Nonadiabatic corrections and detrapping for variable parameter free electron lasers

In analytical treatments of free electron laser (FEL) problems an adiabatic, approximation is used in which the electron orbits are calculated as though the wiggler and signal fields were constant. The slow spatial variations of the field amplitudes are then reinstated at the end of the calculation. A check to see if secular effects are present in the corrections to the adiabatic approximation for FEL problems is described. The nonstatic action angle transformation produces a new Hamiltonian in which the nonadiabatic effects are explicitly given by the term g kappa. Perturbation expansions in g are valid in the interior of the trapping region with the exception of an exponentially small neighborhood of the separatrix. By using Lie transform perturbation theory it was found that secular effects are absent to O(g), and the O(g2) secular effects are not important in the FEL application.