Inhomogeneous Dielectric Slabs with Analytic Frequency-dependent Permittivities

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Abstract
We consider a general class of inhomogeneous dielectric slabs, where the dielectric permittivity (l) is assumed to depend on the frequency (y) at each position, through the Kramers-Krönig dispersion relation. The physical basis for this assumption is discussed and the following general consequences of it are established. First, using integral relations for propagation modes, we prove the absence of the latter for physically unwanted values of y and of the propagation constant. Secondly, we treat the transmission and reflection of electromagnetic waves by the slab through rigorous eikonal methods, and prove that improvement of convergence does occur at high y. Thirdly, we derive analytic representations for the phases of the transmission and reflection amplitudes in terms of their moduli and their complex zeros in the y -plane, at normal incidence. Finally, we present a formal construction of l at high y, in terms of the reflection amplitude at normal incidence, thus giving a partial mathematical solution for the inverse scattering problem. Most of the previous results break down if l is y -independent. We also study the dispersion relation for the propagation mode, when the slab is a monomode optical waveguide.