  We theoretically investigate the influence of a longitudinal laser
polarization component from beam focussing on spin dynamics in Kapitza-Dirac
scattering by solving the relativistic Dirac equation with time-dependent
perturbation theory. The transverse spacial dependence of the longitudinal beam
polarization component is accounted for, by approximating a Gaussian beam with
plane-wave components. We find that corrections from a longitudinal laser beam
polarization component approximately scale with the second power of the
diffraction angle $\epsilon$, from which we conclude that a related influence
from beam focussing can be made negligibly small for sufficiently low beam
foci.
