On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients

TL;DR

This paper studies electromagnetic wave scattering by plasmonic devices with corners, addressing mathematical well-posedness issues and proposing a novel numerical method using PMLs at corners to handle singularities.

Contribution

It introduces a systematic approach to define outgoing solutions in corner scattering problems and develops an original PML-based numerical method for singular fields.

Findings

Well-posedness depends on the device shape and permittivity interval.

Corner singularities require specialized radiation conditions.

PMLs at corners effectively capture singular electromagnetic fields.

Abstract

We investigate in a $2$D setting the scattering of time-harmonic electromagnetic waves by a plasmonic device, represented as a non dissipative bounded and penetrable obstacle with a negative permittivity. Using the $\textrm{T}$-coercivity approach, we first prove that the problem is well-posed in the classical framework $H^1_{\text{loc}} $ if the negative permittivity does not lie in some critical interval whose definition depends on the shape of the device. When the latter has corners, for values inside the critical interval, unusual strong singularities for the electromagnetic field can appear. In that case, well-posedness is obtained by imposing a radiation condition at the corners to select the outgoing black-hole plasmonic wave, that is the one which carries energy towards the corners. A simple and systematic criterion is given to define what is the outgoing solution. Finally, we propose an original numerical method based on the use of Perfectly Matched Layers at the corners. We emphasize that it is necessary to design an $\textit{ad hoc}$ technique because the field is too singular to be captured with standard finite element methods.