Coherent x-ray source due to the quantum reflection of an electron beam from a laser-field phase lattice.

Within the scope of relativistic quantum theory for electron-laser interaction in a medium and using the resonant approximation for the two degenerated states of an electron in a monochromatic radiation field, a nonperturbative solution of the Dirac equation is obtained. The multiphoton cross sections of the electrons' coherent scattering on the plane monochromatic wave at the Cherenkov resonance are obtained, taking into account the specificity of an induced Cherenkov process and spin-laser interaction as well. As a result of coherent reflection from the "phase lattice" of a slowed plane wave in a medium, electron beam quantum modulation at high frequencies occurs. So, a coherent x-ray source in an induced Cherenkov process is expected, since such a beam is a potential source of coherent radiation itself.