Optical-mode trapping in the free-electron laser

Transverse-optical-mode trapping (``active guiding'') in the free-electron laser (FEL) is discussed via a variational approach based on classical field theory. The variational principle for the field leads to a set of simple evolution equations for the mode parameters which generalizes the discussion in A. Amir and Y. Greenzweig, Phys. Rev. A 34, 4809 (1986), where a parabolic expansion of the interaction term was used. The method can be applied to calculate the evolution of the optical-beam parameters under the interaction with an arbitrary electron-beam profile. Electron-beam inhomogeneities are included by using a distribution function. In the linear regime, the stationary condition of these equations results in a simple algebraic equation for the trapped-mode parameters. The values obtained in this method are compared with the results of several numerical simulations. The equations of motion for the optical-beam parameters are also derived for the full nonlinear FEL problem using the same principles.