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

Lide Cai, Junqing Chen, Yanpeng Gao

TL;DR

This work addresses the computation of propagating (guided) wave numbers in open periodic structures by recasting the unbounded quasi-periodic scattering problem into a bounded domain via a Dirichlet-to-Neumann operator and then into a quadratic eigenvalue problem using a perfectly matched layer (PML) truncation. The authors prove exponential convergence of the PML-augmented DtN operator to the original problem and apply Gohberg–Sigal–Ammari theory to quantify eigenvalue perturbations under truncation. They develop a finite element discretization framework, formulate a block-matrix representation, and obtain provable discretization-error bounds, corroborated by numerical experiments that show second-order convergence and rapid exponential decay of guided modes outside the periodic medium. The approach provides a robust, accurate route to compute propagative wave numbers in open periodic waveguides with rigorous perturbation control and practical finite-element implementation.

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.

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

TL;DR

This work addresses the computation of propagating (guided) wave numbers in open periodic structures by recasting the unbounded quasi-periodic scattering problem into a bounded domain via a Dirichlet-to-Neumann operator and then into a quadratic eigenvalue problem using a perfectly matched layer (PML) truncation. The authors prove exponential convergence of the PML-augmented DtN operator to the original problem and apply Gohberg–Sigal–Ammari theory to quantify eigenvalue perturbations under truncation. They develop a finite element discretization framework, formulate a block-matrix representation, and obtain provable discretization-error bounds, corroborated by numerical experiments that show second-order convergence and rapid exponential decay of guided modes outside the periodic medium. The approach provides a robust, accurate route to compute propagative wave numbers in open periodic waveguides with rigorous perturbation control and practical finite-element implementation.

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.

Paper Structure

This paper contains 14 sections, 18 theorems, 158 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem formulation on a bounded domain
  3. Variational problem with DtN operator
  4. Properties of the propagative wave numbers
  5. Truncation by perfectly matched layer
  6. The PML formulation
  7. Analysis of the PML DtN operator
  8. Multiplicity of the propagative wave numbers
  9. Analysis of the eigenvalue perturbation
  10. Discretization of the PML problem
  11. Block matrix representation of the PML problem
  12. Finite element discretization
  13. Discretization error
  14. Numerical experiments

Key Result

Theorem 1

For $k > 0$ and $\alpha \notin \bigl\{z \in \mathbb{C} \mid \Re(k^{2} - (z + \frac{2\pi n}{\Lambda})^{2}) = 0,\, \Im(k^{2} - (z + \frac{2\pi n}{\Lambda})^{2}) < 0 \text{ for some } n \in \mathbb{Z}\bigr\}$, there is a compact operator $K(\alpha)$ in $\mathcal{L}(H^1_{per}(\Omega_{H}))$, such that fo Here $I$ denotes the identity operator.

Figures (5)

  • Figure 1: The description of the periodic structures and the guided mode.
  • Figure 2: Computational domain in Experiment 1.
  • Figure 3: Computational domain in Experiment 2.
  • Figure 4: Numerical guided modes calculated in Experiment 2. Left: $\alpha_3=0.4637$; Right: $\alpha_4=0.2728$.
  • Figure 5: $\Vert u^{(3)}_{prop}(\cdot ,x_2) \Vert_{L^{\infty}}$(Left) and $\Vert u^{(4)}_{prop}(\cdot ,x_2) \Vert_{L^{\infty}}$(Right) calculated in Experiment 2.

Theorems & Definitions (36)

  • Definition 1
  • Theorem 1
  • proof
  • Lemma 1
  • Corollary 1
  • Theorem 2
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • ...and 26 more