††thanks: These authors have contributed equally to this work.††thanks: These
authors have contributed equally to this work.

# Efficiency and beam quality for positron acceleration in loaded plasma
wakefields

C. S. Hue  LOA, ENSTA Paris, CNRS, Ecole Polytechnique, Institut Polytechnique
de Paris, 91762 Palaiseau, France  G. J. Cao  LOA, ENSTA Paris, CNRS, Ecole
Polytechnique, Institut Polytechnique de Paris, 91762 Palaiseau, France
Department of Physics, University of Oslo, NO-0316 Oslo, Norway  I. A.
Andriyash  LOA, ENSTA Paris, CNRS, Ecole Polytechnique, Institut Polytechnique
de Paris, 91762 Palaiseau, France  A. Knetsch  LOA, ENSTA Paris, CNRS, Ecole
Polytechnique, Institut Polytechnique de Paris, 91762 Palaiseau, France  M. J.
Hogan  SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA  E.
Adli  Department of Physics, University of Oslo, NO-0316 Oslo, Norway  S.
Gessner  SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA  S.
Corde<EMAIL_ADDRESS>LOA, ENSTA Paris, CNRS, Ecole
Polytechnique, Institut Polytechnique de Paris, 91762 Palaiseau, France

###### Abstract

Accelerating particles to high energies in plasma wakefields is considered to
be a promising technique with good energy efficiency and high gradient. While
important progress has been made in plasma-based electron acceleration,
positron acceleration in plasma has been scarcely studied and a fully self-
consistent and optimal scenario has not yet been identified. For high energy
physics applications where an electron-positron collider would be desired, the
ability to accelerate positrons in plasma wakefields is however paramount.
Here we show that the preservation of beam quality can be compromised in a
plasma wakefield loaded with a positron beam, and a trade-off between energy
efficiency and beam quality needs to be found. For electron beams driving
linear plasma wakefields, we have found that despite the transversely
nonlinear focusing force induced by positron beam loading, the bunch quickly
evolves toward an equilibrium distribution with limited emittance growth.
Particle-in-cell simulations show that for $\mathrm{\SIUnitSymbolMicro
m}$-scale normalized emittance, the growth of uncorrelated energy spread sets
an important limit. Our results demonstrate that the linear or moderately
nonlinear regimes with Gaussian drivers provide a good trade-off, achieving
simultaneously energy-transfer efficiencies exceeding $30\text{\,}\%$ and
uncorrelated energy spread below $1\text{\,}\%$, while donut-shaped drivers in
the nonlinear regime are more appropriate to accelerate high-charge bunches at
higher gradients, at the cost of a degraded trade-off between efficiency and
beam quality.

##  I Introduction

Particle accelerators based on radio-frequency technology are being used in a
very broad range of applications, from free-electron lasers or medicine to
particle colliders for high-energy physics. For the latter, the limited
accelerating gradient of this technology renders the footprint and the cost
for future machines prohibitively expensive. Plasma-based acceleration, driven
by particle beams Veksler (1956); Faĭnberg (1960, 1968); Chen  _et al._ (1985)
or laser pulses Tajima and Dawson (1979); Esarey  _et al._ (2009), is a
potential candidate to considerably increase the accelerating gradient and to
provide efficient high-energy particle accelerators. In the last decades,
substantial progress has been made in electron acceleration using plasma-based
accelerators in the nonlinear bubble or blowout regime Rosenzweig  _et al._
(1991); Pukhov and Meyer-ter Vehn (2002); Malka  _et al._ (2002); Faure  _et
al._ (2004); Geddes  _et al._ (2004); Mangles  _et al._ (2004); Lu  _et al._
(2006); Blumenfeld  _et al._ (2007), a regime that is particularly well suited
for high energy efficiency Tzoufras  _et al._ (2008); Litos  _et al._ (2014);
Lindstrøm  _et al._ (2021) and whose field structure is ideal for beam quality
preservation Clayton  _et al._ (2016). However, for applications towards high-
energy colliders, it is imperative for plasma-based accelerators to be capable
of accelerating positrons. Because of the need for high luminosity, positron
beams must also be accelerated with high energy efficiency and have very small
final normalized emittances, on the order of $10$ to
$100\text{\,}\mathrm{n}\mathrm{m}$ Behnke  _et al._ (2013); Aicheler  _et al._
(2012). Moreover, the positron beam energy spread needs to be kept below
percent level so that the beam can be focused to the interaction point by a
final focus system Raimondi and Seryi (2001). This requires keeping both the
correlated energy spread (different longitudinal slices having different
energies) and the uncorrelated (or slice) energy spread under control.
Finally, the acceleration process needs to be stable and to reach full
depletion of the driver.

The above criteria, namely energy efficiency, emittance, energy spread and
stability, must be optimized for simultaneously in a high-energy collider
application. This is particularly challenging in the case of plasma-based
positron acceleration. Experimentally, positron acceleration in plasma has
been successfully demonstrated for positrons at the rear of a single drive
bunch Blue  _et al._ (2003) with high energy efficiency Corde  _et al._
(2015), as well as for distinct trailing positron bunches in both linear and
nonlinear regimes Doche  _et al._ (2017), but the question of beam quality and
emittance preservation was not addressed. Positron bunches were also
accelerated in hollow plasma channels Gessner  _et al._ (2016); Gessner
(2016), however, the stability of the hollow plasma accelerator was shown to
be compromised by the presence of strong transverse wakefields Lindstrøm  _et
al._ (2018) that lead to the beam breakup instability Schroeder  _et al._
(1999).

Novel methods, such as the use of finite-radius plasma columns Diederichs  _et
al._ (2019, 2020) or donut-shaped electron Jain  _et al._ (2015) and laser
Vieira and Mendonça (2014) drivers, have also been proposed to excite plasma
wakefields with an extended region that can be focusing and accelerating for
positrons, usually obtained by the creation of a long plasma electron filament
in the vicinity of the propagation axis. Apart from the hollow plasma channel
accelerator, methods to accelerate positrons in plasma typically have in
common the presence of an excess of plasma electrons within the accelerated
positron bunch. Such plasma wakefields can then be used to accelerate low-
charge bunches while preserving their quality, for example with weak beam
loading in the linear regime Brinkmann  _et al._ (2017), but the energy
efficiency with weak beam loading is too low to be of interest for a collider
application. When increasing positron beam charge and energy efficiency,
positron beam loading becomes the key challenge because of the quick response
of the plasma electrons to the positron load. This quick response can lead to
a transversely nonlinear focusing force that can potentially drive emittance
growth Muggli  _et al._ (2008), as well as a transversely non-uniform
accelerating field that can induce growth in uncorrelated energy spread. The
problem of positron beam loading in the presence of plasma electrons within
the bunch has similarities to the physics of ion motion for electron
acceleration in the blowout regime Rosenzweig  _et al._ (2005); Gholizadeh
_et al._ (2010); An  _et al._ (2017); Benedetti  _et al._ (2017); Mehrling
_et al._ (2018), but is considerably more challenging, as plasma electrons are
much more mobile than ions. A first insight into the physics of positron beam
loading was reported in the nonlinear regime Corde  _et al._ (2015), where
beam loading allows the flattening of $E_{z}(\xi)$ and induces a filament of
plasma electrons, thereby reaching high energy efficiencies and providing
focusing to the accelerated positrons. There are also detailed studies in the
specific context of the finite-radius plasma column Diederichs  _et al._
(2020), and in the case of nonlinear and asymmetric plasma wakefields in
hollow channels Zhou  _et al._ (2020), where plasma electrons from the channel
wall cross the axis to provide focusing to the positron beam.

In this article, we focus on plasma wakefields driven by Gaussian or donut-
shaped electron beams in uniform plasmas, from the linear to the nonlinear
regime. Positron beam loading is shown to induce a trade-off between energy
efficiency and beam quality, and that for $\mathrm{\SIUnitSymbolMicro
m}$-scale normalized emittance, an important limit arises from the growth of
uncorrelated energy spread driven by the transversely nonuniform accelerating
field associated with the strong positron load. In Sec. II, we illustrate the
positron beam loading problem for linear plasma wakefields. Section II.1
presents an analytical model and simulation results for the energy-transfer
efficiency in three dimensions, and shows that a mismatch in transverse size
of the wakefield from the electron driver and the wakefield from the positron
bunch can considerably reduce the energy efficiency. The latter suggests that
higher efficiencies can be obtained with linear plasma wakefields for beam
sizes smaller than the plasma skin depth. In Sec. II.2, the evolution of the
transverse phase space of the positron bunch is discussed, and it is shown
that when starting from quasi-matched conditions, the beam evolves rapidly
toward an equilibrium, with limited emittance growth during this initial
evolution, and emittance preservation afterwards. Section II.3 discusses the
evolution of the longitudinal phase space of the positron bunch, and shows
that the growth of slice energy spread can pose serious limitations. In Sec.
III, the blowout regime with a donut-shaped electron driver is considered,
where the donut allows for an excess of plasma electrons to be present near
the propagation axis in the blowout cavity, thus providing focusing to the
positron bunch. The properties of the plasma wakefields and the induced slice
energy spread for the accelerated positron beam are presented and discussed.
Section IV presents a comparison of different regimes, considering similar
initial positron beam parameters but optimizing the driver separately for each
specific regime (see Sec. IV.1), so that their performance can be determined
in terms of energy efficiency and uncorrelated energy spread, illustrating the
trade-off between these two quantities (see Sec. IV.2). Conclusions are
finally presented in Sec. V.

##  II Linear plasma wakefields

###  II.1 Energy-transfer efficiency

The energy-transfer efficiency $\eta$ is defined as the ratio of energy gained
by the trailing bunch $W_{t,\mathrm{gain}}$ and energy lost by the driver
$W_{d,\mathrm{loss}}$,

$\eta=\frac{W_{t,{\text{gain}}}}{W_{d,{\text{loss}}}}=-\frac{Q_{t}\langle
E_{z}\rangle_{t}}{Q_{d}\langle E_{z}\rangle_{d}},$ (1)

with $\langle E_{z}\rangle_{t,d}$ the longitudinal electric field averaged
over particles in the bunch and over the propagation distance in the plasma,
and $Q_{t,d}$ the bunch charge. It represents the efficiency in the transfer
of energy from the drive to the trailing bunch through the plasma, and can
also be thought as the energy-extraction efficiency from the plasma to the
trailing bunch ($W_{d,\mathrm{loss}}$ being the energy given to the plasma by
the drive bunch). Importantly, this energy-transfer efficiency $\eta$ is a
figure of merit of the wakefield and does not take into account the ratio
between $W_{d,\mathrm{loss}}$ and the total initial energy in the drive bunch,
which would represent the efficiency from drive to plasma and would depend on
the acceleration distance that can be achieved. In the linear regime, beam
loading can be understood from the fact that the total wakefield is simply the
superposition of drive and trailing plasma wakefields, and in one dimension
one finds for the energy-transfer efficiency Katsouleas  _et al._ (1987):

$\eta=\frac{N_{t}}{N_{d}}\left(2-\frac{N_{t}}{N_{d}}\right),$ (2)

where $N_{t,d}$ is the number of particles in each bunch, and we have assumed
that the drive and trailing particle charges satisfy $|q_{d}|=|q_{t}|$, the
drive-trailing bunch separation is $\Delta\xi=\lambda_{p}/2\pmod{\lambda_{p}}$
for $q_{d}=q_{t}$ [respectively $\Delta\xi=\lambda_{p}\pmod{\lambda_{p}}$ for
$q_{d}=-q_{t}$], and either $\sigma_{dz}=\sigma_{tz}$ (same drive and trailing
bunch lengths) or both bunches are short: $k_{p}\sigma_{z}\ll 1$, with
$k_{p}=\omega_{p}/c$ and $\omega_{p}=\sqrt{n_{0}e^{2}/(m_{e}\epsilon_{0})}$
the plasma frequency for a plasma of density $n_{0}$. The parabolic
relationship in Eq. (2) shows that the efficiency is maximized and equal to 1
when $N_{t}=N_{d}$, that is in the case of perfectly-destructive interference
between drive and trailing wakefields corresponding to a vanishing plasma wave
behind the trailing bunch, when all the energy in the plasma is extracted by
the trailing bunch.

In three dimensions, the energy-transfer efficiency not only depends on
$N_{t,d}$, but also on the size and shape of drive and trailing plasma
wakefields. In particular, one cannot expect to reach perfectly-destructive
interference ($\eta=1$) if the size and shape of drive and trailing wakefields
differ. In general, $\eta$ will also depend on drive and trailing beam sizes
($\sigma_{dr}$ and $\sigma_{tr}$), bunch lengths ($\sigma_{dz}$ and
$\sigma_{tz}$) and on the plasma skin depth $1/k_{p}$. In the linear regime,
analytical calculations can be performed for separable bunch shapes, as shown
in Appendix A and Eq. (17), and can be simplified for large
($k_{p}\sigma_{r}\gg 1$) Gaussian bunches to

$\eta=\frac{N_{t}}{N_{d}}\frac{\sigma_{dr}^{2}}{\sigma_{tr}^{2}}\left[\frac{4}{1+\frac{\sigma_{dr}^{2}}{\sigma_{tr}^{2}}}-\frac{N_{t}}{N_{d}}\right],$
(3)

under the same assumptions as Eq. (2), and also takes the form of a parabolic
relationship between $\eta$ and $N_{t}/N_{d}$, with parameters now depending
on beam sizes.

Figure 1:  Analytical (solid lines) and simulation (circles and dashed lines)
results for energy-transfer efficiencies in 3D linear regime, for large beams
$\sigma_{r}\gg 1/k_{p}$ (a) and small beams $\sigma_{r}<1/k_{p}$ (b). The
plasma density is set to
$n_{0}=$4\text{\times}{10}^{17}\text{\,}{\mathrm{cm}}^{-3}$$ and the peak
drive bunch density is $n_{b}=0.042\>n_{0}$ for all simulations. The drive
bunch has $\sigma_{dr}=$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$\sigma_{dz}=$4\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ in (a) and
$\sigma_{dr}=\sigma_{dz}=$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ in (b).
The trailing bunch has a beam size $\sigma_{tr}=0.25\>\sigma_{dr}$ (blue),
$\sigma_{tr}=0.5\>\sigma_{dr}$ (orange) or $\sigma_{tr}=\sigma_{dr}$ (green),
a bunch length $\sigma_{tz}=\sigma_{dz}$, and its charge is varied according
to $N_{t}/N_{d}$. The simulated wakefields corresponding to the annotated
points 1-3 are shown in Fig. 2.

In Fig. 1, the analytical results of Eqs. (3) and (17) are compared to
numerical simulations performed using the Open Source quasi-static Particle-
In-Cell (PIC) code QuickPIC Huang  _et al._ (2006); An  _et al._ (2013); Qui .
The plasma wakefield generated by both bunches is obtained from a single-step
quasi-static simulation (no beam evolution), and the energy-transfer
efficiency is computed from the simulated wakefield by averaging $E_{z}$ over
each bunch [see Eq. (1)], showing good agreement with the analytical results.
For large beams, $k_{p}\sigma_{r}\gg 1$, the maximum efficiency [see Eq. (23)]
and the corresponding value of the trailing charge [see Eq. (24)] both rapidly
decrease when the trailing beam size is reduced with respect to that of the
driver [see Fig. 1(a)]. This low efficiency can be understood by the strong
mismach between the size of the wakefields of the drive and trailing bunches,
as can be seen in Fig. 2(e). Indeed, in such a situation, the trailing
wakefield can only overlap and cancel the drive wakefield over a small region
of the drive wakefield, leaving a large amount of energy in the plasma wave
behind the trailing bunch, thus leading to low efficiency, with
$\eta_{\mathrm{max}}\simeq$23\text{\,}\%$$ for $\sigma_{tr}=0.25\>\sigma_{dr}$
in Fig. 1(a). In contrast to large beams, for which the size of the wakefield
is determined by the size of the beam, the wakefield for small beams typically
extends over a plasma skin depth. As a result, a better overlap between drive
and trailing wakefields is found for $\sigma_{tr}\neq\sigma_{dr}$ in the
small-beam case, leading to higher maximum efficiencies corresponding to
higher trailing charge, as seen in Fig. 1(b), with
$\eta_{\mathrm{max}}\simeq$74\text{\,}\%$$ for
$\sigma_{tr}=0.25\>\sigma_{dr}$.

Figure 2:  Simulated wakefields [$E_{z}$ left column, $e(E_{y}+cB_{x})$ right
column] for the annotated points 1-3 in Fig. 1. Point 1 [(a)-(b)] corresponds
to the small beam case with near-100% efficiency ($\sigma_{tr}=\sigma_{dr}$),
point 2 [(c)-(d)] to the small beam case with maximum efficiency for
$\sigma_{tr}=0.25\>\sigma_{dr}$, and point 3 [(e)-(f)] to the large beam case
with maximum efficiency for $\sigma_{tr}=0.25\>\sigma_{dr}$. Dashed ellipses
show the location of drive and trailing bunches with the $1\sigma$ contours of
their bunch densities.

Figure 2 shows simulated wakefields (longitudinal and transverse fields) for a
few cases of interest. For $\sigma_{tr}=\sigma_{dr}$, the drive and trailing
wakefields have exactly the same extent and shape, and it is therefore
possible to approach $\eta=1$ with near-cancellation of the wakefield behind
the trailing bunch [see Fig. 2(a)-(b)]. For the case of small beams and
$\sigma_{tr}=0.25\>\sigma_{dr}$, the wakefield can still be significantly
weakened by the trailing bunch at transverse positions $|x|\gg\sigma_{tr}$
despite different beam sizes and different wakefield shapes for the drive and
trailing bunches. This is because the fields extend typically over a distance
$1/k_{p}\gg\sigma_{tr}$ [see Fig. 2(c)-(d)], thereby ensuring an efficient
beam loading and high energy-transfer efficiencies. For beams with
$k_{p}\sigma_{r}\gg 1$, the transverse size of the wakefield is determined by
the transverse beam size, and beam loading effects are localized within the
trailing beam cross section, which limits the energy-transfer efficiency [see
Fig. 2(e)-(f)].

As it is generally desired to accelerate low emittance positron beams,
especially for high energy physics applications, and given that the matched
beam size for the trailing bunch is very small at low emittance and high
energy (see Sec. II.2), one can expect to have $\sigma_{tr}\ll\sigma_{dr}$. In
this case, given the results presented above, the best strategy for reaching
high energy-transfer efficiencies in the linear regime is therefore to
consider small beams, and thus to work with a drive bunch with
$k_{p}\sigma_{dr}\lesssim 1$. In addition, while a linear wakefield can be
driven by the drive bunch and used for positron acceleration, a matched
positron bunch at low emittance and high energy can have a density easily
exceeding that of the plasma, resulting in nonlinear beam loading (wakefield
superposition no longer holds), which is nonlocal and acts on plasma electrons
at distances much larger than $\sigma_{tr}$, which is favorable for high
efficiencies.

###  II.2 Transverse phase space and equilibrium

Figure 3:  Evolution of the normalized transverse emittance of the trailing
positron bunch, $\varepsilon_{n}=\sqrt{\langle x^{2}\rangle\langle
p_{x}^{2}\rangle-\langle xp_{x}\rangle^{2}}/mc$, during its propagation in the
plasma. For all simulations, the plasma density is set to
$n_{0}=$5\text{\times}{10}^{16}\text{\,}{\mathrm{cm}}^{-3}$$, the peak density
of the electron drive bunch is $n_{b}=0.35\>n_{0}$, the drive beam size and
bunch length are $\sigma_{dr}=$14.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
and $\sigma_{dz}=$16.7\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, the drive beam
evolution is turned off and the drive to trailing bunch separation is
$\Delta\xi=$143\text{\,}\mu\mathrm{m}$$. The initial trailing positron bunch
energy is $1\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ in (a)-(b) and
$100\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ in (c). The matching condition is
varied in (a), while the trailing positron beam is quasi-matched for
increasing values of $n_{b}/n_{0}$ in (b)-(c). All initial trailing positron
beam parameters and corresponding values for efficiency and emittance growth
are listed in Table 1.

While high energy efficiencies are desirable, it must be achieved with high
quality positron beams and the acceleration process must preserve this
quality. For the prospect of a plasma-based high-energy collider with a
luminosity exceeding
$1\text{\times}{10}^{34}\text{\,}\mathrm{c}\mathrm{m}^{-2}\mathrm{s}^{-1}$, a
normalized emittance on the order of $10\text{\,}$ to
$100\text{\,}\mathrm{n}\mathrm{m}$ (in one transverse direction) is generally
targeted, which represents a considerable challenge for the acceleration of
positron beams in plasmas. Emittance preservation is possible in a
transversely linear focusing force, which is the case for the acceleration of
a matched electron beam in the blowout regime in the absence of ion motion
Clayton  _et al._ (2016). However, a transversely nonlinear focusing force,
for example in the case of ion motion for electron acceleration in the blowout
regime or plasma electron suck-in for positron acceleration, can induce
emittance growth An  _et al._ (2017); Muggli  _et al._ (2008). In general, a
non-Gaussian transverse equilibrium can be quickly reached by the beam, as is
the case for a single decelerated beam in a linear wakefield Lotov (2017), an
electron beam in the blowout regime with ion motion Benedetti  _et al._
(2017); An  _et al._ (2017), or for a step-like focusing force Diederichs  _et
al._ (2019). While the equilibrium for a single decelerated beam has unusual
and unwanted properties due to the absence of an external focusing field from
a driver, such as on-axis singularity and very large transverse tails Lotov
(2017), the case of a trailing bunch in the focusing force from a driver has
much more favorable equilibrium distributions, as shown in Ref. Benedetti  _et
al._ (2017) for ion motion and Ref. Diederichs  _et al._ (2019) for the step-
like focusing force of the finite-radius plasma column scheme. In particular,
the emittance growth from an initially Gaussian-shaped trailing bunch to the
equilibrium can be limited to about 10 to $20\text{\,}\%$ for ion motion
Benedetti  _et al._ (2017); An  _et al._ (2017), and to approximately
$6\text{\,}\%$ for the step-like focusing force Diederichs  _et al._ (2019),
and this emittance growth can be further reduced or eliminated by head-to-tail
bunch shaping Benedetti  _et al._ (2017). Although perfect emittance
preservation is not possible for an initially Gaussian trailing bunch, the
minimum emittance growth is obtained for a specific value of the beam size,
which is referred to as the quasi-matched beam size.

|  $\sigma_{tr}\>($\mathrm{\SIUnitSymbolMicro m}$)$ |  $\varepsilon_{n}\>($\mathrm{\SIUnitSymbolMicro m}$)$ |  $\beta\>($\mathrm{cm}$)$ |  $\sigma_{tz}\>($\mathrm{\SIUnitSymbolMicro m}$)$ |  $n_{b}/n_{0}$ |  $k_{b}\sigma_{tz}$ |  $E\>($\mathrm{GeV}$)$ |  $\eta$(%) |  $\Delta\varepsilon_{n}(\%)$
---|---|---|---|---|---|---|---|---|---
Fig. 3(a) |  0.7 |  0.5 |  0.20 |  2.14 |  1 |  0.09 |  1 |  0.30 |  27.61
0.8 |  0.5 |  0.26 |  2.14 |  1 |  0.09 |  1 |  0.39 |  11.58
1.0 |  0.5 |  0.40 |  2.14 |  1 |  0.09 |  1 |  0.61 |  1.74
1.6 |  0.5 |  1.02 |  2.14 |  1 |  0.09 |  1 |  1.55 |  55.40
Fig. 3(b) |  1.01 |  0.5 |  0.41 |  2.14 |  0.25 |  0.045 |  1 |  0.16 |  1.74
1.00 |  0.5 |  0.40 |  2.14 |  2.5 |  0.14 |  1 |  1.52 |  2.64
0.80 |  0.5 |  0.26 |  2.14 |  25 |  0.45 |  1 |  9.15 |  5.83
Fig. 3(c) |  0.327 |  0.5 |  4.28 |  2.14 |  1 |  0.09 |  100 |  0.07 |  2.73
0.288 |  0.5 |  3.33 |  2.14 |  25 |  0.45 |  100 |  1.63 |  3.67
0.189 |  0.5 |  1.43 |  2.14 |  250 |  1.4 |  100 |  5.24 |  30.01

Table 1: Initial trailing positron beam parameters and corresponding values of
efficiency and emittance growth for simulations presented in Fig. 3.

Here, we study, by means of PIC simulations, the emittance evolution of an
initially Gaussian trailing positron bunch in a quasilinear wakefield from the
driver, which exhibits a transversely nonlinear focusing force. As we are
interested in reaching high energy efficiencies, the trailing positron bunch
can strongly load the wakefield and thus substantially modify the focusing
force, and this positron beam loading can be nonlinear if $n_{b}/n_{0}>1$ for
the trailing bunch. To model the potential emittance growth solely driven by
the non-ideal properties of the wakefield and most importantly by positron
beam loading, the driver evolution is turned off in the PIC simulations. By
doing so, we do not consider other potential limits such as driver and
wakefield stability over long propagation distances, and focus on the study of
the evolution of the trailing bunch towards an equilibrium. Figure 3 shows the
results of QuickPIC-OpenSource simulations performed for the same drive beam
that excites a quasilinear wakefield, varying the initial parameters of the
trailing positron bunch. The initial emittance of the trailing positron bunch
is kept the same to $0.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in all
simulations. Quasi-matching is, as expected, critically important, as can be
observed in Fig. 3(a) for a 1 GeV initial trailing-bunch energy, where the
emittance evolution is shown for different initial trailing beam sizes
$\sigma_{tr}$, that is for different initial beta functions. Because of the
nonlinear focusing force, a mismatched beam undergoes substantial emittance
growth and quickly saturates after a few betatron periods, while the quasi-
matched beam, with $\sigma_{tr}=$1.0\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
and $\beta=$0.4\text{\,}\mathrm{cm}$$, reaches the equilibrium state with only
negligible emittance growth.

Figure 3(b) illustrates the emittance evolution for different initial peak
densities $n_{b}/n_{0}$ of the trailing positron bunch, by increasing its
charge. Each simulation is optimized to determine the quasi-matched beam size
and minimize emittance growth. Interestingly, the results show that the
emittance growth of a quasi-matched beam does not substantially increase when
$n_{b}/n_{0}$ becomes larger than one (comparing $n_{b}/n_{0}=0.25$ and
$n_{b}/n_{0}=2.5$ with emittance growth of about $2\text{\,}\%$), that is when
beam loading becomes nonlinear. To test much higher values of $n_{b}/n_{0}$,
increasing further the trailing charge completely overloads the wakefield
(with part of the trailing bunch being decelerated), however, higher values of
$n_{b}/n_{0}$ can be reached by increasing the trailing beam energy, which
reduces the quasi-matched beam size and increases $n_{b}/n_{0}$. In Fig. 3(c),
the same emittance evolution for increasing values of $n_{b}/n_{0}$ is shown
but at a much higher initial trailing bunch energy of
$100\text{\,}\mathrm{GeV}$, instead of $1\text{\,}\mathrm{GeV}$ previously,
corresponding to smaller quasi-matched beam sizes for similar trailing charges
and energy efficiencies. Expressed in number of betatron periods, the
propagation distance required to reach the equilibrium is larger at higher
values of $n_{b}/n_{0}$, as well as the emittance growth, which is
$30\text{\,}\%$ for the largest simulated trailing bunch density,
$n_{b}/n_{0}=250$. Here, a transition in the characteristic regime of the
plasma electron response to the positron beam occurs when $k_{b}\sigma_{tz}$
becomes larger than one, where $k_{b}$ is the plasma wave number associated
with the positron beam density
[$k_{b}=\sqrt{n_{b}e^{2}/(m_{e}\epsilon_{0})}/c$]. This transition was
discussed in the case of a trailing electron beam Lehe  _et al._ (2014), in
which case a blowout is formed within the bunch itself when
$k_{b}\sigma_{tz}\gtrsim 1$. In the case of a trailing positron beam, plasma
electrons are initially sucked in and will cross the propagation axis within
the bunch itself if $k_{b}\sigma_{tz}\gtrsim 1$, or even execute multiple
oscillations if $k_{b}\sigma_{tz}\gg 1$. In contrast, for $k_{b}\sigma_{tz}\ll
1$, positron beam loading is not as severe as plasma electrons are sucked in,
thereby modifying the transverse wakefield, but do not cross the axis within
the bunch. In the simulations shown in Fig. 3(c), $k_{b}\sigma_{tz}$ increases
from 0.09 to 1.4 when $n_{b}/n_{0}$ is increased from 1 to 250, explaining the
difference in the qualitative evolution of the emittance discussed above.

These results demonstrate that, for a driver exciting a quasilinear wakefield,
the trailing positron bunch rapidly evolves towards an equilibrium with
limited emittance growth during this initial evolution, and emittance
preservation afterwards. Emittance growth is found to be negligible at the
few-percent level for linear and nonlinear positron beam loading when
$k_{b}\sigma_{tz}\ll 1$, and moderate for longer bunch lengths and/or higher
beam densities with $k_{b}\sigma_{tz}\gtrsim 1$.

###  II.3 Longitudinal phase space

High-quality beams have not only low transverse emittances, that need to be
preserved as discussed in Sec. II.2, but also low longitudinal emittances,
that characterize the area occupied by the beam in the longitudinal phase
space (LPS). The longitudinal beam quality is most often discussed in terms of
total energy spread (including correlated and uncorrelated components) and
bunch length, rather than longitudinal emittance.

One important challenge in plasma-based accelerators lies in the large energy
chirp that can be imprinted on the accelerated bunch, due to the
$\xi$-dependent longitudinal electric field $E_{z}$, which leads to a large
correlated energy spread. This problem has been extensively studied, and can
be addressed by either flattening $E_{z}$ (having a uniform $E_{z}$ along the
bunch) Tzoufras  _et al._ (2008); Lindstrøm  _et al._ (2021) or by using a
mechanism to dechirp the beam Brinkmann  _et al._ (2017); Manahan  _et al._
(2017); Döpp  _et al._ (2018); D’Arcy  _et al._ (2019); Ferran Pousa  _et al._
(2019); Shpakov  _et al._ (2019); Wu  _et al._ (2019a, b); Pompili  _et al._
(2021). While the correlated energy spread can be compensated, the
uncorrelated component (or slice energy spread) is an intrinsic feature of the
beam that cannot be removed by dechirping. Electron acceleration in the
blowout regime and in the absence of ion motion has the key advantage that
$E_{z}$ is independent of the transverse coordinate, and therefore the slice
energy spread is not degraded during acceleration. This property no longer
holds when accounting for ion motion in the blowout regime, and in the case of
positron acceleration, the presence of mobile plasma electrons within the
accelerated positron bunch induces a transversely non-uniform accelerating
field, contributing to an increase on the slice energy spread. Because this
degradation of the LPS via the slice energy spread poses a more fundamental
limit, we focus the discussion on the uncorrelated energy spread, assuming
that the correlated energy spread can be compensated by other means.

Figure 4:  Simulated LPS of the accelerated trailing positron bunch [(a)-(c)]
and corresponding longitudinal wakefield map in the vicinity of the positron
bunch [$E_{z}$, (d)-(f)] for different initial peak densities: $n_{b}/n_{0}=6$
[(a), (d)], $n_{b}/n_{0}=20$ [(b), (e)] and $n_{b}/n_{0}=50$ [(c), (f)]. The
slice-energy-spread-to-gain ratio $\delta(\xi)$ is represented as orange solid
lines on each LPS distribution. $\Delta E$ is the difference between final and
initial energy. All plasma, drive and trailing parameters are the same as for
simulations in Fig. 3(b), keeping the trailing beam size fixed at
$\sigma_{tr}=$0.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and varying only the
trailing charge and $n_{b}/n_{0}$.

To better assess the growth of slice energy spread in an accelerator stage, we
normalize it to the energy gain and define the slice-energy-spread-to-gain
ratio $\delta(\xi)$ as

$\delta(\xi)=\frac{\Delta E_{\mathrm{final}}(\xi)}{\langle
E_{\mathrm{final}}\rangle(\xi)-E_{\mathrm{init}}},$ (4)

where $\Delta E_{\mathrm{final}}(\xi)$ and $\langle
E_{\mathrm{final}}\rangle(\xi)$ are the absolute energy spread and mean energy
of slice $\xi$ in the final state, and we have assumed that all particles have
an energy $E_{\mathrm{init}}$ in the initial state. The slice energy spread
can vary from head to tail of the bunch and thus generally depends on the
slice longitudinal position $\xi$. Assuming a stable wakefield, $\delta(\xi)$
can also be calculated from the knowledge of the longitudinal wakefield
$E_{z}(x,y,\xi)$ and of the bunch density $n_{b}(x,y,\xi)$:

$\delta(\xi)=\frac{1}{\langle
E_{z}\rangle(\xi)}\left[\frac{\int{[E_{z}(x,y,\xi)-\langle
E_{z}\rangle(\xi)]^{2}n_{b}dxdy}}{\int{n_{b}dxdy}}\right]^{1/2},$ (5)

with

$\langle
E_{z}\rangle(\xi)=\frac{\int{E_{z}(x,y,\xi)n_{b}dxdy}}{\int{n_{b}dxdy}}.$ (6)

Figure 4 illustrates how the beam LPS, after $23\text{\,}\mathrm{cm}$ of
acceleration in the quasilinear plasma wakefield excited by an electron drive
bunch, and the longitudinal wakefield map of $E_{z}$, are modified when
increasing the peak density of the trailing positron beam, and thus the
energy-transfer efficiency. It is shown that at high beam density, positron
beam loading induces a transversely non-uniform accelerating field $E_{z}$
[see Fig. 4(d)-(f)], especially at the rear of the bunch, leading to an
increase of the slice energy spread [see Fig. 4(a)-(c)], with $\delta(\xi)$
approaching $1.5\text{\,}\%$ at the rear of the bunch for $n_{b}/n_{0}=50$.
The results show that the degradation of the LPS via the uncorrelated energy
spread can be severe, considering a $1\text{\,}\mathrm{\char 37\relax}$
uncorrelated energy spread as an upper bound for the acceptance of a collider
final focus system Raimondi and Seryi (2001). While the drive parameters are
kept fixed here as $n_{b}/n_{0}$ and $\eta$ are increased, it is possible to
optimize the driver, thereby minimize the uncorrelated energy spread at a
given level of efficiency $\eta$ (see Sec. IV.1). Subsequently, the best
performance at each value of $\eta$ can be determined and the trade-off that
exists between uncorrelated energy spread and energy-transfer efficiency can
be assessed (see Sec. IV.2).

##  III Blowout regime using a donut-shaped electron beam driver

Figure 5:  Loaded nonlinear plasma wakefield driven by a hollow electron
driver. The donut electron driver parameters are
$Q_{d}=$8.6\text{\,}\mathrm{n}\mathrm{C}$$, $k_{p}r_{0}=1$,
$k_{p}\sigma_{r}=0.39$, $k_{p}\sigma_{z}=0.7$ and $\xi_{0}=0$. A Gaussian
trailing positron bunch is located at $k_{p}\xi=-0.55$ with
$Q_{t}=$189\text{\,}\mathrm{p}\mathrm{C}$$, $k_{p}\sigma_{r}=0.035$, and
$k_{p}\sigma_{z}=0.09$. The plasma density is set to
$n_{0}=$5\text{\times}{10}^{16}\text{\,}{\mathrm{cm}}^{-3}$$. (a) Cross
section of the donut driver bunch density in the $(x,y)$ plane, (b) map of the
plasma electron density in the $(\xi,x)$ plane, with dashed red lines showing
the 1$\sigma$ contours of drive and trailing bunch densities and dashed green
lines showing the positron central slice and slices at $2\sigma_{z}$ from the
center.

The traditional blowout regime with uniform ion density is known to provide
ideal features for electron acceleration: high acceleration gradient,
transversely linear focusing force and uniform accelerating field. For
positrons, this regime is not favorable unless a uniform electron filament is
added inside the blowout cavity, which provides focusing and minimizes the
slice energy spread. An electron filament can be present on the propagation
axis in the blowout regime when using a donut-shaped electron Jain  _et al._
(2015); Vieira  _et al._ (2016) or laser Vieira and Mendonça (2014) driver.
Here, we focus on the donut electron beam driver, and define the bunch density
of the hollow electron driver as follows Jain  _et al._ (2015):

$n_{b}(r,\xi)=n_{\mathrm{peak}}\exp\left[-\frac{(r-r_{0})^{2}}{2\sigma_{r}^{2}}-\frac{(\xi-\xi_{0})^{2}}{2\sigma_{z}^{2}}\right],$
(7)

where the peak of the bunch density, $n_{\mathrm{peak}}$, is located at
$(r_{0},\xi_{0})$, off the propagation axis, and $r_{0}$ and $\sigma_{r}$
represent respectively the radius and thickness of the donut ring. With such
hollow structure for the electron driver, plasma electrons with initial radii
$r>r_{0}$ are blown out similarly to the case of a Gaussian-shaped driver,
thereby generating a strong wakefield with high accelerating fields, but inner
plasma electrons tend to be pushed inward Jain  _et al._ (2015). This
separation of plasma-electron trajectories allows for the formation of a
region with an excess of plasma electrons near the propagation axis that can
provide focusing for positrons. A cross section of such a hollow electron
driver, with $k_{p}r_{0}=1$ and $k_{p}\sigma_{r}=0.39$, and the generated
blowout structure with the trailing positron bunch and the on-axis electron
filament are shown in Fig. 5. The simulation was performed using QuickPIC-
OpenSource. The quality of the plasma electron uniformity near the propagation
axis, as experienced by the trailing positron bunch, can be manipulated by
driver optimization (see Sec. IV.1) and by placing the positron bunch at the
appropriate phase of the accelerating field.

Figure 6:  Plasma electron density in the vicinity of the positron bunch with
the same set of parameters given in Fig. 5: map of $n_{p}$ in the $(\xi,x)$
plane (up), on-axis value of $dn_{p}/d\xi$ (bottom left) and transverse line-
outs of $n_{p}$ (bottom right) for different $\xi$ slices (shown as dashed
lines in bottom left). The inset of the top plot shows the slice-energy-
spread-to-gain ratio $\delta(\xi)$ from $-2\sigma_{z}$ to $2\sigma_{z}$ of the
positron bunch.

Similarly to the discussion in Sec. II.3, we would like to find an optimal
situation minimizing the slice energy spread across all bunch slices.
According to Panofsky-Wenzel theorem Panofsky and Wenzel (1956),
$\partial_{\xi}F_{r}=\partial_{r}F_{z}$. Neglecting ion motion, the
contribution of ions and blown-out electrons to the transverse force is a
simple $\xi$-independent and transversely linear term $k_{p}r/2$, and thus the
shape of $F_{r}$ is mainly determined by the plasma electron filament source
term. The best situation can be expected when the plasma electron density is
uniform on-axis, and has the same dependence in $r$ off-axis for all
longitudinal slices. This can be reasonably achieved over the size of a short
and small positron bunch in unloaded or lightly loaded cases. When the
wakefield is moderately or heavily loaded, the positron bunch induces a strong
response from plasma electrons near the propagation axis, leading to non-
uniformity in their density, both radially and longitudinally, resulting in
sizeable slice energy spread, especially near the bunch center. Figure 6 shows
the electron plasma density in the vicinity of the positron bunch and the
corresponding slice-energy-spread-to-gain ratio $\delta(\xi)$, for the same
simulation as the one presented in Fig. 5. The plasma density is uniform in
$\xi$ and has a slow dependence in $r$ at the head, that is, when the beam
loading is still weak. As a result, the slice energy spread is low at the
bunch head. Moving backward along the bunch, the non-uniformity in the plasma
density then increases and peaks right after the bunch center, and then drops
slightly towards the bunch tail. This behavior in plasma electron uniformity
is directly reflected in the slice-energy-spread-to-gain ratio $\delta(\xi)$
of the positron bunch (see inset of upper plot in Fig. 6).
In order to maintain the slice-energy-spread-to-gain ratio below the percent
level, the trailing-to-drive bunch charge ratio as well as the energy-transfer
efficiency $\eta$ remain low at the few percent level. Continuing to increase
the positron bunch charge results in overloading the wakefield and degrading
the beam quality. For the parameters of Figs. 5-6, the charge ratio is
$2.2\text{\,}\mathrm{\char 37\relax}$, the energy-transfer efficiency is
$2.9\text{\,}\mathrm{\char 37\relax}$, and the slice energy spread is at the
percent level.

##  IV Uncorrelated energy spread and energy efficiency

To determine optimal scenarios for positron acceleration in different schemes
and regimes, we focus on two figures of merit: the energy-transfer efficiency
and the uncorrelated energy spread, as they can be important limits (see Secs.
II and III) for a high-energy collider application. While we have considered
so far the slice-energy-spread-to-gain ratio $\delta(\xi)$, which depends on
the slice $\xi$, we will use here a single parameter $\delta$ quantifying the
longitudinal quality, and defined as

$\delta=\frac{1}{\langle
E_{z}\rangle}\left[\frac{1}{N_{b}}\int{[E_{z}(x,y,\xi)-\langle
E_{z}\rangle(\xi)]^{2}n_{b}dxdyd\xi}\right]^{1/2},$ (8)

with

$\langle E_{z}\rangle=\frac{1}{N_{b}}\int{E_{z}(x,y,\xi)n_{b}dxdyd\xi}.$ (9)

This parameter $\delta$ describes the ratio between the total energy spread of
the beam after removal of the chirp induced by $\langle E_{z}\rangle(\xi)$ and
the energy gain, and will be referred to as the uncorrelated-energy-spread-to-
gain ratio from now on.

###  IV.1 Driver optimization

For a given set of positron beam parameters, the driver, and in particular its
$\sigma_{r}$ value, can be optimized in order to minimize $\delta$. Here, we
keep the drive beam charge constant so that $\eta$ is not being strongly
modified in this optimization. Figure 7(a) highlights the process of drive
beam optimization in the linear regime. All simulations were performed using
QuickPIC-OpenSource. Looking at the transverse dependence of the longitudinal
electric field, $E_{z}$ vs. $x$, and comparing the unloaded (dashed line) to
the loaded case (solid line), it is found that beam loading can transversely
flatten $E_{z}$, thereby minimizing the uncorrelated energy spread. This
result is valid when the flattening of $E_{z}$ vs. $x$ is done for the central
beam slice, which has the largest effect on the overall beam quality. At a
given positron bunch charge, the fields produced by a specific drive bunch is
either overloaded, flattened or insufficiently loaded at the central bunch
slice. For instance, a large positron charge can quickly overload the fields
for drivers with large $\sigma_{r}$. Therefore, optimizing the drive beam size
allows the transverse flattening of $E_{z}$ which leads to an optimized
$\delta$ [see inset of Fig. 7(a)]; this optimization is required for each
value of the positron charge. When increasing the positron bunch charge,
drivers with smaller $\sigma_{r}$ are required to maintain a transversely
flattened $E_{z}$ and to keep $\delta$ to a minimum.

Figure 7: The central-slice accelerating field $E_{z}$ as a function of the
transverse coordinate $x$ for the linear regime (a) and for the donut regime
(b), for different values of $k_{p}\sigma_{r}$ for the driver, and for loaded
(solid lines) and unloaded (dashed lines) cases. The insets in (a) and (b)
show the uncorrelated-energy-spread-to-gain ratio $\delta$. (c) Loaded
accelerating field $\langle E_{z}\rangle_{t}$ as a function of the positron
beam charge $Q_{t}$ in the donut regime, for drivers with different values of
$k_{p}\sigma_{r}$. The plasma density is
$n_{0}=$5\text{\times}{10}^{16}\text{\,}{\mathrm{cm}}^{-3}$$ in all
simulations.

This optimization, however, is ultimately limited if one wants to remain in
the linear regime, as the decrease of $\sigma_{r}$ eventually leads to
$n_{b}>n_{0}$ for the driver. Here, we will refer to the linear or quasilinear
regime when the driver density satisfies $n_{b}/n_{0}\leq 0.5$. Nevertheless,
it can be interesting to leave the linear regime and further decrease
$\sigma_{r}$ and increase $n_{b}$ with a partial blowout forming, as long as a
good performance is achieved. This is the case in the moderately nonlinear
regime with $n_{b}/n_{0}$ going up to 2, above which a complete blowout
structure is observed with a degraded longitudinal quality for the accelerated
positron bunch. In this moderately nonlinear regime, characterized by a driver
with $n_{b}/n_{0}\sim 1-2$ and a normalized current
$\Lambda=2I_{\mathrm{peak}}/I_{A}=k_{p}^{2}\sigma_{r}^{2}n_{b}/n_{0}<1$ (non-
relativistic response of plasma electrons Lu  _et al._ (2010),
$I_{\mathrm{peak}}$ being the driver peak current and
$I_{A}\simeq$17\text{\,}\mathrm{k}\mathrm{A}$$ the Alfvén current), the
drivers are optimized the same way as in the linear regime, yet this
optimization only provides small changes in $\delta$. An additional
optimization in trailing positron bunch position is performed to ensure that
the positron bunch stays in a focusing field.

Figure 7(b) shows a similar process for the donut regime. On the one hand,
when the donut-shaped electron driver has a large hollow region [i.e. with a
small value of $\sigma_{r}$ in Eq. (7)], allowing an excessive amount of
plasma electrons to enter in the blowout cavity, it leads to non-uniform
plasma electron densities near the propagation axis. In addition, such a
driver is more susceptible to beam loading, as indicated in Fig. 7(c), showing
the stronger reduction of the loaded accelerating field with the trailing
positron charge, for small values of $\sigma_{r}$. In such a regime with the
usual plasma electron sheath orbiting around the blowout cavity and with the
on-axis plasma electron filament, the positron beam load acts on both plasma
species. When decreasing $\sigma_{r}$ and increasing the number of on-axis
plasma electrons, the relative contribution of the latter in the beam loading
process becomes more prominent, leading to a stronger reduction of $E_{z}$
when the positron charge is increased [see Fig. 7(c)]. On the other hand, a
relatively uniform on-axis plasma electron density is created by a driver with
a large value of $\sigma_{r}$, allowing sufficient number of plasma electrons
to enter the blowout cavity and to provide focusing for the positron bunch,
while at the same time keeping non-uniformity in the plasma electron density
in the vicinity of the positron bunch at a low level. This results in a much
flatter $E_{z}$ field (transversely) at large $\sigma_{r}$ in both unloaded
and loaded cases [see Fig. 7(b)], and in a lower uncorrelated energy spread
[see inset of Fig. 7(b)]. However, the donut thickness $\sigma_{r}$ cannot be
increased indefinitely, as for $k_{p}\sigma_{r}\gtrsim 0.4$, the plasma
electron density on the propagation axis becomes insufficient to compensate
for the background ions, resulting in a defocusing field for the positron
bunch.

###  IV.2 Trade-off $\delta-\eta$ for different regimes

Generally, the most straightforward way to increase the energy-transfer
efficiency $\eta$ is to increase the trailing positron charge, in order to
extract more energy from the plasma wakefield. For electron acceleration in
the blowout regime, apart from limits associated with ion motion Rosenzweig
_et al._ (2005); Gholizadeh  _et al._ (2010); An  _et al._ (2017); Benedetti
_et al._ (2017); Mehrling  _et al._ (2018) and hosing instability Huang  _et
al._ (2007); Mehrling  _et al._ (2017); Lebedev  _et al._ (2017), this
increase of charge and efficiency can be done without compromising beam
quality, or even with an improved total energy spread by flattening $E_{z}$
longitudinally and thus minimizing the energy chirp along the accelerated
bunch Tzoufras  _et al._ (2008). In the case of positron acceleration with
plasma electrons providing focusing fields in the vicinity of the trailing
bunch, the situation is different and typically higher positron charge and
higher $\eta$ comes at the cost of a degraded beam quality and a higher
uncorrelated-energy-spread-to-gain ratio $\delta$, because of the strong and
fast response of those plasma electrons to the positron load. As a result,
there is a clear trade-off between high efficiency and low uncorrelated energy
spread, and different regimes can perform differently in this manner.

Figure 8: (a) Uncorrelated-energy-spread-to-gain ratio $\delta$ vs. energy-
transfer efficiency $\eta$ for different regimes of positron acceleration. The
dashed black line depicts the constraint
$\delta\lesssim$1\text{\,}\mathrm{\char 37\relax}$$ explained in the text. For
each regime, the different data points are obtained by increasing $Q_{t}$ and
optimizing the driver to minimize $\delta$. (b) Loaded accelerating field
$E_{z}$ vs. trailing positron charge $Q_{t}$ for the same data points as in
(a). Beam and plasma parameters or parameter range for these simulation
results can be found in Table 2.

As higher efficiency is achieved by stronger beam loading, it directly impacts
$\delta$. Yet, as discussed in Sec. IV.1, as we increase $\eta$ by taking a
higher positron charge, the plasma wakefield and its driver need to be re-
optimized to provide the lowest uncorrelated energy spread for this higher
positron charge and efficiency. This is especially important in the linear
regime, as the optimum for $\delta$ is obtained by transverse flattening of
$E_{z}$, which can be controlled by the driver beam size $\sigma_{r}$.

In order to make relevant comparisons between different schemes, we keep
similar parameters for the trailing positron bunch, namely a beam size
$\sigma_{r}$ of around $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$, and the same
bunch length, $\sigma_{z}=$2.14\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The
drive bunch length is also kept fixed at
$\sigma_{z}=$16.7\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, as well as the
plasma density at
$n_{0}=$5\text{\times}{10}^{16}\text{\,}{\mathrm{cm}}^{-3}$$. All parameters
or parameter range used in the simulations are summarized in Table 2. The
determination of $\eta$ and $\delta$ with single-step QuickPIC-OpenSource
simulations does not depend on the positron beam energy and emittance, but
given the positron beam sizes considered here, quasi-matching (see Sec. II.2)
would typically require $\mathrm{\SIUnitSymbolMicro m}$-scale normalized
emittance for a $1\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ beam. Simulating
scenarios relevant to lower emittances and higher energies require much
smaller beam sizes and thus extremely high resolution, which is beyond the
scope of this paper.

The result of this comparison is presented in Fig. 8(a), where for each regime
increasing values of $\eta$ are obtained by increasing the trailing positron
charge $Q_{t}$, and the optimization described in IV.1 is performed to
minimize $\delta$ for each individual data point. Figure 8(a) thus provides a
direct representation of the trade-off between $\delta$ and $\eta$ for
different regimes.

In the linear regime with a low drive beam charge of
$38\text{\,}\mathrm{p}\mathrm{C}$ (referred to as “Linear Low Charge” in Fig.
8), one finds the trailing positron charge and energy-transfer efficiency can
be increased to about $5\text{\,}\mathrm{p}\mathrm{C}$ and
$30\text{\,}\mathrm{\char 37\relax}$, while keeping $\delta$ below
$1\text{\,}\mathrm{\char 37\relax}$. Here, we will consider this
$1\text{\,}\mathrm{\char 37\relax}$ limit in $\delta$, shown as a dashed line
in Fig. 8(a) as an upper bound for an acceptable uncorrelated energy spread in
view of the requirements of a final focus system Raimondi and Seryi (2001).
Because of the small drive charge, it’s possible to keep decreasing the driver
beam size $\sigma_{r}$ as $Q_{t}$ and $\eta$ are increased, and to continue to
excite a linear wakefield with $n_{b}/n_{0}\leq 0.5$ for the driver. However,
with such low drive charge, the loaded accelerating field only slightly
exceeds $1\text{\,}\mathrm{GV}\text{\,}{\mathrm{m}}^{-1}$ [see Fig. 8(b)], and
the accelerated positron charge is only $5\text{\,}\mathrm{p}\mathrm{C}$.

| Linear Low Charge | Linear High Charge | Moderately Non-Linear | Donut Driver
---|---|---|---|---
|  Driver |  Trailing |  Driver |  Trailing |  Driver |  Trailing |  Driver |  Trailing
$\sigma_{r}\>($\mathrm{\SIUnitSymbolMicro m}$)$ |  6.09–19.27 |  1.19 |  12.19–14.56 |  1.19 |  6.28–8.22 |  1.19 |  9.4 |  0.85
$\sigma_{z}\>($\mathrm{\SIUnitSymbolMicro m}$)$ |  16.7 |  2.14 |  16.7 |  2.14 |  16.7 |  2.14 |  16.7 |  2.14
$n_{b}/n_{0}$ |  0.05–0.5 |  0.25–15.5 |  0.35–0.5 |  1–75 |  1.1–1.88 |  25–70 |  2.97 |  35–15000
$k_{p}\xi$ |  0 |  -6.2 |  0 |  -6.2 |  0 |  -6.25 – -5.90 |  0 |  -0.55

Table 2: Beam parameters or parameter range for the simulation results
presented in Fig. 8. The plasma density is
$n_{0}=$5\text{\times}{10}^{16}\text{\,}{\mathrm{cm}}^{-3}$$ for all
simulations, and $k_{p}r_{0}=1$ for “Donut Driver” simulations. In all
regimes, for a trailing positron beam with an initial energy
$E=$1\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$, quasi-matched conditions
generally correspond to $\mathrm{\SIUnitSymbolMicro m}$-scale normalized
emittances.

One can naturally seek to accelerate higher trailing positron charges at
higher fields by increasing the drive beam charge. Considering the linear
regime again but with a higher drive beam charge of
$152\text{\,}\mathrm{p}\mathrm{C}$ (referred to as “Linear High Charge” in
Fig. 8), and repeating the same optimization process, one finds that the limit
of $n_{b}/n_{0}\leq 0.5$ for the driver to continue to excite linear
wakefields prevents any further optimization or decrease of the drive beam
size $\sigma_{r}$ beyond the first two “Linear High Charge” data points in
Fig. 8. Because of this lack of optimization, $\delta$ quickly increases for
the following data points with higher $Q_{t}$ and $\eta$. Using the same
constraint $\delta\lesssim$1\text{\,}\mathrm{\char 37\relax}$$ as before,
$Q_{t}$ and $\eta$ are limited to about $10\text{\,}\mathrm{p}\mathrm{C}$ and
$20\text{\,}\mathrm{\char 37\relax}$, while the loaded accelerating field
$E_{z}$ reaches $3\text{\,}\mathrm{GV}\text{\,}{\mathrm{m}}^{-1}$.

In fact, we can continue to decrease the drive beam size $\sigma_{r}$ despite
leaving the linear regime, as long as the acceleration performance is
satisfactory, which is found to be the case in the moderately nonlinear regime
characterized by $n_{b}/n_{0}\sim 1-2$ and $\Lambda<1$, and introduced in Sec.
IV.1. In Fig. 8, the data points for the moderately nonlinear regime share the
same drive charge of $152\text{\,}\mathrm{p}\mathrm{C}$ as the “Linear High
Charge” case, but the smaller value of $\sigma_{r}$ being used for the driver
makes it possible to considerably improve the transverse uniformity of
$E_{z}$, resulting in lower $\delta$. Figure 8 shows that in the moderately
nonlinear regime we can achieve $\delta\lesssim$1\text{\,}\mathrm{\char
37\relax}$$ with trailing positron charge and energy-transfer efficiency of up
to $25\text{\,}\mathrm{p}\mathrm{C}$ and $40\text{\,}\mathrm{\char 37\relax}$,
and an accelerating field of
$E_{z}\simeq$5\text{\,}\mathrm{GV}\text{\,}{\mathrm{m}}^{-1}$$.

Finally, if one aims for even higher accelerating field and higher positron
charge, Fig. 8 shows that the nonlinear blowout regime with a donut-shaped
electron driver is the best suited, at the cost of a degraded trade-off
between $\delta$ and $\eta$. This regime is indeed compatible with much higher
drive charge ($Q_{d}=$8.6\text{\,}\mathrm{n}\mathrm{C}$$ for the “Donut
Driver” data points in Fig. 8), and thus higher $E_{z}$ and $Q_{t}$, typically
one to two orders of magnitude higher than in the previous cases. However,
above $5\text{\,}\mathrm{\char 37\relax}$ energy-transfer efficiency, $\delta$
degrades beyond $1\text{\,}\mathrm{\char 37\relax}$, so the donut regime can
be used for high fields and high trailing charges with a compromise on either
an energy-transfer efficiency limited to the few-percent range, or on an
uncorrelated energy spread exceeding the percent level.

##  V Conclusion

Our results show the importance of beam loading in plasma-based positron
acceleration, whose properties differ significantly from beam loading for
electron acceleration in the blowout regime. Indeed, when plasma electrons are
present in the vicinity of the accelerated positron bunch to provide focusing
fields, positron beam loading can then alter the focusing properties of the
plasma wakefield as well as the transverse uniformity of the accelerating
field, because of the strong and fast response of those plasma electrons. Yet,
beam loading is a prerequisite for good energy-transfer efficiency, which is
highly desirable for a high energy physics application. The results presented
in the previous section show that one needs to make a compromise between the
uncorrelated energy spread described by the parameter $\delta$ and the energy-
transfer efficiency $\eta$.

In the linear regime, we have found that while generally the positron bunch
quickly evolves towards an equilibrium with limited emittance growth, the
uncorrelated energy spread can set an important limit. To maximize the energy-
transfer efficiency in the linear regime, both the electron driver and the
trailing positron bunch should have small beam sizes, $k_{p}\sigma_{r}<1$, to
avoid leaving energy in the plasma wave by poor matching of the transverse
size of the drive and trailing plasma wakefields. Interestingly, driving a
linear plasma wakefield with a Gaussian-shaped electron driver with
$n_{b}/n_{0}<1$ and extracting its energy with nonlinear beam loading by a
trailing positron bunch with $n_{b}/n_{0}>1$ is fully relevant and provides
good performance with $\eta$ going up to $30\text{\,}\mathrm{\char 37\relax}$
while keeping $\delta\lesssim$1\text{\,}\mathrm{\char 37\relax}$$. The limited
positron charge and accelerating field of the linear regime can be increased
in the moderately nonlinear regime, with $\eta$ going up to
$40\text{\,}\mathrm{\char 37\relax}$ for
$\delta\lesssim$1\text{\,}\mathrm{\char 37\relax}$$. The nonlinear blowout
regime driven by a donut-shaped electron driver is found to allow acceleration
of much higher positron charge at much higher accelerating gradients, but the
energy-transfer efficiency needs to be kept at the few-percent level to
fulfill $\delta\lesssim$1\text{\,}\mathrm{\char 37\relax}$$.

Further research will aim at extending the results to lower emittances,
asymmetric beams and higher energies that are relevant in the route towards a
plasma-based collider design, and to provide systematic comparisons of
existing positron acceleration regimes. It is important to note that a
$e^{+}e^{-}$ collider does not necessarily require the same performance for
electrons and positrons, and in particular for a plasma-based collider, the
requirements for positrons could be not as stringent as those for electrons,
and thus may be somewhat lowered in comparison to the parameters of linear
collider designs based on RF accelerators Behnke  _et al._ (2013); Aicheler
_et al._ (2012). Given the challenges of positron acceleration in plasmas,
having asymmetrical requirements for electrons and positrons could be
crucially important towards realistic designs of plasma-based colliders Chen
_et al._ (2020).

###### Acknowledgements.

The work at LOA was supported by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (Miniature
beam-driven Plasma Accelerators project, Grant Agreement No. 715807). SLAC was
supported by U.S. DOE Contract DE-AC02-76SF00515. The work at University of
Oslo was supported by the Research Council of Norway, Grant No. 313770.
Numerical simulations were performed using HPC resources from GENCI-TGCC
(Grants No. 2020-A0080510786 and No. 2020-A0090510062) with the IRENE
supercomputer, and using the Open Source version Qui  of the QuickPIC code
Huang  _et al._ (2006); An  _et al._ (2013), a 3D parallel (MPI & OpenMP
Hybrid) Quasi-Static Particle-In-Cell code.

##  Appendix A energy-transfer efficiency in three-dimensional linear plasma
wakefields

In a two-bunch acceleration scheme, energy is transferred from the driver to
the plasma wave, and the trailing bunch takes the energy from the plasma wave
excited by the driver. Apart from energy-transfer efficiency calculated using
the particle point of view shown in Eq. (1), the efficiency can also be
obtained by calculating the energy in the plasma waves generated by the drive
and trailing bunches:

$\eta=1-\frac{\int E_{z0,\mathrm{tot}}^{2}d^{2}\mathbf{r}_{\perp}+\int
E_{r0,\mathrm{tot}}^{2}d^{2}\mathbf{r}_{\perp}}{\int
E_{z0,d}^{2}d^{2}\mathbf{r}_{\perp}+\int
E_{r0,d}^{2}d^{2}\mathbf{r}_{\perp}},$ (10)

where $E_{z0,d}$ and $E_{r0,d}$ are the amplitudes of the $z$ and $r$
components of the electric field of the drive beam plasma wave,
$E_{z0,\mathrm{tot}}$ and $E_{r0,\mathrm{tot}}$ are the amplitudes for the
total plasma wave from both beams, and the integral is performed over the
transverse coordinates. Importantly, these amplitudes need to be evaluated in
the wake of the beams.

In the linear regime, the longitudinal and radial components of the electric
field for the plasma wave excited by a particle beam with cylindrical symmetry
can be written as Keinigs and Jones (1987):

$\displaystyle E_{z}(r,\xi)=$
$\displaystyle\frac{qk_{p}^{2}}{\epsilon_{0}}\int_{0}^{+\infty}r^{\prime}dr^{\prime}K_{0}(k_{p}r_{>})I_{0}(k_{p}r_{<})$
$\displaystyle\times\int_{\xi}^{+\infty}d\xi^{\prime}n_{b}(r^{\prime},\xi^{\prime})\cos{k_{p}(\xi-\xi^{\prime})},$
(11) $\displaystyle E_{r}(r,\xi)=$
$\displaystyle-\frac{qk_{p}}{\epsilon_{0}}\int_{0}^{+\infty}r^{\prime}dr^{\prime}K_{1}(k_{p}r_{>})I_{1}(k_{p}r_{<})$
$\displaystyle\times\int_{\xi}^{+\infty}d\xi^{\prime}\frac{\partial
n_{b}(r^{\prime},\xi^{\prime})}{\partial
r^{\prime}}\sin{k_{p}(\xi-\xi^{\prime})},$ (12)

where $n_{b}(r,\xi)$ is the bunch density, $q$ is the particle charge, $r_{<}$
is the smaller of $r$ and $r^{\prime}$ and $r_{>}$ the larger. $I_{n}$ and
$K_{n}$ are the $n^{\mathrm{th}}$ order modified Bessel functions of the first
and second kind respectively.

###  A.1 Efficiency for separable beams

When the beam density is mathematically separable in the coordinates $r$ and
$\xi$, that is $n_{b}(r,\xi)=N\mathcal{R}(r)\mathcal{Z}(\xi)$ with $N$ the
number of particles in the beam, $E_{z}(r,\xi)$ and $E_{r}(r,\xi)$ can also be
written in separable forms. We use the following convention for $\mathcal{R}$
and $\mathcal{Z}$:

$\displaystyle\int_{0}^{+\infty}2\pi rdr\mathcal{R}(r)=1,$
$\displaystyle\int_{-\infty}^{+\infty}d\xi\mathcal{Z}(\xi)=1,$ (13)

for which the condition $\int n_{b}d^{3}\mathbf{r}=N$ is satisfied. In this
case, $E_{z}(r,\xi)$ and $E_{r}(r,\xi)$ can be written as:

$\displaystyle
E_{z}(r,\xi)=\frac{Nqk_{p}^{2}}{\epsilon_{0}}\mathbb{L}_{z}(\xi)\mathbb{T}_{z}(r),$
$\displaystyle
E_{r}(r,\xi)=-\frac{Nqk_{p}^{2}}{\epsilon_{0}}\mathbb{L}_{r}(\xi)\mathbb{T}_{r}(r),$
(14)

with

$\displaystyle\mathbb{L}_{z}(\xi)=\int_{\xi}^{\infty}d\xi^{\prime}\mathcal{Z}(\xi^{\prime})\cos{k_{p}(\xi-\xi^{\prime})},$
$\displaystyle\mathbb{L}_{r}(\xi)=\int_{\xi}^{\infty}d\xi^{\prime}\mathcal{Z}(\xi^{\prime})\sin{k_{p}(\xi-\xi^{\prime})},$
$\displaystyle\mathbb{T}_{z}(r)=\int_{0}^{\infty}r^{\prime}dr^{\prime}K_{0}(k_{p}r_{>})I_{0}(k_{p}r_{<})\mathcal{R}(r^{\prime}),$
$\displaystyle\mathbb{T}_{r}(r)=\frac{1}{k_{p}}\int_{0}^{\infty}r^{\prime}dr^{\prime}K_{1}(k_{p}r_{>})I_{1}(k_{p}r_{<})\frac{\partial\mathcal{R}(r^{\prime})}{\partial
r^{\prime}},$ (15)

where $\mathbb{L}_{z,r}$ describes the longitudinal shape and only depend on
$\xi$, and $\mathbb{T}_{z,r}$ describes the transverse shape and only depend
on $r$. $\mathbb{L}_{z,r}$ and $\mathbb{T}_{z,r}$ are all dimensionless
functions.

In Eq. (10), we need the amplitude of the fields in the wake of the beams,
which can be mathematically evaluated in the limit $\xi\rightarrow-\infty$
when beams have finite lengths. In this limit, both $\mathbb{L}_{z}$ and
$\mathbb{L}_{r}$ are sinusoidal functions of $\xi$, which are
$90\text{\,}\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{s}$
out of phase with each other, but share the same maximum, which we will simply
denote as $\mathbb{L}$ in the following. We have for this longitudinal factor
$\mathbb{L}$:

$\displaystyle\mathbb{L}$
$\displaystyle=\max_{\phi}\int_{-\infty}^{+\infty}d\xi^{\prime}\mathcal{Z}(\xi^{\prime})\cos{k_{p}(\phi-\xi^{\prime})}$
$\displaystyle=\max_{\phi}\left[\mathrm{Re}\int_{-\infty}^{+\infty}d\xi^{\prime}\mathcal{Z}(\xi^{\prime})e^{ik_{p}(\phi-\xi^{\prime})}\right]$
$\displaystyle=\max_{\phi}\left[\mathrm{Re}\;e^{ik_{p}\phi}\int_{-\infty}^{+\infty}d\xi^{\prime}\mathcal{Z}(\xi^{\prime})e^{-ik_{p}\xi^{\prime}}\right]$
$\displaystyle=|\tilde{\mathcal{Z}}(k=k_{p})|,$ (16)

where $\tilde{\mathcal{Z}}$ is the Fourier transform of $\mathcal{Z}$:

$\displaystyle\tilde{\mathcal{Z}}(k)=\int_{-\infty}^{+\infty}dx\mathcal{Z}(x)e^{-ikx}.$

In particular, for short bunches with $k_{p}\sigma_{z}\ll 1$,
$\mathbb{L}\simeq 1$.

Using Eqs. (10) and (14)-(15), and under the same assumptions as in Eq. (2)
for the drive-trailing bunch separation $\Delta\xi$ and particle charges
$q_{d}=\pm q_{t}$, the efficiency for separable beams in the 3D linear regime
can be calculated, and reads:

$\displaystyle\eta=$ $\displaystyle
2\frac{N_{t}\mathbb{L}_{t}}{N_{d}\mathbb{L}_{d}}\frac{\int
rdr\left(\mathbb{T}_{z,d}\mathbb{T}_{z,t}+\mathbb{T}_{r,d}\mathbb{T}_{r,t}\right)}{\int
rdr\left(\mathbb{T}_{z,d}^{2}+\mathbb{T}_{r,d}^{2}\right)}$
$\displaystyle-\left(\frac{N_{t}\mathbb{L}_{t}}{N_{d}\mathbb{L}_{d}}\right)^{2}\frac{\int
rdr\left(\mathbb{T}_{z,t}^{2}+\mathbb{T}_{r,t}^{2}\right)}{\int
rdr\left(\mathbb{T}_{z,d}^{2}+\mathbb{T}_{r,d}^{2}\right)},$ (17)

where the subscripts $d$ and $t$ specify driver and trailing for each
quantity.

###  A.2 Efficiency for 3D Gaussian beams

The beam density for a 3D Gaussian beam writes as:

$n_{b}(r,\xi)=\frac{N}{\sqrt{2\pi}^{3}\sigma_{r}^{2}\sigma_{z}}\exp{\left(-\frac{r^{2}}{2\sigma_{r}^{2}}\right)}\exp{\left(-\frac{\xi^{2}}{2\sigma_{z}^{2}}\right)}.$
(18)

It is a particular case of a separable beam with:

$\displaystyle\mathcal{R}(r)=\frac{1}{2\pi\sigma_{r}^{2}}\exp{\left(-\frac{r^{2}}{2\sigma_{r}^{2}}\right)},$
$\displaystyle\mathcal{Z}(\xi)=\frac{1}{\sqrt{2\pi}\sigma_{z}}\exp{\left(-\frac{\xi^{2}}{2\sigma_{z}^{2}}\right)}.$
(19)

The functions $\mathbb{T}_{z}$ and $\mathbb{T}_{r}$ are given by Eq. (15), the
longitudinal factor $\mathbb{L}$ simplifies to:

$\displaystyle\mathbb{L}=\exp\left(-\frac{k_{p}^{2}\sigma_{z}^{2}}{2}\right),$
(20)

and the energy-transfer efficiency can be calculated using Eq. (17).

When the beam transverse size is far larger than the plasma skin depth,
$\sigma_{r}\gg 1/k_{p}$, the function $\mathbb{T}_{z}$ approximately takes a
Gaussian shape
$\mathbb{T}_{z}\sim\exp{(-r^{2}/2\sigma_{r}^{2})}/\sigma_{r}^{2}$, and
$\mathbb{T}_{r}\ll\mathbb{T}_{z}$. When both drive and trailing bunches
satisfy $k_{p}\sigma_{r}\gg 1$, the expression for $\eta$ can be simplified
using the asymptotic approximation for the transverse shape and reads:

$\displaystyle\eta=\frac{N_{t}\mathbb{L}_{t}}{N_{d}\mathbb{L}_{d}}\frac{\sigma_{dr}^{2}}{\sigma_{tr}^{2}}\left[\frac{4}{1+\frac{\sigma_{dr}^{2}}{\sigma_{tr}^{2}}}-\frac{N_{t}\mathbb{L}_{t}}{N_{d}\mathbb{L}_{d}}\right].$
(21)

For either short bunches ($k_{p}\sigma_{z}\ll 1$ and $\mathbb{L}\simeq 1$) or
for equal bunch length ($\sigma_{dz}=\sigma_{tz}$ and thus
$\mathbb{L}_{d}=\mathbb{L}_{t}$), we finally obtain Eq. (3):

$\eta=\frac{N_{t}}{N_{d}}\frac{\sigma_{dr}^{2}}{\sigma_{tr}^{2}}\left[\frac{4}{1+\frac{\sigma_{dr}^{2}}{\sigma_{tr}^{2}}}-\frac{N_{t}}{N_{d}}\right].$
(22)

In this case, $\eta$ is maximum for the following values:

$\displaystyle\eta_{\mathrm{max}}=\frac{\sigma_{dr}^{2}}{\sigma_{tr}^{2}}\left(\frac{2}{1+\frac{\sigma_{dr}^{2}}{\sigma_{tr}^{2}}}\right)^{2},$
(23)
$\displaystyle\left(\frac{N_{t}}{N_{d}}\right)_{\mathrm{max}}=\frac{2}{1+\frac{\sigma_{dr}^{2}}{\sigma_{tr}^{2}}}.$
(24)

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