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Limiting absorption principle for time-harmonic acoustic and electromagnetic scattering of plane waves from a bi-periodic inhomogeneous layer

Guanghui Hu, Andreas Kirsch, Yulong Zhong

Abstract

The Rayleigh expansion is widely used as a formal radiation condition in the analysis and numerical treatment of grating diffraction problems for incoming plane waves. However, the Rayleigh expansion does not always lead to uniqueness of open waveguide scattering problems, due to the existence of surface/guided waves (in other words, Bound States in the Continuum (BICs)) which exponentially decay in the direction perpendicular to the periodicity. In this paper we suppose that a bi-periodic inhomogeneous medium supports BICs at some real-valued incident wavenumber. Based on singular perturbation arguments, we justify the Limiting Absorption Principle (LAP) for both time-harmonic acoustic and electromagnetic scattering of plane waves from bi-periodic structures. Replacing the wavenumber $k$ with $k+iε$, we prove that the unique solution with $ε>0$ converges to a solution of the original diffraction problem that additionally satisfies an orthogonal identity. This constraint condition together with the classical Rayleigh expansion leads to a sharp radiation condition to ensure uniqueness of time-harmonic scattering of plane waves by BIC-supporting bi-periodic materials.

Limiting absorption principle for time-harmonic acoustic and electromagnetic scattering of plane waves from a bi-periodic inhomogeneous layer

Abstract

The Rayleigh expansion is widely used as a formal radiation condition in the analysis and numerical treatment of grating diffraction problems for incoming plane waves. However, the Rayleigh expansion does not always lead to uniqueness of open waveguide scattering problems, due to the existence of surface/guided waves (in other words, Bound States in the Continuum (BICs)) which exponentially decay in the direction perpendicular to the periodicity. In this paper we suppose that a bi-periodic inhomogeneous medium supports BICs at some real-valued incident wavenumber. Based on singular perturbation arguments, we justify the Limiting Absorption Principle (LAP) for both time-harmonic acoustic and electromagnetic scattering of plane waves from bi-periodic structures. Replacing the wavenumber with , we prove that the unique solution with converges to a solution of the original diffraction problem that additionally satisfies an orthogonal identity. This constraint condition together with the classical Rayleigh expansion leads to a sharp radiation condition to ensure uniqueness of time-harmonic scattering of plane waves by BIC-supporting bi-periodic materials.
Paper Structure (13 sections, 14 theorems, 149 equations, 3 figures)

This paper contains 13 sections, 14 theorems, 149 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Well-posedness for acoustic scattering problem
  3. Problem formulation
  4. Variational formulation
  5. The limiting absorption argument
  6. Uniqueness and existence results
  7. Well posedness for an electromagnetic scattering problem
  8. Problem formulation
  9. Transform to space of periodic functions
  10. The limiting absorption argument
  11. Appendix
  12. A singular perturbation result
  13. Compact embedding theorem of $X_{\alpha,0}$

Key Result

Lemma 2.2

If $\alpha=k\tilde{\theta}$, then $\inf\{\operatorname{Im}\beta_n(k+i\varepsilon):n\in{\mathbb Z}^2\}>0$ for all $k,\varepsilon>0$.

Figures (3)

  • Figure 1: Diffraction of an acoustic plane wave from a penetrable bi-periodic layer in ${\mathbb R}^3$.
  • Figure 2: The sets of cut-off vectors (blue) and propagative wave vectors (red) for Example \ref{['ex:1']}
  • Figure 3: Diffraction of an electromagnetic plane wave from a bi-periodic inhomogeneous layer in $x_3>0$.

Theorems & Definitions (31)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Example 2.4
  • Lemma 2.6
  • Remark 2.7
  • Lemma 2.8
  • proof
  • Remark 2.9
  • ...and 21 more