End field formation in planar hybrid undulators to ensure gap independence of the first and second field integrals

Magnetic field amplitude in pure permanent magnet (PPM) or hybrid undulators is varied by changing the gap. It is necessary to ensure that for all gaps the deviation of the first and second field integrals from ideal be sufficiently small, especially in the XFEL project where dozens of undulator sections, each about 5-m long, are to be used. Since a high field is required, a hybrid undulator should be used. In this paper, field calculations for such an undulator were carried out using the “Radia” code. Unlike a PPM undulator scheme, hybrid ones require more sophisticated schemes of minimizing field integrals because of complicated redistribution of magnetic flux at the undulator ends with changing gap. Saturation of iron poles makes solution of the problem more difficult. Both effects were studied and several compensating methods were found. Parameters of the permanent magnets and iron poles of the first and last end-field-forming undulator periods were varied in order to change the form and mean value of the first and second integrals versus undulator gap.