Pulse propagation and optical-klystron free-electron-laser devices.

We derive the equation governing the evolution of the optical field amplitude of a free-electron-laser oscillator operating in the optical-klystron configuration. The equation includes short pulse effects, due to the finite length of the electron bunches, and are valid under the assumption of small signal regime and low gain. Stationary solutions of the supermode type, analogous to those of conventional free-electron-laser configurations, are shown to exist for this case, too. Analytical expressions for the optical pulse shape are derived, without any assumption of dominance of the dispersive section, in the case of the long bunch regime, namely when the bunch length is larger than the slippage length. We present useful gain formulas, including the combined effect of dispersive section and pulse effects, and discuss the limits of validity of our treatment.