Influence of higher order modes on the beam stability in the high power superconducting proton linac

Higher order modes (HOMs) can severely limit the operation of superconducting cavities in a linear accelerator with high beam current, high duty factor, and complex pulse structure. The full HOM spectrum has to be analyzed in order to identify potentially dangerous modes already during the design phase and to define their damping requirements. For this purpose a dedicated beam simulation code simulation of higher order mode dynamics (SMD) focused on beam-HOM interaction was developed, taking into account important effects like the HOM frequency spread, beam input jitter, different chopping patterns, as well as klystron and alignment errors. Here, SMD is used to investigate the influence of HOMs in detail in the superconducting proton linac at CERN and their potential to drive beam instabilities in the longitudinal and transverse plane.

Figure 1

Period layout in the medium beta section.

Figure 2

Period layout in the high beta section.

Figure 3

$(R/Q{)}_n(\beta )$ maps of the monopoles and dipoles with high $(R/Q{)}_n$ values in the analyzed symmetric SPL cavities.

Figure 4

2D phase space histogram of one pulse at the end of the linac, which is not disturbed by HOMs. This distribution will be used as a reference later on. Settings: no HOM, $N=1$, see Table .

Figure 5

2D phase space histogram at the exit of 1000 linacs, where always the same pulse was tracked through, but the seed for the HOM frequency distribution was varied. Settings: $N=1000$, see Table .

Figure 6

Average (black) and maximum (red) HOM voltage present in the cavities after two pulses. The $(R/Q{)}_n(\beta )$ dependency is clearly visible in the medium beta section. Settings: $N=1000$, see Table .

Figure 7

2D phase space histogram at the exit of 1000 linacs with rf errors, but no HOM present. The phase space area is increased significantly compared to Fig. 5. Settings: uniform distributed rf errors (0.5° in phase and 0.5% in magnitude), no HOM, $N=1000$, see Table .

Figure 8

Average longitudinal phase space increase and deviation of 100 linacs against the damping $Q_{\mathrm{ex}}$ for different $⟨I_b⟩$. ${ϵ}_{\mathrm{HOM}}/ϵ$ is proportional to $⟨I_b{⟩}^2$. The plateau around $Q_{\mathrm{ex}}\approx 4\times {10}^7$ is due to the pulse period structure. Settings: $N=100$, see Table .

Figure 9

Average longitudinal phase space increase and deviation as a function of the Gaussian bunch to bunch charge scatter. The phase space increases approximately with ${\sigma }_q^2$. Settings: $N=100$, see Table .

Figure 10

Average (solid, blue) and maximum (dashed, red) longitudinal phase space increase against the HOM frequency spread for 100 linacs. Settings: $N=100$, see Table .

Figure 11

Average (solid, blue) and maximum (dashed, red) longitudinal phase space increase as a function of the HOM frequency spread at the 4th machine line. Losses occur in the case of ${ϵ}_{\mathrm{HOM}}/ϵ>100$. Settings: $⟨f_{\mathrm{HOM}}⟩=1408.8\mathrm{MHz}$, $⟨I_b⟩=40\mathrm{mA}$, $N=100$, see Table .

Figure 12

Average (solid, blue) and maximum (dashed, red) longitudinal phase space increase as a function of the distance to 4th machine line. Settings: $⟨f_{\mathrm{HOM}}⟩$ shifted, $⟨I_b⟩=40\mathrm{mA}$, $N=100$, see Table .

Figure 13

Average (hollow dots) and maximum (solid dots) longitudinal phase space increase as a function of the damping, where a HOM is resonantly excited. The average increase due to rf errors is indicated with the dashed line. Settings: $⟨f_{\mathrm{HOM}}⟩=1408.8\mathrm{MHz}$, $N=100$, see Table .

Figure 14

Longitudinal phase space increase as a function of the cavity, where the HOM at the 8th machine line is located for different damping. The average increase due to rf errors is indicated with the dashed line. An oscillating phase space increase is observed, which decreases further downstream in the linac. The oscillation is due to the used injection noise pattern. Settings: $f_{\mathrm{ML}}=2817.6\mathrm{MHz}$, $(R/Q{)}_{\mathrm{ML}}=1\Omega$, $N=1$, other cavities use default HOM configuration, see Table .

Figure 15

Longitudinal phase space increase against the HOM frequency, where a HOM falls on a chopping machine line at nominal current and $Q_{\mathrm{ex}}={10}^7$ for different chopping patterns. The rf induced growth (black dashed line) is indicated and the found monopole modes (violet dashed lines) in the investigated frequency range. Settings: chopping, $⟨f_{\mathrm{HOM}}⟩$ shifted, $⟨I_b⟩=40\mathrm{mA}$, $N=1$, see Table .

Figure 16

Same as Fig. 15, but with strong HOM damping. Settings: chopping, $⟨f_{\mathrm{HOM}}⟩$ shifted, $⟨I_b⟩=40\mathrm{mA}$, $Q_{\mathrm{ex}}={10}^5$, $N=1$, see Table .

Figure 17

2D phase space histogram at the exit of 1000 linacs, where rf errors and one HOM is present. There is no significant change in the phase space histogram compared to Fig. 7. Settings: uniform distributed rf errors (0.5° in phase and 0.5% in magnitude), $N=1000$, see Table .

Figure 18

Average longitudinal phase space increase as function of the rf error with and without a HOM. Settings: uniform distributed rf errors, $N=1000$, see Table .

Figure 19

Average (hollow dots) and maximum (filled dots) phase space increase against the damping $Q_{\mathrm{ex}}$ at nominal current for different cases, where rf errors (rf) and/or a HOM at a machine (ML) or off resonance (HOM) is present. The average (dashed line) and maximum (solid line) increase due to pure rf errors is also included. Settings: uniform distributed rf errors (0.5° in phase and 0.5% in magnitude), $⟨I_b⟩=40\mathrm{mA}$, $⟨f_{\mathrm{HOM},\mathrm{ML}}⟩=1,408.8\mathrm{MHz}$, $N=100$, see Table .

Figure 20

$(R/Q{)}_n(\beta )$ map of the ${\mathrm{TM}}_{010}$ modes in the studied cavity. The dashed line is the $(R/Q{)}_n$ value of the mode at the geometrical beta, which can differ several orders of magnitude from the maximum $(R/Q{)}_n$ in the covered velocity range.

Figure 21

Average longitudinal phase space increase and upper RMS error against the damping $Q_{\mathrm{ex}}$ caused by the ${\mathrm{TM}}_{010,4/5\pi }$ modes and different currents. Above the rf limit (dashed line) lossy runs occur. Settings: ${\mathrm{TM}}_{010,4/5\pi }$ mode, ${\sigma }_{f_n}=10\mathrm{kHz}$, $N=100$, see Table .

Figure 22

Average longitudinal phase space increase and upper RMS error against the damping $Q_{\mathrm{ex}}$ caused by the ${\mathrm{TM}}_{010,3/5\pi }$ mode and ${\mathrm{TM}}_{010,2/5\pi }$ mode and a chopped beam, where the mode frequency is shifted to the chopping machine line. The phase space increase due to rf errors is indicated with the dashed line. Settings: passband modes, $⟨f_n⟩=699.998\mathrm{MHz}$, ${\sigma }_{f_n}=10\mathrm{kHz}$, chopping pattern: $50/80$, $N=10$, see Table .

Figure 23

Average longitudinal phase space increase and RMS error against the damping $Q_{\mathrm{ex}}$ caused by the ${\mathrm{TM}}_{010,3/5\pi }$ mode and ${\mathrm{TM}}_{010,2/5\pi }$ mode and a chopped beam. Settings: passband modes, ${\sigma }_{f_{\mathrm{HOM}}}=10\mathrm{kHz}$, chopping pattern: $50/80$, $N=10$, see Table .

Figure 24

Average longitudinal phase space increase against the damping $Q_{\mathrm{ex}}$ caused by the ${\mathrm{TM}}_{010,3/5\pi }$ mode and ${\mathrm{TM}}_{010,2/5\pi }$ mode. Settings: passband modes, ${\sigma }_{f_n}=10\mathrm{kHz}$, $N=10$, see Table .

Figure 25

2D phase space plot of the second pulse at the exit of the linac without dipole modes present. This distribution is used as a reference for subsequent studies. Settings: no HOM, $N=1$, see Table .

Figure 26

2D phase space plot of the second pulse at the exit of the linac with one dipole mode present for 1000 linacs. Settings: $N=1000$, see Table .

Figure 27

Average (black) and maximum (red) transverse HOM voltage present in the cavities after two pulses for 1000 linacs. Settings: $N=1000$, see Table .

Figure 28

Average transverse phase space increase of the beam against the damping $Q_{\mathrm{ex}}$ and different beam currents $⟨I_b⟩$ where one HOM per cavity is present for 100 linacs. The influence of the HOM rises above $Q_{\mathrm{ex}}={10}^5$ and is proportional to $⟨I_b{⟩}^2$. The plateau around $Q_{\mathrm{ex}}={10}^7$ is due to the pulse period structure. Settings: $N=100$, see Table .

Figure 29

Average effective transverse phase space growth of the beam as a function of the Gaussian bunch to bunch charge scatter. Settings: $N=100$, see Table .

Figure 30

Average and maximum transverse phase space increase dependency on the HOM frequency spread. Settings: $N=100$, see Table .

Figure 31

Average transverse phase space increase against the damping $Q_{\mathrm{ex}}$ and for different chopping patterns on (ML) and off a resonance. The error bars are only plotted for the $50/80$ chopping pattern and about the same for the other cases. Settings: chopping, $⟨f_{\mathrm{HOM}}⟩$ partly shifted (see Table ), $N=100$, see Table .

Figure 32

Average transverse phase space increase against cavity alignment errors (red crosses with error bars). A second order polynomial is fitted through the data (dashed line). For comparison the phase space increase without cavity displacement (blue line) is added. Settings: $N=100$, see Table .

Figure 33

Average ${ϵ}^{*}$ growth in case of an off-axis injected beam with a HOM present and normalized with same off-axis beam without HOMs. The phase space increases with the injection offset, but is fare less than the deviation $\sigma ({ϵ}^{*})=0.28\%$ introduced by different HOM frequency patterns. Settings: $N=100$, see Table .

Figure 34

Maximum observed HOM voltage in the high beta section as function of the injection error and a linear fit through the data. Settings: $N=100$, see Table .

Figure 35

Cavity with labels of the shape parameters used in Table .