Uniqueness results on phaseless inverse scattering with a reference ball

TL;DR

This paper proves the unique determination of obstacles and media in phaseless inverse scattering problems using a reference ball and superposition of waves, applicable to various boundary conditions at a single frequency.

Contribution

Introduces a novel reference ball technique combined with superposed incident waves to achieve uniqueness in phaseless inverse scattering problems.

Findings

Unique determination of obstacle shape and boundary conditions.

Applicable to both obstacle and medium scattering problems.

Works with a single frequency and general boundary conditions.

Abstract

This paper is devoted to the uniqueness in inverse acoustic scattering problems for the Helmholtz equation with phaseless far-field data. Some novel techniques are developed to overcome the difficulty of translation invariance induced by a single incident plane wave. In this paper, based on adding a reference ball as an extra artificial impenetrable obstacle (resp. penetrable homogeneous medium) to the inverse obstacle (resp. medium) scattering system and then using superpositions of a fixed plane wave and some point sources as the incident waves, we rigorously prove that the location and shape of the obstacle as well as its boundary condition or the refractive index can be uniquely determined by the modulus of far-field patterns. The reference ball technique in conjunction with the superposition of incident waves brings in several salient benefits. First, the framework of our theoretical analysis can be applied to both the inverse obstacle and medium scattering problems. Second, for inverse obstacle scattering, the underlying boundary condition could be of a general type and be uniquely determined. Finally, only a single frequency is needed.