Figotin A., Klein A.
Localization of Classical Waves II: Electromagnetic Waves
(109K, LaTeX)
ABSTRACT. We consider electromagnetic waves in a medium described by a position
dependent dielectric
constant $\varepsilon (x)$. We assume that $\varepsilon (x)$ is a random
perturbation
of a periodic function $\varepsilon_{0}(x)$ and that the periodic
Maxwell operator ${\bf M}_{0} =\nabla^{\times} \frac 1{\varepsilon_0
(x)}\nabla^{\times} $
has a gap in the spectrum, were $\nabla^{\times} \Psi =\nabla {\times}
\Psi $.
We prove the existence of localized waves, i.e., finite energy solutions
of Maxwell's equations with the property that
almost all of the wave's energy remains in a fixed bounded region of
space at
all times.
Localization of electromagnetic waves is a consequence of Anderson
localization
for the self-adjoint operators
${\bf M} =\nabla^{\times} \frac 1{\varepsilon(x)}\nabla^{\times}$.
We prove that, in the random medium
described by $\varepsilon(x)$, the random operator ${\bf M}$ exhibits
Anderson
localization
inside the gap in the spectrum of ${\bf M}_{0}$ . This is shown even in
situations when the gap is totally
filled by the spectrum of the random operator; we can prescribe random
environments that ensure
localization in almost the whole gap.