# Energy recovery in filament-regime plasma wakefield acceleration of positron
beams

Max Varverakis1, Robert Holtzapple1, Severin Diederichs2, Carl Schroeder3 and
Spencer Gessner4 1 Department of Physics, California Polytechnic State
University, 1 Grand Ave, San Luis Obispo, 93407, CA, USA<EMAIL_ADDRESS>2 Deutsches Elektronen-Synchroton DESY, Notkestraße 85, 22607, Hamburg,
Germany  3 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley,
94720, CA, USA  4 SLAC National Accelerator Laboratory, 2575 Sand Hill Road,
Menlo Park, 94025, CA, USA<EMAIL_ADDRESS>
###### Abstract

Plasma wakefield acceleration using an electron filament offers stable, high-
gradient, high-quality acceleration of positron beams analogous to the
acceleration of electrons in the blowout regime. However, low energy-transfer
efficiency is currently a limiting factor for future collider applications. We
explore the addition of a secondary electron bunch in the electron filament
plasma wakefield acceleration scheme to recover additional energy from the
wake. Particle-in-cell simulations using HiPACE++ are used to demonstrate
various energy recovery schemes. In addition to confirming the energy
efficiency gains with a recovery electron beam, we also develop energy
recovery schemes in the context of future plasma colliders.

††:

Keywords: Positron, Collider, Plasma

##  1 Introduction

Plasma-based accelerators offer extremely high accelerating gradients, which
may enable a compact future linear collider at the TeV scale [1, 2, 3, 4, 5].
In recent decades, progress in plasma wakefield acceleration (PWFA) [6, 7] has
addressed many challenges on the path to a future plasma collider. For
electron-positron colliders, the ability to accelerate both electrons and
positrons is imperative. In the non-linear blowout regime, high-efficiency and
high-quality PWFA can be achieved for electrons [8] and has been demonstrated
experimentally [9]. However, high-efficiency and high-quality positron
acceleration remains a challenge in the non-linear regime [10, 11, 12, 13,
14].

In the non-linear blowout case, the region of the wakefield that is both
accelerating and focusing for positron bunches is relatively small. Recently,
new ideas have been proposed to elongate the region of high plasma electron
density at the back of the bubble, enabling better loading and acceleration of
trailing positron bunches [15, 16, 17]. Of these approaches, the plasma column
case has been studied in the most detail [15]. The wake formed in the plasma
column regime not only focuses and accelerates positrons, but it also
preserves beam quality [18] and stabilizes under misalignment [19].

The plasma column regime is a promising candidate for positron acceleration in
a future plasma collider, but the energy-transfer efficiency from the drive
beam to witness beam is limited to roughly 5% due to beam loading effects
[18]. This indicates that most of the energy in the wake is not used to
accelerate the witness beam. In the ongoing efforts towards minimizing the
carbon footprint of particle accelerators, energy efficiency remains a crucial
factor in determining the feasibility of a future plasma collider [20, 21].
Therefore, efficiency enhancements for positron acceleration in the plasma
column regime must be realized before this scheme can be utilized as part of a
future plasma collider.

Other PWFA schemes that elongate the electron filament at the back of the
bubble are currently under investigation, including the elongated bubble
regime [16] and the uniform non-linear regime [17]. In the uniform non-linear
regime, the placement of a positron beam just behind the bubble of a blowout
wake elongates the on-axis plasma electrons into an electron-dense filament
capable of focusing, accelerating, and preserving the positron beam. In an
optimized drive-witness configuration, efficiencies of up to 35% have been
demonstrated in simulations, which is comparable to the efficiency required
for future plasma colliders [2]. Once again, minimizing energy losses is
pertinent for a sustainable future collider, and maximizing energy efficiency
remains an important step in the realization of a future plasma collider.

Energy recovery from laser-driven plasma wakefields using trailing laser
pulses has been proposed [22]. Extending this concept to beam-driven plasma
wakefields, we propose the addition of a second electron beam to absorb
additional energy from the wakefield. We use the 3D quasi-static particle-in-
cell (PIC) code HiPACE$++$ [23] to show that the presence of an electron
recovery beam can result in a significant increase in energy extraction from
the wake. Additionally, we develop future collider concepts with the energy
recovery electron filament schemes.

##  2 High efficiency in the linear regime

Before addressing energy recovery in the plasma column and uniform non-linear
regimes, it is instructive to review energy recovery in the linear regime
[24]. Perturbatively solving for the plasma response to an electron drive
beam, it is possible to tailor a witness beam profile such that the fields are
nearly zero behind the trailing bunch [25]. Given a drive beam density
$\rho(\xi)$ and trailing bunch head at $\xi=\xi_{0}$, the requirement of a
constant accelerating field $E_{a}$ across the witness beam is described by

$E_{a}=E_{0}\cos{k_{p}\xi}-4\pi\int_{\xi_{0}}^{\xi}d\xi^{\prime}\rho(\xi^{\prime})\cos
k_{p}(\xi-\xi^{\prime}),$ (1)

where the relevant variables are defined in A. As a result, for any positive
integer $n$, loading an identical electron witness beam $(2n-1)\pi$ plasma
skin depths behind the drive beam results in near-100% energy-transfer
efficiency from the wake to the witness beam. A positron beam can be loaded in
a similar fashion, with $2n\pi$ plasma skin depth separation leading to ideal
energy transfer.

Figure 1: Normalized plasma charge density (top row) and accelerating field
(bottom row) for a loaded (left column) and unloaded (right column) wake in
the linear regime in the $x\xi$-plane. The on-axis fields from HiPACE$++$ are
outlined in black and are in agreement with the analytical solutions indicated
by red dashed lines. Gray dashed lines give the longitudinal current density
profiles for the beam(s). The witness beam is loaded $-3\pi\,k_{p}^{-1}$
behind the drive beam. The simulation domain contains 1 plasma electron macro
particle per cell and is $(-150,150)\times(-150,150)\times(-350,100)~{}\mu$m3
in $x\,\times\,y\,\times\,\xi$. The resolution is $0.59\times 0.59\times
0.89~{}\mu$m3.

In HiPACE$++$, we simulate PWFA in the linear regime with two identical bi-
Gaussian electron beams separated by $3\pi k_{p}^{-1}$ in $\xi$. The
simulation consisted of a plasma electron density of $n_{0}=8\times 10^{16}$
cm-3, $N_{b}=4\times 10^{7}$ macro particles per beam, and beam size
$\sigma_{x,y}=20$ $\mu$m and $\sigma_{z}=15$ $\mu$m. Beams are injected at an
energy of 10 GeV and a peak density of $4.23\times 10^{14}$ cm-3.

Both simulation and numerical analysis show a full depletion of the wakefield
at 100% efficiency when properly loaded, as illustrated in Figure 1.
Unfortunately, the linear regime is not suitable for collider applications,
because the low-emittance, high-charge, and high-energy beams are extremely
dense and will drive non-linear wakes [24, 26].

##  3 Energy recovery in the Plasma Column regime

As thoroughly studied in [15, 18, 19, 27], a plasma column that is radially
smaller than the blowout radius causes a spread in the plasma electron
trajectories near the boundary of the ion bubble. The plasma wake forms a
high-density plasma electron filament that is elongated on-axis. The filament
region not only contains the necessary focusing and accelerating fields for
positrons, but it also preserves beam quality and provides stability. To study
energy recovery in the plasma column regime, we first simulate positron PWFA
in the plasma column without energy recovery modifications to establish a
baseline efficiency. Next, we explore increasing the energy efficiency by
placing an electron recovery bunch either in front of or behind the positron
witness bunch. There are focusing and accelerating fields for a second
electron bunch in both regions of the wake. We note that the parameter space
for energy recovery is large, with bunch charge, bunch length, and transverse
emittance all free parameters. The goal of this study is to maximize energy
extraction from the wake with a triple-beam configuration, without addressing
all beam quality considerations.

Helium plasma is simulated with 400 electron and 16 ion macro particles per
cell with a column radius of $2.5~{}k_{p}^{-1}$ and background density
$n_{0}=10^{17}$ cm-3. A plasma temperature of $k_{B}T=15$ eV is used, where
$k_{B}$ is the Boltzmann constant. All three simulations use a -3.38 nC bi-
Gaussian electron drive beam with $10^{6}$ macro particles, an initial energy
of 5.11 GeV, and a $\beta$-matched normalized transverse emittance of
$\epsilon_{x,y}=2.96~{}\mu$m rad. The drive beam is centered at $\xi=0$ with
$\sigma_{x,y}=0.05~{}k_{p}^{-1}$ and $\sigma_{z}=1.41~{}k_{p}^{-1}$. The
positron witness and electron recovery beams are simulated with $1.25\times
10^{6}$ and $10^{6}$ macro particles, respectively, and at energies
approximately equal to 1 GeV. Both trailing beams are radially Gaussian with
$\sigma_{x,y}=0.029~{}k_{p}^{-1}$ and $\epsilon_{x,y}=0.45~{}\mu$m rad for the
positron beam and $\sigma_{x,y}=0.05~{}k_{p}^{-1}$ and
$\epsilon_{x,y}=1.33~{}\mu$m rad for the recovery electron beam. Additional
beam properties are listed in Table 1.

Table 1: Trailing beam parameters for plasma column simulations. Subscripts $p$ and $r$ correspond to the positron and electron (recovery) beam, respectively. Simulation | (a) | (b) | (c)
---|---|---|---
$k_{p}\xi_{p,\,\textrm{\footnotesize head}}$ | $-10.5$ | $-10.5$ | $-12.9$
$k_{p}\xi_{r,\,\textrm{\footnotesize head}}$ | – | $-20.0$ | $\hbox{\phantom{\footnotesize\rm 0}}\hbox{\phantom{\footnotesize\rm 0}}\hbox{\phantom{\footnotesize\rm 0}}\\-7.6$
$Q_{p}$ [pC] | 182 | 181 |  064
$Q_{r}$ [pC] | – | $\\-517$ | $\\-707$
$\eta$ [%] |  03.8 |  012.0 |  027.4

Compared to Figure 1, which shows near-100% efficiency in the linear regime,
Figure 2 illustrates an imperfect energy transfer due to the fact that the
wakefield persists behind the trailing beams. Nevertheless, the energy-
transfer efficiency increases with the addition of a recovery bunch. Based on
these simulations, we find a baseline efficiency of $\eta=3.8\%$. The
additional electron recovery beams placed behind and in front of the positron
beam resulted in efficiencies of $\eta=12.0\%$ and $\eta=27.4\%$,
respectively. See A for more detail on efficiency calculations.

An iterative search was used to determine the electron recovery beam
parameters that maximize the efficiency of the three-bunch configuration. The
trailing bunch profiles were optimized to flatten the average accelerating
fields with the SALAME algorithm as part of the HiPACE++ code [18]. Figure
2(c) shows large variation of the on-axis $E_{z}$ field for the trailing
recovery bunch, but this variation is minimized by averaging $E_{z}$
transversely across the bunch. According to our simulations, transverse
witness beam properties did not significantly affect energy-transfer
efficiency for the initial particle injection. However, efficiency losses are
expected to occur when propagating the beams with inappropriate transverse
characteristics, such as when using a positron bunch transversely larger than
the electron filament.

We note that our recovery beam parameter search was non-exhaustive, but
initial results from our simulations indicate that the efficiency of the
three-bunch scheme is unlikely to improve far beyond what we have demonstrated
here.

Figure 2: Normalized focusing wakefield (in $\xi$-$x$) and on-axis normalized
accelerating field (magenta curve) in the plasma column filament regime for
(a) no energy recovery, (b) recovery behind the positron beam, and (c)
recovery in front of the positron beam. Longitudinal currents of electron
beams are plotted in red and positron beams in blue. On-axis beam current
density profiles are displayed on the $\xi$-axis in gray. Simulations were
executed in a $(-16,16)\times(-16,16)\times(-24,6)~{}k^{-3}_{p}$ domain in
$x\,\times\,y\,\times\,\xi$. The corresponding mesh resolution is $0.031\times
0.031\times 0.029~{}k^{-3}_{p}$.

##  4 Energy recovery in the Uniform Nonlinear regime

Generating an electron filament in uniform plasma involves a slightly
different procedure than in the plasma column case. In a traditional blowout
wake, a region of highly dense plasma electrons appears behind the ion bubble,
but is longitudinally short. Similar to the elongated bubble scheme [16], a
method was studied in [17] in which the presence of a high-intensity positron
beam causes an electron filament to form in the region behind the bubble of
the blowout wake. The forces induced by the positron beam bring sheath
electrons back on-axis to produce the desired electron filament.

As before, there are three situations that we explore. First, we simulate this
scheme without the use of an energy recovery beam to obtain a baseline in
energy-transfer efficiency. Next, we add an electron recovery beam behind the
positron beam. Finally, we simulate an electron recovery beam ahead of the
positron beam.

In the cases presented in Figure 3, we use a helium plasma with background
density $n_{0}=7.8\times 10^{15}$cm-3 and with 25 macro particles per cell for
both plasma electrons and ions. All beams were simulated with $10^{6}$ macro
particles and at 2.5 GeV. The beam parameters were chosen to match the
parameters used in Ref. [17]. The drive beam follows a bi-Gaussian profile
with $\sigma_{x,y}=5~{}\mu$m and $\sigma_{z}=40~{}\mu$m. Both the positron
beam and recovery beam are radially Gaussian with $\sigma_{x,y}=2~{}\mu$m and
$\sigma_{x,y}=3~{}\mu$m, respectively. The total charge of the drive beam is
$Q_{d}=-534$ pC. The drive beam is simulated with a normalized transverse
emittance of 6 $\mu$m rad, the positron beam with 2.5 $\mu$m rad, and the
recovery beam with 7.5 $\mu$m rad. See Table 2 for trailing beam placements
and the corresponding energy-transfer efficiency.

Table 2: Trailing beam parameters for uniform plasma simulations. Subscripts $p$ and $r$ correspond to the positron and electron (recovery) beam, respectively. Simulation | (a) | (b) | (c)
---|---|---|---
$k_{p}\xi_{p,\,\textrm{\footnotesize head}}$ | $-5.1$ | $-5.1$ | $-5.3$
$k_{p}\xi_{r,\,\textrm{\footnotesize head}}$ | – | $-8.7$ | $-3.5$
$Q_{p}$ [pC] | 102 | 102 |  063
$Q_{r}$ [pC] | – | $\\-310$ | $\\-177$
$\eta$ [%] |  025.9 |  045.0 |  073.5

As before, the presence of a wakefield behind the trailing beam in Figure 3
indicates that the wake energy is not fully transferred to the witness beams.
Nevertheless, the addition of the recovery bunch significantly increases
energy extraction from the wake. The base case with no energy recovery
resulted in an energy-transfer efficiency of $\eta=25.9\%$, which is
consistent with the findings in [17]. Energy-transfer efficiency increases to
$\eta=45.0\%$ with recovery beam placement behind the positron beam, and
$\eta=73.5\%$ with the recovery beam in front.

Since we use the same optimization methods as in the plasma column case, we
acknowledge the potential for slightly higher efficiencies than those
identified in our study. Further simulations are required to establish the
uniformity of $E_{z}$ and stability of the wake in this regime.

Figure 3: Normalized focusing wakefield (in $\xi$-$x$) and on-axis normalized
accelerating field (magenta curve) in the uniform plasma filament regime for
(a) no energy recovery, (b) recovery behind the positron beam, and (c)
recovery in front of the positron beam. Longitudinal currents of electron
beams are plotted in red and positron beams in blue. On-axis beam current
density profiles are displayed on the $\xi$-axis in gray. We use a simulation
window of $(-6,6)\times(-6,6)\times(-15,6)~{}k^{-3}_{p}$ and a resolution of
$0.012\times 0.012\times 0.021~{}k^{-3}_{p}$ in $x\times y\times\xi$.

##  5 Collider Concepts

The simulations performed in this study show that it is possible to extract an
appreciable fraction of the drive beam’s energy from the wake by accelerating
both a positron bunch and an electron recovery bunch. This process is repeated
over many stages to achieve the desired positron beam energy for collisions
[28]. The energy stored by the electron recovery bunches must also be utilized
to minimize the power consumption of the facility.

One option for utilizing the energy in the electron recovery bunches is to
pass the recovery bunches through an Energy Recovery Linac (ERL) [29]. The ERL
serves as both a decelerator of the recovery bunches and an accelerator of new
electron drive bunches. This scheme is likely feasible with existing
technology, but might not be viable for two reasons. First, an elaborate
beamline is required to separate the drive electron bunch from the trailing
positron and electron recovery bunches at the end of a plasma stage. The large
number of return beamlines, along with associated vacuum chambers and magnets,
will increase the cost of the collider facility. The second challenge with the
ERL approach is that even if the recovery bunches can be extracted and their
energy used to accelerate new drive bunches, there is still the issue of
creating new recovery bunches, which limits the overall efficiency of the
scheme.

As an alternative, we may consider the trailing electron bunch as a colliding
bunch rather than a recovery bunch. In this scenario, both the electron and
positron bunches will be delivered to the IP, albeit with somewhat different
beam parameters. For very high energy collisions, the electron and positron
bunches should have the same energy (note that this is not necessarily the
case for lower energy Higgs Factories [30]), but it is possible that other
beam parameters such as charge, bunch length and emittance might differ
between the two beams. A detailed study using beam-beam codes such as GUINEA-
PIG or WarpX is required to better understand both the efficiency and the
luminosity-per-power optimization in this scenario.

Once the beams reach their final energy after many stages of acceleration,
they may be delivered to a single IP using an SLC-like configuration [31]. To
preserve the beam emittance, the size of the arcs should scale as $E^{2}$,
which is an unfavorable scaling for multi-TeV collision energies. A dual-IP
collider is considered in Figure 4. Previous collider designs, such as the
NLC, have also considered dual IPs [32]. Dual IP designs enable multiple
general-purpose detectors to be operated simultaneously, which allows for
competition and validation of high-profile particle physics measurements.

Figure 4: Schematic of a collider with two IPs, with each linac arm of the
collider accelerating both electron and positron bunches for collisions. Note
that the electron and positron bunches will not have the same properties at
the collision point (e.g. charge, bunch length, emittance) and that further
studies with beam-beam simulation codes are needed to assess the luminosity-
per-power of such a scheme.

##  6 Conclusion

Electron filament PWFA shows promise towards high-quality positron
acceleration for a future plasma collider. The realization of such a collider
is contingent upon minimizing its environmental impact [20, 21] and is
challenged by energy efficiency limitations. Our simulations indicate that the
introduction of a secondary electron beam into an electron filament scheme
results in a net gain in energy-transfer efficiency. In some cases, specially
tailored beam profiles for the trailing beams allowed the average accelerating
field to be flattened over the beams. However, the sensitivities experienced
with $E_{z}$ implies that beam quality preservation remains a challenge.
Developing a theory on recoverable energy in the blowout regime requires
further research. Moreover, additional simulations are necessary to better
understand beam preservation in the energy recovery schemes described in this
paper. Lastly, future studies will examine the stability of these energy
recovery schemes by subjecting them to beam offsets at finer resolution.

##  7 Data Availability

Input scripts and analysis can be found at
https://github.com/MaxVarverakis/PositronPWFA.

## Acknowledgments

Work supported by the U.S. Department of Energy, United States under Contracts
DE-AC02-76SF00515 and DE-AC02–05CH11231, and the National Science Foundation,
United States (Grants No. PHY-1535696 and No. PHY-2012549).

##  Appendix A Relevant definitions and notation

For the purposes of this paper, we define the following variables and
constants: $\xi=z-ct$ is the co-moving coordinate in the speed-of-light frame,
$c$ being the speed of light in vacuum, $k_{p}^{-1}=c/\omega_{p}$ the plasma
skin depth, and $\omega_{p}=\sqrt{n_{0}e^{2}/m\varepsilon_{0}}$ the plasma
frequency for $n_{0}$ plasma electron density, $e$ the electron charge, $m$
electron mass, and $\varepsilon_{0}$ vacuum permittivity. Electric and
magnetic fields are normalized to the cold, nonrelativistic wave-breaking
field $E_{0}=\omega_{p}mc/e$.

We define the energy-transfer efficiency $\eta$ as the ratio between the
energy change of the beams [24],

$\eta=-\left(\frac{Q_{p}\langle E_{z}\rangle_{p}+Q_{r}\langle
E_{z}\rangle_{r}}{Q_{d}\langle E_{z}\rangle_{d}}\right),$ (2)

where $Q_{p,r,d}$ is the corresponding positron, electron recovery, and drive
beam charge and $\langle E_{z}\rangle_{p,r,d}$ the longitudinal electric field
averaged over the beam profiles. In practice, we compute each $Q\langle
E_{z}\rangle$ term as the dot product of the longitudinal electric field with
the per-slice charge of the beam. We ignore off-axis contributions to
efficiency due to the cylindrical symmetry of the fields in the blowout
regime.

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