Visibility, invisibility and unique recovery of inverse electromagnetic problems with conical singularities

TL;DR

This paper investigates the inverse electromagnetic scattering problem with conical singularities, establishing conditions for invisibility and proving that visibility guarantees unique support recovery through local and global uniqueness results.

Contribution

It introduces new local and global uniqueness theorems for inverse scattering with conical singularities, linking invisibility to the vanishing of characteristic parameters.

Findings

Invisibility occurs when characteristic parameters vanish locally.

Visibility implies the support of the scatterer can be uniquely recovered.

Established local vanishing of electromagnetic transmission eigenfunctions near conical points.

Abstract

In this paper, we study time-harmonic electromagnetic scattering in two scenarios, where the anomalous scatterer is either a pair of electromagnetic sources or an inhomogeneous medium, both with compact supports. We are mainly concerned with the geometrical inverse scattering problem of recovering the support of the scatterer, independent of its physical contents, by a single far-field measurement. It is assumed that the support of the scatterer (locally) possesses a conical singularity. We establish a local characterisation of the scatterer when invisibility/transparency occurs, showing that its characteristic parameters must vanish locally around the conical point. Using this characterisation, we establish several local and global uniqueness results for the aforementioned inverse scattering problems, showing that visibility must imply unique recovery. In the process, we also establish the local vanishing property of the electromagnetic transmission eigenfunctions around a conical point under the H\"older regularity or a regularity condition in terms of Herglotz approximation.