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 ε is given to handle both low and high frequency ranges. Indeed, it is shown analytically that the choice of the standard optimal splitting parameter ε₀ will cause overflow errors at high frequencies. Numerical examples illustrate the results and the sensitivity of the Ewald representation to the splitting parameter ε.