An open waveguide may be defined as any physical device, with longitudinal axial symmetry and unbounded cross-section, capable of guiding electromagnetic waves. In contrast to closed waveguides, the electromagnetic field in the cross-section is not confined to a limited region of space, thus extending up to infinity. Examples of closed waveguides are well-known from basic electromagnetic courses; they include coaxial cables, metallic waveguides (with rectangular or circular cross section), etc. Their spectrum is also well-known, being made by an infinite, discrete set of modes, orthogonal and complete over the waveguide cross-section. On the other hand, any waveguide not entirely enclosed in a metal shielding, may be taken as an example of an open waveguide. It is also worth noting that, by cutting a slot parallel to the longitudinal axis in the metal enclosure of a closed waveguide, again an open waveguide is obtained. An open waveguide possesses a spectrum which is no longer discrete, but is constituted by a few bound modes, when present, plus a continuum of modes.

This chapter deals with planar dielectric waveguides, also called slab waveguides, which are possibly the most simple and yet practically useful open waveguides. In this introduction we briefly recall how a bound mode can occur; we also discuss why it is natural to look for the complete spectrum of slab waveguides and what features it should possess.

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 continuum of an open waveguide describes radiation as simply, in principle, as the discrete spectrum of a classical guide describes any physical field in it. In fact, the knowledge of the modal spectra for open waveguides allows us to apply to discontinuity problems in open environments the same techniques originally developed for closed waveguides. However, this part of the spectrum, the continuum, has received little attention for most open guides. Therefore, this chapter is devoted to an introduction of the continuum. The chapter can be considered to be divided into two parts. In the first part we introduce the main characteristics of this part of the spectrum. In the second part we exemplify the procedure by deriving the continuum for some practical transmission lines such as coplanar, rib and inset waveguides.