{['t:simple']} or \ref{['t:noStates']}.
  • Figure 3: Lemma \ref{['l:plasmonic']} in fact shows that, modulo $O(|\lambda_j|^{-\infty})$, all surface plasmons are as pictured here (with $\Omega_{{_{\mathcal{I}}}}=B(0,1)$). These plasmons concentrate in a $|\lambda_j|^{-1}$ neighborhood of the boundary, $\partial\Omega$ and oscillate at frequency $\sim |\lambda_j|$ in $\partial\Omega$. The functions plotted here are the real parts of the resonant state corresponding to $n|_{B(0,1)}\equiv 3$ with resonance $\lambda_L \approx 8.4647\times 10^0-1.0396\times 10^{-2}\mathrm{i}$ (on the left) and $\lambda_R\approx 13.145-8.5412\times 10^{-4}\mathrm{i}$ (on the right).
  • Figure 4: The figure shows the resonances for \ref{['e:nProblem']} with $n|_{\partial\Omega}>1$. Non-plasmonic resonances are denoted with x's and plasmonic resonances with o's. The resonance free regions are those determined by Theorem \ref{['t:simple2']} or \ref{['t:states']}. Theorems \ref{['t:simple4']} and \ref{['t:count']} determine the asymptotic number of plasmonic resonances.

Theorems & Definitions (101)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 91 more