A one-sided version of the Poisson sum formula for semi-infinite array Green's functions

The Poisson sum formula provides an efficient method for transforming many slowly converging infinite summations into equivalent, but more rapidly converging infinite summations. Electromagnetic applications for this result come in the analysis of infinite arrays of periodic scatterers, such as frequency selective surfaces. However, in some applications, such as when one desires the radiation of a semi-infinite array of periodically spaced currents, the original form of the Poisson sum formula is inappropriate. For such applications, we derive a one-sided version of the formula and apply it to the radiation from a semi-infinite array of line sources with currents dictated by Floquet's theorem. The one-sided Poisson sum transformation yields enhanced convergence characteristics for certain regions of application as a result of the inverse bandwidth relationship between Fourier transform pairs. The application of it to a semi-infinite line source array also provides a plane wave representation for the fields, which makes for an extension of the solution to geometries with stratified dielectric media.