The effects of transverse gradient of static magnetic wigglers on the free electron laser

The static magnetic wiggler fields of the free electron laser possesses transverse gradients, which cause betatron oscillations. We derived the electron trajectories in the wiggler field. The betatron oscillations cause an effective axial velocity spread. We obtained the condition for the input radiation field to initially trap all the electrons. We found that the minimum requirement on the input radiation power density to initially trap all electrons for a helical wiggler is one quarter of that for a linear wiggler for beams possessing identical emittance.