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.>