Gain equation for a free-electron laser with a helical wiggler and ion-channel guiding

The theory of ion-channel guiding in a helical wiggler is presented. Electron motion in the combined ion electrostatic and wiggler magnetostatic fields is analyzed in the absence of the radiation field, The Phi function that determines the rate of change of axial velocity with energy is derived and studied numerically. A detailed analysis of the pendulum equation and the gain equation in the low-gain-per-pass limit are presented. It is shown that the gain for stable group I orbits is positive, while for group Pi orbits the gain is negative in the negative mass regime and positive in the positive mass regime.