Electron trajectories in a helical free-electron laser with or without an axial guide field

An earlier study of the motion of electrons in a helical wiggler with an axial guide field is extended to allow the axial field to be zero. The present treatment, like its predecessor, provides analytic but approximate expressions for the trajectories. In particular, it is able to adequately describe those trajectories that do not encircle the axis. When the axial field is zero, the new treatment allows us to obtain closed form expressions for the trajectories, which depend parametrically on the ratio of the helical invariant to the on-axis wiggler field. In the limit of high energy or low wiggler field, further simplifications occur, and we find a simple yet detailed description that agrees with exact numerical calculations of the trajectories.