On time-harmonic Maxwell’s equations in periodic media
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Date
2019
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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.
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Keywords
Homogenisierung, Maxwellgleichungen, Limiting absorption principle