LONG PULSES AND SIDEBAND INSTABILITY IN FREE ELECTRON LASER OSCILLATORS

A number of designs for free electron laser oscillators are to produce optical pulses which are long compared to the slippage distance s (the distance by which the optical field moves ahead of the electrons during a single pass through the undulator). For such pulses, modulation of wavelength χ ≈ s may be amplified due to the synchrotron oscillations of the electrons, since these oscillations are then coperiodic with the optical field modulation encountered by the electrons due to slippage. Thus a sideband in the pulse optical spectrum may grow at a fractional frequency shift ≈ N-l, where N is the number of periods in the undulator. We study the evolution of such long pulses over many passes by considering a single section of length χ of the pulse and imposing periodic boundary conditions (Fig. 1). We therefore consider only sideband frequencies which differ from the carrier frequency ωc by Δω/ωc = ±2πnc/χωc = ±ns/Nχ, n=1,2,3... We consider a range of values of χ to determine the relative importance of different sideband frequencies. The coupled one-dimensional electron and wave equations are used and the transverse optical mode neglected. The undulator is conventional and untapered. We find for typical parameters that the optical field is unstable against the growth of modulation when near saturation. The instability is greatest for χ ≈ s, although the exact value depends on device parameters (Fig. 2). The instability has a finite width in χ (and thus Δω) since the interaction is of finite duration and the electrons have a range of synchrotron periods. Either the lower or upper sideband may dominate, although both are present along with their harmonics due to nonlinearities. When sideband growth is complete up to 50% of the optical power may reside in the sidebands. Similar sideband growth is predicted for short optical pulses by Colson and by Dattoli et al. For weak fields the synchrotron oscillations are small so that modulation does not grow until stronger fields develop. This synchrotron instability is similar to that found for tapered undualtor amplifiers by Kroll and Rosenbluth. The instability may be suppressed by use of frequency-selective resonator mirrors or by introducing an electron energy spread. Future work will consider the sideband instability for oscillators using other resonator designs