Limiting absorption principle for time-harmonic acoustic and electromagnetic scattering of plane waves from a bi-periodic inhomogeneous layer

TL;DR

This paper establishes a rigorous limiting absorption principle for acoustic and electromagnetic scattering from bi-periodic inhomogeneous layers supporting bound states in the continuum, ensuring uniqueness of solutions.

Contribution

It provides a novel justification of the limiting absorption principle for BIC-supporting structures, combining singular perturbation methods with a new radiation condition.

Findings

Proves convergence of solutions as absorption parameter tends to zero.

Derives a sharp radiation condition for uniqueness.

Ensures well-posedness of scattering problems with BICs.

Abstract

The Rayleigh expansion is widely used as a formal radiation condition in the analysis and numerical treatment of grating diffraction problems for incoming plane waves. However, the Rayleigh expansion does not always lead to uniqueness of open waveguide scattering problems, due to the existence of surface/guided waves (in other words, Bound States in the Continuum (BICs)) which exponentially decay in the direction perpendicular to the periodicity. In this paper we suppose that a bi-periodic inhomogeneous medium supports BICs at some real-valued incident wavenumber. Based on singular perturbation arguments, we justify the Limiting Absorption Principle (LAP) for both time-harmonic acoustic and electromagnetic scattering of plane waves from bi-periodic structures. Replacing the wavenumber $k$ with $k+i\epsilon$, we prove that the unique solution with $\epsilon>0$ converges to a solution of the original diffraction problem that additionally satisfies an orthogonal identity. This constraint condition together with the classical Rayleigh expansion leads to a sharp radiation condition to ensure uniqueness of time-harmonic scattering of plane waves by BIC-supporting bi-periodic materials.