Laser-induced phase shift of swift electrons

We revisit the calculation of the phase shift experienced by swift electrons on passing through the electromagnetic field of a laser. Such phase shifts are now utilized in the form of `laser phase plates' in transmission electron microscopes (TEMs), for example. We calculate the phase shift using three different methods, namely, perturbation theory applied to the Dirac equation, the Volkov solution to the Klein-Gordon equation, and the relativistic Hamilton-Jacobi equation. We find that all three methods are in agreement, and that the calculated phase shift is independent of the relative orientation of the electron and laser beams. The agreement between the quantum and classical theories is explained. Our Lorentz invariant result for the phase shift differs from certain results published in the literature.