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Limiting absorption principle of Helmholtz equation with sign changing coefficients under periodic structure

Wenjing Zhang, Yu Chen, Yixian Gao

TL;DR

This work addresses the Helmholtz equation with sign-changing coefficients modeling interfaces between conventional and negative-index materials in 2D periodic structures. The authors combine the complementing boundary condition with the limiting absorption principle, transform the unbounded periodic problem to a bounded domain via a Dirichlet-to-Neumann map, and reduce the sign-changing system to an elliptic problem on a half-plane with transparent boundary conditions. They prove existence and uniqueness of the perturbed solution $u_\sigma$ for almost all frequencies outside a discrete set and establish convergence to a limiting solution $u_0$ as $\sigma\to0$, along with stability estimates. The results extend rigorous well-posedness theory for sign-changing Helmholtz problems to periodic metamaterial structures and provide a principled framework for analyzing wave propagation in such media.

Abstract

Negative refractive index materials have attracted significant research attention due to their unique electromagnetic response characteristics. In this paper, we employ the complementing boundary condition to establish rigorous a priori estimates for the Helmholtz equation, from which the limiting absorption principle is analytically derived. Within this mathematical framework, we conclusively establish the well-posedness of the electromagnetic transmission problem at the interface between conventional materials and negative refractive index materials in two-dimensional periodic structures.

Limiting absorption principle of Helmholtz equation with sign changing coefficients under periodic structure

TL;DR

This work addresses the Helmholtz equation with sign-changing coefficients modeling interfaces between conventional and negative-index materials in 2D periodic structures. The authors combine the complementing boundary condition with the limiting absorption principle, transform the unbounded periodic problem to a bounded domain via a Dirichlet-to-Neumann map, and reduce the sign-changing system to an elliptic problem on a half-plane with transparent boundary conditions. They prove existence and uniqueness of the perturbed solution for almost all frequencies outside a discrete set and establish convergence to a limiting solution as , along with stability estimates. The results extend rigorous well-posedness theory for sign-changing Helmholtz problems to periodic metamaterial structures and provide a principled framework for analyzing wave propagation in such media.

Abstract

Negative refractive index materials have attracted significant research attention due to their unique electromagnetic response characteristics. In this paper, we employ the complementing boundary condition to establish rigorous a priori estimates for the Helmholtz equation, from which the limiting absorption principle is analytically derived. Within this mathematical framework, we conclusively establish the well-posedness of the electromagnetic transmission problem at the interface between conventional materials and negative refractive index materials in two-dimensional periodic structures.
Paper Structure (7 sections, 6 theorems, 181 equations, 3 figures)

This paper contains 7 sections, 6 theorems, 181 equations, 3 figures.

Table of Contents

  1. introduction
  2. problem formulation
  3. the complementing boundary condition
  4. Definition of the complementing boundary condition
  5. Application of the complementing boundary condition and the $H^2$ regularity
  6. preliminary work
  7. proof of theorem \ref{['TM']}

Key Result

Theorem 2.1

Under the assumption that where $K=\dfrac{\langle\nabla\mathcal{R}(u_{\sigma}|_{S}),\nabla\mathcal{R}(u_{\sigma}|_{S})\rangle_{\Omega_2}}{\langle\varepsilon_1^{-1}\nabla u_{\sigma},\nabla u_{\sigma}\rangle_{\Omega_1}}$, the following hold:

Figures (3)

  • Figure 1: Problem geometry
  • Figure 2: Problem geometry with TBC
  • Figure 3:

Theorems & Definitions (13)

  • Theorem 2.1
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • Lemma 4.3
  • ...and 3 more