Scattering of relativistic electron beam by two counter-propagating laser pulses: A new approach to Raman X-ray amplification

We present a detailed study of the properties of electron beam injected and trapped in an high intensity optical lattice. By using the hydrodynamic and kinetic approaches, we identified the beam trapping conditions, the high-frequency- longitudinal beam eigenm.odes and their dependence on the electron angular and energy spread. The coupling of these beam eieenmodes to the laser waves is also considered. This corresponds to the convective parametric instability: a stimulated scattering of two laser beams creating the optical lattice on the trapped electron beam mode. The amplification coefficients for the up-scattered Raman modes propagating parallel to the electron beam are calculated and their dependence on the beam characteristics is analyzed.