Investigation of mode interaction on planar dielectric waveguides with loss and gain

On lossless isotropic planar waveguides the discrete proper modes of propag ation form independent transverse electric and transverse magnetic sets suc h that there is no mode coupling or interaction between modes. In the event of material loss or gain, mode interactions are possible, leading to a com plicated spectrum and apparent nonuniqueness of the modes. In this paper we analyze for the first time the cause of these modal interactions by studyi ng the simplest canonical planar waveguide which exhibits these effects, th e symmetric-slab waveguide. We show that mode interactions are due to the m igration of complex-frequency-plane branch points associated with specific wave phenomena, with varying loss or gain. As these singularities move near the real-frequency axis they influence the modal behavior for time-harmoni c (real-valued) frequencies, crossing the real axis at some critical value of loss or gain. It is shown that as time-harmonic frequency varies, passin g above, below, or through these branch points results in different modal b ehavior. Passing above or below, and near to, the branch point yields mode coupling behavior, while passing through the branch point results in modal degeneracy. The result of this branch point migration is that the associati on of a particular mode with a certain branch of the dispersion function de pends not only on the value of material loss or gain, but also on the order in which physical parameters of the problem are varied. Three different br anch point types are identified and discussed, which leads to an understand ing of the relevant wave phenomena and to a method for organizing the mode spectrum in a consistent and unique manner. While many of the observations described here are based on careful numerical analysis of the transverse ma gnetic modes existing on a certain symmetric-slab waveguide, the described phenomena are reasonably expected to be generally found in other open diele ctric waveguiding structures.