TITLE
	Non-linear dynamics of a storage-ring free-electron laser

AUTHOR
	Giovanni DE NINNO

ABSTRACT 
	In a Storage Ring Free Electron Laser (SRFEL) an ultra-relativistic electron 
	beam interacts with the magnetostatic periodic field generated by an 
	undulator. The electrons are forced to move along sin-like trajectories, 
	thus emitting synchrotron radiation. Once stored in the optical cavity, the 
	light is amplified during successive passes of the electrons in the ring. 
	The laser dynamics depends strongly on the value of the temporal light-
	electron beam synchronism, i.e. the difference between the electron beam 
	revolution period and the round trip of the photons in the cavity. For a 
	small detuning, the laser intensity displays a “continuous wave” (“cw”) 
	behaviour, while regular oscillations are found for larger detuning amounts. 
	If an external modulation is applied to the detuning parameter the system is 
	shown to produce chaotic regimes. Given this setting and aiming at improving 
	the performance of the device, it is of paramount importance to investigate 
	the non-linear dynamics of the system and achieve a comprehensive 
	understanding of the mechanism of saturat  ion. In particular, it is crucial 
	for user applications to obtain a stable and extended “cw” region.
	In [1,2] a new model for the longitudinal dynamics of a SRFEL was introduced 
	and shown to be particularly suitable for analytic calculations. Within this 
	framework explicit expressions for the asymptotic values of the main 
	statistical parameters of the laser distribution are provided as function of 
	the detuning amount. Moreover, the transition between stable and unstable 
	regimes is characterized as a Hopf bifurcation and an explicit expression 
	for the critical detuning is given. These findings open up the perspective 
	of introducing a derivative self-controlled feedback to enlarge the region 
	of stable signal.
	A feedback of this type has been implemented on the Elettra SRFEL and fully 
	confirmed our theoretical predictions. In this paper we review these results 
	and present new experiments carried out at ELETTRA. 
	[1] G. De Ninno, D. Fanelli, Phys. Rev. Lett., 92, 094801 (2004) 
	[2] G. De Ninno, D. Fanelli, Phys. Rev. E 70, 016503 (2004)