Inhomogeneous Slab and Cylindrical Dielectric Waveguides with Discontinuous Permittivities: Propagation Modes

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We study monomode slab and cylindrical waveguides, in which the dielectric permittivity l has both a graded spatial variation and finite discontinuities at the boundaries. New integral equations for the H-type propagation mode in the slab and for the one in the cylinder are obtained, which display separate contributions due to both the graded variations and the discontinuities, and in which the boundary value problem arising from the latter is solved. Based on the specific long wavelength singularities of the free-space Green's functions in one and two dimensions, exact dispersion relations are presented for the slab and cylinder propagation modes, respectively. We give approximate explicit solutions of those dispersion relations, which are correct up to and including second order (in some parameter measuring the deviation of l from unity), and present applications for parabolic permittivity profiles in both geometries. The discontinuities of l at the boundary give rise to contributions of second and higher orders in the dispersion relations, but never at first order. Electromagnetic scattering by the cylinder in general and, in particular, its long wavelength behaviour, are studied briefly.
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