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A finite element contour integral method for computing the resonances of metallic grating structures with subwavelength holes

Yingxia Xi, Junshan Lin, Jiguang Sun

Abstract

We consider the numerical computation of resonances for metallic grating structures with dispersive media and small slit holes. The underlying eigenvalue problem is nonlinear and the mathematical model is multiscale due to the existence of several length scales in problem geometry and material contrast. We discretize the partial differential equation model over the truncated domain using the finite element method and develop a multi-step contour integral eigensolver to compute the resonances. The eigensolver first locates eigenvalues using a spectral indicator and then computes eigenvalues by a subspace projection scheme. The proposed numerical method is robust and scalable, and does not require initial guess as the iteration methods. Numerical examples are presented to demonstrate its effectiveness.

A finite element contour integral method for computing the resonances of metallic grating structures with subwavelength holes

Abstract

We consider the numerical computation of resonances for metallic grating structures with dispersive media and small slit holes. The underlying eigenvalue problem is nonlinear and the mathematical model is multiscale due to the existence of several length scales in problem geometry and material contrast. We discretize the partial differential equation model over the truncated domain using the finite element method and develop a multi-step contour integral eigensolver to compute the resonances. The eigensolver first locates eigenvalues using a spectral indicator and then computes eigenvalues by a subspace projection scheme. The proposed numerical method is robust and scalable, and does not require initial guess as the iteration methods. Numerical examples are presented to demonstrate its effectiveness.
Paper Structure (10 sections, 1 theorem, 49 equations, 9 figures, 1 table)

This paper contains 10 sections, 1 theorem, 49 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. The eigenvalue problem
  3. Finite element discretization
  4. Multi-step eigensolver based on the contour integral
  5. Numerical examples
  6. Perfectly conducting metal
  7. Sheetmetal grating with rectangular slits
  8. Metallic grating with Drude-Sommerfeld model
  9. Metallic grating with trapezoidal slit holes
  10. Discussions

Key Result

Theorem 4.2

(Theorem 3.1, Beyn2012) Let $\tilde{V}\in\mathcal{C}^{(N+J),L_1}(M\leq L_1\leq N+J)$ be chosen randomly. In a generic sense, the volumn vectors of $\tilde{V}$ are linearly independent. Then, it holds that $\rm{rank}(W^H\tilde{V})=M$ and $\rm{rank}(V)=M$. Assume that and the singular value decomposition where $V_0\in \mathcal{C}^{(N+J)\times M}$, $\Sigma_0=diag(\sigma_1,\sigma_2,\cdots,\sigma_M)$

Figures (9)

  • Figure 1: Geometry of the periodic metallic structure. The slits $S$ are arranged periodically with the size of the period $d$. The domains above and below the metallic slab are denoted as $\Omega^{+}$ and $\Omega^{-}$ respectively, and the domain of the metal is denoted as $\Omega$.
  • Figure 2: Initial mesh $\mathcal{T}_{h_0}$ in the computational domain. The mesh is uniformly refined to obtain a series of mesh $\{\mathcal{T}_{h_j} \}_{j=0}^5$ for discretizing \ref{['eq:weak_main']}.
  • Figure 3: Indicators of sub-regions in Step (1) of Algorithm 1.
  • Figure 4: Real parts of eigenfunctions when the Bloch wavenumber $\kappa=\frac{\pi}{4d}$ for the sheetmetal grating with rectangular slits.
  • Figure 5: The band structure for the sheetmetal grating, where the Drude model \ref{['FreeElectron']} is used for the permittivity of the metal.
  • ...and 4 more figures

Theorems & Definitions (4)

  • Remark 2.1
  • Remark 4.1
  • Theorem 4.2
  • Remark 4.3