Averaged equations of electron motion in the field of two electromagnetic waves of unequal frequencies under beat resonance conditions

Truncated equations of motion are derived for a relativistic electron moving in the field of two inhomogeneous electromagnetic waves of unequal frequencies. The equations are used to study scattering of quasi-optical beams and TE and TM waveguide modes by relativistic electron beams. These equations are specified by an averaged high-frequency potential which is sensitive to the polarization of the waves. It is shown that TE-TM conversion during scattering in a waveguide is constrained if the angle between the wave vector of the TM wave and V, the translational velocity of the electrons, is equal to 1/gamma, where gamma is the Lorentz factor.