# Effective Maxwell's equations in general periodic microstructures

## TL;DR

This paper derives effective Maxwell's equations for periodic meta-materials with perfect conductors and voids, revealing how topology influences wave propagation in the homogenized limit.

## Contribution

It introduces a new effective system for Maxwell's equations in periodic meta-structures and links topological properties to wave propagation capabilities.

## Key findings

- Certain field components vanish in the homogenized limit due to topology.
- The effective system predicts wave propagation based on topological features.
- The analysis applies to geometries with non-trivial topology.

## Abstract

We study the time harmonic Maxwell equations in a meta-material consisting of perfect conductors and void space. The meta-material is assumed to be periodic with period $\eta > 0$; we study the behaviour of solutions $(E^{\eta}, H^{\eta})$ in the limit $\eta \to 0$ and derive an effective system. In geometries with a non-trivial topology, the limit system implies that certain components of the effective fields vanish. We identify the corresponding effective system and can predict, from topological properties of the meta-material, whether or not it permits the propagation of waves.