Beam focus and longitudinal polarization influence on spin dynamics in the Kapitza-Dirac effect

We theoretically investigate the influence of a longitudinal laser polarization component from beam focusing 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 focusing can be made negligibly small for sufficiently low beam foci.