Asymptotic behaviour of the spectra of systems of Maxwell equations in periodic composite media with high contrast

TL;DR

This paper investigates the asymptotic spectral behavior of Maxwell equations in high-contrast periodic media as the period tends to zero, revealing new phenomena beyond scalar wave theory.

Contribution

It introduces a detailed analysis of the spectral limits of Maxwell systems with increasing contrast, highlighting phenomena absent in scalar wave models.

Findings

Spectral limits include the spectrum of a homogenized Maxwell system.

Demonstrates phenomena not present in scalar wave theory.

Shows the impact of high contrast on spectral behavior.

Abstract

We analyse the behaviour of the spectrum of the system of Maxwell equations of electromagnetism, with rapidly oscillating periodic coefficients, subject to periodic boundary conditions on a "macroscopic" domain $(0,T)^d, T>0.$ We consider the case when the contrast between the values of the coefficients in different parts of their periodicity cell increases as the period of oscillations $\eta$ goes to zero. We show that the limit of the spectrum as $\eta\to0$ contains the spectrum of a "homogenised" system of equations that is solved by the limits of sequences of eigenfunctions of the original problem. We investigate the behaviour of this system and demonstrate phenomena not present in the scalar theory for polarised waves.