Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Time-harmonic scattering of plane waves from an infinite periodically inhomogeneous medium

Guanghui Hu, Andreas Rathsfeld, Jiayi Zhang, Ruming Zhang

TL;DR

This work develops a Floquet-based framework to address time-harmonic scattering by an open, horizontally periodic grating above an infinitely bi-periodic substrate. A downward radiation condition is constructed from upward/downward Floquet modes, enabling an explicit downward Dirichlet-to-Neumann map to truncate the unbounded substrate and a bounded-cell variational problem. The authors prove the downward DtN map is well-defined and bounded, establish strong ellipticity of the resulting sesquilinear form, and prove unique solvability for all wavenumbers $k>0$ except a discrete set of exceptional values. The approach yields an explicit DtN operator without LAP, providing a robust basis for analysis and numerical schemes for open periodic waveguides with vertical periodicity in the refractive index.

Abstract

We propose a new radiation condition for an infinite inhomogeneous two-dimensional medium which is periodic in the vertical direction and remains invariant in the horizontal direction. The classical Rayleigh-expansion radiation condition does not apply to our case, because this would require the medium to be inhomogeneous in a half plane. We utilize the Floquet theory to derive upward/downward wave modes and define radiation conditions by expansions w.r.t. these modes. The downward radiation conditions leads to a downward Dirichlet-to-Neumann map which can be used to truncate the infinite inhomogeneous domain in the vertical direction. So we prove mapping properties of the upward/downward Dirichlet-to-Neumann maps based on the asymptotic behavior of high-order wave modes. Finally, we verify the strong ellipticity of the sesquilinear form corresponding to the new scattering problem and show the unique solvability for all wavenumbers with the exception of a countable set of numbers bounded below by a small positive constant.

Time-harmonic scattering of plane waves from an infinite periodically inhomogeneous medium

TL;DR

This work develops a Floquet-based framework to address time-harmonic scattering by an open, horizontally periodic grating above an infinitely bi-periodic substrate. A downward radiation condition is constructed from upward/downward Floquet modes, enabling an explicit downward Dirichlet-to-Neumann map to truncate the unbounded substrate and a bounded-cell variational problem. The authors prove the downward DtN map is well-defined and bounded, establish strong ellipticity of the resulting sesquilinear form, and prove unique solvability for all wavenumbers except a discrete set of exceptional values. The approach yields an explicit DtN operator without LAP, providing a robust basis for analysis and numerical schemes for open periodic waveguides with vertical periodicity in the refractive index.

Abstract

We propose a new radiation condition for an infinite inhomogeneous two-dimensional medium which is periodic in the vertical direction and remains invariant in the horizontal direction. The classical Rayleigh-expansion radiation condition does not apply to our case, because this would require the medium to be inhomogeneous in a half plane. We utilize the Floquet theory to derive upward/downward wave modes and define radiation conditions by expansions w.r.t. these modes. The downward radiation conditions leads to a downward Dirichlet-to-Neumann map which can be used to truncate the infinite inhomogeneous domain in the vertical direction. So we prove mapping properties of the upward/downward Dirichlet-to-Neumann maps based on the asymptotic behavior of high-order wave modes. Finally, we verify the strong ellipticity of the sesquilinear form corresponding to the new scattering problem and show the unique solvability for all wavenumbers with the exception of a countable set of numbers bounded below by a small positive constant.
Paper Structure (13 sections, 19 theorems, 164 equations, 3 figures)

This paper contains 13 sections, 19 theorems, 164 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Quasiperiodic boundary value problem in an inhomogeneous half space
  3. Downward radiation condition for $q = q\left(x_{2}\right)$ in $x_2<b$.
  4. Downward and upward radiation conditions in $\mathbb{R}^2$.
  5. Special instance: $q(x_{2}) \equiv 1$
  6. Asymptotics of $\eta_{n}$ and $\mu_{n}$ as $\vert n\vert \to +\infty$.
  7. Uniqueness and existence results
  8. Dirichlet-to-Neumann (DtN) maps
  9. Variational formulation
  10. Well-posedness results
  11. Floquet theory
  12. Proof of Lemma \ref{['lem:allEtan']} (ii)
  13. Asymptotic behavior of $\frac{v_{n,1}'(d)}{v_{n,1}(d)}$ for $k \!\to\! +0$

Key Result

lemma 1

The characteristic multipliers and exponents of (eq:hill) satisfy the relations

Figures (3)

  • Figure 1: Geometry of the diffraction problem.
  • Figure 2: Illustration of a periodic cell $C \coloneqq (0, {\tt p}) \times (b,d)$, which is covered by a homogeneous medium in $x_2>d$ and sits above an inhomogeneous half plane $x_2<b$.
  • Figure 3: An illustration of dependence of $\eta$ on $\lambda$.

Theorems & Definitions (44)

  • remark 1
  • definition 1
  • definition 2
  • lemma 1
  • lemma 2
  • Proof 1
  • remark 2
  • definition 3
  • definition 4: Radiation Conditions
  • lemma 3: Comparison Theorem
  • ...and 34 more