Two unique Identifiability results for inverse scattering problems within polyhedral geometries

TL;DR

This paper establishes two new unique identifiability results for inverse scattering problems involving polyhedral obstacles and diffraction gratings in three dimensions, allowing simultaneous recovery of shape and impedance parameters from a single measurement.

Contribution

The paper introduces novel geometric properties of Laplacian eigenfunctions that enable unique determination of obstacle shape and impedance parameters in complex polyhedral geometries.

Findings

Two new unique identifiability theorems for inverse scattering in polyhedral geometries.

Ability to recover obstacle shape and impedance parameters simultaneously.

Extension of geometric eigenfunction properties to more general polyhedral cases.

Abstract

We consider the unique determinations of impenetrable obstacles or diffraction grating profiles in $\mathbb{R}^3$ by a single far-field measurement within polyhedral geometries. We are particularly interested in the case that the scattering objects are of impedance type. We derive two new unique identifiability results for the inverse scattering problem in the aforementioned two challenging setups. The main technical idea is to exploit certain quantitative geometric properties of the Laplacian eigenfunctions which were initiated in our recent works [8,9]. In this paper, we derive novel geometric properties that generalize and extend the related results in [9], which further enable us to establish the new unique identifiability results. It is pointed out that in addition to the shape of the obstacle or the grating profile, we can simultaneously recover the boundary impedance parameters.