General Mapping Between Complex Spatial and Temporal Frequencies by Analytical Continuation

Abstract

This article introduces a general technique for intermapping the complex spatial frequency (or propagation constant) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\gamma =\alpha +j\beta $ </tex-math></inline-formula> and the temporal frequency <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\omega = \omega _{\text {r}}+j\omega _{\text {i}}$ </tex-math></inline-formula> of arbitrary electromagnetic media and structures. This technique is based on the analytic property of complex functions that describe physical phenomena; it invokes the analytic continuity theorem to assert the unicity of the mapping function and builds this function from known data within a restricted domain of its analyticity by curve fitting to a generic polynomial expansion. It is not only applicable to canonical problems admitting an analytical solution but also to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">general</i> medium or structure problems, from eigen- or driven-mode full-wave simulation results. The proposed technique is demonstrated for several systems, namely, an unbounded lossy medium, a dielectric-filled rectangular waveguide, a periodically loaded transmission line, a 1-D photonic crystal, and a series-fed patch (SFP) leaky-wave antenna (LWA), and it is validated either by analytical results or by full-wave simulated results.