Analytical and numerical comparison of different approaches to the description of SAse in high gain FELs

Correlation function theory which has been developed recently gives rigorous statistical description of the SASE FEL operation. It directly deals with the values averaged over many shots. There are two other approaches which are based either on Vlasov equation or on direct solution of particle motion equations. Both of them perform calculations for some particular initial conditions. After that one can either consider the result as a “typical” sample, or repeat calculations for other initial conditions and then average the results. To check the validity of these three approaches it might be interesting to compare them with each other. In this paper we present the results of such comparison obtained for the 1-D FEL model. We show that two-particle correlation function approximation is equivalent to the quasilinear approximation for the Vlasov equation approach. These two approximations are in a good agreement with the results of direct solution of particle motion equations at linear and early saturation stages. To obtain this agreement at strong saturation, high order harmonics in Vlasov equation have to be taken into account, which corresponds to taking into account of three and more particle correlations in the correlation function approach.