## Abstract

The status and evolution of the electron beam-driven Plasma Wakefield Acceleration scheme is described. In particular, the effects of the radial electric field of the wake on the drive beam such as multiple envelope oscillations, hosing instability and emission of betatron radiation are described. Using ultra-short electron bunches, high-density plasmas can be produced by field ionization by the electric field of the bunch itself. Wakes excited in such plasmas have accelerated electrons in the back of the drive beam to greater that 4 GeV in just 10 cm in experiments carried out at the Stanford Linear Accelerator Centre.

Although there were four distinct plasma-particle acceleration schemes based on the type of driver pulse used to excite the plasma wake (Joshi & Katsouleas 2003), a consensus is emerging that the highly nonlinear, but also the most robust, blow-out or bubble regime (Barov *et al*. 1998; Joshi *et al*. 2002) is probably best suited for reproducible particle acceleration. This regime was first studied in connection with a highly relativistic particle beam driver where the beam density exceeded the plasma electron density. When such a beam propagates through a plasma, the plasma electrons are blown predominantly transversely outward. If the beam is approximately *πc*/*ω*_{p} long, then these transversely blown out electrons rush back behind the electron beam because of the Coulomb attraction provided by the relatively immobile ions. Here *ω*_{p} is the plasma frequency. The electrons converge on the beam axis to produce a density spike, overshoot and set up a wakefield oscillation. The electron density contours in three dimensions resemble a series of balloons or bubbles; hence, the name the bubble regime (see figure 1). Eventually, the wake becomes turbulent because of phase mixing of the electron orbits.

The same process can occur when an intense but short laser pulse propagates through the plasma (Pukov & Meyer-ter-Vehn 2002). In this latter case, it is the transverse ponderomotive force of the laser pulse that expels the plasma electrons rather than the space-charge or Coulomb force of the beam. Since the ponderomotive force acts locally on the plasma electrons, as opposed to the Coulomb force, which can act over a distance of roughly *c*/*ω*_{p}, the bubble radius is approximately the laser beam radius. In the electron beam case, the bubble radius can be much larger than the electron beam radius.

In this paper, we summarize the results obtained thus far on the beam-driven Plasma Wakefield Accelerator (PWFA) scheme. Many phenomena that have been studied in developing this scheme have counterparts in the laser-driven wakefield accelerator scheme. In particular, we focus on three of the important transverse effects that affect both the driver and the accelerating beam:

The necessity of matching the beam beta function in vacuum to its value in plasma.

Transverse stability of the drive beam against the so-called hosing instability.

Emission of betatron radiation and its eventual implication at ultra-high energies.

## 1. Beam matching

As mentioned earlier when the beam density, *n*_{b}, exceeds the plasma density, *n*_{p}, all the plasma electrons are expelled by the beam. The resulting ion column has a radius , where *σ*_{r} is the r.m.s. transverse size of the beam and *n*_{bo} is the peak beam density. For a Gaussian beam, , where *σ*_{z} is the r.m.s. longitudinal length of the beam and *N* is the total number of particles in the beam.

This ion channel exerts a radial focusing force that is linear in *r*. The radial electrostatic field is given by . Here *e* and *ϵ*_{o} are charge on the electron and permittivity of free space, respectively. This electric field will tend to focus the rest of the electron beam that resides in the ion channel. If the density length product of the plasma is large enough, the electron beam can focus within the plasma itself and indeed undergo multiple focusing oscillations known as betatron oscillations (see figure 2*a*) of its envelope (Clayton *et al*. 2002). Clearly, this is an undesirable situation, because the beam exiting the plasma can suffer emittance growth and radiate a significant amount of its energy as betatron or synchrotron radiation, as we shall see later. It is therefore important to match the beam to the plasma such that its radius remains constant as it propagates over many betatron wavelengths. This happens if the beam expansion due to its emittance force is exactly balanced by the focusing force of the ion column. The behaviour of the electron beam with a normalized emittance *ϵ*_{N} is described by the beam envelope equation(1.1)where *k*=*ω*_{p}/(2*γ*)^{1/2}*c* is the restoring constant of the plasma, or equivalently the betatron wavenumber *k*_{b}. *γ* is the relativistic Lorentz factor and *c* is the speed of light. The beam is matched if the beam radius is such that *β*_{beam}=1/*k*=*β*_{plasma}. This matched beam radius *r*_{bm} is found by setting *σ*″(*z*)=0 leading to *r*_{bm}=(*ϵ*_{N}/*γk*_{p})^{1/2}. For a plasma of a given length, if the emittance force initially is larger than the plasma focusing force, the beam radius will oscillate but the amplitude of these oscillations will be reduced, as the plasma density and therefore the plasma focusing force (or equivalently the restoring constant in equation (1.1)) is increased. Figure 2*b* shows the experimental results on the variation of spot size of the 28.5 GeV electron beam containing 1.4×10^{10} electrons after it propagated through an approximately 1.4 m long lithium plasma as the plasma density is increased. The spot size oscillations measured on an external screen damp out as expected, indicating that beam beta function is being matched to that of the plasma at this point. The radiation loss as well as initial noise level for hosing is minimized at this density.

## 2. The hosing instability

Both the laser wakefield and the plasma wakefield accelerators are robust against most parametric type laser–plasma or beam–plasma instabilities. This is because the drive beam is only about half a plasma wavelength (period) long. However, in both cases there is one transverse instability that can affect the stable propagation of the drive beam. This is the so-called hosing instability (Geraci & Whittum 2000). In the case of the electron beam driver, the hosing instability can lead to the growth of transverse perturbations on the beam due to the nonlinear coupling of the beam electrons to the plasma electrons at the edge of the ion channel through which the beam propagates. As a result, these perturbations grow nonlinearly along the beam leading to transverse break-up of the beam. The differential equations that describe the coupling between the centroid offsets of the beam slice *x*_{b} and the centroid offset of a pre-formed ion channel *x*_{c} at a position *ξ* within the beam are:(2.1)where *ξ*=*z*/*c*−*t*, *s*=*z* and . In the asymptotic limit, the displacement *x*_{b} (*s*,*ξ*) of the longitudinal slice of the beam with an initially linear head-to-tail tilt *x*_{o} is given by(2.2)where the factor . For instance, for the beam parameters of the ongoing PWFA experiments at Stanford Linear Accelerator Centre (SLAC), *γ*=56 000, *n*_{e}=3×10^{17} cm^{−3}, *ξ*=10^{−13} *s* and *s*=30 cm, we get *x*_{b}/*x*_{o}=155, a very substantial growth indeed. In fact, even if *x*_{o}=10^{−2}*σ*_{r}, this amount of growth is larger than the size of the blow-out radius *r*_{i} given earlier for *n*_{bo}/*n*_{p}∼1.

Fortunately, in the PWFA experiments the hosing growth is found to be substantially less than the predictions of the theory for a preformed ion channel (Dodd *et al*. 2002). There are several reasons for this, including formation of the plasma by field ionization, non-constant ion channel radius, asymmetric beam sizes and emittances in the two planes, etc. However, all these effects can be modelled using three-dimensional, particle-in-cell (PIC) code simulations. One finds that the hosing growth is substantially less than expected from idealized theory (M. Zhou 2005, personal communication).

Figure 3*a*,*b* shows two energy spectra of the beam taken using an imaging spectrometer, with and without the plasma. The beam has both a head-to-tail energy chirp (head having greater energy than the tail) and a head-to-tail tilt in the transverse direction. After traversing just a 30 cm long, approximately 6×10^{16} cm^{−3} density plasma, the beam energy spectrum is considerably modified. More important, however, in this context is the fact that the ‘tail’ of the beam not only gains energy, but is also being amplified in its *x*-tilt. Current experiments are aimed at quantifying the hosing growth and comparing it with theory.

## 3. Emission of betatron radiation

As mentioned earlier, an unmatched beam undergoes betatron oscillations of its spot size if the plasma density–length product is sufficiently large. In a plasma with density *n*_{p}, the beam with energy *γmc*^{2} will radiate frequencies *ω*_{r} given by(3.1)where *q* is the harmonic number, and *Ω*≪1 is the observation angle from the beam axis. The radiation is emitted in a cone angle approximately *α*/*γ*, where *α* is the effective wiggler strength given by *α*=*γk*_{β}*r*_{o} which for an ultra-relativistic electron beam can be much less than 1. Furthermore, particles at different radial locations *r*_{o} have different *α* and therefore radiate different frequencies. Consequently, for a beam of electrons with *α*≫1, the spectrum typically resembles bending magnet or synchrotron spectrum with a cut-off or critical frequency given by(3.2)

When a 30 GeV electron beam is propagated through a plasma with 10^{13}<*n*_{e}(cm^{−3})<10^{17}, the radiated peak lies in the range 10<*hν* (keV)<10 MeV energy range (Wang *et al*. 2002; see figure 4).

The electron energy loss is found from the relativistic Larmor formula. Using the betatron orbits due to the radial electrostatic potential, the energy loss as a function of distance is given by(3.3)

In the ongoing beam–plasma wakefield accelerator experiments at SLAC, *n*_{e}<3×10^{17} cm^{−3}, transverse r.m.s. spot size, *σ*_{r}=10 μm and the beam energy is 28.5 GeV. Using these parameters and setting *r*_{o}=*σ*_{r}, we find that *α*=173 with *ω*_{c}≅50 MeV on-axis and an energy loss rate to betatron radiation of 4.3 GeV m^{−1}.

It is clear that for any future high-energy physics applications of either the beam or the laser–plasma wakefield accelerator, the accelerating beam must be extremely narrow, such that the rate at which the particles energy gain far exceeds the rate at which they radiate away the energy.

Recently, we have been considering how the betatron radiation loss might be advantageously used in a future collider. Since one can copiously produce photons in the 10 MeV range, it is natural to ask if these photons can be efficiently converted into e^{+}e^{−} pairs in a thin, high *z* target. Our calculations show that it is indeed possible to produce a high yield of positrons (up to several e^{+} per incident electrons) using a target that is roughly 0.5 radiation thickness thick and that the conversion efficiency is largely independent of the material (Chao 1998). This idea is currently being tested in our experiments at SLAC.

## 4. High gradient acceleration of electrons

Using extremely short electron bunches, *σ*_{z}∼35 μm, containing up to 1.5×10^{10} electrons per bunch, we were able to generate plasma by using the electric field of the bunch (see figure 5) itself (O'Connell *et al*. 2005). At the optimum density of 3×10^{17} cm^{−3}, the excited wakefield had a short enough wavelength to accelerate the particles at the back of the bunch by more than 4 GeV in less than 10 cm of plasma length (Hogan *et al*. 2005). Up to 8% of the beam particles, containing approximately 240 pC of charge, were accelerated to greater than 1.5 GeV in these experiments. The maximum energy gain was limited by the plasma length, which in turn, had to be made short enough so that the beam line downstream of the magnetic spectrometer could still handle the dispersed beam.

## 5. Future prospects

In the upcoming experiments, the energy acceptance aperture of the beam line has been increased to accept ±10 GeV energy changes to the beam. Now it is possible to explore if the energy gain continues to scale linearly with the length of the plasma or if the hosing instability would reduce the energy gain of particles in the tail of the beam. If these experiments continue to show the linear scaling of energy gain with plasma length, we expect to do a two bunch experiment where a distinct second bunch will be injected behind the drive bunch (as shown in figure 1) to demonstrate the high-gradient acceleration with a narrow energy spread.

## Acknowledgments

The authors are indebted to all their colleagues from the E157, 162, 164, 164X and 167 experiments for their contributions to the work described here. This work was partially supported by DOE grant DE-FG02-ER40727.

## Footnotes

One contribution of 15 to a Discussion Meeting Issue ‘Laser-driven particle accelerators: new sources of energetic particles and radiation’.

- © 2006 The Royal Society