Closed form evaluation of the Green´s function in the discrete time domain using the Catalan triangle

The FDTD algorithm and its popular Yee grid implementation are vastly used to discretize the differential equation representation of the electromagnetic field. This representation can also be converted into an integral formulation with an appropriate Green´s function within the context of the same grid (Holtzman, R. and Kastner, R., IEEE Trans. Anten. and Propag., vol.49, no.7, p.1079-93, 2001). The evaluation of the Green´s function is a crucial step in the integral formulation. One may wish to simply discretize the continuous Green´s function, however this form of discretization is incompatible with the Yee difference equations. In order to produce a self consistent formulation, one that preserves the second order accuracy and includes FDTD artifacts, such as numerical dispersion, this Green´s function needs to be evaluated from the difference equations as first principles. The paper presents this derivation, leading to a closed form of the Green´s function in one dimension. Combinatorial considerations bring the Catalan triangle, an 18th century generalization of the Pascal binomial triangle, into the picture. This time domain procedure is duly verified by comparison to the frequency domain, multi-dimensional inverse Z-transform presented by Kastner at this symposium, and the direct numerical evaluation for the z=z´ case of Holtzman and Kastner.