Electron Phase Slip in an Undulator with Dipole Field and BPM Errors

A statistical analysis of a corrected electron trajectory through a planar undulator is used to predict the optimal beam position monitor (BPM) spacing. The undulator is composed of multiple modular sections, each containing many dipoles with random field strength and roll angle errors. Located between each section are inaccurate BPMs, steering correctors, and quadrupole magnets. An analytical formula for electron-to-photon phase errors is derived and is also used to estimate the optimum BPM spacing. The rms trajectory amplitude is also predicted and the results are applied to the LCLS FEL undulator where the requirements on electron trajectory straightness are very demanding.