Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Interface conditions for Maxwell's equations by homogenization of thin inclusions: transmission, reflection or polarization

Ben Schweizer, David Wiedemann

TL;DR

The paper analyzes the homogenization of time-harmonic Maxwell equations in a domain with thin, periodically distributed perfectly conducting inclusions along a surface, deriving effective interface conditions on the macroscopic interface $\Gamma$ that depend on the micro-structure's asymptotic connectivity. A cell-function framework distinguishes asymptotic dis-connectedness (E-type) and connectivity (H-type), enabling precise limit passage and three qualitative regimes: perfect transmission (inactive interface), perfect reflection (reflecting), and polarization (directionally selective). The authors develop explicit constructions for wire-like geometries, including a rigorous treatment of 2D-to-3D cell functions and a critical radius scaling $r_{\eta}$ that governs connectivity, revealing how topology and geometry drive macroscopic EM response. These results provide a rigorous, topology-aware mechanism for designing metasurfaces and Faraday-cage-like structures with targeted transmission, reflection, or polarization effects.

Abstract

We consider the time-harmonic Maxwell equations in a complex geometry. We are interested in geometries that model polarization filters or Faraday cages. We study the situation that the underlying domain contains perfectly conducting inclusions, the inclusions are distributed in a periodic fashion along a surface. The periodicity is $η>0$ and the typical scale of the inclusion is $η$, but we allow also the presence of even smaller scales, e.g. when thin wires are analyzed. We are interested in the limit $η\to 0$ and in effective equations. Depending on geometric properties of the inclusions, the effective system can imply perfect transmission, perfect reflection or polarization.

Interface conditions for Maxwell's equations by homogenization of thin inclusions: transmission, reflection or polarization

TL;DR

The paper analyzes the homogenization of time-harmonic Maxwell equations in a domain with thin, periodically distributed perfectly conducting inclusions along a surface, deriving effective interface conditions on the macroscopic interface that depend on the micro-structure's asymptotic connectivity. A cell-function framework distinguishes asymptotic dis-connectedness (E-type) and connectivity (H-type), enabling precise limit passage and three qualitative regimes: perfect transmission (inactive interface), perfect reflection (reflecting), and polarization (directionally selective). The authors develop explicit constructions for wire-like geometries, including a rigorous treatment of 2D-to-3D cell functions and a critical radius scaling that governs connectivity, revealing how topology and geometry drive macroscopic EM response. These results provide a rigorous, topology-aware mechanism for designing metasurfaces and Faraday-cage-like structures with targeted transmission, reflection, or polarization effects.

Abstract

We consider the time-harmonic Maxwell equations in a complex geometry. We are interested in geometries that model polarization filters or Faraday cages. We study the situation that the underlying domain contains perfectly conducting inclusions, the inclusions are distributed in a periodic fashion along a surface. The periodicity is and the typical scale of the inclusion is , but we allow also the presence of even smaller scales, e.g. when thin wires are analyzed. We are interested in the limit and in effective equations. Depending on geometric properties of the inclusions, the effective system can imply perfect transmission, perfect reflection or polarization.

Paper Structure

This paper contains 25 sections, 16 theorems, 79 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Asymptotic connectedness and disconnectedness of conductors
  3. Interface conditions of the limit system
  4. Outlook regarding highly conducting obstacles
  5. Literature
  6. Content layout
  7. Main results
  8. Geometry and notations
  9. Weak form of the eta-problem
  10. Weak form of the limit systems
  11. Homogenization result
  12. Cell functions and asymptotic (dis-)connectedness
  13. E-type cell functions and asymptotic disconnectedness
  14. H-type cell functions and asymptotic connectedness
  15. Rescaling of the cell functions
  16. ...and 10 more sections

Key Result

Theorem 2.1

Let the setting be that of Section sec:Geometry: The macroscopic sets are $\Omega$ and $\Gamma$, the sets for the $\eta$-problem are given by $\Sigma^\eta_Y$, $\Omega_\eta$ and $\Sigma_\eta$ for a sequence $\eta \to 0$. Let $(E^\eta, H^\eta) \in L^2(\Omega_\eta, \mathbb{C}^3) \times L^2(\Omega_\eta, Then, the equations for $E^\text{hom}$ and $H^\text{hom}$ are given as follows, depending on the co

Figures (3)

  • Figure 1: Geometry of the conductor
  • Figure 2: Visualization of the cell functions in $Y$ for $i=1$, $j=2$ and $\Sigma^\eta_Y = T^{(1)}_{0.15}\,$. The illustration in (B) does not show the cell function $\Psi_\eta^{(1)}$, but sketches $2\Psi_\eta^{(1)} - e_2$.
  • Figure 3: Geometries for the construction of $\phi_\eta$. The vector $(1,0) \in \mathbb{R}^2$ in the $2$D picture corresponds with $e_2 = (0,1,0)$ in the $3$D picture and $(0,1)\in \mathbb{R}^2$ with $e_3 = (0,0,1)\in \mathbb{R}^3$. When performing calculations in the $2$D context, we also write $e_1$ for $(1,0) \in \mathbb{R}^2$ and $e_2$ for $(0,1)\in \mathbb{R}^2$.

Theorems & Definitions (42)

  • Theorem 2.1: Main result
  • Lemma 2.2: Trivial part of the limit system
  • proof
  • Remark 2.3: Compactly contained obstacles are disconnected
  • Theorem 2.4: Main result for wires
  • proof
  • Remark 2.5: On the weak solution concept
  • Remark 2.6: $L^2$-boundedness of the solution sequence
  • Remark 2.7: Density, symmetry properties of the limit system
  • Definition 3.1: Asymptotically disconnected obstacles
  • ...and 32 more