# A laser-plasma platform for photon-photon physics

B. Kettle1, D. Hollatz2,3, E. Gerstmayr1, G.M. Samarin4, A. Alejo4, S.
Astbury5, C. Baird6, S. Bohlen7, M. Campbell8, C. Colgan1, D. Dannheim8, C.
Gregory5, H. Harsh2,3, P. Hatfield9, J. Hinojosa10, Y. Katzir5, J. Morton11,
C.D. Murphy6, A. Nurnberg8, J. Osterhoff7, G. Pérez-Callejo12, K. Põder7, P.P.
Rajeev5, C. Roedel2, F. Roeder3, F.C. Salgado2,3, G. Sarri4, A. Seidel2,3, S.
Spannagel8,7, C. Spindloe5, S. Steinke13, M.J.V. Streeter4,14, A.G.R.
Thomas10,14, C. Underwood6, R. Watt1, M. Zepf 2,3, S.J. Rose1 and S.P.D.
Mangles1 1The John Adams Institute for Accelerator Science, Imperial College
London, London, SW7 2AZ, UK  2Helmholtz Institut Jena, Fröbelstieg 3, 07743
Jena, Germany  3Institut für Optik und Quantenelektronik, Friedrich-Schiller-
Universität, 07743 Jena, Germany  4School of Mathematics and Physics, Queen’s
University of Belfast, BT7 1NN, Belfast, UK  5Central Laser Facility, STFC
Rutherford Appleton Laboratory, Didcot OX11 0QX, UK  6Department of Physics,
University of York, York, YO10 5DD, UK  7Deutsches Elektronen-Synchrotron
DESY, Notkestr. 85, 22607 Hamburg, Germany  8CERN, Geneva, Switzerland
9Department of Physics, University of Oxford, Oxford, OX1 3PU, UK  10Center
for Ultrafast Optical Science, University of Michigan, Ann Arbor, MI
48109-2099, USA  11AWE Aldermaston, Reading RG7 4PR, UK  12Université de
Bordeaux-CNRS-CEA, Centre Lasers Intenses et Applications (CELIA), UMR 5107,
F-33405 Talence, France  13Lawrence Berkeley National Laboratory, Berkeley, CA
94720, US  14Physics Department, Lancaster University, Lancaster LA1 4YB, UK
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We describe a laser-plasma platform for photon-photon collision experiments to
measure fundamental quantum electrodynamic processes such as the linear Breit-
Wheeler process with real photons. The platform has been developed using the
Gemini laser facility at the Rutherford Appleton Laboratory. A laser wakefield
accelerator and a bremsstrahlung convertor are used to generate a collimated
beam of photons with energies of hundreds of MeV, that collide with keV x-ray
photons generated by a laser heated plasma target. To detect the pairs
generated by the photon-photon collisions, a magnetic transport system has
been developed which directs the pairs onto scintillation-based and hybrid
silicon pixel single particle detectors. We present commissioning results from
an experimental campaign using this laser-plasma platform for photon-photon
physics, demonstrating successful generation of both photon sources,
characterisation of the magnetic transport system and calibration of the
single particle detectors, and discuss the feasibility of this platform for
the observation of the Breit-Wheeler process. The design of the platform will
also serve as the basis for the investigation of strong-field quantum
electrodynamic processes such as the nonlinear Breit-Wheeler and the Trident
process, or eventually, photon-photon scattering.

##  1 Introduction

###  1.1 Background

Gregory Breit and John A. Wheeler predicted in 1934 that when two photons
collide, they can produce an electron and a positron in a process called
“inelastic photon-photon scattering” [1]. This Breit-Wheeler (BW) process
$\gamma+\gamma\rightarrow e^{-}+e^{+}$ has been studied in detail using the
theoretical framework of quantum electrodynamics (QED) [2]. In the linear BW
process, two photons must have sufficient energy to create an electron and a
positron, i.e. the invariant mass of the collision must satisfy
$\sqrt{s}>2m_{\rm e}c^{2}$, where $\sqrt{s}$ is the total energy measured in
the centre of momentum frame. For two photons of energy $E_{1}$ and $E_{2}$
interacting at angle $\theta$, $s=[1-\cos(\theta)]E_{1}E_{2}$. To date, the
linear BW process has not been observed in the laboratory with real photons
due to the lack of suitably bright sources of photons with sufficient energy.
There have been proposals for $\gamma-\gamma$ colliders using conventional
particle accelerators [3, 4], but none have yet to be realised.

Indeed, in their original paper, Breit and Wheeler even stated: “It is
hopeless to try to observe the pair formation in the laboratory” as in 1934
there was no conceivable way of generating such photon sources. Today however,
bright electron beams with GeV energies can be produced over a few centimetres
by laser wakefield acceleration (LWFA) [5, 6]. These electron beams can be
used to produce brilliant $\gamma$ ray beams by bremsstrahlung [7, 8, 9, 10]
and inverse Compton scattering [11, 12, 13]. Colliding such brilliant $\gamma$
ray photons with a suitable x-ray photon field could be a viable route to
study two-photon physics in the laboratory for the first time [14, 15].

The BW process plays a pivotal role in a range of astrophysical phenomena
[16]. The linear BW process is the key process in Pair-Instability Supernovae
(PIS) and the closely related pulsational pair-instability supernova, the fate
of massive stars where the BW process softens the equation-of-state at the
core, leading to collapse and ignition [17, 18]. PIS are of particular
interest as they might be the only stars observable in the epoch of
reionization when the first stars are formed, meaning that they may be the
only view into star formation in the metal-poor early Universe possible with
the James Webb Space Telescope [19, 20]. The BW process also occurs in the
coronae of accretion disks in Active Galactic Nuclei [21], shaping particle
spectra [22]. Finally, the BW process occurs when $\gamma$ rays interact with
the extragalactic background light (EBL), producing a cut-off of $\gamma$ rays
at $\sim 10^{13}$ eV [23]. There has been some suggestion of the possibility
of non-standard model physics in the spectra of blazars, which some studies
suggest have anomalously hard high-energy $\gamma$ ray spectra after the BW
process is accounted for, implying less pair-production from QED than
anticipated [24, 25]. This problem could well be solved with conventional
physics (e.g. corrections to models of the EBL [26]) as happened for the
‘Compactness Problem’, a contradiction that arose from $\gamma$ ray burst
light curve data which was eventually solved by including relativistic motion
in the calculations [27]. Nonetheless this illustrates the importance of
confirming that QED physics behaves as expected in this regime for this
important process.

The existence of a photon-photon experimental platform studying the BW effect
will serve as a basis for the investigation of other QED processes including
elastic photon-photon scattering $\gamma\gamma\rightarrow\gamma\gamma$ and the
nonlinear BW process. The nonlinear BW process is closely related to the BW
process but involves the interaction of one high-energy photon with many $(n)$
low energy ones, i.e. $\gamma+n\omega\rightarrow e^{-}+e^{+}$. The generation
of matter via the nonlinear BW process plays a large role at the poles of
rotating black holes [28]. In 1997, an experiment at SLAC collided a 48 GeV
electron beam with a then state-of-the art laser pulse [29, 30], demonstrating
the nonlinear BW process, thus studying the perturbative regime of strong-
field quantum electrodynamics [31].

Elastic photon-photon scattering, $\gamma\gamma\rightarrow\gamma\gamma$, is
important in models of primordial abundances and may also affect the observed
spectra from distant gamma-ray bursts [32, 33]. Photon-photon scattering
involving virtual photons has previously been observed. Delbrück scattering,
$\gamma\gamma*\rightarrow\gamma\gamma*$, where a real photon scatters from a
virtual photon in the electric field of an ion, was observed in 1975 [34] and
scattering between two virtual photons in the Coulomb field of two high energy
ion beams $\gamma*\gamma*\rightarrow\gamma\gamma$ was observed at the Large
Hadron Collider in 2017 [35]. However, photon-photon scattering is most
relevant in astrophysics with real photons close to threshold for pair
production energy ($\sqrt{s}\leqslant 2m_{e}c^{2}$) as at higher energies pair
production dominates. Photon-photon scattering with real photons has yet to be
observed in the laboratory.

###  1.2 Laser-plasma approach

In 2014, Pike et al. [14] proposed an ambitious scheme that could enable the
observation of the linear BW process by combining a laser-driven $\gamma$ beam
source with a laser-driven x-ray source. They proposed using a megajoule laser
pulse, e.g. the National Ignition Facility [36], coupled to a petawatt short
pulse laser to produce 10,000 pairs in a single shot. However, no such
combination of high-power lasers exists worldwide. Ribeyre et al. (2016) [37]
proposed the interaction between two high-power laser generated $\gamma$-ray
beams. However this method relies on extremely precise spatial and temporal
overlap of the two beams, and due to the approximately equal incoming photon
energies, would generate pair particles into a wide range of angles, making
detection difficult. In this article we present an alternative laser-based
approach that can operate at existing laser facilities, benefits from directed
pair production, and which can make use of higher shot repetition rates to
provide the statistics needed for any given measurement.

Our approach is based on the interaction of a single $\gamma$ beam colliding
with an x-ray bath, similar in nature to that of Pike et al. [14], but with a
few hundred terawatt laser system providing tens of joules (as opposed to a
megajoule) at a repetition rate of over 100 shots an hour (as opposed to a few
shots a day). A major difficulty that a laser based approach overcomes is the
spatial and temporal overlap that goes hand-in-hand with colliding intense
photon sources. Using the same laser system to drive the photon sources means
that they can be inherently synchronised in time (within tens of femtoseconds)
and do not suffer from significant temporal jitter. Making use of a large
x-ray radiation field ($>$ $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$
$\times$ $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and tens of picoseconds
in duration) means that the spatial alignment of the two photon sources is
also relatively straightforward. In addition to the overlap, the strong
asymmetry of the photon energy of the two sources also results in a strongly
directional emission of the created electron-positron pairs. This is
particularly important considering that for many experimental configurations
the BW process can be obscured by other more efficient pair creation
processes, such as the Bethe-Heitler process, when high energy photons
interact with the matter in the vicinity of the photon-photon collision. A
strategic and tailored detection system that can decouple the BW signal from
background pairs from other processes is required, and a clean particle-free
interaction volume is desirable. Two decoupled photon sources with a spatially
directed signal is a solution to this problem and presents a major advantage
over other suggested schemes that collide two sources of photons that are
equal in energy [38].

Although the predicted pair creation numbers per shot are lower than that of
Pike et al. (around unity, as discussed later), many shots can be taken per
day to build a statistical significance. It is also worth noting that the
platform we present here can also be modified to study the nonlinear BW
process by colliding the $\gamma$ ray photons directly with the intense
optical laser field that would otherwise drive the x-ray bath.

In this paper we detail the design and commissioning of a laser plasma
platform for photon-photon physics. Our characterisation and modelling of the
background levels and expected signal level indicate that a measurement of the
linear BW process is feasible within a reasonable timeframe (1000s of laser
shots), but show that this is very sensitive to the performance of the photon
sources. Further developments of the platform are discussed, which should
bring the number of required shots to less than a few hundred.

##  2 Experimental platform overview

The platform was developed at the Gemini laser system for strong field QED
experiments [39] at the Central Laser Facility, UK and adapted to the specific
requirements of this experiment. It is an optically synchronised two beam
system, where each beam contains up to 15 J in $\approx 45$ fs (FWHM), with a
shot rate of once every 20 seconds. The synchronisation of the two beams
allows for a temporal overlap and jitter of less than 10 fs [40]. One of the
beams was used in a long focal length $f$/40 configuration to drive a laser-
wakefield accelerator. The high energy electron bream produced in this
interaction is subsequently used to generate bremsstrahlung MeV $\gamma$
photons. The second beam was stretched in duration to 40 ps (FWHM), and used
to drive the laser-plasma generated keV x-ray field (using an $f$/2 focusing
optic). These two sources are made to overlap spatially within the collision
interaction point and a magnetic transport system relays any created electron-
positron pairs to a set of single particle detectors. An overview of the
experimental layout can be seen in figure 1 and a full description of the
setup is as follows [41].

Figure 1: The experiment setup. An f/40 laser pulse was used to drive a laser-
wakefield accelerated electron beam in a gas cell. This high-energy electron
beam was used to create on axis $\gamma$-rays before being swept away by a
magnet. A second laser pulse was incident on a germanium target tape, driving
an x-ray bath through which the $\gamma$ beam passed. Any particle pairs
created in this interaction region propagate on axis and are swept away to
single particle detectors using a magnetic transport system.

The long focal length arm of the laser, which drives the laser-wakefield
accelerator, is focused onto a gas cell. In this case the cell was 17.5 mm
long and filled with helium and 2% nitrogen as a dopant at an electron density
of $(2.6\pm 0.4)\times 10^{18}\,\mathrm{cm}^{-3}$. The typical measured laser
spot size was $44\text{\,}\mathrm{\SIUnitSymbolMicro m}$ $\times$
$53\text{\,}\mathrm{\SIUnitSymbolMicro m}$ FWHM at a pulse duration of $45\pm
5$ fs and $5.5\pm 0.6$ J on target, which corresponds to a normalised vector
potential $a_{0}=1.1\pm 0.2$. As the laser pulse travels through the gas it
strips electrons from the ions, forming a plasma. The ponderomotive force of
the laser pulse pushes these electrons away from the intense pulse, while the
ions remain stationary, forming a positively charged cavity that trails in the
wake of the laser pulse [42]. The electrons can then be trapped and
accelerated in the cavity to relativistic energies, travelling in the
direction of the laser pulse. After driving the laser-wakefield accelerator,
the remaining drive laser exits the gas cell and is removed from the beamline
using a refreshable tape drive.

The generated high-energy electron beam continues along the beamline where it
passes through a 1 mm thick bismuth ‘converter’ target. While passing through
the high-Z atoms of the converter, the electrons generate bremsstrahlung
radiation, emitting $\gamma$ photons with over 100 MeV energy along the
propagation direction. The $\gamma$ beam is collimated to $\sim$ 8 mrads
divergence using a 10 cm long tungsten cylinder with a 2 mm aperture. One half
of the $\gamma$ beam is also blocked by a 5 cm thick tungsten shield. The
remaining footprint of the $\gamma$ beam is semi-circular in shape. This D
shape is designed to allow the tape target of the x-ray bath source (further
downstream) to be placed in close proximity to the centre of the $\gamma$
beam, but hidden from direct photon collisions at the interaction point. See
figure 2 for a depiction of the overlap geometry.

Figure 2: (a) Simplified view of the photon collision geometry. The tungsten
half-beam block and collimator shape the $\gamma$ beam to the desired profile
before, much further downstream, colliding with the pre-expanded x-ray bath.
(b) On-axis view of the overlap geometry. The x-ray bath radiates in all
directions from the germanium target dot, providing a suitable particle-free
collision volume for the D-shaped $\gamma$ beam to pass through.

After the $\gamma$ beam has been tailored to the desired shape, a $\int
Bdx=0.4~{}\mathrm{Tm}$ dipole magnet is used to sweep away the initial
electron beam as well as any secondary particles generated from collisions
with the tungsten shield elements. This combination of $\gamma$ beam tailoring
and particle removal provides a clean environment for photon-photon
collisions.

The second laser beam is incident upon a plastic tape containing regularly
spaced germanium targets, which are placed 1 mm from the $\gamma$ beam
footprint. The laser pulse rapidly heats and ionises the metal, which emits
M-L shell transition radiation in the 1 - 2 keV region; see section 4 for more
details. This emission radiates in all directions, but we are concerned with
the x-ray plume that penetrates through the plastic and out of the rear of the
target towards the interaction region. The two drive laser pulses should be
timed so that the $\gamma$ beam passes through the x-ray field at its highest
photon density. The x-ray driver is also short enough in duration that no ions
from the germanium plasma plume have penetrated through to the collision
volume by the time the $\gamma$ beam is incident. The
$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick Kapton tape substrate helps
prevent plasma from the front (laser-irradiated) side of the germanium target
penetrating through to the rear (where photon collisions occur). For ions to
propagate from the plasma to the interaction volume (1 mm away) within 100 ps
of the laser driver, they would need to move at $\sim 10^{7}$ m/s, or $\sim
3\%$ the speed of light. However calculations of the laser-plasma interaction
indicate a peak electron temperature of $\sim 2.5$ keV, near the peak of the
pulse, with an associated ion sound speed ($c_{s}\sim 3.5-6.3\times 10^{5}$
m/s) that is a fraction of that required to reach the interaction by the
collision time. Additional hydrodynamic simulations support this, showing that
the Kapton substrate of the targets has not been burnt through by the time the
$\gamma$ beam collides with the interaction volume. Because the $\gamma$-rays
are much higher energy than the x-rays, any electrons and positrons produced
in the photon-photon collisions travel along the initial $\gamma$-ray beam
direction. A second dipole magnet $\int Bdx=0.35~{}\mathrm{Tm}$ with a large
aperture sweeps the generated positrons off axis. It is at this point the
positrons exit the vacuum chamber through a
$125\text{\,}\mathrm{\SIUnitSymbolMicro m}$ Kapton window. They pass through a
third dipole magnet with opposite polarity to the second, redirecting the
particles along a focused beam path. This particle transport system is
discussed in detail in section 5.

Single particle detectors (SPDs) are placed on this beam path to measure the
particle flux. Electrons generated within the interaction region are swept by
the second dipole in the opposite direction, and a fourth magnet (identical to
the third one) directs the electrons to a beam dump. The profile and spectrum
of the on-axis $\gamma$-ray beam are diagnosed using scintillator-based
detectors at the end of the beam line. The $\gamma$-ray diagnostics and single
particle detectors are discussed in more detail in section 3 and section 6
respectively. Lead shielding is used extensively throughout the setup to
mitigate the influence of unwanted high energy particle noise on the
diagnostics, details of the shielding and the effect on background levels are
discussed in section 8.

##  3 $\gamma$-ray photon source

During the commissioning of the platform the laser energy used to drive the
accelerator was limited to 5 J, which is significantly below the maximum that
can be achieved in Gemini (12-15 J on target). With 5 J laser energy the best
electron beam performance achieved (data set ‘A’) reached on average a maximum
energy of $710\pm 50$ MeV and contained a charge of $50\pm 7$ pC. The data set
contained 10 shots and the average spectra of these beams is presented in
figure 3 (a), labelled dataset A (blue). The measurement had a low-energy cut
off at $\sim 350$ MeV due to the yoke of the spectrometer magnet. The beams
had a divergence of $2.3\pm 0.3$ mrad and a shot-to-shot pointing standard
deviation of $0.6$ mrad. Additionally, during the commissioning phase of the
platform (discussed later), the LWFA electron beam performed in a reduced
capacity, with lower maximum energies achieved and less than half the total
charge (data set ‘B’). There were 79 shots in total and the average spectra
for these runs are provided in figure 3, labelled dataset B (purple).

Figure 3: Properties of the electron beam used to generate the $\gamma$-ray
photon source: (a) Average electron spectra with standard deviation from mean
(shaded). (b) Average electron beam properties with standard deviations (error
bars). A (blue) and B (purple) represent the optimum beam performance and
performance during commissioning, respectively.

The $\gamma$ profile diagnostic took the form of an array of $20\times 20$
caesium-iodide (CsI) crystals, each of dimensions $2\,\mathrm{mm}\times
2\,\mathrm{mm}$ and $20\,\mathrm{mm}$ thick, positioned normal to the $\gamma$
beam and $3.39\,\mathrm{m}$ downstream of the $\gamma$ source. The
scintillation of the CsI crystals upon exposure to high energy photons is
observed with an optical camera. An example $\gamma$ beam profile from this
diagnostic can be seen in figure 4 (a), the D shaped profile due to the
combination of the collimator and tungsten block shielding the x-ray target
are clearly apparent. Another larger array of $33\times 47$ crystals, each
$5\,\mathrm{mm}\times 5\,\mathrm{mm}\times 50\,\mathrm{mm}$, is positioned
along the beam axis behind the profile diagnostic and measures the decay of
signal in the propagation direction to deduce the spectrum [43]. The $\gamma$
spectrum is inferred using a forward model to calculate the energy deposited
in each crystal, finding the model parameters which most closely reproduce the
observed signal using a least squares approach. In this case our forward model
is based on Geant4 simulations of the signal produced due to the $\gamma$ rays
produced by a wide range of experimentally observed electron spectra. Both the
profile and spectrum scintillator arrays are observed by cooled 14-bit EMCCD
cameras (Andor iXon), with single photon sensitivity to the scintillation
light.

Figure 4: Properties of the $\gamma$-ray photon source: (a) $\gamma$ beam
profile example. The D-shaped footprint produced by the collimator and half-
block shadows is clear. (b) Simulated $\gamma$ spectra for both data sets
(solid lines) and $1\sigma$ interval of the first 10 spectra of the
corresponding data sets measured with the $\gamma$ spectrometer (shaded
regions).

The converter target scatters the incident electron beam such that the
properties of the $\gamma$ beam and the electrons cannot be measured
simultaneously. Therefore the electron beam was characterised without the
converter in place, and this was subsequently used to simulate the expected
average $\gamma$ spectrum. The $\gamma$ spectra that were measured with the
larger crystal array immediately after the characterisation of the electron
beam (figure 4 (b), shaded regions) agreed well with the simulated spectra
(figure 4 (b), solid lines).

Geant4 simulations using dataset A indicate the emission of $(3.5\pm
0.5)\times 10^{8}$ photons or $(7\pm 1)\times 10^{6}$ photons/pC, distributed
in an exponentially falling spectrum with a high-energy cutoff at the maximum
electron energy (see figure 4 (b)). $(7\pm 3)\times 10^{7}$ of these photons,
or about $20\%$ of the total photon yield, have an energy above $100$ MeV. By
matching the average electron beam and the average yield on the $\gamma$
profiler, we obtain a calibration factor of $(5.0\pm 0.4)\times 10^{6}$
$\gamma$-counts/pC. This calibration allows us to estimate the electron beam
charge on shots where the convertor target is in place (and hence no electron
spectrometer measurement is possible). The variations in the $\gamma$ yield
indicate stronger fluctuations in the photon numbers from shot to shot than
expected from the electron charge. This is believed to be due to pointing
fluctuations and the limited size of the $\gamma$ detector occasionally
missing outer portions of the beam. As described previously, for photon-photon
collisions, the $\gamma$ beam is collimated and one half is blocked to prevent
interactions with the x-ray target (see figure 2 and 4 (a)). This reduces the
footprint of the beam and the effective yield of photons by a factor $3.6\pm
1.2$.

Figure 5:  (a) Average x-ray spectra emitted by the germanium target. Measured
using a TlAP crystal spectrometer over 47 shots. The standard deviation is
given by the black shaded area. (b) Example median-filtered x-ray pinhole
camera image of the germanium target. The aspect ratio of the x-ray source
closely resembles the $f/2$ focal spot.

##  4 x-ray photon source

The second arm of the laser system was used to generate the dense keV x-ray
bath at the interaction point. The drive laser was focused onto a refreshable
Kapton tape target system using an $f/2$ off-axis parabola (OAP). Prior to
reflecting from the final focussing optic, the 150 mm diameter beam passed
through a distributive phase plate [44]. This allowed the laser energy to be
deposited in a large quasi-flat-top focal spot
$210\text{\,}\mathrm{\SIUnitSymbolMicro m}$ $\times$
$90\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (major and minor diameters at
$1/e^{2}$). Approximately $72\%$ of the laser energy is contained in this
spot, with the remaining light being diffracted away into higher orders. The
laser pulse was also stretched to $40$ ps duration, resulting in an intensity
of $2\times 10^{15}$ Wcm-2 on target. The targets themselves were $\sim 100$
nm coating of germanium metal on a 5 $\pm$
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ etched square window ($\sim
1~{}\textrm{mm}^{2}$) of the Kapton tape (which is otherwise nominally
$25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick) [45]. As the laser pulse is
incident on the germanium coating, a hot dense plasma is created on the front
surface. In this highly-ionised plasma, M-L shell transitions in the germanium
efficiently produce photons in the range of $1-2$ keV [46]. These x-rays are
emitted into a 4$\pi$ sphere, such that a few milimetres from the target a
dense x-ray bath exists. A similar technique has previously been used to
rapidly heat targets to extreme conditions [47]. The Kapton target substrate
thickness was optimised to increase the transmission of these keV photons,
while suppressing sub-keV photons and preventing any material contamination to
the interaction region.

A Thallium Acid Phthalate (TlAP) crystal spectrometer viewed the front of the
tape targets at angle of $\approx 55^{\circ}$ (to normal) to diagnose the
x-ray yields. Bragg reflections from the flat TlAP crystal onto an in-vacuum
x-ray CCD (Andor DX-420-BN) allowed the x-ray spectrum to be measured between
$1.3-1.5$ keV with a spectral resolution of 4 eV. The average measured spectra
from 47 shots can be seen in figure 5 (a). The absorption edge from an
aluminium filter at $1.56$ keV provided a spectral marker allowing the
dispersion across the CCD to be calculated. A variety of other filters were
also used, allowing the signal to be corrected for background contributions
from direct CCD irradiation, higher order Bragg reflections and crystal-
substrate fluorescence.

In addition to the crystal spectrometer, a 2-channel pinhole imaging system
was employed. By imaging the x-ray producing plasma, the source size and
target alignment were monitored on-shot. An example image from this diagnostic
can be seen in figure 5 (b). While the entire emission collected by the
pinhole camera passed through a $25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ Be
filter (acting as a $600$ eV high-pass filter), both channels of the imaging
system had different spectral filters. This allowed for a broad x-ray yield
quantification, which correlated well with the total energy measured by the
crystal spectrometer.

It was found that the total conversion efficiency from laser energy to
$1.3-1.5$ keV x-rays was $(2.4\pm 0.3)\%$. This corresponds to $(3.7\pm
0.4)\times 10^{10}$ photons/eV/J/srad emitted normal to the front surface of
the germanium target. Taking into account the absorption in the kapton layer
on the rear side of the targets, at the interaction region (1 mm from the
tape) this corresponds to a photon density of ${(1.4\pm 0.5)\times
10^{12}~{}\textrm{mm}^{-3}}$. These photon numbers assume the Kapton tape
remains cold throughout the interaction, and are supported by simulations of
the X-ray source directionality and duration ($97\pm 11~{}\textrm{ps}$, FWHM).
The average transmission of the Kapton tape is 47% over the range
$1.3-1.5~{}\textrm{keV}$. The flux stability from shot-to-shot was found to
have a standard deviation of $28\%$. A tape drive was used to place a fresh
target at the focal plane of the x-ray drive laser. The tape targets were able
to be aligned at a repetition rate of 1 shot per minute. Note however that it
should be possible to increase this rate with improvements to the target
alignment system.

##  5 Positron Transport System

In order to measure the produced pairs, a system composed of permanent magnets
and single particle sensitive detectors has been developed [39] and
specifically adapted for the energy range in this experiment. The overall
geometry is shown in figure 6. The generated pairs initially follow the
$\gamma$-ray beam path, but after passing through an aperture in a lead wall,
they were separated from the beam using a dipole magnet. This magnet was 60 cm
long with a field strength of 0.6 T ($\int Bdx=0.36~{}\mathrm{Tm}$) and is
referred to as the separator magnet. To avoid scattering of $\gamma$-rays on
the poles, the magnet had a gap of 10 cm. The pairs were dispersed and exited
the vacuum chamber through a $125\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick
Kapton window. The pairs were then detected by two different types of single
particle detectors, which have a limited spatial extent and are presented in
section 6. To increase the collection efficiency of the generated pairs, two
additional dipole magnets were used to focus the electrons and positrons to
individual detectors. These magnets are referred to as the collimator magnets.
They had an aperture of 9 cm, were 70 cm long and provided a magnetic field
strength of 0.5 T ($\int Bdx=0.35~{}\mathrm{Tm}$). The combination of
separator and collimator magnets is called the analyser magnet system (AMS).
The efficiency of this transport system is limited by various apertures along
the beam path, especially the vacuum exit window frame which has three
openings, one for each particle beam and a central one for $\gamma$-rays, see
figures 1 and 6. The AMS transport efficiency was evaluated using simulations
performed with Radia [48][49] for Wolfram Mathematica. The simulations model
the magnetic fields and calculate the trajectory of positrons through the
system for a range of possible source positions and initial momenta. The
collection of all trajectories represents a 5-dimensional function (the source
point is fixed along the beam axis) that maps any positron to the detector
plane depending on its initial phase space coordinate at the time of pair
creation. Combined with a check if individual trajectories are blocked by
apertures, this function, which is called the Radia filter, determines if and
where a positron hits the detector. This is used to evaluate the transmission
efficiency and the spatial SPD signal distribution for any incident positron
population. This filter was used to map positrons produced by a 1 mm thick
plastic PTFE calibration target to the SPD plane as a detector calibration,
which is described in greater detail in section 6. Since it is a 5-dimensional
filter, the transport efficiency is hard to visualise due to correlations
between all 5 initial coordinates and also depends on the phase space
distribution of the incident positrons. A projection of the full filter to the
energy axis as well as source point and divergence planes is shown on figure
7, which exhibits a narrow spectral acceptance between 200 and 420 MeV as well
as a narrow angular acceptance between -1 and +7 mrad relative to $\gamma$
beam axis. The source point does not affect the transmission significantly
over the considered range from -6 to 6 mm relative to $\gamma$ beam axis. The
overall transport efficiency for positrons from the PTFE calibration target
above 200 MeV is 28 % and the transport efficiency for produced BW positrons
above 200 MeV is 38 %. For all BWpositrons in the full spectral range, the
transmission is 14 %. The accuracy of calculated positron impact locations is
approximately 1 cm at the detector plane, limited by the positioning accuracy
of magnets, detectors and apertures inside the laboratory.

Figure 6: Overview of the analyser magnet system (AMS) to transport generated
pairs to the single particle detectors (SPD). The combination of separator and
collimator magnets was adapted for the energy range in this experiment by
adjusting the length of the magnets and their separation. The electron and
positron paths were adjusted to avoid the SPDs being directly exposed to
radiation from noise sources inside the vacuum chamber (such as the electron
source and convertor target). Figure 7: Transmission efficiency of positrons
through the AMS to the CsI SPD depending on positron energy (a), source point
(b) and divergence (c). For BW positrons, the integrated efficiency is 38%
above 200 MeV and 14% overall.

##  6 Single Particle Detectors

###  6.1 Caesium Iodide Single Particle Detector

A scintillation based single particle detector (SPD) system was employed. It
was composed of a thallium doped caesium iodide (CsI) crystal array coupled to
a 4Picos ICCD from Stanford Computer Optics. The crystals were coated with
titanium dioxide to avoid optical crosstalk between neighbouring crystals. The
surface facing towards the camera was polished. The 4Picos camera achieve
single photon sensitivity through a multi-channel-plate with photoelectron
multiplication factors up to $10^{6}$. This combination of CsI crystals and
4Picos camera gives the SPD single particle sensitivity.

The CsI SPD was calibrated with a PTFE calibration target in the beam close to
the BW pair source point to produce a known positron signal on the SPD. As the
$\gamma$ beam passes through the PTFE, electron-positron pairs are generated
in a narrow on-axis cone via the Bethe-Heitler process. The PTFE positron
population was calculated with Geant4 using the measured LWFA beam as input
and simulating the bremsstrahlung process, $\gamma$-ray beam shaping and
Bethe-Heitler pair production. The measured electron beam performance (and
subsequent Geant4 input) for these calibration shots are given in figure 3
dataset B. The obtained positron population was filtered and projected to the
detector plane using the AMS phase space filter calculated with Radia as
described in section 5. The simulated spatial distribution of positron impact
locations on the SPD plane is shown on figure 8 together with the measured
signal. The square box indicates the detector aperture and the dashed lines
show positron cut-off’s, one due to lead shielding that intentionally shadows
the lower portion of the SPD and another one due to the AMS field
distribution. The regions where no positrons impact the SPD are used to
measure background radiation other than positrons transported through the AMS,
which is described in section 7.

The simulated positron yield integrated over the detector aperture is $(200\pm
7)$ positronsams, where the error is the standard error of the mean over 3
identical Geant4 simulations. The subscript $ams$ indicates that we are
referring to positrons transmitted through the AMS with the associated
properties specified in figure 7. The LWFA beam spectral shape is assumed to
be constant for all calibration shots, which allows us to assume a linear
dependency between positron yield and LWFA charge. This normalisation to LWFA
beam charge assuming the reference spectral shape is denoted with pCr. The
Geant4 simulations were performed with a total charge of 9.2 pCr, so the
positron yield produced by the PTFE target is obtained as $(21.7\pm
0.8)~{}\mathrm{positrons_{ams}}/\mathrm{pC_{r}}$.

The mean measured signal integrated over the detector area is $(8.44\pm
0.69)\times 10^{8}$ countsn, where the subscript $n$ denotes counts that are
corrected for camera and imaging system effects. The PTFE data was taken with
an average charge of $(19.3\pm 1.1)~{}\mathrm{pC_{r}}$, so the total measured
PTFE positron yield is obtained as $(4.77\pm 0.17)\times
10^{7}~{}\mathrm{counts}_{n}/\mathrm{pC_{r}}$ with errors being standard
errors of the mean. This is then corrected for background radiation by taking
shots without the PTFE target in the beam, which was measured to be $(0.109\pm
0.004)\times 10^{7}~{}\mathrm{counts_{n}}/\mathrm{pC_{r}}$, and gives the BG
corrected yield of $(4.66\pm 0.17)\times
10^{7}~{}\mathrm{counts_{n}}/\mathrm{pC_{r}}$. This measured signal is now
directly correlated to the simulated number of incident positrons to obtain
the SPD calibration of $(2.15\pm 0.03)\times
10^{6}~{}\mathrm{counts_{n}}/\mathrm{positron_{ams}}$.

Figure 8: (a) The simulated positron hit map on the detector plane produced by
the PTFE calibration target on a 9.2 pCr shot. (b) The measured detector
response with the PTFE calibration target in the beam. The combined
information from both images is used as a detector calibration.

###  6.2 Timepix3 Single Particle Detectors

A second SPD system relied on two Timepix3-based hybrid detectors [50]
arranged in a stacked geometry, with one 40 mm behind the other. The rear was
offset laterally by 4 mm, closer to the beam axis. Both detectors were placed
before the CsI SPD. Timepix3 are CMOS based hybrid pixel detectors with an
active area of $14\times 14~{}\textrm{mm}^{2}$ ($256\times 256$ pixels, each
$55\text{\,}\mathrm{\SIUnitSymbolMicro m}$ $\times$
$55\text{\,}\mathrm{\SIUnitSymbolMicro m}$). Timepix3 assemblies with silicon
sensors were chosen with the front detector having a
$300\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick sensor and the rear
$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick. Any minimum ionising
particles (MIPs) passing through the silicon deposit $\sim 2900$ eV per
micron, registering a trail of charge across the pixels. A MIP is considered
to be a charged particle that has kinetic energy twice its rest mass, i.e. any
positron particles over 1 MeV. As charged particles pass through the detector,
each pixel records the time-over-threshold at a rate of up to 640 Mhz. If
fully calibrated the temporal resolution can be as low as $\sim 1$ ns, but in
our case we integrate the signal over one second during the shot period. A
significant advantage of these detectors is that all charged particles of
interest will register a signal, where as due to the silicon nature of the
detector, they are relatively insensitive to high energy photons, i.e. they
will be insensitive to background $\gamma$ photon noise. For example, the
probability of $\gamma$-ray absorption in
$150\text{\,}\mathrm{\SIUnitSymbolMicro m}$ silicon is $\sim 8\times
10^{-4}-2\times 10^{-3}$ [51]. As an additional step to confirm that we are
witnessing primarily charged particles rather than photons, the stacked
Timepix3 setup was simulated in Geant4 with either an incoming particle beam,
or an incoming photon beam. In the experiment calibration shots (discussed
further below), an approximately equal number of strikes was measured on the
front and rear detectors, but with 0.4 times the total energy absorbed in the
rear (the sensor of half the thickness). This aligns with the Geant4
simulations for charged particle strikes, as opposed to photons which
estimated approximately 2.5 times the energy should be deposited in the rear
detector due to secondaries created from photon absorption in the first
detector.

We deem each contiguously connected group of pixel charge registered on the
detectors (a “cluster”) as a single charged particle or photon hit. In order
to reduce unwanted noise, we can filter these clusters by their characteristic
traits, to select only those that are consistent with positrons that have been
transported through the AMS (as opposed to stray background particles). We do
so by training an event identification criteria on a set of calibration data
with an artificially created positron signal at the collision point (in a
similar manner to the PTFE calibration of the CsI SPDs). For this, a 1 mm
tungsten cube is placed on the $\gamma$ beam axis, within a few cm of the
collision point. The generated particle pairs are similar in energy to those
created by our photon-photon collision (100’s MeV). We collated the cluster
statistics on 28 shots with the tungsten cube in place, and 28 null shots
immediately before this, with the tungsten cube removed. The latter shots act
as a comparable null reference for the background present on the Timepix3. The
shots with the artificial positron source show that the clusters have a higher
average charge per pixel, as well as narrower distributions for the width and
height, see figure 9 (a) and (b). These criteria can be applied to all
clusters to identify AMS positron events and remove background hits. To build
a complete AMS positron specification, we take the distribution for each
metric (e.g. cluster width) individually and subtract the null shot series
distribution. The limits for that metric are then taken as the 95% interval of
the remaining distribution. The final AMS positron specification for our setup
is given in table 1. It was observed on subsequent data and background shots
that the cluster filtering was rejecting up to 60% of the total hits on the
Timepix3 detectors. See figure 9 (c) for an example of this AMS positron
identification.

Figure 9: TimePix3 training data. Average charge per pixel (a) and cluster width (b) distributions for shots with and without an on-axis positron source (Tungsten). The difference in the distributions with and without the positron source form the basis of a positron event specification. (c) Examples of the positron specification being implemented and rejecting non-conforming hits (highlighted with a negative colour value).  Detector |  Charge Sum |  Num. Pixels |  Height |  Width
---|---|---|---|---
Front $300\text{\,}\mathrm{\SIUnitSymbolMicro m}$ |  $40\leq X\leq 145$ |  $\leq 5$ |  $\leq 2$ |  $\leq 3$
Rear $150\text{\,}\mathrm{\SIUnitSymbolMicro m}$ |  $10\leq X\leq 85$ |  $\leq 3$ |  $\leq 2$ |  $\leq 2$
Table 1: Timepix3 on-axis positron specifications. Applying these criteria to
each TimePix3 charge cluster allows the identification of positrons travelling
through the AMS, and rejects events that are background noise.

The AMS positron shot series was also used to infer the particle trajectory
between the two Timepix3 detectors. With a known stream of incident particles
passing through both detectors, it is possible in theory to track the
particles as they travel ballistically through the front detector and onto the
rear. To do so, on each shot a 2D map of positron hits was generated for each
Timepix3 detector. A raster of horizontal and vertical positions of one
detector relative to the other was then calculated, and a correlation value
given to the overlapping region. Accounting for the fact the rear detector was
offset laterally by 4 mm closer to the beam axis, it was found that particles
had a range of trajectories of 4∘-12∘ between the two detectors. This is
supported by the Radia tracking simulations for the AMS. Unfortunately this
variation in angular spread (stemming from the variation in particle energy
through the AMS) meant it was not possible to track particles from the front
Timepix3 through to the rear. It was possible however to treat the area of the
second TimePix3 outside of these trajectories as an additional detection area,
without risking double counting positrons passing through the first TimePix3.
This increased the total detection area to 308 mm2 (a 57% increase).

##  7 Commissioning

###  7.1 Characterisation of background sources

The radiation background was characterised using the CsI SPD and found to be
composed of two distinct particle types, high energy positrons transported
through the analyser magnet system and low energy diffuse radiation from the
$\gamma$-ray beam dump. The discrimination between these two was made by
analysing the yield on different spatial regions on the SPD shown on figure 8,
where AMS positrons can only impact part of the SPD aperture with the
remaining area being shielded. The measured background in the shielded region
below the lead brick shadow and right of the AMS cut-off was equivalent to
$BG_{diffuse}=1.3\pm 0.1\times
10^{-3}~{}\mathrm{positrons_{ams}}/\mathrm{pC_{r}}/\mathrm{crystal}$. This is
assumed to originate from diffuse low energy radiation leaking through the
$\gamma$-ray beam dump due to its homogeneous distribution over the detector
aperture. The average background level in the region open for positrons was
equivalent to $BG_{open}=2.5\pm 0.1\times
10^{-3}~{}\mathrm{positrons_{ams}}/\mathrm{pC_{r}}/\mathrm{crystal}$, almost
twice as large compared to the shielded region. The difference,
($BG_{open}-BG_{diffuse})=BG_{AMS}={1.2\pm 0.1\times
10^{-3}~{}\mathrm{positrons_{ams}}/\mathrm{pC_{r}}/\mathrm{crystal}}$ is
therefore assumed to originate from high energy positrons that propagate
through the analyser magnet system. For context, this corresponds to 1
background positron for every 4 pCr of reference LWFA beam charge integrated
over the full 208 crystal large CsI SPD region that is open for AMS positrons.

Sources of background AMS positrons are Bethe-Heitler pairs produced by the
$\gamma$-ray beam in collisions with residual gas inside the vacuum chamber
induced by the LWFA target or with apertures close to the beam line, for
example the vacuum exit window or lead shielding in between the BW interaction
point and the separator magnet shown on figure 1. Geant4 simulations were used
to understand these noise sources and help design an appropriate shielding
configuration. Figure 10 depicts the combined photon and charged particle flux
estimated throughout the final shielded setup. The main goal of any shielding
is to minimise the flux through the SPD location. The Geant4 simulations
identify that the main sources of background positrons which are transported
through the AMS are the $\gamma$ beam collimator and the aperture in the lead
wall before separator magnet. The aperture size was chosen to minimise the
background generated by particles (both $\gamma$ rays and electrons from the
LWFA) striking the edge of the aperture while maintaining a low background
caused by particles striking the pole pieces of the separator magnet. The
primary source of photon background at the detector was identified as the
$\gamma$ beam profile and $\gamma$ spectrum detectors and the beam dump.

Figure 10: Combined photon and charged particle background noise flux
throughout the setup. Simulated in Geant4. Lead walls shield the SPD location
from direct sight lines to heavy noise sources such as the $\gamma$ collimator
and first lead wall aperture (labelled for clarity).

###  7.2 Gamma dependence of background noise

To assess the capability of the photon-photon platform to measure the BW
process it is important to quantify the background level of the SPDs on data
shots. This measurement was made by taking a series of shots where the
$\gamma$ beam was operating and the x-ray foil was at the collision position,
but the second laser beam which generates the x-rays was not fired. Since all
the background sources are generated by the interaction of the $\gamma$ beam
with components of the experiment, including the x-ray generation foil, these
shots represent a direct measurement of the background that will be present on
shots where the two photon sources are colliding. We call these type of shots
_null shots_.

In a series of null shots it was found that the number of background events
detected by both SPDs is linearly proportional to the effective charge in the
electron beam, as measured on the calibrated $\gamma$ profile diagnostic (see
figure 11). A linear regression, assuming that the number of background events
follows Poisson statistics, with the expected number of background events
$\langle N_{\rm bg}\rangle$ varying linearly with the electron charge, $C$,
i.e. $\langle N_{\rm bg}\rangle=\sigma_{\rm bg}C$ shows that the signal on the
CsI detector corresponds to $11.2\pm 0.6$ positron events on a shot with the
average charge. The number of events counted on the Timepix3 detector
corresponds to $3.8\pm 0.3$ events on an average shot. The CsI detector area
(5200 mm2) is much larger than the Timepix3 area ($308$ mm2), so we might
expect the CsI detector signal to be 17 times higher than the Timepix3
detector. However the spatial non-uniformity of the background on the CsI
detector is expected to reduce this to a ratio of 9. The observed ratio is in
fact close to 3.

The two detectors have very different sensitivity to the different background
sources. The CsI detector analysis removes the spatially diffuse background
caused by low energy $\gamma$ events generated in the $\gamma$ beam dump. This
background subtraction is possible on the CsI detector because part of the
detector is shielded from positrons transported by the analyser magnet system.
However, this is not possible for the smaller Timepix3 detector. This
difference in the background subtraction method on the detectors could be
responsible for the difference in the observed rate on null shots. However,
the thin silicon Timepix3 detector is quite insensitive to $\gamma$ rays (at
most $10^{-3}$ absorption), and the track shape detection is able to discount
low energy events and events incident at an angle that is inconsistent with
positrons transported through the analyser magnet system. This suggests that
both detectors should be fairly insensitive to the diffuse $\gamma$ background
caused by the beam dump.

Figure 11: Variation of number of background events measured with electron
beam charge on the Timepix3 and CsI detectors. Orange: CsI detector, Blue:
Timepix3 detector. Shaded regions represent $1\sigma$ confidence interval on
the linear fit.

###  7.3 Commissioning with low yield positron source

To test the detection capability of the SPDs, a series of commissioning shots
were performed. These involved inserting a
$25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick Kapton tape target into the
$\gamma$ beam at the photon-photon collision location (while not firing the
x-ray drive laser). The tape target was 1 cm wide, and spooled in the vertical
plane, presenting $\approx 20$ cm of tape length. This meant it covered the
entire vertical length of the $\gamma$ footprint, but some fraction of the
footprint in the horizontal plane. By translating the target so that it was
only partially intercepting the $\gamma$ beam profile we could vary the number
of positrons produced and transported to the detectors. Only a limited number
of shots ($\lesssim 10$) were taken at each position. The $\gamma$ background
was subtracted from the CsI data using the method describe in section 6. The
background on both detectors owing to particles generated away from the
collision point (e.g. in the lead aperture) being transported to the detectors
was characterised by completely removing the Kapton target from the $\gamma$
beam (null shots). The measured positron signal is shown in figure 12 for both
detectors. After subtraction of the positron background, the CsI detector
shows agreement within experimental uncertainty with the predicted number of
positron reaching the detector calculated using Geant4, confirming the
calibration procedure in this low signal regime. The larger area CsI detector
was capable of measuring a signal above the positron background when only 20%
of the $\gamma$ beam was intercepted by the Kapton tape, corresponding to
signal level of approximately 0.1 positrons per picocoulomb reaching the
detector. The smaller area Timepix3 detector was able to discern a signal
above background when 50% of the $\gamma$ beam was intercepted, which is also
a signal level of approximately 0.1 positrons per picocoulomb reaching the
detector. Even with the limited amount of data taken in these commissioning
shots, it is clear that the single particle detectors are capable of measuring
small numbers of positrons per picocoulomb. However the detection limit of
approximately 0.1 positrons per pC above background is very dependent on the
number of shots taken to characterise both the signal and the background. In a
data run with many shots we would be able to achieve a significantly higher
sensitivity.

Figure 12: Commissioning of the single particle detectors. Positrons are
generated by colliding a fraction of the $\gamma$ beam with a
$25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick Kapton target. (a) Positron
signal measured (after gamma background subtraction) on the CsI single
particle detector. The shaded region is the measured background on a series of
shots with the Kapton removed. (b) Positron signal measured on the Timepix3
detector. The shaded region is the measured background on a series of shots
with the Kapton removed. (c) Positrons above null shot background measured on
the CsI detector. The shaded region is the predicted number of positrons
transported to the detector under the same experimental conditions. (d)
Positrons above null shot background measured on Timepix3 detector. The dashed
line is an estimate of predicted number based on scaling the estimated number
of positrons on the larger CsI detector.

##  8 Modelling

The key factor which determines whether a BW detection experiment is
successful is the signal-to-noise ratio. In order to model this parameter and
interpret it correctly, accurate predictions involving large scale numerical
simulations are required. During the design and optimisation phase of this
experiment, the Monte-Carlo platform Geant4 was used to perform background
noise calculations. It is desirable to also perform signal calculations within
this same framework, enabling faster design and optimisation of the
experimental setup. However, the BW process is not included within any of the
standard Geant4 physics packages. Therefore, a new photon-physics package was
developed for this purpose which includes the BW process. Here a brief
overview of the package is provided, however, a more in depth review can be
found in reference [52].

###  8.1 Geant4 Breit-Wheeler Module

Geant4 is a package for simulating the passage of particles through matter. To
include the BW process within Geant4, it must be treated in a similar way.
This has been achieved by handling the two interacting photon sources
differently where one is treated as a dynamic beam of photons, and the other
is treated as a static field fully defined by a spectral photon density per
unit volume per unit solid angle, $n(\omega,\theta,\phi)$. Individual photons
from the dynamic source are sampled and tracked through the static field.
Calculations are then performed to determine whether the BW process has
occurred, and if so, the properties of the emitted electron-positron pair are
found.

The interaction between the dynamic particle beam and the static photon field
occurs at a constant average rate making it a Poisson process. Therefore, the
probability that a dynamic particle travels a length, $x$, is given by the
following exponential distribution

$P(x)=\lambda_{d}^{-1}e^{-x/\lambda_{d}}$ (1)

where $\lambda_{\mathrm{d}}$ is the mean free path of the dynamic particle. As
this is an interaction between massless particles, the mean free path is given
by [53]

$\frac{1}{\lambda_{d}}=\displaystyle\int_{0}^{2\pi}\mathrm{d}\phi\int_{0}^{\pi}\mathrm{d}\theta\int_{0}^{\infty}\mathrm{d}\omega\,\sigma\textrm{(}s)\,n(\omega,\phi,\theta)\,(1-\cos{\theta})$
(2)

where the BW cross-section is

$\sigma(s)=\frac{\pi
r_{e}^{2}(1-\beta^{2})}{2}\Bigg{[}(3-\beta^{4})\mathrm{log}\frac{1+\beta}{1-\beta}-2\beta(2-\beta^{2})\Bigg{]},$
(3)

$r_{e}$ is the classical electron radius, $\beta=\sqrt{1-s^{-1}}$, and
$s=(1-\cos\theta_{s})E_{1}E_{2}$ for two photons of energy $E_{1}$ and $E_{2}$
interacting at angle $\theta_{s}$. For each dynamic photon that passes into
the static field, $x$ is sampled from equation 1. If this is longer than the
static field, it will propagate through unaffected, if it is shorter, then it
will propagate $x$ before interacting.

If an interaction occurs, the dynamic photon is removed from the simulation
and an electron-positron pair is added. However, before adding the new
particles their momentum must be obtained, which can be defined through each
particle’s energy $E_{\pm}$ and polar, $\theta_{\pm}$ and azimuthal,
$\phi_{\pm}$ scattering angles, where the $\pm$ indicates a positron or
electron. It is easiest to first obtain these properties in the centre-of-mass
frame. Here, both the electron and positron receive an energy of
$E_{\pm}=\sqrt{s}/2$. The positron polar scattering angle is then obtained by
sampling from the differential cross-section for the BW process

$\frac{d\sigma}{d\Omega}=\frac{r_{e}^{2}\beta}{s}\bigg{[}\frac{1+2\beta\mathrm{sin}^{2}\theta-\beta^{4}-\beta^{4}\mathrm{sin}^{4}\theta}{(1-\beta^{2}\mathrm{cos}^{2}\theta)^{2}}\bigg{]},$
(4)

whereas the positron azimuthal scattering angle is obtained by sampling from
the uniform distribution $\mathcal{U}(0,2\pi)$. Due to conservation of
momentum, this also defines the scattering angles of the electron, which are
given by $\theta_{-}=\pi-\theta_{+}$ and $\phi_{-}=\phi_{+}-\pi$. After
applying a Lorentz boost, these properties are obtained in the laboratory
frame, allowing the electron-positron pair to be added to the simulation.

This method does not account for either spatial gradients or a temporal
evolution of the static photon field. However, if the full field is
constructed by defining multiple sub fields, each with a different
$n(\omega,\theta,\phi)$, then spatial gradients can be accounted for. It is
challenging to account for a temporal evolution due to the static nature of
the Geant4 computational domain. Therefore, this module will only provide
accurate predictions if the time for the dynamic photons to traverse the
static field is much shorter than the characteristic timescale of the static
field. Due to the large difference in the duration of the sources, this is a
good approximation for this experiment, where the x-ray source is designated
as the static field and the $\gamma$-ray beam is treated dynamically.

###  8.2 Predictions

By developing a BW module for Geant4, start-to-end simulations of this
experiment can be performed within a single framework, allowing for efficient
calculations of the signal-to-noise ratio. Such calculations have been carried
out using the electron beam and x-ray field measured during the experiment and
discussed in sections 3 and 4 respectively. The results of these calculations
are summarised in table 2, which show the predicted number of BW signal
positrons, background leptons, and background photons which pass through the
Timepix3 detectors. The background has been separated into leptons and photons
as the former produces the same signature as the BW signal positrons on the
Timepix3 detectors whereas the latter produces a different signature.
Therefore, photon hits can be easily removed as a noise source, whereas lepton
hits cannot. Note that the quoted numbers refer to the number of particles
through the TimePix3 detectors, not the larger CsI SPD. For estimates of the
number of particles through the CsI these numbers need to be scaled by the
increased area, however, the BW signal is non-uniform over this increased
area. From the Radia simulations described in section 5, it is estimated that
the CsI would cover approximately 9 times more BW signal than the TimePix3.

Table 2: Predicted number of background particles and signal Breit-Wheeler pairs reaching the TimePix3 detectors, per pC of primary electrons in the LWFA beam. The “experiment” row shows predictions using the experimental measurements whereas the “optimum” row shows predictions using the best obtained Gemini parameters. |  BW pairs / pC |  BG leptons / pC |  BG photons / pC
---|---|---|---
Experiment |  $(1.87\pm 0.69)\times 10^{-6}$ |  $(1.97\pm 0.25)\times 10^{-2}$ |  $1.37\pm 0.06$
Optimal |  $(1.37\pm 0.05)\times 10^{-3}$ |  $(10.36\pm 1.44)\times 10^{-2}$ |  $4.06\pm 0.25$

To make an observation of the BW process, a statistically significant
difference between the mean number of detected positrons on full shots (both
laser drivers with $\gamma$ and x-rays colliding) and null shots (no
collisions viable as no x-rays present) is required. Using the values in table
2, a coarse estimate of the number of shots needed to observe this difference
can be made. Assuming an electron beam charge of $50\,\mathrm{pC}$
(corresponding to $\sim 3\times 10^{8}$ primary particles), the mean number of
events detected on full and null shots are $\mu_{\mathrm{F}}=0.95+(0.9\times
10^{-4})$ and $\mu_{\mathrm{N}}=0.95$ respectively. The number of events
detected on a shot is a Poisson variable, so after $N$ shots the error on
$\mu_{\mathrm{N/F}}$ is $\sigma_{\mathrm{N/F}}=\sqrt{\mu_{\mathrm{N/F}}/N}$.
If a $2\sigma$ significance observation is desired, then it is required that
$\mu_{\mathrm{F}}-2\sigma_{\mathrm{F}}>\mu_{\mathrm{N}}+2\sigma_{\mathrm{N}}$.
Figure 13 (a) shows this condition occurs after $\sim 10^{9}$ shots. This is
far more shots than can be achieved over the duration of a Gemini experiment.

Figure 13: Mean number of positrons and $2\sigma$ error against number of
shots, for both full and null shots. (a) Predictions based on measurements
made during the experiment. (b) Predictions based on optimum photon sources
and the existing shielding configuration. (c) Predictions based on optimum
photon sources, but with an extended setup that reduces the background noise
by a factor of ten.

The signal-to-noise ratio obtained using the measured experimental parameters
is far lower than what had been predicted from coarse calculations carried out
prior to the experiment. This is due to both the electron beam and x-ray field
behaving sub-optimally during the experiment. For example, figure 14 shows
electron beam spectra obtained during a previous Gemini experiment (see ref
[54]). These spectra extend to far higher energies than figure 3, with higher
beam charges, in excess of $100\,\mathrm{pC}$, also obtained. On top of this,
the energy in the laser pulse used to drive the x-ray field was also lower
than expected. In previous experiments $15\,\mathrm{J}$ on target has been
achieved, a factor of $\sim 1.5$ times higher than this experiment. Including
these factors leads to a significant increase in the BW yield as demonstrated
in table 2. Using the same approach as before, the number of shots required to
observe the BW process with optimum conditions can be estimated. Assuming a
beam charge of $160\,\mathrm{pC}$ (corresponding to $10^{9}$ primary
electrons), figure 13 (b) shows $\sim 5000$ shots are required to make a
$2\sigma$ observation with optimum experimental parameters.

Figure 14: High energy electron beams obtained during a previous Gemini
experiment. Taken from ref [54]

##  9 Conclusions and Future Directions

A laser-plasma platform for investigating photon-photon physics has been
presented. A high brilliance $\gamma$ source with associated diagnostics for
yield and spectrum was commissioned. It was found that $(7\pm 3)\times 10^{7}$
$\gamma$ photons/shot can be produced with energies over 100 MeV. In tandem, a
high repetition rate tape-based keV x-ray source was also commissioned.
Approximately $(1.4\pm 0.5)\times 10^{12}$ x-ray photons/mm3 in the energy
range 1.3-1.5 keV were produced in the proximity of the photon collision
interaction region. A magnetic transport system was implemented to
successfully relay an estimated 14% of any pair particles produced in the
interaction region to two SPD systems (38% of particles over 200 MeV). A CsI
SPD was tested and was able to detect a signal level of $0.08\pm
0.03~{}\mathrm{positrons_{ams}}/\mathrm{pC_{r}}$ over a background level
equivalent to $0.32\pm 0.01~{}\mathrm{positrons_{ams}}/\mathrm{pC_{r}}$ This
agreed, within uncertainty, with the simulated signal level. Additionally a
Timepix3 SPD system was also used, recording a signal level of $(0.08\pm
0.07)~{}\mathrm{positrons_{ams}}/\mathrm{pC_{r}}$ over a background of
$(0.07\pm 0.04)~{}\mathrm{positrons_{ams}}/\mathrm{pC_{r}}$.

These detection limits where achieved over only $\lesssim 10$ shots with the
sensitivity expected to increase with the number of shots.

We estimated that with the achieved performance of the platform, over $10^{9}$
shots would be needed to observe the linear BW process with $2\sigma$
significance. However, it is clear that the electron beam performance is
crucial to the capabilities of the setup. Given optimal performance of the
laser facility in question, electron beam energies of over 1.5 GeV are
possible. Simulations indicate that the increased BW signal level associated
with this increase in electron energy reduces the number of required shots to
approximately 5000. With the repetition rate of the laser system used, this is
achievable in under 28 hours of operation, which would be an achievable feat.
Note it has been shown recently that the stability and performance of LWFA
electrons beams can be improved using machine learning techniques such as
Bayesian optimisation [55, 56, 57].

In addition, we have identified possible improvements to the experiment setup
to significantly increase the efficiency of the platform. Firstly, the
Timepix3 SPD setup in this instance was limited to covering approximately 3
cm2 of the signal area (just over one sensor). Increasing this detector area
with the implementation of more sensors will reduce the number of required
shots as more of the pair signal is collected. For example doubling the number
of sensors will halve the amount of data required. It is estimated that five
sensors could cover the majority of the pair signal at the detector plane and
reduce the number of required shots for $2\sigma$ significance to $\sim 1000$;
less than 6 hours of data taking. With more investment and development it
should also be possible to increase the efficiency of the SPDs further. This
might include (but is not limited to) a fully functioning telescope system
[58] for the Timepix3 detectors to enable particle tracking, or tilting the
chips to increase the capability of filtering by angular trajectory [59].
Fully calibrated Timepix3 sensors (with close to nanosecond time resolution
[60]) could also employ time-of-flight distinction between signal and noise
sources. An upgraded Timepix4 sensor will also soon be available which not
only has a larger detection area ($\sim$7 cm2 area) but can be interconnected
to tile large areas [61].

Secondly, by making the setup more compact and bringing the interaction point
closer to the $\gamma$ source, a greater fraction of the $\gamma$ rays can
overlap with the regions of highest $x$ ray density, increasing the pair
yield. Alternatively an active plasma lens [62] could be used to focus the
electron beam, creating a narrower $\gamma$-ray beam at the collision point,
allowing a greater fraction of the $\gamma$ rays to interact with the regions
of highest $x$-ray density.

Finally it is clear that reducing the detector background noise can be as
important as increasing the number of pairs created or the platform
efficiency. The number of shots required to make a significant measurement
scales $\sim$ noise / signal2. Undertaking a more detailed design study or
shielding construction to reduce the detector noise could have a major impact
on the length of time that needs to be dedicated to data taking. As an
example, figure 13 (c) depicts the number of shots required to make a
$2\sigma$ significance measurement with the optimum performance presented
previously, but with a x10 reduction in background level. The number of shots
required is now only approximately 500.

It should be noted that future opportunities will be provided by increased
laser capabilities coming online in the near future at other facilities across
the globe [63, 64, 65, 66, 67, 68]. For example, with 1 PW it is possible to
produce electron beams of over 8 GeV energy [6]. As described previously the
achievable electron energy is critical to the production rate of the pairs.
The planned facilities are also in general closer to a repetition rate of 1
Hz, and will have increased laboratory space. The magnetic transport system
and SPDs described for this platform were designed to fit within the size
constraints of the Gemini target area which is less than 8 m long. With more
available space it would be possible to separate the electron and $\gamma$
beam dumps from the SPDs, and to introduce more shielding, reducing the
background level further, which as mentioned above plays a major role in
reducing the number of shots required for a measurement. More space would
remove limitations in the spectral selection and focusing capabilities of the
magnetic transport system.

These developments of the platform lead to exciting opportunities for
investigating other photon-photon physics. This includes testing the nonlinear
BW process by replacing the x-ray bath source with the direct high-intensity
laser pulse. It may also be possible to use the system to measure or reduce
the upper bound on the cross section for photon-photon scattering [69, 70].

## Acknowledgements

We wish to acknowledge the support of the staff at the Central Laser Facility,
the help from Robin Pitman and Vernon Gibson, and also to thank the Medipix3
Collaboration for providing the Timepix3 sensors. This project has received
funding from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme (grant agreement no 682399) and
STFC (grant No: ST/P002021/1). JH and AT acknowledge support from the US
National Science Foundation grant #1804463. GS would like to acknowledge
support from EPSRC (grant No: EP/N027175/1, EP/P010059/1).

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