Electron orbits and stability in realizable and unrealizable wigglers of free-electron lasers

Abstract

Magnetostatic wiggler fields $\hat{x}coskz+\hat{y}sin\mathrm{kz}$, commonly used to model the undulators of free-electron lasers, violate Maxwell's equations and are "unrealizable." Realizable wigglers that approximate the unrealizable ones near the axis have a radial variation and an axial field component, both of which affect electron motion. Exact helical equilibrium orbits are given for relativistic electrons in a combined uniform guide field and realizable wiggler, in cylindrical geometry. The parameter $\mathrm{ka}$ that measures the size of the helix also measures the imparted quiver motion, on which the gain of the laser depends. Hence, wigglers that impart substantial quiver motion necessarily have electrons far from the axis, for which the unrealizable wiggler model is not valid. A linearized stability analysis shows that the equilibrium helical orbits are either strongly unstable or else exhibit a secular growth, linear in time. The trajectory of an electron that starts from given position and velocity in combined guide field and wiggler is also found from the perturbation analysis, with corrections for realizability and for harmonics of practical wigglers, such as a bifilar winding. Although the helical orbits are either strongly or weakly unstable, a class of nonhelical bounded orbits is found when the secular behavior is suppressed.