3-D Green's functions in 1-D periodic structures: A comparative analysis of the acceleration techniques

In this work we treat in detail the acceleration of the free-space scalar Green's function in three dimensional (3D) structures with one dimensional (1D) periodicity. We analyze the methods able to deal with a complex phase shift, necessary to study complex waves in periodic structures. We present a spectral Kummer-Poisson's decomposition; then we extend the application of the Ewald's method by performing an optimization of the relevant splitting parameter, and we also find an integral representation. Comparisons among the various acceleration methods are performed, thus providing fundamental information on their actual efficiency.