# Reconstruction of a Local Perturbation in Inhomogeneous Periodic Layers from Partial Near Field Measurements

## TL;DR

This paper develops a method to reconstruct local perturbations in inhomogeneous periodic layers from near field measurements, overcoming challenges posed by the perturbation preventing reduction to a single periodic cell.

## Contribution

It introduces a reformulation of the inverse scattering problem into quasi-periodic problems on bounded domains, enabling unique solution existence and numerical reconstruction of local perturbations.

## Key findings

- Unique reconstruction of perturbations demonstrated in 2D and 3D.
- Numerical algorithms show effective approximation of the perturbed medium.
- Analysis of measurement operators highlights ill-posedness of the inverse problem.

## Abstract

We consider the inverse scattering problem to reconstruct a local perturbation of a given inhomogeneous periodic layer in $\mathbb{R}^d$, $d=2,3$, using near field measurements of the scattered wave on an open set of the boundary above the medium, or, the measurements of the full wave in some area. The appearance of the perturbation prevents the reduction of the problem to one periodic cell, such that classical methods are not applicable and the problem becomes more challenging. We first show the equivalence of the direct scattering problem, modeled by the Helmholtz equation formulated on an unbounded domain, to a family of quasi-periodic problems on a bounded domain, for which we can apply some classical results to provide unique existence of the solution to the scattering problem. The reformulation of the problem is also the key idea for the numerical algorithm to approximate the solution, which we will describe in more detail. Moreover, we characterize the smoothness of the Bloch-Floquet transformed solution of the perturbed problem w.r.t. the quasi-periodicity to improve the convergence rate of the numerical approximation. Afterward, we define two measurement operators, which map the perturbation to some measurement data, and show uniqueness results for the inverse problems, and the ill-posedness of these. Finally, we provide numerical examples for the direct problem solver as well as examples of the reconstruction in 2D and 3D.