Electron trajectories and gain in a free-electron laser with realizable helical wiggler and ion-channel guiding

A theory is developed for a free-electron laser (FEL) with a three-dimensional helical wiggler and ion-channel guiding. The relativistic equation of motion for a single electron in the combined wiggler and ion-channel fields is solved in the rotating wiggler frame. With the aid of the conservation of energy, equations for the axial velocity and the $Φ$ function (which determines the rate of change of axial velocity with energy) are studied numerically. An analysis of the electromagnetic radiation copropagating with the electron beam in the FEL interaction region is also presented. The gain formula is derived and calculations indicate that the gain of the realizable wiggler is considerably greater than the gain of the idealized one, and the gain enhancement increases with increasing wiggler magnetic field. It is shown that the gain for group-I orbits is positive, while for group-II orbits, the gain is negative in the negative mass regime (i.e., $Φ<0$⁠) and positive in the positive mass regime.