基于高增益自由电子激光饱和状态的统计物理方法，在不同的电子束相空间初始分布下对原有理论进行了发展和推广，借助于多重数值积分技术解决了粒子束一般动量初始分布带来的计算困难，并以更接近实际的高斯动量分布为例进行计算。作为对比验证编写高斯初始动量分布下的直接数值模拟程序，将高斯初始分布同“水袋”两种初始分布的“暖电子束”入射情形下自由电子激光达到饱和时系统的光强、聚束因子和粒子相空间分布均进行相应计算和比较。结果表明：数值模拟和统计计算结果相一致；在目前选取的参数范围内，两种初始分布的相应结果差异并不显著。

Julien Barré 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 technics. 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.