An analytical method for construction of single particle electron trajectories in free electron lasers

Abstract

We apply a method of Linshtedt, also called improved expansion, to solve the equations of motion and obtain single-particle trajectories of electrons moving in crossed static magnetic fields of a hybrid non-relativistic free electron laser. Making use of a natural small parameter, the ratio of the amplitude of spatially periodic magnetic field and the guide magnetic field, one can re-write the motion equations for an electron in a form, which allows their solution by an asymptotic series. In such a way the non-linear frequency shifts and renormalized mean electron velocity are calculated analytically. The analytical results are in a good compared with numerical simulations of the electron trajectories.