# Narrow bandwidth, low-emittance positron beams from a laser-wakefield
accelerator

M.J.V. Streeter  School of Mathematics and Physics, Queen’s University Belfast
, BT7 1NN, Belfast UK  C. Colgan  The John Adams Institute for Accelerator
Science, Imperial College London, London, SW7 2AZ, UK  J. Carderelli  Center
for Ultrafast Optical Science, University of Michigan, Ann Arbor, MI
48109-2099, USA  Y. Ma  Center for Ultrafast Optical Science, University of
Michigan, Ann Arbor, MI 48109-2099, USA  N. Cavanagh  School of Mathematics
and Physics, Queen’s University Belfast , BT7 1NN, Belfast UK  E.E. Los  The
John Adams Institute for Accelerator Science, Imperial College London, London,
SW7 2AZ, UK  H. Ahmed  Central Laser Facility, STFC Rutherford Appleton
Laboratory, Didcot OX11 0QX, UK  A.F. Antoine  Center for Ultrafast Optical
Science, University of Michigan, Ann Arbor, MI 48109-2099, USA  T. Audet
School of Mathematics and Physics, Queen’s University Belfast , BT7 1NN,
Belfast UK  M.D. Balcazar  Center for Ultrafast Optical Science, University of
Michigan, Ann Arbor, MI 48109-2099, USA  L. Calvin  School of Mathematics and
Physics, Queen’s University Belfast , BT7 1NN, Belfast UK  B. Kettle  The John
Adams Institute for Accelerator Science, Imperial College London, London, SW7
2AZ, UK  S.P.D. Mangles  The John Adams Institute for Accelerator Science,
Imperial College London, London, SW7 2AZ, UK  Z. Najmudin  The John Adams
Institute for Accelerator Science, Imperial College London, London, SW7 2AZ,
UK  P.P. Rajeev  Central Laser Facility, STFC Rutherford Appleton Laboratory,
Didcot OX11 0QX, UK  D.R. Symes  Central Laser Facility, STFC Rutherford
Appleton Laboratory, Didcot OX11 0QX, UK  A.G.R. Thomas  Center for Ultrafast
Optical Science, University of Michigan, Ann Arbor, MI 48109-2099, USA
G.Sarri<EMAIL_ADDRESS>School of Mathematics and Physics, Queen’s
University Belfast , BT7 1NN, Belfast UK

###### Abstract

The rapid progress that plasma wakefield accelerators are experiencing is now
posing the question as to whether they could be included in the design of the
next generation of high-energy electron-positron colliders. However, the
typical structure of the accelerating wakefields presents challenging
complications for positron acceleration. Research in plasma-based acceleration
of positrons has thus far experienced limited experimental progress due to the
lack of positron beams suitable to seed a plasma accelerator. Here, we report
on the first experimental demonstration of a laser-driven source of ultra-
relativistic positrons with sufficient spectral and spatial quality to be
injected in a plasma accelerator. Our results indicate, in agreement with
numerical simulations, selection and transport of positron beamlets containing
$N_{e+}\geq 10^{5}$ positrons in a 5% bandwidth around 600 MeV, with
femtosecond-scale duration and micron-scale normalised emittance. Particle-in-
cell simulations show that positron beams of this kind can be efficiently
guided and accelerated in a laser-driven plasma accelerator, with favourable
scalings to further increase overall charge and energy using PW-scale lasers.
The results presented here demonstrate the possibility of performing
experimental studies of positron acceleration in a plasma wakefield.

††preprint: APS/123-QED

Plasma-based wakefield accelerators [1] have been gathering significant
attention in recent years, mainly thanks to the ultra-high accelerating
gradients (in the region of 10s - 100 GV/m) that they are able to sustain,
providing a promising platform for the miniaturisation of particle
accelerators. In addition, electron beams from a plasma accelerator naturally
possess unique properties such as intrinsic femtosecond-scale duration,
micron-scale source size, and sub-micron normalised emittances at the GeV
level (see, e.g., Ref. [2, 3]). Progress in laser and plasma technology can
now also enable stable operation of these accelerators over long periods [4].

This rapid scientific and technological progress is now posing the realistic
question as to whether plasma-based acceleration could be a viable
complementary technology for the next generation of particle colliders,
proposed to break the TeV barrier [5]. Several international consortia and
proposed large-scale facilities are now actively addressing this question
(e.g., [6, 7]), which is also identified by several national and international
roadmaps as a central area of research (see, e.g., Refs. [8, 9, 10]).

While plasma-based acceleration of electrons has achieved a relatively high
level of maturity, plasma-based acceleration of positrons presents harder
fundamental challenges. This is due to the inherent structure of the wakefield
accelerating structures [11] as well as the stringent requirements on the
temporal and spatial properties of the seed positron beam. For example, quasi-
linear acceleration in a plasma with a density of $n_{e}=10^{17}$ cm-3 (as
proposed in baseline stuides, see e.g., Ref.[12]), would require longitudinal
and transverse seed beam dimensions $\sigma_{z},\sigma_{x}\lesssim 10$ µm.
While several positron acceleration schemes have been theoretically proposed
[13, 14, 15, 16, 17, 18, 19] and first proof-of-principle experiments have
demonstrated potential in this direction [20, 21, 22, 23, 24], progress in
this area has been hampered by the scarcity of facilities capable of providing
positron beams with these demanding characteristics. To date, only FACET-
II[25] at SLAC could be in principle suited in the future for proof-of-
principle experiments in this area.

In order to enable experimental studies of positron acceleration in a
wakefield, it is thus necessary first to provide positron witness beams with
high spatial and spectral quality at the GeV level. To achieve this goal, it
would be desirable to avoid storage rings, so that the intrinsic femtosecond-
scale duration of laser-driven positrons could be preserved. Laser-driven
generation of ultra-relativistic positron beams is thus currently being
actively studied, with several landmark results already reported, including
maximum positron energies in the region of 100 MeV [26, 27, 28], the
generation of high-density and quasi-neutral electron-positron beams [29], and
first experimental observation of pair-plasma dynamics [30]. However, to
effectively enable plasma acceleration of positrons in the laboratory, it is
necessary to produce beams that simultaneously have %-level energy spreads,
femtosecond-scale duration, and micron-scale normalised emittance [5, 7]. To
date, properties of this kind have only been predicted numerically [31, 27,
32].

Figure 1:  Illustration of the experimental setup showing the electron plasma
accelerator, the converter, the emittance mask, scintillators for electrons
(LE1 and LE2) and positrons (LP1 and LP2). Electron (red) and positron (blue)
trajectories are also shown to guide the eye.

Here, we experimentally demonstrate that positron beams of this kind can be
generated with a laser system of relatively modest peak power. Crucially, we
demonstrate that the obtained positron beams are of sufficient quality to be
energy-selected, with our results showing the isolation of positron beamlets
containing $\geq$ $10^{5}$ positrons in a 5% bandwidth at energies exceeding
500 MeV. These beams present femtosecond-scale duration and micron-scale
normalised emittance, and are thus of sufficient spectral and spatial quality
to efficiently act as a witness beam in a positron wakefield accelerator, as
we demonstrate with proof-of-principle particle-in-cell simulations.
Favourable scalings in the beam spectral and spatial characteristics already
indicate that even higher quality can be achieved with laser systems of higher
peak power [31, 32]. These results represent a critical milestone towards the
realisation of plasma-based particle accelerators and their potential
implementation in the next generation of particle colliders.

## Results

Experimental setup. The experiment was performed using the Gemini laser at the
Central Laser Facility [33], (setup sketched in figure 1). The laser pulses
contained a mean (and RMS variation) of $7.9\pm 0.5$ J in a FWHM pulse length
of $48\pm 7$ fs (peak power $P_{0}=156\pm 9$ TW) and a central wavelength of
800 nm. The pulses were focused with an $f/40$ off-axis parabola into a gas
jet from a 15 mm exit diameter nozzle, to generate high-energy electron beams
via laser wakefield acceleration (LWFA). The gas was a mixture of 2% nitrogen
and 98% helium and had an electron density of $n_{e}=(1.5\pm 0.2)\times
10^{18}$ cm-3, as measured via optical interferometry. The laser focal spot,
measured in vacuum, was $(37\pm 3)\times(52\pm 4)$ µm in the transverse $x$
and $y$ directions respectively ($1/e^{2}$ radius), giving a peak intensity
$I_{0}=(3.1\pm 0.3)\times 10^{18}$ Wcm-2.

The residual laser exiting the LWFA was removed by reflection from a self-
generated plasma mirror on the surface of a 125 µm polyimide tape which was
replenished after every shot. The tape target was kept for all the
experimental data shown here. A movable lead converter target (placed at a
distance $z_{D}$ = 50 mm from the LWFA exit plane of the gas jet) was used to
generate electron-positron beams through a two-step bremsstrahlung induced
Bethe-Heitler process [34]. The converter was a 45-degree wedge, such that
translating it perpendicularly to the electron beam axis allowed the effective
converter thickness to be varied continuously over the range $1\geq L\geq 25$
mm.

A shielding lead wall with an on-axis 10 mm diameter aperture was placed to
allow only particles emitted from the converter within a 12.6 mrad half-angle
to propagate to the detectors. A permanent magnetic dipole (Dipole 1 in figure
1, with integrated strength of $B_{x}z=0.3$ Tm) was placed behind the lead
wall to sweep electrons and positrons onto the primary scintillator screens
(LP1 and LE1, both Kodak LANEX) either side of the central axis, allowing
observation of particles with kinetic energy $E\geq 200$ MeV. Due to
experimental limitations, the two screens were placed at slightly different
distances from the dipole and angles from the main axis: 134 cm and 41.1∘ for
the positron side of the spectrometer (LP1 in Fig. 1) and 137 cm and 45.4∘ for
the electron side of the spectrometer (LE1 in Fig. 1). A second scintillator
screen (LE2) was placed $1$ m behind the first in order to increase the
measurement accuracy of the high energy electrons (more details in the Methods
section). A second identical magnetic dipole (Dipole 2 in figure 1) with a
$25$ mm wide lead slit placed at its entrance was positioned in the dispersed
positron beam, in a dog-leg configuration. The slit performed energy selection
on the dispersed positron beam, which was then collimated onto an additional
scintillator screen (LP2) by the magnet.

A 5.0 mm thick tungsten mask composed of horizontal slits with a period of
1100 µm (550 µm gaps) was placed into the beam axis 290 mm behind the rear
face of the converter (see figure 1) to perform energy-resolved emittance
measurements on the generated positrons and the scattered electrons (details
on the emittance retrieval in the Methods section).

Experimental results.

Figure 2: Primary electron beam characteristics. Typical a) angularly resolved
and b) angularly integrated electron spectra of the LWFA electron beams (red).
The average (black) and standard deviation (grey) of the integrated electron
spectra are shown along with the approximation used as an input for the
positron generation simulations (blue).

The electron spectra produced by the LWFA were first characterised with the
tape drive in place but without the converter. Ten shots were taken with
nominally identical conditions, with variations in the electron spectrum due
to inherent variations in laser and plasma source parameters. The five shots
with the highest total beam energy are shown in figure 2a. The angularly
integrated electron spectra for each of these shots are plotted in figure 2b
along with their average. When analysing the electron-positron beams, shots
with the highest total charge (3-8 shots out of 10) were used for each
converter length, and so the average shown in figure 2b was taken as the
expected LWFA spectrum for those shots. The average (and RMS variation) total
beam charge for electron energies above 200 MeV was $Q_{b}=1.4\pm 0.2$ nC and
the total beam energy was $W_{b}=0.8\pm 0.1$ J, giving a laser-to-electron
beam energy efficiency of $\eta\approx 10\%$. The angular distribution of the
energy integrated electron spectra is partially affected by the propagation
through the tape target [35] and is closely approximated by the square of a
Lorentzian function with a FWHM of $\theta_{x}=(3.8\pm 0.4)$ mrad.

To demonstrate energy selection of laser-driven positron beams, a lead
converter thickness of $5.0$ mm ($0.9$ radiation lengths) was used and the
recorded positron spectra after the second dipole magnet are shown in figure 3
for different transverse positions of the slit. For central energies of
$E>500$ MeV, more than $10^{5}$ positrons per shot were transmitted within a
FWHM bandwidth of $\Delta E/E\leq 5\%$, demonstrating the possibility of
performing efficient energy selection and capture of laser-driven positron
beams of this kind. Similar results were obtained for different target
thicknesses (not shown).

Figure 3:  Narrow energy spread positron beams. Typical single-shot positron
spectra measured after energy selection for different positions of the energy
selection slit. Raw data has been background-subtracted and smoothed with a 10
MeV Gaussian filter, with the shaded region representing the local RMS scatter
of the data. The central energy and FWHM bandwidth of each spectrum is
indicated in the figure legend.

The energy-resolved emittance of the electrons and positrons exiting the
converter target were characterised for different converter thicknesses by
inserting the tungsten mask in front of the lead sheilding wall (see Fig. 1).
This insertion resulted in a modulated beam profile onto the scintillators, as
shown in figure 4. The modulation had a lower spatial frequency at low energy,
since particles with an energy $E\lesssim 300$ MeV exited the side of the
dipole field and experienced defocusing as their longitudinal momentum
($p_{z}$) coupled with the transverse fringe field ($B_{y}$). In order to
correct for the fringe field defocusing effect in the analysis, the measured
signals were re-scaled in the non-dispersion direction such that the
magnification of the grid pattern was kept constant for all energies. This
results in an overestimation of the source size by $\sim 5\%$ for energies
$E<300$ MeV. In addition, the finite spectral resolution of the spectrometer
caused blurring of the grid pattern where the magnification varied most
strongly. As a result, the source size has a total systematic uncertainty of
$\sim 30$% for $E<350$ MeV. For $E>350$ MeV, the blurring effect had a
negligible effect on the measured source size.

Figure 4:  Raw images of energy-resolved beam profiles with the emittance
mask. Example modulated a) positron and b) electron spatial charge density as
a function of position on the screens ($x_{p},y_{p},x_{e},x_{e}$) for a single
shot with a converter thickness of 8.0 mm and the emittance mask in the beam-
line. The positions corresponding to the given particle energies in MeV are
shown as vertical red dashed lines. The slight difference between the electron
and positron raw data is due to the slightly different position of the
scintillator screens (discussed in the text).  Figure 5:  Positron properties
as function of energy and converter thickness. Measured electron and positron
beam properties as functions of particle energy for different converter
thicknesses. The a) spectra (charge per 5% bandwidth), b) source size, c)
divergence and d) geometric emittance are given for each converter length as
shown by the color-bars at the side of the figure. For each converter
thickness, only the shots resulting in the highest charge of the positron
beams were used for the analysis; the lines shown are thus an average of 4, 6,
3 and 8 shots for converter lengths of 1.0, 2.0, 4.0 and 8.0 mm, respectively
(RMS variation of $\approx$ 43% in the spectrum and $\approx$ 20% in
emittance, respectively). Each line results from a Gaussian weighting of each
measurement point using a kernel width of $\sigma_{E}=25$ MeV. The typical
input electron spectrum (black-dashed) is shown in a) for comparison.

The beam divergence $\sigma_{\theta}=\sqrt{\langle x^{\prime 2}\rangle}$ was
found for each energy slice by fitting the envelope of the background
subtracted signal and dividing by the source-to-screen distance. The non-zero
source size (assumed to be Gaussian [31]) of the electron and positron beams
resulted in a convolution of the detector resolution limited grid pattern with
a Gaussian distribution with an RMS width of $\sigma_{s}=\sigma_{x}M$. The
value $\sigma_{s}$ was found by iterative deconvolution, and then divided by
the source plane magnification, $M$, to yield the source size (RMS)
$\sigma_{x}=\sqrt{\langle x^{2}\rangle}$.

The geometric emittance of a particle beam is defined as
$\epsilon=\sqrt{\langle x^{2}\rangle\langle x^{\prime 2}\rangle-\langle
xx^{\prime}\rangle}$, where $\langle xx^{\prime}\rangle$ is the angle-position
correlation term. In the drift space between the LWFA and the converter, the
primary electron beam develops a strong correlation term. However, the
relatively large scattering angles in the converter dominate so that, at the
exit of the converter, the positron beam is largely uncorrelated. Monte-Carlo
simulations (discussed in the following) indicate that the small level of
remaining correlation implies that the positron beam is equivalent to an
uncorrelated beam originating from $100-200$ µm inside the rear surface of the
converter. Therefore, the correlation term was neglected and the geometric
emittance was calculated as the product of the measured divergence and source
size, i.e. $\epsilon=\sigma_{x}\sigma_{\theta}$.

The electron and positron beam properties were measured as functions of energy
for converter lengths $L=1.0,2.0,4.0,8.0$ mm (0.2, 0.4, 0.7 and 1.4 radiation
lengths) and are plotted in figure 5. The number of observed electrons was
seen to decrease with converter thickness (figure 5a), while the number of
positrons was maximised for $L=4.0$ mm. The electron and positron source size
(figure 5b) was observed to weakly decrease with energy, from 140$\pm$10 µm at
300 MeV down to 110$\pm$10 µm at 600 MeV, with fluctuations between 200 and
300 MeV due to the systematic uncertainties described before. The electron and
positron RMS divergence (figure 5c) was approximately constant at
$\sigma_{\theta}=5.3\pm 0.3$ mrad, as it was limited by the aperture in the
lead wall. Due to the fixed beam divergence, the emittance trends were largely
determined by the variation in source size. The positron geometric emittance
exhibited a gradual linear decrease as a function of energy with values of
$\epsilon=640$ nm at $E=200$ MeV and $\epsilon=480$ nm at $E=600$ MeV for a
1.0 mm converter. As numerically predicted previously [27], the positron
geometric emittance (figure 5d) was consistently lower than that of the
scattered electrons. The results had an RMS variation of $\approx 43\%$ in the
spectrum and $\approx 20\%$ in emittance, due to the shot-to-shot variation in
the primary electron beam.

In order to ascertain the effect of the beam-line geometry on the measured
positron beam characteristics, simulations were performed using the particle
physics Monte-Carlo code FLUKA [36, 37] (details in the Methods section).
Electrons were initialised from the LWFA electron spectrum approximation shown
in figure 2b and the momenta and position of all electrons and positrons were
recorded as they exited the rear surface of the converter. The divergence of
the primary LWFA electron beam was modelled by applying randomised shifts
(matching the measured LWFA divergence) to the position and propagation angles
of each generated particle. The transverse particle positions were also
modified according to the expected LWFA electron source size of 1 µm [38, 39,
40], although this contribution was observed to be negligible. The effect of
the aperture in the beamline was simulated by removing all particles which had
radial positions greater than 5 mm at the aperture plane.

Figure 6:  Comparison with numerical simulations. Positron beam a) spectrum
(charge per 5% bandwidth), b) source size, c) divergence and d) geometric
emittance plotted for a 1.0 mm thick lead converter. The experimental data
(red) is plotted alongside FLUKA simulations for: zero drift distance for the
primary electron beam (black); including the drift distance and the primary
electron beam divergence and source size (cyan dashed); and including the 12.6
mrad shielding aperture (blue).

For example, the results of the numerical simulation for a converter thickness
of $L=1.0$ mm show good agreement with the experimental data (blue and red
lines in figure 6, respectively) with an RMS average difference for all
converter lengths of 15%, 3.5%, 2.8% and 0.5% for the spectrum, emittance,
source size and divergence, respectively.

Due to the higher initial divergence for lower energy particles, the aperture
transmits fewer lower energy positrons, modifying the detectable spectrum
(figure 6a). The aperture also constrains the beam divergence to an
approximately constant value of $\sigma_{\theta}\approx 5$ mrad (figure 6c).
Including the finite divergence of the LWFA does not affect the positron
spectrum or divergence but has a strong effect on the source size (figure 6b)
and, subsequently, the emittance of the positron beam (figure 6d). A
significant reduction in positron source size can be readily obtained by
placing the converter closer to the exit of the LWFA, immediately after the
tape drive or, alternatively, by using a beam transport system to minimise the
electron beam size on the converter. A replenishing tape (such as the one used
for this experiment) can be used to extract the post-plasma laser pulse and
protect the converter from damage. Operations of tapes of this kind up to the
kHz has already been demonstrated [41]. As observed in our experiment, the
laser-plasma interaction at the plasma mirror surface causes a small ($\sim$
mrad) increase in electron beam divergence [35], but this would be negligible
compared to the inherent divergence of the pair production process.

For the measured LWFA electron beam FWHM divergence of 3.8 mrad, a typical
beam waist at the LWFA source of 1 µm, and a converter thickness of 1 mm, the
positron source size at 600 MeV could be reduced to 2.7 µm by minimising the
free drift distance of the electron beam. In this case the positron beam would
have a divergence $\sigma_{\theta}=$5.5 mrad, a geometric emittance
$\epsilon=$15 nm, and a normalised emittance
$\bar{\epsilon}=\gamma\beta_{z}\epsilon=18$ µm at 600 MeV. All positron beam
parameters for different target thicknesses can then be readily obtained using
known scaling laws (see, e.g., Refs. [31, 32]). For a converter target placed
right after the LWFA, our simulations indicate that different electron source
size or divergence will have a small effect on the positron beam
characteristics, which can anyway be taken into account by adding it in
quadrature to the obtained values. For example, doubling the electron source
size only increases the positron source size by approximately 10%. The
simulations also show a spreading in the duration of the positron beam of 0.1
fs for positrons within a 5% bandwidth of 600 MeV implying that the positron
beam duration will be similar to that of the primary electron beam (i.e.,
$\sigma_{z}\lesssim\lambda_{p}/2=14$ µm for $n_{e}=1.5\times 10^{18}$ cm-3,
corresponding to $\tau\lesssim 50$ fs).

## Positron Post-Acceleration Simulations

Simulations were performed using the particle-in-cell code FBPIC [42] to
assess the suitability of the generated positron beams for post-acceleration
in a plasma wakefield. As an example following a baseline study on plasma
accelerators [12], the laser-wakefields were produced by the interaction of a
Gaussian laser pulse with radius $r_{L}=70$ µm, pulse length $\tau_{L}=56$ fs
and a normalised vector potential of $a_{0}=1.5$, with a pure helium plasma
with an electron density of $2\times 10^{17}$ cm-3 and a length of 10 cm
enclosed by two 0.5mm-long linear ramps.

The initial positron particle distribution was obtained from FLUKA simulations
of the interaction of the LWFA electron beam with a 1 mm converter. A
longitudinal spread of 10 µm (uniform distribution) was added to the particles
to take the positron bunch duration into account. The positron bunch was
initialised at the peak of the positive accelerating field of the plasma
wakefield. Further details of the simulation setup are given in the Methods
section.

Fig. 7 shows a summary of the main results of the simulation. For these laser
and plasma conditions, a quasi-linear wakefield is generated behind the laser
pulse (Fig. 7a) with a peak accelerating field of 13 GVm-1 and a period of 75
µm. After 96 mm of propagation in the plasma, 31% of the charge remains in a
focused positron beam (figure 7b) which is accelerated to an average of 1.0
GeV with a relative rms energy spread of 20% (7c-d). The beam is chirped,
however, and so the energy spread could be reduced further using a plasma
dechirper [43]. After an initial increase, the normalised emittance of the
trapped positron beam remains approximately constant throughout the
acceleration, at $\bar{\epsilon}=33.0\pm 0.2$ µm.

These results can in principle be experimentally achieved by driving a laser
wakefield accelerator directly behind the converter, as could be achieved with
compact plasma mirror staging. This configuration can be directly implemented
in existing and near-term laser facilities, enabling experimental studies of
wakefield acceleration of positrons. Alternatively, a magnetic beam-line, as
considered for example in EUPRAXIA [6], could be used to perform energy
selection and controlled focusing to increase the experimental capabilities.
As an example of this capability, we show as supplementary material FBPIC
simulations of wakefield acceleration of the energy-selected positron bunch
(energy of 500 MeV with a 5% energy spread) in the same conditions as those
discussed above [44]. In this case, approximately 50% of the positron bunch is
trapped and accelerated up to an energy of $1.2\pm 0.3$ GeV with a normalised
emittance of 57 µm.

Figure 7: Simulated post-acceleration of a laser-generated positron beam. a&b
show the longitudinal electric fields of the plasma wakefield generated by the
laser pulse (yellow orb) and the trailing positron bunch (density in a
logarithmic colorscale) at the beginning of the plasma and after 96 mm of
propagation. Panels c&d show the positron longitudinal phase space before an
after acceleration, with the energy spectra indicated by the red lines. e
shows the average energy and normalised emittance of the trapped bunch,
defined as being comprised of particles which remain within $\pm 50$ µm of the
central axis.

## Discussion and conclusions

With the minimisation of the drift length for the LWFA electrons, numerical
simulations show that the obtained positron beam characteristics are now well
suited for efficient capture and post-acceleration as a witness beam in a
plasma wakefield accelerator. This result is also consistent with other
studies reported in the literature. For example, Silva _et al._ [19]
numerically demonstrated plasma acceleration for a positron bunch with a
radius of $\sigma_{x}=5$ µm and a normalised emittance of 10 µm using an
electron beam driver, while Vieira _et al._ [15] numerically demonstrated
effective wakefield acceleration of a $\sigma_{x}=6$ µm positron beam in a
Laguerre-Gaussian laser mode. Proof-of-principle experiments on wakefield
acceleration of positrons reported in the literature [20, 21, 22, 23], used
witness positron beams with similar geometric emittance to our inferred beam
properties, though at a higher energy (see Table 1), demonstrating the
suitability of the positron source reported here to provide witness beams for
wakefield acceleration studies, such as proposed for EuPRAXIA [6]. Most
notably, the inherent short duration of the positron beam (inferred to be of
the order of $\lesssim$ 50 fs, corresponding to $\sigma_{z}\lesssim 14$ µm) is
naturally suited for injection in a positron-accelerating wakefield structure
without the need for complex beam manipulation, as also confirmed by the
proof-of-principle simulations reported in this article.

| Measured | Inferred | [20] | [21] | [22]
---|---|---|---|---|---
E (GeV) | $0.6$ | 0.6 | 28.5 | 20.3 | 20.3
$\sigma_{x}$ (µm) | 100 | 2.7 | 25 | $<100$ | 50
$\sigma_{z}$ (µm) | - | $\lesssim 14$ | 730 | 30 - 50 | 35
$\epsilon$ (nm) | 480 | 15 | $14\times 3$ | $5\times 1$ | 7
$\bar{\epsilon}$ (µm) | 560 | 18 | $390\times 80$ | $200\times 50$ | 300
Table 1: Summary of measured and inferred positron parameters, compared with
sources used for previously reported proof-of-principle positron wakefield
acceleration experiments.

In conclusion, direct and comprehensive spatial and spectral characterisation
of GeV-scale laser-driven positron beams is reported. Experimental results
show that minimising the free propagation of the primary electron beam to the
converter results in the production of GeV-scale positron beams with micron-
scale source size and normalised emittance, using a 100 TW-class laser system.
The beam is also shown to be of sufficient quality to undergo energy
selection, with beamlets containing $\approx 10^{5}$ positrons in 5%
bandwidths around 500 MeV being isolated, and of sufficient quality to be
injected in a plasma wakefield accelerator. These results demonstrate the
possibility of experimentally studying laser-wakefield acceleration of
positrons, a critical milestone towards the realisation of the next-generation
of plasma-based particle accelerators and colliders. For instance, an
experimental platform of this kind can be implemented in future laser
facilities with dual beam capability [45] or in beam-driven wakefield
facilities with laser capability (e.g. FLASHForward [46] and SPARC_LAB [47])
to study beam-driven methods, without the need for an emittance damping
storage ring.

In terms of future developments, increasing the energy of the primary electron
beam is expected to further improve the positron beam characteristics. Single
stage LWFA has been demonstrated beyond 5 GeV [48], which along with a
minimised drift distance for the LWFA electrons is expected to readily
provide, for instance, nm-scale geometric emittances and a normalised
emittance of 10 µm at 3 GeV [32, 31]. As a final remark, we note that
different mechanisms for the laser-driven generation of positron beams based
on the Breit-Wheeler pair production process have been numerically proposed
(see, e.g., [49, 50]). However, these mechanisms require next-generation
multi-PW laser facilities and generally result in positron beams of different
characteristics. Studies on direct irradiation of solid foils with the next
generation of ultra-high intensity laser systems has also been numerically
reported [51].

Methods
Two-screen electron spectrometer: two scintillator screens were placed in the
electron beam after the magnetic dipole, to detect electrons with kinetic
energy in the range of $200\geq\gamma m_{e}c^{2}\geq 2500$ MeV and propagation
angles relative to the laser propagation axis of $-15\geq\theta_{y}\geq 15$
mrad. The 3D 3-vector field distribution of the dipole magnets was
experimentally mapped and found in agreement with numerical simulations of the
magnet setup using RADIA [52]. The screen dispersion functions were calculated
by numerically solving the particle trajectories using the Boris-pusher and
recording the particle position on each screen as a function of the initial
particle 3-momentum. This was used to produce a look-up table which gave the
particle energy as a function of its position on each screen and its initial
propagation angle $\theta_{y}$. The electron beam spectrum was determined by
finding the coefficients of a third-order polynomial function
$\theta_{y}(\gamma)$ that minimised the mean squared difference between the
retrieved angularly integrated electron spectra from each screen. Charge
calibration of the electron spectrometers were performed by measuring electron
spectra on an absolutely calibrated image plate placed in front of the LANEX
screen and comparing to the images recorded on the CCD over the same shots.
The image plate used was BAS-TR2040, with a sensitivity of 1 PSL per 350
electrons.

Source size, divergence, and emittance retrieval: The beam profile after the
beam aperture was modelled as an azimuthally symmetric clipped Gaussian
distribution, such that only particles with
$x_{i,\mathrm{ap}}^{2}+y_{i,\mathrm{ap}}^{2}\leq R_{\mathrm{ap}}^{2}$ were
transmitted, where $x_{i,\mathrm{ap}}$ and $y_{i,\mathrm{ap}}$ are the
transverse spatial coordinates of the $i^{th}$ particle at the aperture plane
$z_{\mathrm{ap}}$, with aperture radius $R_{\mathrm{ap}}$. This profile was
dispersed according to the individual particle energies onto the spectrometer
screen. The particle distribution $S^{\prime}_{y}(x)$ was measured at the
detector plane $z_{\mathrm{det}}$ where $x$ and $y$ are transverse coordinates
perpendicular and parallel to the dispersion plane of the spectrometer
respectively. Due to the combination of energy spread and divergence, the
profile $S^{\prime}_{y}(x)$ is due to particles over a range of different
energies where their initial propagation angle $\theta_{i,y}$ and energy
$E_{i}$ result in the particle intersecting the detector plane at the position
$y$. With the assumption that the spectrum $N(E)$ is slowly varying, then each
slice measurement $S^{\prime}_{y}(x)$ represents the integral of the beam
profile over $y$, i.e.,

$\displaystyle S^{\prime}_{y}(x)$
$\displaystyle=2A_{y,0}\int_{0}^{\sqrt{R^{2}-x^{2}}}\exp\left[{-\frac{x^{2}+y^{2}}{2\sigma_{x}^{2}}}\right]\mathrm{d}y$
$\displaystyle S^{\prime}_{y}(x)$
$\displaystyle=\sqrt{2\pi}A_{y,0}\sigma_{x}\operatorname{erf}\left(\sqrt{\frac{R^{2}-x^{2}}{2\sigma_{x}^{2}}}\,\right)\exp\left[{-\frac{x^{2}}{2\sigma_{x}^{2}}}\right]$
(1)

where $A_{y,0}$ is the amplitude of the particle distribution,
$R=R_{\mathrm{ap}}z_{\mathrm{det}}/z_{\mathrm{ap}}$ is the projected size of
the aperture at the detector plane and $x\leq R$. The functional form of
equation 1 was used to fit the amplitude of the modulated signal $S_{y}(x)$
when retrieving the apertured beam properties as described below.

Several steps were followed to extract the particle emittance from the
spectrometer signals. Firstly, a variable threshold filter was used to remove
hard-hits caused by stray photons hitting the CCD directly. Secondly, the
defocusing of the effect of the magnetic dipole fringe fields was removed by
re-scaling the measured signal in non-dispersion direction such that the
spatial frequency of the grid pattern was made constant for all energies.
Vertical slices were then taken through the resultant image, averaging over 4
mm in the dispersion direction to produce the signal modulation $S_{y}(x)$ as
a function of $x$ at a given $y$ position. The scattered particles from the
grid formed a smooth background on the detector which was removed by fitting a
Gaussian to the values at the minima of the observed modulations. The envelope
of the signal was similarly found by fitting the beam profile function
(equation 1) to the signal maxima. The RMS width of the fitted envelope was
then divided by the source-to-screen distance to obtain the beam divergence
$\sigma_{\theta}$.

An ideal zero source size beam would produce a sharp step-function within the
bounds of the scattering signal and the beam envelope, with the spatial period
of the magnified grid size. Blurring of this pattern was observed due to
contributions of the finite spatial resolution of the diagnostic (215 µm) and
the source size $\sigma_{x}$ of the beam, which was found by iterative
minimisation of the mean squared error between the measured signal and the
calculated signal for a given source size. The geometric emittance was
calculated as the product of the measured divergence and source size, i.e.
$\epsilon=\sigma_{x}\sigma_{\theta}$.

In order to benchmark the retrieval process, synthetic data was created by
numerically propagating results from a FLUKA simulation and removing particles
that would hit the solid bars of the emittance measurement grid. The
dispersion of the magnet was added by shifting the particles transversely
according to their energy using the same dispersion function as for the
experimental spectrometer. To create the modulated signal $S_{x}(y)$ for a
given energy band, the particles are selected according to their position on
the spectrometer. Due to the significant beam divergence, there is some
trajectory crossing such that some particles of different energies are
selected, and some of the correct energy are omitted. The synthetic signals
were analysed with the same procedure as for the experimental data and
compared to the values directly calculated from the particle distributions.
The retrieved beam properties closely agree with the directly computed values
for the apertured beam, verifying the analysis procedure.

Monte-Carlo simulations: simulations of the bremsstrahlung induced pair-
production process were performed using the particle physics Monte-Carlo code
FLUKA with the EM-cascade defaults. $10^{6}$ primary electrons for each
converter thickness were simulated and the resultant particle number was then
scaled up by 8738 to match the higher charge of the experimental LWFA electron
beam. A lead converter of variable thickness $L$ was placed in the path of the
electrons, and the momenta and position of all electrons, positrons and
photons were recorded as they exited the rear surface of the converter. In
order to simulate the effects of the finite divergence of the electron beam,
each particle was assigned random angular shifts ($\Delta x^{\prime}_{i}$ and
$\Delta y^{\prime}_{i}$) from the probability density function
$f(x^{\prime})=f_{0}[(x^{\prime}/\theta_{w})^{2}+1)]^{-2}$, which was seen to
approximate the experimentally measured transverse profile of the primary
electron beam with $\theta_{w}=2.9\pm 0.3$ mrad ($f_{0}$ is the normalisation
constant). The particle transverse momenta and positions were then altered
according to these shifts and using the experimental drift length between the
LWFA exit and the converter rear face of 50 mm. The transverse particle
positions were also modified according to the expected LWFA electron source
size of 1 µm, although this contribution was negligible. Each particle was
shifted 10 times from the value taken from the FLUKA simulation, with the
shifted particle properties recorded each iteration to produce a final list
with 10 times the number of particles as were produced by the FLUKA
simulations. Particle distribution properties were then calculated at the
longitudinal plane for which the correlation term $\langle xx^{\prime}\rangle$
was minimised.

FBPIC simulations: The FBPIC Particle-In-Cell code [42] was used to run
simulations of injection and acceleration of the output positrons from FLUKA
Monte-Carlo simulations in a laser-driven wakefield. As FBPIC employs a
cylindrical grid with azimuthal decomposition, two cylindrical modes were used
to capture the physics of the wakefield formation and acceleration. The
simulation was performed in the Lorentz boosted frame with a Lorentz factor of
$\gamma=10$. The wavelength of the laser was $\lambda_{L}=1000\ \mathrm{nm}$,
with a normalized vector potential of $a_{0}=1.5$. The background plasma
profile consisted of a 0.5 mm linear up-ramp starting at the right-edge of the
initial simulation window from vacuum to a plasma density of $n_{e0}=2\times
10^{17}$ cm-3 which was then constant for 100 mm. The plasma terminates with a
0.5 mm long down-ramp of back to vacuum. The particles per cell set for each
coordinate direction were 2 along $\hat{z}$, 2 along $\hat{r}$, and 6 along
$\hat{\theta}$. The simulation box lengths were
$z_{length}=\left(\frac{6c}{\omega_{p0}}+2\lambda_{p0}\right)\approx 220.6$ µm
and $r_{length}=2\times r_{max}=2\times\frac{15c}{\omega_{p0}}\approx 2\times
178.2$ µm for $\hat{z}$ and $\hat{r}$ respectively, where
$\omega_{p0}=\sqrt{\frac{e^{2}n_{e0}}{\epsilon_{0}m_{e}}}$ is the plasma
frequency of the peak initial density in the accelerator, and
$\lambda_{p0}=\frac{2\pi c}{\omega_{p0}}$ is the plasma wavelength. The number
of cells in each direction were for $\hat{z}$
$N_{z}=\lfloor\frac{z_{length}}{\Delta z}\rfloor=1764$ where $\Delta
z=\lambda_{L}/8$, and for $\hat{r}$ $N_{r}=\lfloor\frac{r_{max}}{\Delta
r}\rfloor=89,$ where $\Delta r=2\times\lambda_{L}$. The time step size of the
simulation was set to be $\Delta t=r_{max}/(2\times\gamma\times
N_{r})/c\approx 0.334$ fs. The positrons simulated in figure 7 were imported
into the FBPIC simulation from the output files given a random longitudinal
spatial spread of 10 µm to match the inferred produced beam duration from this
experimental campaign.

Authors Contribution G.S. devised and proposed the experiment, which was
numerically modelled and designed by T.A., L.C., J.C. and M.J.V.S.. The
experiment was carried out by N.C., E.E.L., A.F.A., M.D.B., L.C., H.A., B.K.,
P.P.R., and D.R.S., under the overall coordination of M.J.V.S., C.C., and G.S.
The data analysis was predominantly performed by M.J.V.S and the particle-in-
cell simulations were carried out by J.C. and Y.M., under the supervision of
A.G.R.T. The manuscript was written by G.S., M.J.V.S, and J.C. with input from
S.P.D.M., Z.N., and A.G.R.T.

Acknowledgments The authors wish to acknowledge support from EPSRC
(EP/V044397/1, EP/N027175/1, EP/V049577/1), STFC (ST/V001639/1), US NSF (grant
#2108075) and from the staff of the Central Laser Facility.

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