Efficient Evaluation of the 3-D Periodic Green´s Function Through the Ewald Method

The Ewald method is efficiently applied for the computation of the Green´s function for a 3-D array of phased point sources. A best splitting parameter E, which avoids the loss of accuracy occurring at high frequencies while maintaining rapidly converging properties, is derived. The loss of significant digits in the calculation of the modified spatial and spectral series involved in the Ewald summation technique can be set by the user, thus limiting the total error in the computation to a controllable level. Particular care is also given to the best way of summing the series that appear in the Ewald method and a new efficient algorithm for the reduction of the involved three-index series to one-index series is proposed, which allows for a further acceleration of the computation. Numerical results are presented to show the validity and efficiency of the proposed formulation.