Closed waveguiding structure is reported. In this chapter a few examples of waveguiding structures are considered which are characterised by perfectly reflecting walls or boundaries. Such waveguides form a class of closed waveguides since all fields are confined inside the boundaries which are taken as either perfectly electric or perfectly magnetic walls.

Open waveguides are characterised by imperfectly reflecting, or open, boundaries. Thus, beside power guided along the waveguide, a certain portion can escape away from the guiding structure. The natural modes of propagation on an open waveguide fall into two main categories: guided modes whose power flow is confined to the near vicinity of the guide and radiation modes which account for radiated power. Examples of open waveguides include the single constant impedance surface, the dielectric planar slab and the dielectric rod. A vast variety of planar waveguides particularly used in millimetric wave circuits are identified as open waveguides. Besides, most natural waveguides such as atmospheric ducts, the earth ionosphere waveguide and tunnels within the earth's medium can be considered as open waveguides. The simplest open waveguide structure to start with is the single planar impedance surface, considered earlier in section 3.3. Besides the surface wave mode studied earlier, we shall derive radiation modes which are often called pseudomodes. Pseudomodes form a continuous spectrum over the transverse wavenumber and account for radiation as will be seen below. Together, the surface wave mode and the continuous spectrum of pseudomodes constitute the complete spectrum of modes in terms of which the fields of an arbitrary source can be expanded. In general, there is more than one surface wave mode on an open waveguide and they form a discrete spectrum over the transverse wavenumber. In section 4.2 we derive the complete spectrum of modes for each of the single planar impedance surface, the planar dielectric slab and the dielectric rod. Orthogonality of modes in a given waveguide and mode coupling at a junction between two different waveguides are treated in section 4.3. In the rest of the chapter we present an analytical account of guided modes in a few examples of millimetric waveguides.

In this chapter we study the electromagnetic wave propagation in free tunnels and in tunnels with axial conductors. Thus the attenuation of free modes in tunnels of circular and rectangular shapes is derived and plotted versus frequency in section 6.2. The effect of curvature of the tunnel walls is then investigated and it is shown that curvature can cause a considerable increase of the mode attenuation. In section 6.3 the theory of modal propagation in a tunnel containing an axial conductor is presented and in section 6.4 the character of the monofilar and bifilar modes is investigated. Mode conversion in tunnels is discussed in section 6.5.

In this appendix, we review some of the basic properties of Bessel functions, Mathieu functions and Airy functions. These functions occur as wave functions in circular and elliptical cylindrical coordinates.