# On the inverse scattering from anisotropic periodic layers and transmission eigenvalues

## TL;DR

This paper develops a factorization method for reconstructing the shape of anisotropic periodic layers from scattering data and investigates the existence of transmission eigenvalues, providing new insights into inverse scattering problems.

## Contribution

It introduces a rigorous factorization method for shape reconstruction and proves the existence of infinitely many transmission eigenvalues for anisotropic periodic layers.

## Key findings

- Factorization method enables fast shape reconstruction.
- Unique determination of scatterer shape from near field data.
- Existence of infinitely many transmission eigenvalues proven.

## Abstract

This paper is concerned with the inverse scattering and the transmission eigenvalues for anisotropic periodic layers. For the inverse scattering problem, we study the Factorization method for shape reconstruction of the periodic layers from near field scattering data. This method provides a fast numerical algorithm as well as a unique determination for the shape reconstruction of the scatterer. We present a rigorous justification and numerical examples for the factorization method. The transmission eigenvalue problem in scattering have recently attracted a lot of attentions. Transmission eigenvalues can be determined from scattering data and they can provide information about the material parameters of the scatterers. In this paper we formulate the interior transmission eigenvalue problem and prove the existence of infinitely many transmission eigenvalues for the scattering from anisotropic periodic layers.