In this Chapter we begin to examine some simple discontinuity problems by using the example of the slab waveguide. The known modal structure of this waveguide and its one-dimensional cross-section allows us to concentrate on the discontinuity problem itself. From a practical viewpoint the problems dealt with in this Chapter are relevant in integrated optics and millimetrics as the slab waveguide may be considered a good approximation of the guides used in common practice.

The methodology applied to deal with discontinuity problems finds a wide application in the following Chapters of this book. In fact, in order to recover the spectra of waveguides with two-dimensional non separable cross-sections, a discontinuity problem must be solved. This holds true not only for the bound modes, but also for the continuous part of the spectrum.

Moreover, once the spectrum is found, the junction of two different waveguides of 2D cross-sections is again a discontinuity problem, which can be considered and solved with the techniques introduced in this Chapter. As an example, the microstrip step discontinuity may be considered as a junction of two microstrips with different metal widths. The scattering parameters of this discontinuity may be obtained by matching their relative spectra at the junction.

The problem of a single step discontinuity is described in Section 5.1. We start from the simplest approach, i.e. we consider the junction between two slabs of different heights as two lines with different impedances. As a refinement of this model we then introduce the variational equivalent...