Three-dimensional statistical theory for analyzing high-gain free-electron laser at saturation

Considering the transverse emittance of electron beam and electron beta oscillation, the statistical physics approach of the analysis to the single pass high-gain free electron laser at saturation is improved and extended to the three-dimensional case. Firstly, a three-dimensional, normalized and simplified model is conducted, then the Vlasov equation of one-dimensional light field including the transverse movement of electrons is derived. In the helical undulator case a three-dimensional statistical physics approach is developed by introducing the conserved quantities of transverse movement, and the relevant code is written for calculation of the saturation intensity, bunching factor and other parameters. For comparison and verification, a direct numerical simulation code of free electron laser which includes N particles is written. The results of numerical simulation and statistical theory are in well accordance with the original one-dimensional simulation and statistical physics approach, and the three-dimensional theory reveals the effect of saturation point parameters introduced by transverse emittance of electron beam and electron beta oscillation.