Unique determination of the index of refraction from phase-less near-field data in two dimensions

TL;DR

This paper proves the unique determination of a medium's index of refraction in two dimensions using phase-less near-field data, employing complex geometric optics solutions and multi-term expansions.

Contribution

It introduces a novel approach to uniquely identify the index of refraction from magnitude-only data in 2D, advancing inverse scattering theory.

Findings

Uniqueness of the inverse problem established for the index of refraction.

Use of multi-term CGO solutions to address phase-less data challenges.

Applicable to fixed frequency and bounded media cross sections.

Abstract

Motivated by non-destructive testing of optical fiber, we consider the problem of determining the index of refraction of a two-dimensional medium from magnitude of the total field resulting from known incident plane waves at a fixed frequency. The finiteness of the fiber cross section allows us to assume that the index of refraction is variable over a compact set. The measurement set is a closed curve, e.g., a circle of fixed radius, containing the fiber cross section. We use complex geometric optics (CGO) solutions to show that the inverse problem for the unknown index of refraction is unique in $C^{k}$ for large enough $k$. The key tool in this work is the multi-term expansion of the CGO solutions which we rely on to address the uniqueness question.