Electron beam equilibria with self-fields for a free electron laser with a planar wiggler

A general formalism for non-neutral cold relativistic planar steady flows is developed and applied to the study of the equilibrium of a sheet electron beam in a planar wiggler free electron laser. The full transverse dependence of the wiggler field as well as the equilibrium self-fields of the beam are included. In particular, the betatron oscillations in the presence of self-fields are studied. For a thick beam equilibrium with a particular density profile it is shown that the betatron oscillations are eliminated. For a thin beam configuration the paraxial approximation is employed and it is also shown that for some critical density there are no betatron oscillations. If the density is larger than this critical density the beam oscillates with the betatron frequency but there are no trajectory crossings and the beam preserves its cold fluid nature. The single-particle equations of motion are also considered in the presence of both planar wiggler and planar self-fields. It is shown that in some cases the particles oscillate with a reduced betatron frequency, in contrast to the previous case of cold fluid motion where the self-fields do not change the betatron frequency. For the study of the betatron oscillations in the thick beam equilibrium a two-space scale method is employed. For the thin beam within the paraxial approximation the Floquet theory for equations with periodic coefficients is used.