Ewald acceleration for the dyadic Green’s functions for a linear array of dipoles and a dipole in a parallel-plate waveguide

In this paper, the hat nland denotes unit vectors and we use the exp(jwt) time convention. The radiated field can be represented as a purely spatial sum of spherical waves or as a purely spectral sum of cylindrical waves. Both these representations are slowly convergent. Furthermore, the purely spatial sum cannot be applied for complex wavenumbers k_z0 while the purely spectral sum can not be applied when the observation point is on the array axis (i.e., p = 0).