Compensation of Variable Skew- and Normal Quadrupole Focusing Effects of Apple-ii Undulators with Computer-aided Shimming

Variable (phaseand gap-dependent) skewand normal-quadrupole focusing effects of APPLE-II undulators on electron beam are reportedly complicating practical use of this type of insertion devices in many synchrotron radiation sources. We show that these undesirable effects, whatever their physical “origin”, can be well controlled and in many cases efficiently compensated during the standard “virtual” shimming of APPLE-II undulators. Our method exploits small variations of the skewand normal-quadrupole focusing components resulting from extra magnetic interaction, introduced by displacements of permanent magnet blocks during the shimming procedure, at different undulator phase and gap values. These variations can be calculated to a high accuracy, included into the corresponding “shim signatures” of magnetic field integrals, and used, along with undulator magnetic measurements data, for calculation of the most efficient magnet displacements. This approach is well suited for a computer-aided (e.g. genetic optimization based) shimming procedure. Practical results obtained with several APPLE-II undulators, which are currently successfully operating on the SOLEIL storage ring, are presented.