Scattering resonances in unbounded transmission problems with sign-changing coefficient

Abstract

Abstract It is well known that classical optical cavities can exhibit localized phenomena associated with 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, e.g. 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 shows 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 close to the real axis are associated with surface plasmons. Examples for several metamaterial cavities are provided.