Efficient computation of the lattice sums for leaky waves using the Ewald method

The efficient calculation of the Lattice Sums (LSs) for evaluating the free-space periodic Green's function (PGF) of an infinite phased array of line sources is addressed here and is extended for the first time to leaky waves. The method is based on the Graf's addition theorem for the 0-th order Hankel function that leads to an expansion in terms of Bessel functions of the relevant PGF. The coefficients of the expansion, called LSs, are slowly convergent series involving higher-order Hankel functions. Their efficient computation is performed by resorting to an extension of the standard Ewald method, in terms of spectral and spatial series having Gaussian convergence even in the case of complex waves and improper harmonics. The adopted approach allows for the analysis of a wide class of periodic waveguides supporting leaky modes, and it is particularly suitable for the treatment of infinite array of dielectric fibers and rods.