Complete Normalized Orthogonal Set Including Evanescent Modes for Circular Dielectric Waveguides

We have established a complete set of normalized orthogonal modes for circular dielectric waveguides, in which all the modes are explicitly included, namely the propagation modes, the radiation modes, and the evanescent modes. In the literature, the orthogonality of the modes is commonly defined by the energy propagation, which brings out a problem when dealing with evanescent modes. In order to accommodate the evanescent modes, a different definition for orthogonality is needed. We have solved the problem by defining a pseudo-transversal admittance, which allows the hybrid modes to be treated equivalently to the transversal modes. The use of this new definition requires a modification of the usual method for calculating the modes. This new method is described and then used to establish an actually complete orthogonal set. Such a complete set has been shown useful for applying the direct moment method to analyze the probes in near-field scanning optical microscopy.