Stability of longitudinal bunch length feedback for heavy-ion synchrotrons

Abstract

feedback systems to increase the stability and to improve the beam quality. In the longitudinal direction, important modes are the coherent longitudinal dipole and quadrupole oscillation. In this paper we present a new and rigorous approach to analyze the longitudinal feedback to damp these modes. The results are applied to the rf feedback loop at GSI that damps the quadrupole mode. The stability analysis is compared with simulations and is in good agreement with results of a beam experiment. Finally, we summarize practical implications for the operation of the feedback system regarding performance and stability. n ¼ 0. Early investigations of feedback for this mode are found in Refs. (6,7). Since then, considerable progress of digital hardware such as FPGAs has been made and most of the newly developed feedback systems rely on digital hardware. Recent works for the damping of the quadrupole mode are Refs. (8-10) for electron machines, Ref. (11) for a proton synchrotron, Ref. (12) for a heavy-ion collider, and Ref. (13) at GSI for a heavy-ion synchrotron. Thenew contributionof thispaper is arigorous approach for the modeling and analysis of the feedback loop, taking into account also nonlinearities due to the accelerating rf voltage. The structure of the paper is as follows. Section II describes the single-particle longitudinal dynamics in heavy-ion synchrotrons and basic notations. The bunch shape oscillations are defined and the control problem is formulated. In Sec. III, a feedback model for the bunch length oscillations is derived. The modeling approach is based on moments and is applicable for quite general non- linear accelerating voltages. Because the moments are not readily available for measurements, it is shown how they can be obtained from the measured beam current signal. The presented results are used to derive the closed-loop dynamics of a specific setup for the heavy-ion synchrotron SIS18 at GSI. The stability and performance of the feed- back is analyzed in Sec. IV. The analytical calculations are compared with nonlinear macro particle tracking simula- tions (cf. (14-16)) and experimental measurements. A tuning rule for the feedback is presented in Sec. V, the practical implications of the results are discussed in Sec. VI, and a conclusion is given in Sec. VII.