ELECTRON TRAJECTORIES IN A HELICAL FREE-ELECTRON LASER WITH AN AXIAL-GUIDE FIELD

Electronic trajectories in a free-electron laser consisting of a helic al wiggler magnetic field and a uniform guide field are studied using a three-dimensional approach. It is well known that, to any orbit, the re corresponds two conserved quantities. One is the energy, while the second, which we call P-z, is a consequence of the screw-displacement symmetry of the wiggler field. Depending on the value of P-z, the Hami ltonian, after a canonical transformation, may be shown to have a fixe d point which represents steady motion on an axially centered helical path of the same pitch as the wiggler. Expanding the Hamiltonian about the fixed point and retaining only quadratic terms, we obtain an appr oximate description of the motion in terms of two harmonic oscillators whose characteristic frequencies and normal modes are determined by t he value of P-z. Despite the simplicity of the dynamics, the nonlinear relations which link our oscillator variables to the Cartesian coordi nates and velocities provide a detailed description of the complex beh avior of the latter. Provided that the magnitudes of the oscillator am plitudes are not too large, our method yields trajectories in close ag reement with those computed numerically. Among the features encountere d is that in both group I, and with reversed-field operation, one of t he frequencies is negative, while in group-II operation a repulsion of the frequencies at a pseudocrossing leads to highly perturbed traject ories when the wiggler frequency is approximately half the cyclotron f requency. In favorable circumstances, which we specify, the transverse motion is accurately described by a superposition of three circular m otions; one corresponds to the fixed point, the second to the cyclotro nic motion, while the third is a very slow motion of the center of gyr ation. The axial velocity then shows ripple at approximately the diffe rence of cyclotron and wiggler frequencies. The spontaneous-forward-em ission spectrum peaks at the Lorentz-boosted wiggler and cyclotron fre quencies. Under less favorable circumstances, the motion we predict is more complicated, and the resulting forward-emission spectrum rather complex.