Accurate analytic off-axis electron trajectories in realistic two-dimensional transverse wigglers with arbitrary magnetic field variation.

Abstract

For several reasons, it is essential to have analytic descriptions of single-electron trajectories for wigglers and undulators. (a) They give a better insight into the complicated dependence of the electron motion on quantities like the injection conditions or the wiggler parameter K; (b) one can hope to calculate the radiation pattern from a particular wiggler or undulator only if at least the single-electron velocity is given in closed form; (c) analytic single-particle trajectories are an essential starting point for investigating advanced problems like stability analyses and free-electron laser performance. Therefore, single-electron motion was studied in wigglers with two-dimensional magnetic fields that are transverse and periodic but otherwise arbitrary on axis, and satisfy Maxwell's equations off axis. In this class of fields, the energy and one component of the canonical momentum are conserved and can be used to reduce the trajectory problem to the integration of a single complicated second-order ordinary differential equation, which, for all realistic electron-beam emittances and for all realistic values of ${\ensuremath{\delta}}_{w}$=K/\ensuremath{\gamma}, is weakly nonlinear and lends itself to a two-level two-scale analysis, through which an accurate analytic description of the complete electron trajectory can be established. This analytic trajectory exhibits all the features that one expects to find in off-axis motion, such as small-amplitude rapid wiggler oscillations superimposed on large-amplitude slowly varying betatron oscillations. The accuracy of this analytic trajectory was tested for a sinusoidal on-axis field variation by comparison with a numerical trajectory, and it was found that even for far-off-axis electrons, the agreement was excellent, typically on the order of one part in a thousand. This high accuracy in the off-axis regime is an important feature, since in many devices real transverse beam dimensions, initial beam velocity spreads, lateral beam injection, and a small undulator period ${\ensuremath{\lambda}}_{0}$ lead to electrons traveling considerably off axis.