  A study is made of the saturation mechanism of a single superradiant spike of
radiation in a Free Electron Laser. A one-dimensional (1D) computer model is
developed using the Puffin, un-averaged FEL simulation code, which allows
sub-radiation wavelength evolution of both the spike radiation field and the
electron dynamics to be modelled until the highly non-linear saturation process
of the spike is observed. Animations of the process from the start to the end
of the interaction are available. The resultant saturated spike duration is at
the sub-wavelength scale and has a broad spectrum. The electrons passing
through the spike can both lose and gain energy many times greater than that of
the normal non-pulsed FEL interaction. A saturation mechanism is proposed and
tested via a simple analysis of the 1D FEL equations. The scaling results of
the analysis are seen to be in good agreement with the numerical results. A
simple model of three dimensional diffraction effects of the radiation is
applied to the results of the 1D simulations. This greatly reduces longer
wavelengths of the power spectrum, which are seen to be emitted mainly after
the electrons have propagated through the spike, and is seen to be in
qualitative agreement with recent experimental results.
