Response of colliding beam-beam system to harmonic excitation due to crab-cavity rf phase modulation

During 2008 and 2009 dedicated beam experiments with crab cavities were performed in the KEKB. The goal was to measure the impact of crab-cavity radio frequency (rf) noise on the beam quality. These experiments were performed as a validation of the crab-cavity beam dynamics models in view of the possible use of crab cavities in the upgrade of the CERN Large Hadron Collider (LHC). An unexpected strong beam-beam instability was observed during the course of the experiments as a kind of frequency response. Understanding this finding required extensive multiparticle and single particle simulations plus an extra experimental session to consolidate the observations.

Figure 1

Luminosity versus HER crab-cavity noise as extrapolated to IP displacement. The excitation frequency (0.522 in tune units) is close to the HER horizontal tune. The HER IP nominal horizontal beam size is $180\mu \mathrm{m}$.

Figure 2

Luminosity versus LER crab-cavity noise as extrapolated to IP displacement. The excitation frequency (0.523 in tune units) is close to the LER horizontal tune. The LER IP nominal horizontal beam size is $160\mu \mathrm{m}$.

Figure 3

Simulated rms beam size of the electron beam distribution versus turn number for different noise amplitudes, clearly revealing the existence of a threshold for the onset of the instability.

Figure 4

Illustration of horizontal spectra of the centroid motion for the different 11 excitation frequencies. The IP equivalent excitation amplitude is $1\mu \mathrm{m}$ for all excitation frequencies.

Figure 5

Simulated relative luminosity versus frequency of the various excitation amplitudes.

Figure 6

Horizontal amplitude response of the positron beam to the harmonic excitation frequency and amplitude.

Figure 7

Response of $\pi$ and $\sigma$ mode amplitudes for the harmonic excitation frequency and amplitude.

Figure 8

Simulated time evolution of the driven spectral lines of the horizontal motion of the electron and the positron beams for an unstable case (CC noise at IP is $5\mu \mathrm{m}$ with frequency 0.523). The phase difference between the driven spectral lines of the two beams is shown versus turn number. The amplitudes of the $e^{-}$ and $e^{+}$ motion at the same frequency are also shown.

Figure 9

Betatron phase difference between positron and electron beams for the excitation frequency and amplitude.

Figure 10

Response of positron beam size for the harmonic excitation frequency and amplitude.

Figure 11

Illustration of the dependence of beam-beam eigenmodes on the beam oscillation amplitude using a rigid bunch model. Top: horizontal spectrum for three cases with increasing initial collision offset showing a reduction of the $\pi$-mode frequency. Other spectral lines generated by the nonlinearity of the beam-beam interaction are observed for the $2.5\sigma$ and $1.0\sigma$ cases. Bottom: $\pi$- and $\sigma$-mode frequencies versus initial collision offset.

Figure 12

Relative luminosity versus frequency of the harmonic excitation for two excitation amplitudes, both from measurements and simulations. The experimental points are connected with lines to guide the eye where considered appropriate.

Figure 13

Observed coherent oscillations versus amplitude of the CC harmonic excitation. Both quantities expressed as equivalents at the IP location. The nominal IP horizontal beam size is $160\mu \mathrm{m}$.

Figure 14

Luminosity versus HER crab-cavity noise (harmonic excitation) as extrapolated to IP displacement. The excitation frequency (0.599 in tune units) is close to the HER vertical tune. The nominal IP horizontal beam size is $180\mu \mathrm{m}$.

Figure 15

Luminosity versus LER crab-cavity noise (harmonic excitation) as extrapolated to IP displacement. The excitation frequency (0.577 in tune units) is close to the LER vertical tune. The nominal IP horizontal beam size is $160\mu \mathrm{m}$.