Quadrupole stability in nonuniform fill patterns

In previous publications, the effect of inhomogeneous beam loading of Landau cavities due to a nonuniform fill pattern in a storage ring has been studied with the aim of predicting the resulting longitudinal profiles of the bunches and their time offsets relative to the main rf [T. Olsson et al., Phys. Rev. Accel. Beams 21, 120701 (2018)]. This work was extended to treat dipolar coupled-bunch modes driven by beam-excited higher-order modes in the rf cavities [F. J. Cullinan et al., Phys. Rev. Accel. Beams 23, 074402 (2020)]. These coupled-bunch modes are of interest because they can become unstable, leading to an increase in the energy spread. The theory has now been extended once again to cover the case of coupled-bunch quadrupole modes where it is the lengths of the bunches that are oscillating, not their centroids. The theory is outlined and its predictions are thoroughly benchmarked against predictions from macroparticle tracking. Observations of the effects of nonuniform fill patterns on coupled-bunch quadrupole instabilities are made and interpreted. Results of measurements at the MAX IV 3 GeV ring, interpreted using theoretical calculations, are then presented to continue the investigation.

Figure 1

Bunch-centroid time offsets with beam loading of the HOM at the threshold shunt impedance of the quadrupole instability compared to the case with no HOM beam loading at all. The results are for a case where 165 out of 176 rf buckets are filled and the Landau cavities are detuned by $+150\mathrm{kHz}$.

Figure 2

RMS oscillation of the bunch length during macroparticle simulations indicating the stability of a HOM-driven coupled-bunch quadrupole instability compared to theoretical predictions of the threshold shunt impedance against the detuning of the Landau cavities. Black indicates that no simulation was run.

Figure 3

Threshold shunt impedances above which a HOM-driven coupled-bunch quadrupole instability develops for different numbers of filled rf buckets as determined by theory compared to the bunch length stability observed in macroparticle tracking shown on a color scale where black indicates that no simulation was run.

Figure 4

Threshold shunt impedances above which a HOM-driven coupled-bunch quadrupole instability develops as determined by theory compared to the bunch length stability observed in macroparticle tracking shown on a color scale for different HOM frequencies and for three cases: a Landau-cavity detuning of 75 kHz and a 160 buckets filled (left), the same fill with a Landau-cavity detuning of 150 kHz (center) and a 150 kHz detuning with 165 buckets filled (right). Black indicates that no simulation was run.

Figure 5

Beam spectra as measured by the phase detection of the bunch-by-bunch feedback system along with the spectrum of coupled-bunch mode 138 and the frequency response of the FIR filter used in the feedback loop.

Figure 6

Threshold Landau-cavity voltage for different fill patterns in the MAX IV 3 GeV ring and calculation of the same for the example HOM and for the HOM shifted in frequency so that it drives coupled-bunch mode 138 when the fill pattern is uniform.

Figure 7

Threshold Landau-cavity voltage for different fill patterns in the MAX IV 3 GeV as calculated for different HOM shunt impedances and quality factors. The measured curve is also included for reference.

Figure 8

Threshold Landau-cavity voltage for different fill patterns in the MAX IV 3 GeV as calculated for different frequency offsets of the HOM from resonance. The measured curve is also included for reference.

Figure 9

Threshold Landau-cavity voltage for different fill patterns in the MAX IV 3 GeV as calculated for HOM parameters chosen to reproduce the measured curve.