Direct and inverse time-harmonic scattering by Dirichlet periodic curves with local perturbations

TL;DR

This paper advances the mathematical understanding of time-harmonic scattering by periodic curves with local perturbations, establishing well-posedness, uniqueness, and inverse problem solutions, including the role of guided waves and BICs.

Contribution

It introduces new well-posedness results and uniqueness proofs for scattering by perturbed periodic curves, incorporating guided waves and BICs, and extends inverse problem solutions with point source and plane wave data.

Findings

Proved well-posedness of scattering problems with guided waves.

Established uniqueness results for inverse defect detection.

Analyzed the role of BICs in scattering and inverse problems.

Abstract

This is a continuation of the authors' previous work (A. Kirsch, Math. Meth. Appl. Sci., 45 (2022): 5737-5773.) on well-posedness of time-harmonic scattering by locally perturbed periodic curves of Dirichlet kind. The scattering interface is supposed to be given by a non-self-intersecting Lipschitz curve. We study properties of the Green's function and prove new well-posedness results for scattering of plane waves at a propagative wave number. In such a case there exist guided waves to the unperturbed problem, which are also known as Bounded States in the Continuity (BICs) in physics. In this paper uniqueness of the forward scattering follows from an orthogonal constraint condition enforcing on the total field to the unperturbed scattering problem. This constraint condition, which is also valid under the Neumann boundary condition, is derived from the singular perturbation arguments and also from the approach of approximating a plane wave by point source waves. For the inverse problem of determining the defect, we prove several uniqueness results using a finite or infinite number of point source and plane waves, depending on whether a priori information on the size and height of the defect is available.