Formation of distributed feedback in an FEL under multi-wave diffraction

The boundary problem for an X-ray crystal quasi-Cherenkov FEL is solved for the case in which distributed feedback is provided by three-wave coplanar diffraction of the emitted photons. In comparison with two-wave distributed feedback, it is shown in the three-wave case that the necessary interaction region between a particle beam and the electromagnetic wave necessary to reach the oscillation threshold, as well as radiation losses inside the medium, can be reduced with an appropriate choice of parameters. This gives rise to the possibility of relaxing the required parameters of a particle beam necessary for lasing in the X-ray spectral range. At the same time it also allows a reduction in the size of an FEL and the construction of more compact coherent radiation sources in any spectral range.