Optimization of the beam configuration in the Raman-type free-electron laser

A theoretical discussion is given for the two-dimensional Raman-type free-electron laser composed of a hollow relativistic electron beam of arbitrary thickness contained in a parallel plate waveguide and an array of permanent magnets for the pumping source. Under the influence of the magnetostatic field pump, the coupling between the scattered wave of the even TE mode (positive-energy wave) and the electron plasma wave of the odd TM mode (negative-energy wave) is investigated in detail. With the aid of numerical illustrations, the electron plasma wave with odd symmetry is found to be split into two modes: one mode is dominant for a thick beam and the other mode dominant for a thin beam. For the case where the scattered wave interacts with the electron plasma wave of the latter mode, an optimum beam thickness is obtained for which the growth rate becomes maximum. The optimum beam thickness is found to be comparable with the wavelength of the scattered wave and yet considerably greater than the reactive skin depth of the electron beam. In addition, the efficiency for energy conversion is found to be greatly improved by utilizing a hollow beam instead of a solid beam.