Reconstruction of Local Perturbations in Periodic Surfaces

TL;DR

This paper introduces a numerical method based on the Floquet-Bloch transform to reconstruct local perturbations in periodic surfaces, effectively reducing an infinite domain problem to a manageable computational task.

Contribution

It develops a two-step algorithm combining initialization and Newton-CG optimization to solve inverse scattering problems with local perturbations in periodic structures.

Findings

Efficient localization of perturbation support

Reduction of inverse problem to a single periodic cell

Numerical examples demonstrate method effectiveness

Abstract

This paper concerns the inverse scattering problem to reconstruct a local perturbation in a periodic structure. Unlike the periodic problems, the periodicity for the scattered field no longer holds, thus classical methods, which reduce quasi-periodic fields in one periodic cell, are no longer available. Based on the Floquet-Bloch transform, a numerical method has been developed to solve the direct problem, that leads to a possibility to design an algorithm for the inverse problem. The numerical method introduced in this paper contains two steps. The first step is initialization, that is to locate the support of the perturbation by a simple method. This step reduces the inverse problem in an infinite domain into one periodic cell. The second step is to apply Newton-CG method to solve the associated optimization problem. The perturbation is then approximated by a finite spline basis. Numerical examples are given at the end of this paper, shows the efficiency of the numerical method.