On Uniqueness in Electromagnetic Scattering from Biperiodic Structures

TL;DR

This paper establishes conditions under which electromagnetic scattering problems involving biperiodic structures have unique solutions, using a Rellich identity and non-trapping assumptions to extend uniqueness results across all wave numbers.

Contribution

The paper derives a Rellich identity for the variational formulation of electromagnetic scattering from biperiodic structures and proves uniqueness under non-trapping conditions for all wave numbers.

Findings

Derived a Rellich identity for the variational problem.

Proved uniqueness of solutions for all wave numbers under non-trapping assumptions.

Extended the understanding of electromagnetic scattering from biperiodic structures.

Abstract

Consider time-harmonic electromagnetic wave scattering from a biperiodic dielectric structure mounted on a perfectly conducting plate in three dimensions. Given that uniqueness of solution holds, existence of solution follows from a well-known Fredholm framework for the variational formulation of the problem in a suitable Sobolev space. In this paper, we derive a Rellich identity for a solution to this variational problem under suitable smoothness conditions on the material parameter. Under additional non-trapping assumptions on the material parameter, this identity allows us to establish uniqueness of solution for all wave numbers.