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.
Highlights
Planar undulators for the production of synchrotron radiation have been built for more than 30 years
Þ k2xsin2ðkxxÞ=ð2kykÞ2: The achieved generating function (GF) is independent on the phase shift term ’; this dependency did vanish by the integration procedure
Undulator fields have to meet two independent requirements: (i) The local on-axis field distribution has to be optimized with respect to the phase error [38] to provide highest photon brightness even at higher harmonics. (ii) The off-axis field integrals have to be minimized for a negligible impact of the undulator on the electron beam dynamics
Summary
Planar undulators for the production of synchrotron radiation have been built for more than 30 years. Most of the third generation storage rings rely on the APPLE II design for variably polarizing photon sources The flexibility of this device provides other operation modes such as linear polarized light with an inclination with respect to the midplane. In contrast to conventional accelerator magnets, undulators have a complicated three-dimensional field distribution and the dynamic behavior is dominated by effects introduced by the oscillating electron trajectory caused by the periodic undulator fields These effects are usually called dynamic kicks or 2nd order kicks because they scale inversely with the square of the particle energy. In low energy rings such as BESSY II the dynamic field integrals are shimmed to minimize the impact of the undulators in terms of lifetime and injection efficiency.
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