Abstract: Based on single particle theory and under the paraxial and adiabatic approximations, Only the wiggler field is considered in the equations of electron trajectories and longitudinal motion equations of electron are averaged over the period of wiggler, λ_w. We derived a set of coupled nonlinear differential equations of both TE_mn and TM_mn modes in a loss-free cylindrical waveguiede. The space charge potential is considered and the longitudinal guiding field is neglected.
Using the data obtained in the EPA-74 FEL experiment, the models are computed by 3-D CAG FEL II code. The computational results show that the saturation position z= 130-150cm, the saturation power P≈1MW and the gain G = 70dB/m, in the scope of experiment parameters.