Emittance growth suppression with a multibunch feedback in high-energy hadron colliders: Numerical optimization of the gain and bandwidth

A transverse feedback system can effectively mitigate the emittance growth caused by injection oscillations and machine noise in hadron beams. However, as its action on the beam depends on beam position measurements of finite accuracy, it introduces additional noise on its own. The machine noise is in general strongest at low frequencies. Hence, the feedback is less needed at high frequencies. In this paper, two theories for the reduction of the machine noise induced emittance growth rate, with a bunch-by-bunch feedback, have been extended to a multibunch feedback. The extended theories show quantitative agreement with sophisticated macroparticle simulations. The emittance growth caused by the beam position measurement noise is numerically found to be only weakly dependent on the feedback’s cutoff frequency, while it is strongly dependent on the single-bunch gain. The ultimate goal of this study is to find the optimal transverse feedback bandwidth and gain, determined by the minimization of the total emittance growth rate. The optimum depends on the ratio between the amplitudes of the beam position measurement error and the machine noise, the power spectrum of the machine noise, the response of the feedback filters, and the magnitude and details of the detuning. For the illustrative case of the Large Hadron Collier during collision in run 2, the optimum is found at the currently lowest possible cutoff frequency of 0.5 MHz, with a single-bunch damping time of approximately 270 turns. Using a chromaticity of 15 units, the minimal emittance growth rate at this cutoff frequency is 72% lower than with a bunch-by-bunch feedback. If the beam position measurement error can be reduced relative to the machine noise, the optimum will shift to larger single-bunch gains, or equivalently shorter single-bunch damping times.

Figure 1

Response functions of two types of filters in time domain in (a) and in frequency domain in (b), assuming a bunch separation of $\mathrm{\Delta }t=25\mathrm{ns}$. The sums of the response functions in time domain, over all neighbors maximally 32 slots away, are given in (c). The EXP filter is an exponential filter, while the ADT filter replicates the transverse feedback in the LHC, both of which are fully symmetric in time in the range $|b-b^{\prime }|\le 32$, although only $b^{\prime }-b\ge 0$ is shown here.

Figure 2

Emittance growth rate for all 128 neighboring bunches affected by only LF noise, using the EXP filter in (a) and the ADT filter in (b). The solid lines are the expected emittance growth rates, calculated with Eqs. (11) and (17).

Figure 3

Average emittance growth for 128 neighboring bunches, affected by either LF noise or BPM noise, using the EXP filter and the ADT filter. The solid lines are the expected emittance growth rates due to LF noise, calculated with Eqs. (11) and (17).

Figure 4

Average emittance growth for 128 neighboring bunches affected by LF noise of various $f_{\max }$, using the EXP filter in (a) and the ADT filter in (b). The solid lines are the expected emittance growth rates, calculated with Eqs. (11) and (17).

Figure 5

Average emittance growth for 128 neighboring bunches affected by LF noise, using the EXP filter in (a) and the ADT filter in (b). The solid lines are the expected emittance growth rates, calculated with Eq. (11) and either Eq. (17) for octupole detuning or Eq. (18) for beam-beam interactions.

Figure 6

Average emittance growth rate for 128 bunches separated by 25 ns, due to LF noise in (a), BPM noise in (b), and both in (c), in a configuration with a beam-beam interaction and $Q^{\prime }=0$. The black crosses mark the simulations. The area marked gray contains configurations $(f_{\text{cutoff}},g)$ for which the feedback loop is unstable. The configurations at the edge of the gray area were stable. The white curves are contours of constant maximum multibunch gain $g\tilde{N}$.

Figure 7

Average emittance growth rate for 128 bunches separated by 25 ns, due to LF noise in (a), BPM noise in (b), and both in (c), in a configuration with a beam-beam interaction and $Q^{\prime }=15$. The black crosses mark the simulations. The area marked gray contains configurations $(f_{\text{cutoff}},g)$ for which the feedback loop is unstable. The configurations at the edge of the gray area were stable. The white curves are contours of constant maximum multibunch gain $g\tilde{N}$.

Figure 8

Average emittance growth rate for 128 bunches separated by 25 ns, after halving the BPM noise currently in the LHC, in a configuration with a beam-beam interaction and $Q^{\prime }=0$ in (a) and $Q^{\prime }=15$ in (b). The black crosses mark the simulations. The area marked gray contains configurations $(f_{\text{cutoff}},g)$ for which the feedback loop is unstable. The configurations at the edge of the gray area were stable. The white curves are contours of constant maximum multibunch gain $g\tilde{N}$.