Recovering piecewise constant refractive indices by a single far-field pattern

TL;DR

This paper demonstrates that, under certain conditions, a single far-field pattern can uniquely determine a piecewise constant refractive index perturbation in an inhomogeneous medium, especially in low-frequency regimes, advancing inverse scattering theory.

Contribution

It establishes the first unique determinacy result showing that a single far-field measurement suffices for certain piecewise constant refractive indices, unlike previous results requiring multiple measurements.

Findings

Single far-field pattern determines the perturbation on support corners.

Injectivity of the far-field map is proven for piecewise constant indices.

Results hold for low-frequency acoustic regimes.

Abstract

We are concerned with the inverse scattering problem of recovering an inhomogeneous medium by the associated acoustic wave measurement. We prove that under certain assumptions, a single far-field pattern determines the values of a perturbation to the refractive index on the corners of its support. These assumptions are satisfied for example in the low acoustic frequency regime. As a consequence if the perturbation is piecewise constant with either a polyhedral nest geometry or a known polyhedral cell geometry, such as a pixel or voxel array, we establish the injectivity of the perturbation to far-field map given a fixed incident wave. This is the first unique determinancy result of its type in the literature, and all of the existing results essentially make use of infinitely many measurements.