A deeper analytical insight into the longitudinal dynamics of a storage-ring free-electron laser

In this paper, a deep characterization is provided of the longitudinal dynamics of a storage-ring free-electron laser. Closed analytical expressions are derived for the main statistical parameters of the system (i.e. beam energy spread, intensity, centroid position and rms value of the laser distribution) as a function of the light-electron beam detuning at each pass inside the interaction region. Moreover, the transition between the stable 'cw' regime and the unstable pulsed behaviour is shown to be a Hopf bifurcation. Finally, a feedback procedure is introduced which suppresses the bifurcation and significantly improves the system stability