Relation between Complex Propagation Constant and Complex Eigenmodes in Lossy Traveling-Wave Structures

A simple procedure to extract the attenuation constant of a lossy traveling-wave structure using eigenmode simulations is presented. For lossy structures, the eigenmode analysis outputs complex eigenvalues which mathematically results in complex frequencies, Ω. Using the proposed relations, these complex frequencies are mapped to the complex propagation constant γ and the corresponding attenuation constant α through the group velocity of the mode. Taking a standard rectangular waveguide as an example, complex eigenmodes are extracted through full-wave simulations, which are then used to correctly predict the the per-unit length attenuation constant under low- loss conditions.