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 Φ 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 II orbits the gain is negative in the negative mass regime and positive in the positive mass regime.