Dynamic Effects

The ILC relies on several different feedback systems to mitigate the impact of dynamic imperfections on the luminosity. These feedback systems act on different timescales. The long ˜1 ms pulse length and relatively large bunch spacing (˜300 ns) makes it possible to use bunch-to-bunch (or intra-train) feedbacks located at critical points, the most important one being the beam-beam feedback at the interaction point which maintains the two beams in collision. Other feedback systems act from train to train (inter-train) at the 5 Hz pulse repetition rate of the machine. Over longer timescales (typically days or more) the beam may have to be invasively re-tuned.

The performance of the feedback systems is governed by the effective loop gain. A large gain (large bandwidth) is desirable to decrease the response time of the feedback; this is particularly true for the intra-train feedback, which reacts to each new pulse. A fast response time minimizes the number of initial bunches over which the feedback converges (normally a few percent effect). A low gain is desirable to reduce the amplification of high-frequency noise in the beam, and to effectively integrate away (average over) monitor resolution. The exact choice of gain is an optimization based on the noise spectrum being corrected (both in the beam and the monitors).

The number, type and location of feedback systems along the machine is also an opti-mization which is currently under study.

Very important sources of dynamic imperfections are ground motion and component vibration. The ground motion depends strongly on the site location. For the ILC-TRC study three ground motion models were developed, all based on measurements at existing sites:

Model A represents a very quiet site (deep tunnel at CERN); Model B a medium site (linac tunnel at SLAC); Model C a noisy site (shallow tunnel at DESY). A fourth model (K) was

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later developed based on measurements at KEK and is roughly equivalent to C. These models have been used in all subsequent simulations of the dynamic behavior of the ILC.

2.8.4.1 Bunch-to-Bunch (Intra-Train) Feedback and Feedforward Systems The damping ring extraction kicker extracts each bunch individually. If this kicker does not fully achieve the required reproducibility, the beam will have bunch-to-bunch variations that cannot be removed by an intra-pulse feedback system (effective white-noise). The feedforward system in the RTML is designed to mitigate this effect. The position jitter of each bunch is measured before the around and then corrected on that very bunch after the turn-around.

Quadrupole vibration in the downstream bunch compressor and (predominantly) in the main linac will induce transverse beam jitter (coherent betatron oscillations). The tolerance on the amplitude of this jitter (and hence on the quadrupole vibration) from the main linac itself is relatively relaxed. Quadrupole vibration amplitudes of the order of 100 nm RMS lead to negligible pulse-to-pulse emittance growth. However the resulting oscillation (one- to two-sigma in the vertical plane) in the BDS could lead to significant emittance degradation from sources such as collimator wakefields. An intra-train feedback at the exit of the linac solves this problem. In addition, this feedback could correct any residual static HOM disturbance in the bunch train. If the main linac quadrupole vibrations are significantly less than 100 nm (e.g. 30 nm RMS, as expected for a typical quiet site), then a intra-train feedback at the exit of the linac may not be required.

Small relative offsets of the two colliding beams, in the range of nanometers, lead to significant luminosity loss. The offsets are particularly sensitive to transverse jitter of the quadrupoles of the final doublet. Fortunately, the strong beam-beam kick causes a large mutual deflection of the offset beams, which can be measured using BPMs just downstream of the final quadrupoles. The intra-train feedback system zeros the beam-beam kick by steering one (or both) beams using upstream fast kickers. The system typically brings the bunch trains into collision within several leading bunches (depending on the gain). The IP fast feedback and the long bunch train also affords the possibility to optimize the luminosity within a single train, using the fast pair monitor as a luminosity signal [?].

Studies performed as part of the TRC indicated that in a quiet site (B or better) the fast beam-beam feedback and a slow orbit correction in the beam delivery system keeps the luminosity loss due to dynamic effects negligible [?, ?]. In a noisy site (e.g. C) some luminosity loss occurs.

2.8.4.2 Train-to-Train (5Hz) Feedback

The exact layout of the train-to-train feedback has not yet been finalized and different options are being studied. A simple but workable option is to use a number of local feedbacks. At certain locations in the machine a few correctors are used to steer the beam back through a few selected BPMs thus keeping the trajectory locally fixed. These feedback systems can be used in a cascaded mode where each of the feedback anticipates the trajectory change due to the upstream feedback systems. Such a system was successfully implemented at SLC.

Since the system corrects only locally, a residual of the dynamic imperfections will remain, due to deterioration of the trajectory between the feedback locations. After longer times this will require a complete re-steering of the machine back the exact trajectory determined from the initial beam-based alignment (gold-orbit).

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Other envisaged options are to perform permanent re-steering with a very low gain; this method avoids the additional layer of steering but may be slower than local feedback. A further option is the use of a MICADO-type correction. In this procedure all BPMs are used to determine the beam orbit. A small number of most effective correctors is identified after each measurement and these are used to correct the trajectory.

2.8.4.3 Feedback Performance (Luminosity Stabilization)

A complete and realistic simulation of the dynamic performance of the collider requires com-plex software models which can accurately model both the beam physics and the errors (e.g.

ground motion and vibration). The problem is further complicated by the various time scales which must be considered, which span many orders of magnitude: performance of the fast intra-train feedbacks requires modelling of the detailed 10 MHz bunch train; fast mechanical vibrations at the ˜Hz level need to be accurately modelled to test the performance of the pulse-to-pulse feedback systems; long-term slow drifts of accelerator components over many days must be studied to determine long-term stability and the mean time between invasive (re-)application of BBA. Ideally all these elements need to be integrated into a single simulation of the complete machine.

Progress towards such complete simulations is on-going. However, many simulations have already been made, which have focused on individual aspects of the problem (time-scales), with varying degrees of sophistication of the feedback models. The results thus far give every indication that the ILC can achieve and maintain the desired performance. For example:

• Extensive simulations have been made of the performance of the fast beam-beam (and other) intra-train feedback using a model of the main linac and BDS to generate realistic bunch trains [95, 97]. For realistic component vibration amplitudes, the results indicate that feedback can maintain the luminosity within a few percent of peak on a pulse-to-pulse timescale (5 Hz). See for example Figure 2.8-2. These results are in agreement with earlier studies [?, ?].

• Drifts of components on the timescale of seconds to minutes have been studied [?, ?].

Simulations of 5 Hz operation with all ground motion models, and assuming the beams are maintained in collision by the fast IP feedback, indicate a slow degradation in luminosity. This can be mitigated by pulse-to-pulse feedback, especially in the BDS, where the tolerances are tightest. Noisy sites (model C) showed the most pronounced effect, and would place most demand on the slower feedback systems.

• Longer term stability has been studied, assuming a variety of configurations for the slower pulse-to-pulse feedbacks. Studies of the main linac [?] using local distributed feedback systems indicate that the time between re-steering ranges from a few hours to a few days for ground motion models C and B respectively. After 10/200 days (models C/B) simple re-steering does not recover the emittance, at which point a complete re-tuning would be necessary.

• Recent dynamic studies integrating the main linac and BDS, again based on distributed local pulse-to-pulse feedback systems (including one in the BDS) and incorporating many error sources and comparing all ground motion models have been made [98]. The noisy sites (models C and K) show a luminosity reduction of up to ˜30%, coming almost entirely from the BDS. Based on results from earlier simulation of other collider designs

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FIGURE 2.8-2. Example of integrated dynamic simulations, showing the performance of the beam-beam intra-train feedback system with realistic beams and beam jitter (simulated from the Main Linac and BDS).

The histograms show performance over 100 seeds of random vibration motion: brown - achieved luminosity for an infinitely fast beam-beam feedback and no bunch-to-bunch variations (3% reduction from ideal);

blue performance including bunch-to-bunch variations (driven by long-range wakefields in the Main Linac);

red as blue but including a finite response time for the feedback (8% reduction from ideal). Taken from [95, 97].

(notably TESLA), it is expected that optimization of the BDS feedback configuration, together with possible additional stabilization of critical magnets, can recover a signif-icant fraction of the loss. By contrast, quiet sites (models A and B) show only a few percent loss for the configuration studied.

Figure 2.8-2 shows the results of running 200 such simulations with differing random seeds. The brown histogram shows the achieved luminosity for an infinitely fast feedback and no bunch-to-bunch variations, it is 3% below the case without dynamic effects. The blue histogram includes the bunch-to-bunch variations while the red one also includes the time to convergence, leading to an average luminosity 8% below nominal. By optimizing the feedback gain and the intra-pulse luminosity tuning strategy, one can hope to recover part of the 5%

additional loss.