On time-harmonic Maxwell’s equations in periodic media

Abstract

In this thesis we study the propagation of time-harmonic electromagnetic waves through periodic media for two different regimes. In the first part, we consider a periodic medium in a bounded domain with a period that is much smaller than the wavelength of the electromagnetic wave hitting the medium. The medium is a periodic assembly of conducting microstructures and void space. In order to describe the effective behaviour of the field propagating through this medium, we homogenise the time-harmonic Maxwell equations. There is a vast literature on homogenising Maxwell's equations under rather restrictive assumptions on the (conducting) microstructure. Using a new averaging method---the so-called geometric average---allows us to consider a large class of microstructures that have not been treated before in the literature. We derive the effective Maxwell equations for two cases: perfectly conducting microstructures and highly conductive microstructures.

The second part of this thesis is concerned with the propagation of electromagnetic waves in a closed, unbounded and periodic waveguide. In this part we assume that the wavelength of the fields and the period of the medium are of the same order. Imposing suitable assumptions on the geometry and the fields, Maxwell's equations reduce to a scalar Helmholtz equation. We truncate the waveguide to obtain a bounded domain and replace the radiation condition at infinity appropriately. In order to establish the existence of a solution to the Helmholtz equation in this bounded periodic waveguide, we derive a limiting absorption principle for sesquilinear forms. Using this principle, we show that up to an at most countable set of singular frequencies there exists a unique solution to the Helmholtz equation in the bounded waveguide that satisfies the replacement of the radiation condition at infinity.