Correction to the adiabatic approximation for variable parameter free-electron lasers

The limits of validity of the adiabatic approximation commonly used in one-dimensional theories of variable parameter free-electron lasers (FEL) are obtained by formulating a Lie-transform perturbation theory in which the zeroth-order problem is the adiabatic approximation, modified to include any secular effects due to the nonadiabatic part of the Hamiltonian. It is shown that secular effects are absent in first order, and that the second-order secular terms can be neglected in practical FEL calculations. The lowest order nonsecular effects are evaluated by perturbation theory and are shown to yield small terms which can be easily included in Maxwell´s equations for the signal field.