Statistical theory of free-electron laser saturation state solution with Gaussian initial distribution

Julien Barr proposed a new approach of the statistical theory to the analysis of high-gain free-electron laser at saturation in 2004. This new approach was developed and generalized in this paper to the case of different initial phase space distribution of the electron beam, and a Gaussian momentum distribution which was more practical was calculated. The calculating difficulties from this more practical momentum initial distribution were resolved by multiple integral techniques. For comparison and verification, a direct numerical simulation code was developed. In the cases of two kinds of initial electron beam energy distribution, analysis was conducted respectively of the radiation intensity, bunching parameters and electron distribution in phase space at saturation of free-electron-laser. It is shown that the results by this new statistical method are in good coincident with the corresponding numerical simulation. It is also shown that under the conditions we investigated the difference between the result of the initial distribution of Gaussian and that of water bag is not significant.