Theory of free electron lasers with pump field focusing

In free-electron lasers which employ stimulated wave scattering and stimulated wiggler radiation, the pump field increases from the axis toward the periphery and forms an average potential variation which focuses the electrons in the transverse plane. Depending on the phase velocity of the combination (beat) wave, scattering may be accompanied either by damping or by pumping of the drift oscillations of the electrons in the potential relief. The analogy between free-electron lasers with a focusing pump field and relativistic cyclotron resonance masers operating under normal or anomalous Doppler effect conditions is pursued.