The PML method for calculating the propagative wave numbers of electromagnetic wave in periodic structures

TL;DR

This paper introduces the PML method for calculating the propagative wave numbers of electromagnetic waves in periodic structures, addressing the challenge of guided mode computation in unbounded domains.

Contribution

It formulates the guided mode calculation as a nonlinear eigenvalue problem, applies PML for domain truncation, and converts it into a quadratic eigenvalue problem solvable by finite element methods.

Findings

Proves the approximation property of PML truncation.

Demonstrates the effectiveness of the quadratic eigenvalue formulation.

Provides numerical examples validating theoretical results.

Abstract

When the electromagnetic wave is incident on the periodic structures, in addition to the scattering field, some guided modes that are traveling in the periodic medium could be generated. In the present paper, we study the calculation of guided modes. We formulate the problem as a nonlinear eigenvalue problem in an unbounded periodic domain. Then we use perfectly matched layers to truncate the unbounded domain, recast the problem to a quadratic eigenvalue problem, and prove the approximation property of the truncation. Finally, we formulate the quadratic eigenvalue problem to a general eigenvalue problem, use the finite element method to discrete the truncation problem, and show numerical examples to verify theoretical results.