Abstract :
The well known definitions of guided and resonant modes (eigenfunctions) of a loss-free system are briefly examined and extensions to lossy systems are discussed. It is argued that the main advantage of the use of proper eigenfunctions lies in the fact that an arbitrary field in a homogeneous and closed waveguide system can be resolved into a spectrum of a complete set of orthonormal proper modes. This is a mathematical concept which is extremely useful in a wide variety of theoretical problems, because various well established and powerful analytical techniques can be used to advantage. It is shown that practical waveguide systems are neither closed nor homogeneous; the fields, therefore, extend to infinity and obey statistical laws. Consequently, in practical problems, serious mathematical difficulties are encountered, which, to some extent, can be circumvented by the introduction of continuous spectra that can frequently be identified with contributions to the radiation field or the scattered field. Furthermore, it is shown that, in all properly formulated physical problems, the discrete spectrum of proper eigenfunctions is absent and the complete solution is given by a continuous spectrum, which is usually interpreted as a radiation field. Yet, physical systems exhibiting distinct resonant or guiding characteristics are common, and it is concluded that such characteristics must be hidden in the continuous spectrum of proper eigenfunctions. It is shown that, in general, it is possible to transform the continuous spectrum into a spectrum of quasimodes (which are suitably chosen portions of improper eigenfunctions) and thereby obtain rapidly convergent field presentations which exhibit the physically observable resonance and guiding characteristics. The method is illustrated by application to a few examples.
