Complex Eigenmodes and Eigenfrequencies in Electromagnetics

Abstract

A comprehensive treatment on complex eigenmodes is presented for general lossy traveling-wave electromagnetic structures. The per unit length propagation phase shift ( β)-dependent complex eigenfrequencies Ω(β) are mapped to frequency-dependent complex propagation constant γ(ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ) for a variety of electromagnetic structures. Rigorous procedures are presented to compute the complex eigenmodes of both uniform and periodic electromagnetic structures, confirmed using full-wave simulations and known analytical results. We further present two mapping procedures for arbitrary uniform and periodic structures, where the known {Ω-β} relationship is expressed using rational polynomial expansions. Consequently, replacing {Ω, jβ} with {ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> , γ} in the known {Ω-β} relation, a characteristic equation is formed, which is then numerically solved for the propagation constant γ, representing the physical dispersion relation ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> (β) and the frequency-dependent attenuation relation α(ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ) of the structure. The mapping procedure is demonstrated for a variety of cases, including an unbounded uniform media, a rectangular waveguide, a Drude dispersive metamaterial, a biased ferrite medium, a periodic dielectric stack, and a periodic rectangular waveguide. The exact propagation characteristics have been successfully retrieved in all cases across both passbands and stopbands across frequency.