Relativistic Raman backscattering in an electron beam with ion-channel guiding and its application in free-electron laser

A theory for Raman backscattering, in the rest frame of an electron beam with ion-channel guiding, is presented when the electrons are making small amplitude betatron oscillations. A relativistic theory of this problem is also presented when the electrons are moving through relativistic and helical trajectories. The relativistic theory is used to find the growth rate of a free-electron laser in Raman regime. The wiggler effects, on the linear dispersion relations of the space-charge wave and radiation, are included in the analysis. The growth rate and free-space wavelength of the radiation are studied numerically at different operating regimes corresponding to small and large ion-channel densities. An improved dispersion relation for a free-electron laser without any guiding is derived.