Instability of relativistic-electron helical trajectories in combined uniform axial and helical wiggler magnetic fields

Abstract

The stability is investigated of helical trajectories of relativistic electrons in combined axial guide and helical wiggler magnetic fields appropriate for studies of free-electron lasers without using the frequently invoked assumption of evaluating the wiggler field on the laser axis. This is the first analysis of such trajectories which is free of this approximation. It is found that the helical trajectories are unstable with the possible exception of one value of the orbital radius. The instabilities are of two types corresponding to different properties of the eigenvalues of the matrix associated with the linearization of the equations of motion. These properties, in turn, depend upon the relative signs and magnitudes of the physical parameters that occur in the problem. At the exceptional radial value the trajectories can be either unstable or linearly stable depending upon the values of these parameters. For the linearly stable situations it is not known whether the trajectories are nonlinearly stable or unstable. We also prove that the trajectories of one of the instability types are of a particular kind called conditionally stable. This type of unstable trajectory has the property that it is approached by some (but not all) solutions of the equations of motion as time approaches positive or negative infinity. This type of instability is found to be more severe for larger values of the orbital radius. Even so, it still appears to be significant for small radii, of the order of $\frac{1}{100}$ of the wiggler wavelength.