Gain parameterization formulas in FEL optical-klystron configurations

We discuss simple formulas useful to analyze the free-electron laser dynamics when an optical-klystron configuration is employed. The small-signal gain dependence on the length of the dispersive section is discussed, as well as the effects of the inhomogeneous broadening contributions. We introduce the concept of saturation intensity for the optical-klystron and study the dependence on the various parameters of the device. We investigate the high gain effects and the saturated behavior. Regarding this last point, we present a formula accounting for the optical gain dependence on the intracavity power.