Accelerating the convergence of series representing the free space periodic Green´s function

A method for improving the convergence of the series representing the doubly periodic free-space Green´s function is presented. The method consists of successively applying three different transformations to the Green´s function spectral representation. Kummer´s transformation is first applied to convert the slowly converging spectral representation into the sum of a rapidly converging series and a slowly converging series. The latter series is recognized as the spectral representation of the original periodic source distribution radiating in a medium with an imaginary wavenumber. Application of the Poisson transformation to this series renders it exponentially convergent since it effectively represents propagation of point source contributions through a medium with imaginary wavenumber. Finally, Shanks´ transform is plotted versus the number of terms taken in the series. Numerical results confirm that an improvement in the convergence rate of the series is achieved for a particular convergence criterion