96-347 Figotin A., Klein A.

Localization of Classical Waves II: Electromagnetic Waves (109K, LaTeX) Jul 26, 96

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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.

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