Complex Eigenmodes and Eigenfrequencies in Electromagnetics

A first comprehensive treatment on complex eigenmodes is presented for general lossy traveling-wave electromagnetic structures where the per unit length propagation phase shift ($β$) dependent complex eigenfrequencies $Ω(β)$ are mapped to the frequency dependent complex propagation constant $γ(ω_0)$ for variety of electromagnetic structures. Rigorous procedures are presented to first compute the complex eigenmodes of both uniform and periodic electromagnetic structures which are confirmed using full-wave simulations and known analytical results. Two mapping procedures are further presented for arbitrary uniform and periodic structures, where the known $\{Ω-β\}$ relationship is expressed using polynomial and Fourier series expansions, respectively. Consequently replacing $\{Ω,~jβ\}$ with $\{ω_0, γ\}$ in the known $\{Ω-β\}$ relation, a characteristic equation is formed which is then numerically solved for the two unknowns, representing the physical dispersion relation $ω_0(β)$ and the frequency dependent propagation loss $α(ω_0)$ of the structure. The mapping procedure is demonstrated for variety of cases including unbounded uniform media, rectangular waveguide, Drude dispersive metamaterial and a periodic dielectric stack, where exact propagation characteristics have been successfully retrieved in all cases across both passbands and stopbands across frequency.

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