1-D Periodic Lattice Sums for Complex and Leaky Waves in 2-D Structures Using Higher Order Ewald Formulation

Abstract

A very efficient and accurate method is proposed to evaluate the lattice sums (LSs) for the analysis of leaky waves in 2-D periodic waveguides. The LSs are the series involving Hankel functions of arbitrary order, which are not convergent for complex wavenumbers. It is shown that by extending Ewald representations to higher order Hankel functions, the LSs can be expressed in terms of spatial and spectral series, granting Gaussian convergence even in the case of complex and leaky waves. The method allows for the appropriate choice of the spectral determination for each space harmonic of a given LS coefficient, thus permitting one to obtain modal solutions that may correspond to physical and nonphysical leaky-wave phenomena. First, the proposed LS calculation is exploited in the evaluation of the free-space 1-D periodic Green’s function for 2-D structures. Then, the same procedure for the LSs is implemented in a cylindrical harmonic expansion method, based on the transition-matrix and the generalized reflection-matrix approach, for the full-wave analysis of leaky modes in 2-D electromagnetic band-gap waveguides formed by layered arrays of cylindrical inclusions. The presented LS formalism is numerically slim, very fast, and thus well suited for the analysis of a significant class of lossy periodic waveguides and leaky-wave antennas.