# Inverse Problem of Diffraction by an Inhomogeneous Solid with a Piecewise Hoelder Refractive Index

## TL;DR

This paper presents a two-step method for reconstructing an unknown piecewise-Hölder refractive index of an inhomogeneous solid from boundary measurements, proving uniqueness and verifying effectiveness through numerical tests.

## Contribution

It introduces a novel two-step approach for inverse diffraction problems with piecewise-Hölder refractive indices, including proof of uniqueness and numerical validation.

## Key findings

- Uniqueness of the current solution in piecewise-constant class.
- Effective numerical reconstruction of the refractive index.
- Validation of the method through test problems with accurate results.

## Abstract

The problem of reconstruction of an unknown refractive index $k(x)$ of an inhomogeneous solid $P$ is considered. The refractive index is assumed to be a piecewise-H\"{o}lder function The original boundary value problem for the Helmholtz equation is reduced to the integral Lippman-Schwinger equation. The incident wave is defeined by a point source located outside $P.$ The solution of the inverse problem is obtained in two steps. First, the "current" $ J = (k^2 - k_0^2)u$ is determined in the inhomogeneity region. Second, the desired function $k (x) $ is expressed via the current $ J (x) $ and the incident wave $u_0.$ The uniqueness of the solution $J$ to the first-kind integral equation is proved in the class of piecewise-constant functions. The two-step method was verified by solving a test problem with a given refractive index. The comparison between the approximate solutions and the exact one approved the efficiency of the proposed method.