  A first comprehensive treatment on complex eigenmodes is presented for
general lossy traveling-wave electromagnetic structures where the per unit
length propagation phase shift ($\beta$) dependent complex eigenfrequencies
$\Omega(\beta)$ are mapped to the frequency dependent complex propagation
constant $\gamma(\omega_0)$ 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 $\{\Omega-\beta\}$ relationship is expressed using polynomial and Fourier
series expansions, respectively. Consequently replacing $\{\Omega,~j\beta\}$
with $\{\omega_0, \gamma\}$ in the known $\{\Omega-\beta\}$ relation, a
characteristic equation is formed which is then numerically solved for the two
unknowns, representing the physical dispersion relation $\omega_0(\beta)$ and
the frequency dependent propagation loss $\alpha(\omega_0)$ 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.
