Free Electron Lasers with Slowly Varying Beam and Undulator Parameters

The performance of a free electron lasers (FEL) is affected when the electron beam energy varies alone the undulator as would be caused by vacuum pipe wakefields and/or when the undulator strength parameter is tapered in the small signal regime until FEL saturation. In this paper, we present a self-consistent theory of FELs with slowly-varying beam and undulator parameters. A general method is developed to apply the WKB approximation to the beam-radiation system by employing the adjoint eigenvector that is orthogonal to the eigenfunctions of the coupled Maxwell-Vlasov equations. This method may be useful for other slowly varying processes in beam dynamics.