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

TL;DR

This paper analyzes the effective electromagnetic behavior of complex geometries with thin, periodically distributed inclusions, revealing conditions that lead to perfect transmission, reflection, or polarization in the limit of vanishing inclusion size.

Contribution

It introduces a homogenization framework for Maxwell's equations with thin inclusions, deriving effective interface conditions that depend on geometric properties.

Findings

Effective equations depend on inclusion geometry

Conditions for perfect transmission, reflection, polarization identified

Homogenization captures multiple scales including thin wires

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 $\eta>0$ and the typical scale of the inclusion is $\eta$, but we allow also the presence of even smaller scales, e.g. when thin wires are analyzed. We are interested in the limit $\eta\to 0$ and in effective equations. Depending on geometric properties of the inclusions, the effective system can imply perfect transmission, perfect reflection or polarization.