Electron trajectories in a free-electron laser with helical wiggler, ion-channel guiding, and parallel/reversed axial magnetic field

An analysis of electron trajectories in a helical magnetic wiggler with a uniform ion channel and a uniform axial magnetic field is presented. The axial field is considered in both the conventional and reversed directions. Equations for the transverse coordinates and velocities of a single relativistic electron in the combined ion electrostatic field and helical and axial magnetic fields are derived. A sixth-degree polynomial equation for the electron velocity and an equation for the function Phi (which determines the rate of change of axial velocity with energy) are derived. Results of some numerical calculations are presented to illustrate the effects of the electrostatic field and the axial magnetic field in each of the two configurations.