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.