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A first comprehensive treatment on complex eigenmodes is presented forgeneral lossy traveling-wave electromagnetic structures where the per unitlength propagation phase shift ($\beta$) dependent complex eigenfrequencies$\Omega(\beta)$ are mapped to the frequency dependent complex propagationconstant $\gamma(\omega_0)$ for variety of electromagnetic structures. Rigorousprocedures are presented to first compute the complex eigenmodes of bothuniform and periodic electromagnetic structures which are confirmed usingfull-wave simulations and known analytical results. Two mapping procedures arefurther presented for arbitrary uniform and periodic structures, where theknown $\{\Omega-\beta\}$ relationship is expressed using polynomial and Fourierseries expansions, respectively. Consequently replacing $\{\Omega,~j\beta\}$with $\{\omega_0, \gamma\}$ in the known $\{\Omega-\beta\}$ relation, acharacteristic equation is formed which is then numerically solved for the twounknowns, representing the physical dispersion relation $\omega_0(\beta)$ andthe frequency dependent propagation loss $\alpha(\omega_0)$ of the structure.The mapping procedure is demonstrated for variety of cases including unboundeduniform media, rectangular waveguide, Drude dispersive metamaterial and aperiodic dielectric stack, where exact propagation characteristics have beensuccessfully retrieved in all cases across both passbands and stopbands acrossfrequency.

Complex Eigenmodes and Eigenfrequencies in Electromagnetics