A Modification of the Kummer´s Method for Efficient Computation of the Green´s Function for Doubly Periodic Structures

An auxiliary function in the form of standing spherical waves with attenuation is proposed in the Kummer´s method for accelerating the convergence of the spectral series representing the Green´s function of doubly periodic structures in free space. Expressions for the amplitude and phase constants of the auxiliary waves versus their attenuation constant are derived, at which the spectral difference series formed as a result of the Kummer´s transformation converges in the worst case as the Floquet mode propagation constant in the power of minus 5, 9, and 13 for the cases of using one, two, and three auxiliary waves, respectively, while the spatial series formed by the auxiliary waves converges exponentially. Some examples allowing determination of optimum values for the attenuation constant are considered, and some comparative results characterizing the effectiveness of the technique proposed are presented and discussed.