Abstract

This is a work in collaboration with Mahran Rihani and Lucas Chesnel. We are interested in time-harmonic electromagnetic waves in a medium where the dielectric permittivity is supposed to be real-valued and piecewise constant, with both positive and negative values. Such model is relevant for instance when considering  metal-dielectric interfaces at optical frequencies. Due to the sign change of the permittivity, so-called plasmonic waves can travel at the surface of the metal. More specifically, this talk concerns configurations where the surface of the metal has a geometric singularity which coincides locally with a conical tip (not necessarily circular). Then a very strange phenomenon occurs. Some plasmonic waves travel towards the tip, slowing down more and more, so that they never reach the tip. These so-called black-hole waves result in a hyper-oscillating behavior of the electric field which is non longer square-integrable, because of the energy accumulated in the vicinity of the tip. Black-hole waves are linked to the solutions of a transmission problem for the Laplace equation for the infinite conical tip, with sign-changing coefficients. After a separation of variables, one has to study an eigenvalue problem set on the sphere. The difficulty comes from the sign-change of the coefficients which makes it non-selfadjoint. During the talk, we will give a survey of the theoretical, analytical and numerical results that have been proved for this unusual spectrum.