Longitudinal Beam Dynamics with a Higher-Harmonic Cavity for Bunch Lengthening
In Sect.1 we review the conditions of operation of an HHC system assuming its stability. We first discuss the longitudinal dynamics with an active HHC and the conditions that lead to a quartic potential in the limit of small oscillations, together with formulae giving the bunch lengthening factor and the synchrotron frequency dependence on the amplitude of the synchrotron oscillations. We discuss next passive HHC operations with uniform fillings, and distinguish the case of normal-conducting HHC, where optimal conditions of operation can be met, from the case of superconducting HHC, where optimal conditions of operation can be satisfied only approximately. In Sect.2 we address the stability of the passive superconducting 3HC system of the NSLS-II storage ring via self-consistent simulations with code SPACE. The case of nominal fractional filling of 80% is compared with the case of fractional filling of 90% and with the uniform filling case. It is shown that for values of the detuning frequency in the neighborhood of the good working point discussed in Sect.2, unstable regimes of oscillation are found, with detuning frequency threshold dependent on the gap in the filling. It is shown that the instability regime is characterized by dipole and quadrupole modes of oscillation exhibiting asymptotically nonlinear saturation. A discussion of the average bunch lengthening, together with the degree of uniformity across the bunch train, achievable under stable conditions as a function of the fractional filling concludes the paper.