Evolution of transverse modes in FEL and the method of approximants

The FEL paraxial wave equation is investigated within the context of the method of the approximants, which has been successfully applied to the one-dimensional FEL dynamics including high gain, inhomogeneous broadening and saturation effects. The method is effective in confining the transverse mode dynamics into a simple analytical model, which is amenable for a fast and reliable numerical procedure, appropriate to small computers. The case of uniform e-beam current profile is considered in some details. Numerical results are presented and compared with those from more sophisticated codes.