Gain calculation of a free-electron laser operating with a non-uniform ion-channel guide

Amplification of an electromagnetic wave by a free electron laser (FEL) with a helical wiggler and an ion channel with a periodically varying ion density is examined. The relativistic equation of motion for a single electron in the combined wiggler and the periodic ion-channel fields is solved and the classes of possible trajectories in this configuration are discussed. The gain equation for the FEL in the low-gain-per-pass limit is obtained by adding the effect of the periodic ion channel. Numerical calculation is employed to analyse the gain induced by the effects of the non-uniform ion density. The variation of gain with ion-channel density is demonstrated. It is shown that there is a gain enhancement for group I orbits in the presence of a non-uniform ion-channel but not in a uniform one. It is also shown that periodic ion-channel guiding is used to reach the maximum peak gain in a low ion-channel frequency (low ion density).