Closure Relations in Macroscopic Fel Equations

Based upon a macro-electron approach, several closure relations are presented to truncate the hierarchy of moments in macroscopic equations for the electron motion in a Compton FEL. In the non-linear and saturation regime of FEL operation, the optical power predicted by the models is compared with results from a multi-electron simulation code and with results from the literature. An idealized macro-electron model shows that the oscillatory non-linear behaviour is completely determined by the sloshing of energy between the electrons and the wave. A more realistic model which includes the slippage between the pulses, describes the behaviour of the optical power in the linear regime as well as the saturation level and the oscillations in the non-linear regime.