Asymptotic electron trajectories and an adiabatic invariant for a helical-wiggler free electron laser with weak self-fields

Abstract

The dynamics of a relativistic electron in the field configuration consisting of a constant-amplitude helical-wiggler magnetic field, a uniform axial magnetic field, and the equilibrium self-fields is described by a near-integrable three-degree-of-freedom Hamiltonian system. The system is solved asymptotically for small ε by the method of averaging, where ε measures the strength of the self-fields. Because the Hamiltonian does not depend on one of the coordinates, it immediately reduces to a two-degree-of-freedom system. For ε=0, this reduced system is integrable, but is not in standard form. The action-angle transformation to standard form is derived explicitly in terms of elliptic functions, thus enabling the application of the averaging procedure. For almost all regular electron trajectories the solution is explicitly derived in asymptotic form and an adiabatic invariant is constructed, both results are in a form that remains uniformly valid over the time interval for electrons to transit the laser. The analytical results are verified by numerical calculations for an example problem.