# The factorization and monotonicity method for the defect in an open periodic waveguide

## TL;DR

This paper addresses the inverse scattering problem in open periodic waveguides, correcting a previous method's mistake and proposing new approaches based on monotonicity to understand defects from different perspectives.

## Contribution

It identifies a mistake in the existing factorization method and introduces monotonicity-based techniques for defect reconstruction in waveguides.

## Key findings

- Corrects the application of the factorization method for this problem.
- Provides new theorems for defect understanding from inside and outside.
- Enhances inverse scattering analysis in periodic waveguides.

## Abstract

In this paper, we consider the inverse scattering problem to reconstruct the defect in an open periodic waveguide from near field data. Our first aim is to mention that there is a mistake in the factorization method of Lechleiter 2008. By this we can not apply it to solve this inverse problem. Our second aim is to give ways to understand the defect from inside (Theorem 1.1) and outside (Theorem 1.2) based on the monotonicity method.