Singular Frequencies in Wave Guides with Inhomogeneous Dielectric

It is shown that in wave guides with inhomogeneous dielectric the well-known modes do not suffice for the modal expansion at frequencies which are limits of an interval in which complex modes occur. At such a singular frequency two different modes become identical and a generalized mode exists, which is indispensable to the modal expansion. Its field increases linearly along the guide and consequently assumes very large values in a long guide. The field assumes large values also at frequencies close to a singular one. Such a field is conveniently described by a combined mode. This is the superposition of the two modes, which become identical at the singular frequency. The generalized mode is the limiting case of the combined mode. The transition from homogeneously to inhomoge-neously filled circular wave guide is treated as an example.