Collective instabilities and high-gain regime free electron laser

The operation of free electron lasers (FEL) in the short wavelength region, (1000 A), requires a large field amplification per undulator pass in order to overcome the large losses of the optical cavity at these wavelengths. Systems based on the combination of a storage ring and of a free electron laser can provide this large amplification. In fact, for these systems small signal gains of the order of 100 to 1000% per pass have been estimated. Of course, at this level of amplification, the small signal gain formula is no longer appropriate and a more accurate description of the FEL is required. FEL studies in the high gain regime have shown that, with an appropriate selection of the electron density, detuning, and undulator length, it is possible to produce an exponential growth of both the radiation field and of the electron bunching. This is the result of the emergence of a collective instability for the electron beam undulator radiation field system. The conditions for the onset of this instability were studied and the characteristic complex frequencies of the FEL system are derived.