Uniqueness in Determining Refractive Indices by Formally-determined Far-field Data

TL;DR

This paper proves new uniqueness results for inverse acoustic scattering problems, showing that refractive indices can be uniquely determined from far-field data, with local results for variable indices and global results for constant indices.

Contribution

It introduces novel uniqueness theorems for inverse problems using far-field data, employing interior transmission eigenvalue analysis.

Findings

Local uniqueness for variable refractive index from fixed incident data

Global uniqueness for constant refractive index from a single measurement

Use of nonlinear interior transmission eigenvalue problems

Abstract

We present two uniqueness results for the inverse problem of determining an index of refraction by the corresponding acoustic far-field measurement encoded into the scattering amplitude. The first one is a local uniqueness in determining a variable index of refraction by the fixed incident-direction scattering amplitude. The inverse problem is formally posed with such measurement data. The second one is a global uniqueness in determining a constant refractive index by a single far-field measurement. The arguments are based on the study of certain nonlinear and non-selfadjoint interior transmission eigenvalue problems.