Scattering resonances in unbounded transmission problems with sign-changing coefficient

TL;DR

This paper investigates scattering resonances in unbounded transmission problems with sign-changing coefficients, such as metamaterial cavities, establishing their existence and asymptotic behavior, and linking certain resonances to surface plasmons.

Contribution

It extends the black-box scattering framework to sign-changing coefficients, proving the existence of resonances in 2D metamaterial cavities and analyzing their asymptotic properties.

Findings

Resonances exist for arbitrary 2D metamaterial cavities.

Resonances near the real axis are linked to surface plasmons.

The framework accommodates sign-changing coefficients in scattering problems.

Abstract

It is well-known that classical optical cavities can exhibit localized phenomena associated to scattering resonances, leading to numerical instabilities in approximating the solution. This result can be established via the ``quasimodes to resonances'' argument from the black-box scattering framework. Those localized phenomena concentrate at the inner boundary of the cavity and are called whispering gallery modes. In this paper we investigate scattering resonances for unbounded transmission problems with sign-changing coefficient (corresponding to optical cavities with negative optical properties, for example made of metamaterials). Due to the change of sign of optical properties, previous results cannot be applied directly, and interface phenomena at the metamaterial-dielectric interface (such as the so-called surface plasmons) emerge. We establish the existence of scattering resonances for arbitrary two-dimensional smooth metamaterial cavities. The proof relies on an asymptotic characterization of the resonances, and showing that problems with sign-changing coefficient naturally fit the black box scattering framework. Our asymptotic analysis reveals that, depending on the metamaterial's properties, scattering resonances situated closed to the real axis are associated to surface plasmons. Examples for several metamaterial cavities are provided.