Symplectic tracking and compensation of dynamic field integrals in complex undulator structures

Abstract

In first approximation storage ring multipole magnets are described as simple two-dimensional magnet structures and many linear and nonlinear beam optic features of a magnet lattice can already be derived from this model. In contrast, undulators, and in particular variably polarizing devices, employ complicated three-dimensional magnetic fields which may have a severe impact on the electron beam, in particular, in low energy third generation storage rings. A Taylor expanded generating function method is presented to generate a fast, flexible, and symplectic mapping routine for particle tracking in magnetic fields. This method is quite general and is based on the solution of the Hamilton-Jacobi equation. It requires an analytical representation of the fields, which can be differentiated and integrated. For undulators of the APPLE II type, an accurate analytic field model is derived which is suitable for the tracking routine. This field model is fully parametrized representing all operation modes for the production of elliptical or linear polarized light with an arbitrary inclination angle or even arbitrary polarization. Based on this field model, analytic expressions for 2nd order kicks are derived. They are used to estimate the influence of APPLE II undulators on the electron beam dynamic. Furthermore, an analytic model for the description of shims is given. The shims are needed for field and performance optimization. Passive and active shimming concepts for the compensation of linear and nonlinear effects of variably polarizing undulators are discussed.