Project Title: Numerical Solution of Complex Eigenvalue Problems


In electrodynamics one describes guided waves and waves in resonators as solutions of nonlinear eigenvalue problems. As long as one has no losses, the eigenvalues are real valued. The introduction of losses is important from the technical point of view. Lossy waveguides are usually described by complex propagation constants and lossy resonators are described by complex resonance frequencies. Although this approach is usual, it causes theoretical problems because a lossy structure is no longer energetically closed. Nonetheless, one can easily design a numerical code for solving such complex eigenvalue problems. If the structure is geometrically simple, one can even find analytic solutions. Analytic solutions of complex eigenvalue problems usually lead to transcendent equations that have to be solved numerically. We have extensively studied lossy structures with rotational symmetry, both with a code based on the analytic solution and with the MMP code for computational electromagnetics. Since the results were often very surprising, the very good agreement of the two solutions (>6 digits) was important. We found different types of transitions of the corresponding modes of a loss-free structure and a transition from guided to evanescent modes. It is well known that evanescent modes (below the cutoff frequency of the corresponding guided mode) have an imaginary propagation constant and exist even in the limit frequency -> 0. However, the correspondent 'evanescent' modes on lossy structures show a cutoff frequency.

Contacts: Ch.Hafner, L. Novotny, IFH, ETZG95, ETH Zentrum, CH-8092 Zurich

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Index Terms: Numerical Methods, Eigenvalue Problems