  We investigate TE-wave propagation in a hollow waveguide with a graded
dielectric layer, described using a hyperbolic tangent function. General
formulae for the electric field components of the TE-waves, applicable to
hollow waveguides with arbitrary cross sectional shapes, are presented. We
illustrate the exact analytical results for the electric field components in
the special case of a rectangular waveguide. Furthermore, we derive exact
analytical results for the reflection and transmission coefficients valid for
waveguides of arbitrary cross sectional shapes. Finally, we show that the
obtained reflection and transmission coefficients are in exact asymptotic
agreement with those obtained for a very thin homogeneous dielectric layer
using mode-matching and cascading. The proposed method is tractable since it
gives analytical results that are directly applicable without the need of
mode-matching, and it has the ability to model realistic, smooth transitions.
