  Theoretical spin properties of the Kapitza-Dirac effect beyond the plane-wave
description are not known in detail. We develop a method for computing electron
diffraction of the two-photon Kapitza-Dirac effect in a two-dimensional
Gaussian beam standing light wave within a relativistic formulation. The
solutions are computed on the basis of time-dependent perturbation theory,
where a momentum space formulation with the use of a Fourier transformation of
the external potential allows for the solving the perturbative time-integrals.
An iteration over each possible quantum state combination leads to a quadratic
scaling of our method with respect to spacial grid resolution, where
time-stepping does not occur in the numeric implementation. The position- and
momentum space grids are adapted to the two-photon interaction geometry at low
resolution, for which our study only finds partial convergence of the simulated
diffraction pattern. Further, the method has the advantage of having an easy
implementable parallelization layout.
