A curious instability phenomenon for a rounded corner in presence of a negative material

TL;DR

This paper investigates an unusual instability in wave transmission at a rounded corner interface between a dielectric and a negative material, showing the solution's critical dependence on the corner's rounding parameter through asymptotic analysis and numerical validation.

Contribution

It introduces a novel instability phenomenon in wave transmission problems involving negative materials with rounded corners, supported by asymptotic expansions and error estimates.

Findings

Solution depends critically on the rounding parameter.

Asymptotic expansion reveals instability behavior.

Numerical illustrations confirm the theoretical results.

Abstract

We study a 2D scalar harmonic wave transmission problem between a classical dielectric and a medium with a real-valued negative permittivity/permeability which models a metal at optical frequency or an ideal negative metamaterial. We highlight an unusual instability phenomenon for this problem when the interface between the two media presents a rounded corner. To establish this result, we provide an asymptotic expansion of the solution, when it is well-defined, in the geometry with a rounded corner. Then, we prove error estimates. Finally, a careful study of the asymptotic expansion allows us to conclude that the solution, when it is well-defined, depends critically on the value of the rounding parameter. We end the paper with a numerical illustration of this instability phenomenon.