Efficient computation of the free-space periodic Green's function for all source-to-observation-point distances

Surface integral equation formulations of periodic structures have received attention because of the inherent efficiency of surface unknowns and automatic satisfaction of radiation condition through the problem's Green's function. These formulations employ the periodic Green's function (PGF); the addition of potentials from all point sources as observed in the unit cell. Unfortunately, the resulting series (1) has slow convergence when direct summation (DS) is employed, which makes its usage in MoM codes rather costly. In this paper a new closed form is derived for efficient computation of the linear one-dimensional and planar (two-dimensional) periodic Green’s function at small source to observation points' distances. When combined with an accelerated modal (Floquet-wave) expression for more distant observation points, an efficient form is obtained for all distances. The efficiency of the proposed formulations have been shown through numerical computation.