Numerical methods for scattering problems from multi-layers with different periodicities

TL;DR

This paper develops a finite element method using the Bloch transform to numerically solve scattering problems involving multi-layered structures with different periodicities, demonstrating convergence through numerical experiments.

Contribution

It introduces a novel approach combining the Bloch transform with finite element methods for multi-periodic layers, handling different periodicities effectively.

Findings

The method converges as shown by numerical experiments.

Approximation of refractive index improves solution accuracy.

The approach effectively reduces complex coupled systems.

Abstract

In this paper, we consider a numerical method to solve scattering problems with multi-periodic layers with different periodicities. The main tool applied in this paper is the Bloch transform. With this method, the problem is written into an equivalent coupled family of quasi-periodic problems. As the Bloch transform is only defined for one fixed period, the inhomogeneous layer with another period is simply treated as a non-periodic one. First, we approximate the refractive index by a periodic one where its period is an integer multiple of the fixed period, and it is decomposed by finite number of quasi-periodic functions. Then the coupled system is reduced into a simplified formulation. A convergent finite element method is proposed for the numerical solution, and the numerical method has been applied to several numerical experiments. At the end of this paper, relative errors of the numerical solutions will be shown to illustrate the convergence of the numerical algorithm.