On electromagnetic scattering from a penetrable corner

TL;DR

This paper proves that electromagnetic waves scatter from a medium with a corner, showing that the presence of a corner ensures non-trivial scattering for most incident fields, extending acoustic scattering results to electromagnetics.

Contribution

It extends the corner scattering theory from acoustic to electromagnetic waves, overcoming challenges related to Maxwell's system and establishing new analytical tools.

Findings

Corners guarantee non-vanishing scattered fields for most incident waves.

Developed novel orthogonality relations for Maxwell solutions.

Constructed CGO solutions with new Lp-estimates for Maxwell systems.

Abstract

This article is concerned with the time-harmonic electromagnetic (EM) scattering from a generic inhomogeneous medium. It is shown that if there is a right corner on the support of the medium, then it scatters every pair of incident EM fields, excluding a possible class of EM fields which are of very particular forms. That is, for every pair of admissible incident EM fields, the corresponding scattered wave fields associated to the medium scatterer cannot be identically vanishing outside the support of the medium. Indeed, we achieve the corner scattering result by establishing a stronger result, that shows the failure of the analytic extension across the corner of certain EM fields satisfying the so-called interior transmission eigenvalue problem. This extends the relevant study in [3] for the acoustic scattering governed by the Helmholtz equation to the electromagnetic case governed by the Maxwell system. Substantial new challenges arise from the corresponding extension from the scalar PDE to the system of PDEs. Our mathematical arguments combine the analysis for interior transmission eigenvalue problems associated to the Maxwell system; the derivation of novel orthogonality relation for the solutions of Maxwell systems; the construction of complex-geometrical-optics (CGO) solutions for the Maxwell system with new Lp-estimates (p > 6) on the remainder terms; and the proof of the non-vanishing property for the Laplace transform of vectorial homogeneous harmonic polynomials.