Electron trajectories in a free-electron laser with planar wiggler and ion-channel guiding

An analysis of quasi-steady-state electron trajectories in a planar wiggler with an ion channel and an axial magnetic field is presented. A tenth-degree polynomial equation for the average axial electron velocity and an equation for the function Φ, which determines the rate of change of axial velocity with energy, are derived. Numerical calculations are made to illustrate the effects of the two electron-beam guiding devices on the trajectories when applied separately and simultaneously. Some interesting effects of their simultaneous application include the existence of three groups of orbits, two singularities of the function Φ, and two negative mass regimes.