Complete and Accurate Discrete Complex Image Approximation of Periodic Green´s Function in Layered Media

The periodic Green´s functions (PGF) in layered media can be expressed as an infinite series in terms of the spectral domain Green´s functions. The discrete complex image method (DCIM) can be used to approximate these spectral domain Green´s functions. In this work, it is demonstrated that the complete and accurate DCIM approximation of the PGF is possible only when the DCIM approximation is obtained through the samples of the spectral domain Green´s function along the real k_ρ axis. This choice of sampling path requires the extraction of surface wave pole contributions prior to the application of DCIM. Different forms of the Ewald method are utilized to efficiently compute the infinite summations associated with the complex images and the contribution of surface wave poles.