Nonlinear resonances in a multi-stage free-electron laser amplifier

A two-beam accelerator (TBA) is a possible candidate of future linear colliders, in which the demanded rf power is provided by a multi-stage free-electron laser (MFEL). After if amplification in each stage, a driving beam is re-accelerated by an induction unit and propagates into the next stage. Recently it has been recognized that the multi-stage character of the MFEL causes resonances between its periodicity and the synchrotron motion in an rf bucket. Since the synchrotron oscillation is strongly modulated by the resonance and at the worst a large fraction of particles is trapped in the resonance islands, the nonlinear resonances in the FEL longitudinal beam dynamics can lead to notable degradation of the MFEL performance, such as output fluctuation and phase modulation which have been big concerns in the accelerator society. The overall efficiency of the MFEL and the quality of the amplified microwave power are key issues for realizing the TBA/FEL Particularly the rf phase and amplitude errors must be maintained within tolerance. One of significant obstacles is an amplification of undesired modes. If a small-size waveguide is employed, the FEL resonance energies for undesired higher order modes shift very far from that for a fundamental mode; so it is possible to prevent higher order modes from evolving. Such a small-size waveguide, however, gives a high power density in the FEL. Simulation results have demonstrated that the nonlinear resonances occur in die FEL longitudinal motion when the power density exceeds some threshold. An analytical method for studying the nonlinear resonance in the TBA/FEL is developed based on the macroparticle model which can describe analytically the drastic behaviors in the evolutions of the phase and amplitude. In the theory the basic 1D-FEL equations are reduced to a nonlinear pendulum equation with respect to the ponderomotive phase