Intrinsic corrections to optical guiding in a free electron laser

The effect on optical guiding of the undulations of an electron beam in an FEL is investigated, and a model for a fully saturated FEL amplifier with no remaining gain is developed. The density of the electrom beam in the model fully includes the effects of both transverse and longitudinal undulation of the beam. The longitudinal density modulation is expressed in terms of the Bessel functions of ζ, where ζ = a w 2 2(1 + a w 2 ) is the shift of the electron phase in the electron bucket caused by its longitudinal undulation. The transverse density modulation is evaluated to second order in the ratio of undulation amplitude δr to beam radial scale length rb. The radiation field from the modeled beam is calculated in terms of spatial modes proportional to exp[i(k + δk + lkw)z − iωt], where l is an arbitrary integer. Here, δk is the change of the wavenumber of the radiation caused by the electron bunches. Radially radiating modes with intensity on the order of ( δk k w )ζ 2 are found. Optical guiding is found to be modified by the transverse undulations of the beam at second order in δr r b , and by the longitudinal undulations to first order in δk k w . For the usual FEL parameters, the correction is quite small.