First- and second-order motion through the fringing field of a bending magnet

Abstract With the assumption that the fringing field of a straight-edged bending magnet is two-dimensional, exact expression are derived for all the first- and second-order matrix elements corresponding to motionin the median symmetry plane. The analogous matrix elements for skew trajectories are related to the solution of an explicitly given differential equation. Each matrix element is expanded to three terms in powers of the reciprocal of the particle stiffness. All expressions are checked by comparison with numerical integrations of the trajectory equations for a typical field distribution.