Self-fields in a planar wiggler and axial magnetic field

A theory for self-fields induced by charge and current densities of the electron beam in a free-electron laser with planar wiggler and axial magnetic field is presented. Mutual influence of the electron velocity and self-magnetic field is considered to account for the total self-magnetic field. The quasisteady-state orbits and their stability under the influence of self-fields are derived and discussed. The function $Φ$ which determines the rate of change of axial velocity with energy is then derived. It is shown that for a planar wiggler, the wiggler-induced self-magnetic has two components. The first one is perpendicular to the wiggler magnetic field and the second is parallel (or antiparallel) to the wiggler magnetic field. The wiggler-induced self-magnetic field has a diamagnetic effect for group I orbits, while for group II orbits it has a paramagnetic effect. Some interesting effects of self-fields on electron dynamics include the generation of a negative singularity for function $Φ$ and creation of new unstable orbits for group II which are not found in the absence of self-fields.