Analysis of the saturation of a high-gain free-electron laser

We study the saturated state of an untapered free-electron laser (FEL) in the Compton regime, arising after exponential amplification of an initially low level of radiation by an initially monoenergetic, unbunched electron beam. The saturated state of the FEL is described by oscillations about an equilibrium state. Using the two invariants of the motion and certain assumptions motivated by computer simulations we provide approximate analytic descriptions of the radiation field and electron distribution in the saturation regime. We first consider a one-dimensional approximation and later extend our approach to treat an electron beam of finite radial extent. Of note is a result on the radiated power in the case of an electron beam with a small radius