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

Electronic trajectories in a free-electron laser consisting of a helical wiggler magnetic field and a uniform guide field are studied using a three-dimensional approach. It is well known that, to any orbit, there 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 Hamiltonian, after a canonical transformation, may be shown to have a fixed 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 approximate description of the motion in terms of two harmonic oscillators whose characteristic frequencies and normal modes are determined by the value of P-z. Despite the simplicity of the dynamics, the nonlinear relations which link our oscillator variables to the Cartesian coordinates and velocities provide a detailed description of the complex behavior of the latter. Provided that the magnitudes of the oscillator amplitudes are not too large, our method yields trajectories in close agreement with those computed numerically. Among the features encountered is that in both group I, and with reversed-field operation, one of the frequencies is negative, while in group-II operation a repulsion of the frequencies at a pseudocrossing leads to highly perturbed trajectories when the wiggler frequency is approximately half the cyclotron frequency. In favorable circumstances, which we specify, the transverse motion is accurately described by a superposition of three circular motions; one corresponds to the fixed point, the second to the cyclotronic motion, while the third is a very slow motion of the center of gyration. The axial velocity then shows ripple at approximately the difference of cyclotron and wiggler frequencies. The spontaneous-forward-emission spectrum peaks at the Lorentz-boosted wiggler and cyclotron frequencies. Under less favorable circumstances, the motion we predict is more complicated, and the resulting forward-emission spectrum rather complex.