Electron orbits in free-electron lasers with helical wiggler and axial guide magnetic fields

An analytic theory of nonsteady-state trajectories in a fully three-dimensional, helical wiggler magnetic field is developed. In contrast to the well-known class of helical steady-state orbits, these orbits are not axicentered. Two limiting cases are considered. In the case of trajectories which are axis-encircling, we obtain a class of orbits by perturbation about the steady-state limit. We also consider the opposite regime in which particle trajectories are not axis-encircling, and derive a guiding-center theory for the limiting case in which the particles remain far from the axis of symmetry. In both limits we obtain trajectories which consist of periodic motions corresponding to both the fast wiggler and Larmor periods, as well as a slow motion corresponding to betatron oscillations arising from the transverse inhomogeneity in the wiggler field.