A kinetic theory is used to investigate the theory of a free-electron laser with a helical wiggler and an ion channel based on the Einstein coefficient method. The laser gain in the low-gain regime is obtained for the case of a cold tenuous relativistic electron beam, where the beam plasma frequency is much less than the radiation frequency, propagating in this configuration. The resulting gain equation is analyzed numerically over a wide range of system parameters.

A theoretical study of electron trajectories and gain in a free electron laser (FEL) with an electromagnetic-wave wiggler and ion-channel guiding is presented based on the Einstein coefficient method. The laser gain in the low-gain regime is obtained for the case of a cold tenuous relativistic electron beam, where the beam plasma frequency is much less than the radiation frequency propagating in this configuration. The resulting gain equation is analyzed numerically over a wide range of system parameters.