Nonlinear inelastic scattering of electrons at an optical standing wave

We investigate higher-order nonlinear inelastic scattering of electrons at an optical standing wave formed by two light waves with the same frequencies in vacuum. Initial conditions necessary for efficient scattering are obtained from the semiclassical energy and momentum conservation laws for all nonlinear orders. The stationary scattering potential in the electrons' rest frame is derived from the perturbative solution of the classical relativistic equation of motion for the four-photon scattering process. The amplitude of the electron velocity modulation introduced during higher-order scattering processes is calculated numerically. The proposed inelastic interaction leads to a periodic modulation of the electrons' energy and longitudinal momentum at a higher harmonic frequency of the driving light waves.