Efficient computation of the 2-D Green's function for 1-D periodic structures using the Ewald method

The Ewald method is applied to accelerate the evaluation of the Green's function of an infinite periodic phased array of line sources. The Ewald representation for a cylindrical wave is obtained from the known representation for the spherical wave, and a systematic general procedure is applied to extend previous results. Only a few terms are needed to evaluate Ewald sums, which are cast in terms of error functions and exponential integrals, to high accuracy. Singularities and convergence rates are analyzed, and a recipe for selecting the Ewald splitting parameter epsilon is given to handle both low and high frequency ranges. Indeed, it is shown analytically that the choice of the standard optimal splitting parameter Eo will cause overflow errors at high frequencies. Numerical examples illustrate the results and the sensitivity of the Ewald representation to the splitting parameter epsilon.