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Time-harmonic scattering by locally perturbed periodic structures with Dirichlet and Neumann boundary conditions

Guanghui Hu, Andreas Kirsch

Abstract

The paper is concerned with well-posedness of TE and TM polarizations of time-harmonic electromagnetic scattering by perfectly conducting periodic surfaces and periodically arrayed obstacles with local perturbations. The classical Rayleigh Expansion radiation condition does not always lead to well-posedness of the Helmholtz equation even in unperturbed periodic structures. We propose two equivalent radiation conditions to characterize the radiating behavior of time-harmonic wave fields incited by a source term in an open waveguide under impenetrable boundary conditions. With these open waveguide radiation conditions, uniqueness and existence of time-harmonic scattering by incoming point source waves, plane waves and surface waves from locally perturbed periodic structures are established under either the Dirichlet or Neumann boundary condition. A DtN operator without using the Green's function is constructed for proving well-posedness of perturbed scattering problems.

Time-harmonic scattering by locally perturbed periodic structures with Dirichlet and Neumann boundary conditions

Abstract

The paper is concerned with well-posedness of TE and TM polarizations of time-harmonic electromagnetic scattering by perfectly conducting periodic surfaces and periodically arrayed obstacles with local perturbations. The classical Rayleigh Expansion radiation condition does not always lead to well-posedness of the Helmholtz equation even in unperturbed periodic structures. We propose two equivalent radiation conditions to characterize the radiating behavior of time-harmonic wave fields incited by a source term in an open waveguide under impenetrable boundary conditions. With these open waveguide radiation conditions, uniqueness and existence of time-harmonic scattering by incoming point source waves, plane waves and surface waves from locally perturbed periodic structures are established under either the Dirichlet or Neumann boundary condition. A DtN operator without using the Green's function is constructed for proving well-posedness of perturbed scattering problems.
Paper Structure (11 sections, 16 theorems, 113 equations, 3 figures)

This paper contains 11 sections, 16 theorems, 113 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Scattering by Dirichlet periodic curves with local perturbations: radiation condition and uniqueness
  3. Notations
  4. The Open Waveguide Radiation Condition And An Energy Formula
  5. A Modified Open Waveguide Radiation Condition
  6. Uniqueness Of Solutions Of The Perturbed And Unperturbed Problems
  7. Construction of the Dirichlet-to-Neumann (DtN) operator
  8. Existence Results For Some Unperturbed Problems
  9. The DtN Operator
  10. Existence of Solutions of the Perturbed Problem
  11. Scattering by Neumann curves and by periodically arrayed obstacles

Key Result

Lemma 2.6

Let $u=\sum_{j\in J}\sum_{\ell=1}^{m_j} a_{\ell,j} \hat{\phi}_{\ell,j}$ for some $a_{\ell,j}\in \mathbb{C}$ and write $q+Q_\infty:=\{x\in D:q<x_1<q+2\pi\}$ for $q\in {\mathbb R}$. Then we have

Figures (3)

  • Figure 1: Illustration of wave scattering from (a) a perfectly reflecting periodic curve and (b) perfectly conducting obstacles. Guided waves might exist in (a)-(b), leading to difficulties in establishing well-posedness of the scattering problem with the classical Rayleigh Expansion radiation condition \ref{['exc:b']}.
  • Figure 2: Illustration of wave scattering from perfectly reflecting periodic curves with a local perturbation.
  • Figure 3: Illustration of the artificial boundary $C:=\partial K\subset D$ (in this case a circle) on which the DtN operator $\Lambda$ (see Definition \ref{['dtn0']}) is defined for scattering by periodically arrayed obstacles with a local defect.

Theorems & Definitions (29)

  • Definition 2.1
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Definition 2.8
  • Remark 2.9
  • Definition 2.10
  • Theorem 2.11
  • Definition 2.12
  • ...and 19 more