Plasma-based accelerators offer high accelerating gradients and scalability through staging or long plasma sources, which makes them good candidates for future accelerator and collider concepts. Proton-driven accelerators in particular have the potential to bring particles to high energy in a single stage. In the quasilinear regime - where the plasma wake is only partially evacuated - a witness bunch of electrons drives a cavitated wake, which acts to preserve the emittance of the portion of the witness inside this self-blowout. In the case of a misalignment between the driver and witness, this behaviour can persist, but its effectiveness is reduced. In this paper, we study transverse witness dynamics in this regime, and develop analytical models to describe the witness motion, and develop a metric to estimate emittance preservation based on a single parameter which estimates the density of the witness after phase mixing. Particle in cell simulations using the AWAKE Run 2c baseline parameters show excellent agreement with the predictive models developed. This work allows alignment constraints to be set both for the AWAKE experiment and other wakefield acceleration schemes operating in the quasilinear regime.
Prospective designs of future particle accelerators look to push the energy frontier forward. In order to do this, the high acceleration gradients of plasma-based acceleraton are attractive [1,2]. While the accelerating gradient of conventional radiofrequency (RF) cavity machines are limited to the 10s of MeV m -1 due to RF breakdown, plasmas can support high accelerating gradients at the GeV m -1 level, and hence have the potential to reduce the real-estate footprint of future accelerator facilities [1,[3][4][5].
In order to use plasma as an accelerator, high electric fields must be set up by separating the electrons and ions. This requires a suitable driver, either an intense short-pulse laser [6] or an energetic particle beam [7]. This driver displaces plasma electrons as it propagates, while the much heavier ions move comparatively much less. This separation of charges induces electric fields which cause the plasma electrons to oscillate, and a travelling electron density wave with a phase velocity equal to the driver velocity follows the driver. This is a plasma wake, into which a ‘witness’ bunch of electrons may be injected and accelerated to high energy.
As the driver propagates through the plasma, energy is gradually transferred to the wake, and ultimately to the witness. If the driver propagates sufficiently far, it will lose a significant fraction of its energy and the wake will collapse, making acceleration no longer possible. Collider-relevant plasma accelerators will need to be hundreds to thousands of metres in length, which exposes them to the possibility of driver depletion.
Two main schemes address this problem. Either, periodically replacing the spent driver in order to continue the acceleration of the witness, in a process called ‘staging’. Staging requires both the extraction of the spent driver, and injection of a new one, whilst simultaneously preserving the witness [8].
Alternatively, by choosing a driver with a high enough energy, the depletion length is guaranteed to be longer than the accelerator. Proton beams are the only practical choice for such a scheme [9,10], with their extremely high mass allowing them to be accelerated with low synchrotron losses in existing circular collider infrastructure.
We consider here the latter, a proton-driven plasma wakefield accelerator.
Proton beams are the only currently-available driver which can reach the energy frontier in a single stage.
The AWAKE experiment [10,11] is a proof-of-principle accelerator using the 400 GeV proton beam from the CERN Super Proton Synchrotron (SPS) to accelerate electrons in plasma. The SPS beam is around 5 cm in length, and the plasma is quite sparse, having a plasma wavelength of ∼ 1 mm, thus the beam spans many plasma periods. The beam is initially too long to effectively drive a strong wake for acceleration, but nontheless the periodic focussing and defocussing action causes it to undergo self-modulation. The beam modulates over several metres into a train of microbunches [12,13] with periodicity on the scale of the plasma wavelength. The modulated beam is then suitable for resonant wakefield excitation.
As the proton beam modulates, the phase of the wake shifts. It is therefore undesireable to attempt acceleration until the wake has stabilised [14]. The AWAKE Run 2c experiment will separate the modulation stage from the acceleration stage physically, in two sequential plasma sources. This allows the proton beam to completely selfmodulate in the first, and then electrons to be injected into a stable wake in the second. Each stage will be 10 m long, with a gap of 1 m to allow for injection [11].
Building upon the proton driven accelerator concept, the ALiVE scheme [5] proposes to use a single short proton bunch [15,16] to drive the wakefield, thus avoiding the need for modulation entirely.
When accelerating electrons, it is typical to operate in the blowout regime. This is where the driver expels all plasma electrons behind it, leaving only the background ions. This is preferable not only because it produces the strongest possible accelerating fields for a given plasma density, but also because the focussing fields on the bunch are linear. By matching the beta function of the witness to the focussing field of the plasma, the witness emittance can be preserved [17]. However, proton-driven wakefields tend to operate in the quasilinear regime, where the wake is not fully evacuated, and so we cannot expect to preserve the witness emittance completely. This being said, the witness itself also drives wakefields and, when its charge density is sufficiently high, can drive its own blowout [18]. This affords some degree of emittance control. A schematic view of this is shown in Figure 1, where we depict a short driver, rather than a train of microbunches.
For an externally injected witness, we consider the effect of imperfect transverse alignment between the driver and witness at injection. This will be prese
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