A laser–plasma platform for photon–photon physics: the two photon Breit–Wheeler process

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 γ + γ → 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 $\sqrt{s} > 2{m}_{\mathrm{e}}{c}^{2}$, where $\sqrt{s}$ is the total energy measured in the centre of momentum frame. For two photons of energy E₁ and E₂ interacting at angle θ, s = 2[1 − cos(θ)]E₁ E₂.

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. 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.

Recent results from the STAR collaboration report the observation of the BW process using quasi-real photons in the peripheral collisions of high energy ions. The high energy of the quasi-real photons involved in those collisions ($\sqrt{s}\approx 2\enspace$ GeV) means the virtuality of the interaction, Q² ≈ (30 MeV)², is sufficiently small to make the measurements indistinguishable from collisions between real photons. There have been proposals for γ–γ colliders using conventional particle accelerators [3, 4], but none have yet been realised and a number of schemes involving multi-PW lasers to produce sufficiently bright γ sources have been proposed [5–7].

Today bright electron beams with GeV energies can be produced over a few centimetres by laser Wakefield acceleration (LWFA) [8, 9]. These electron beams can be used to produce brilliant γ ray beams by bremsstrahlung [10–13] and inverse Compton scattering [14–16]. Colliding such brilliant γ 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 [17, 18].

The BW process plays a pivotal role in a range of astrophysical phenomena [19]. 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 [20, 21]. 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 [22, 23]. The BW process also occurs in the coronae of accretion disks in active galactic nuclei [24], shaping particle spectra [25]. Finally, the BW process occurs when γ rays interact with the extragalactic background light (EBL), producing a cut-off of γ rays at ∼10¹³ eV [26]. 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 γ ray spectra after the BW process is accounted for, implying less pair-production from QED than anticipated [27, 28]. This problem could well be solved with conventional physics (e.g. corrections to models of the EBL [29]) as happened for the 'compactness problem', a contradiction that arose from γ ray burst light curve data which was eventually solved by including relativistic motion in the calculations [30]. 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 γγ → γγ and the nonlinear BW process [31]. 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. γ + nω → e⁻ + e⁺. The generation of matter via the nonlinear BW process plays a large role at the poles of rotating black holes [32]. In 1997, an experiment at SLAC collided a 48 GeV electron beam with a then state-of-the art laser pulse [33, 34], demonstrating the nonlinear BW process, thus studying the perturbative regime of strong-field QED [35].

Elastic photon–photon scattering, γγ → γγ, is important in models of primordial abundances and may also affect the observed spectra from distant γ ray bursts [36, 37]. Photon–photon scattering involving virtual photons has previously been observed. Delbrück scattering, γγ* → γγ*, where a real photon scatters from a virtual photon in the electric field of an ion, was observed in 1975 [38] and scattering between two virtual photons in the Coulomb field of two high energy ion beams γ*γ* → γγ was observed at the large hadron collider in 2017 and 2019 [39–41]. However, photon–photon scattering is most relevant in astrophysics with real photons close to threshold for pair production energy ($\sqrt{s}\leqslant 2{m}_{\mathrm{e}}{c}^{2}$) as at higher energies pair production dominates. Photon–photon scattering with real photons has yet to be observed in the laboratory.

In 2014, Pike et al [17] proposed an ambitious scheme that could enable the observation of the linear BW process by combining a laser-driven γ beam source with a laser-driven x-ray source. They proposed using a megajoule laser pulse, e.g. the National Ignition Facility [42], 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) [5] proposed the interaction between two high-power laser generated γ-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 γ beam colliding with an x-ray bath, similar in nature to that of Pike et al [17], 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 μm × 100 μm 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 [43].

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 γ 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 time frame (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.

A scintillation based 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 10⁶. 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 γ 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 using the measured LWFA beam properties as an input to a Geant4 simulation of the bremsstrahlung process, γ-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.

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The simulated positron yield integrated over the detector aperture is (200 ± 7) positrons_ams, where the error is the standard error of the mean over three 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 pC_r. The Geant4 simulations were performed with a total charge of 9.2 pC_r, so the positron yield produced by the PTFE target is obtained as (21.7 ± 0.8) positrons_ams/pC_r.

The mean measured signal integrated over the detector area is (8.44 ± 0.69) × 10⁸ counts_n , 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 ± 1.1) pC_r, so the total measured PTFE positron yield is obtained as (4.77 ± 0.17) × 10⁷ counts_n /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 ± 0.004) × 10⁷ counts_n /pC_r, and gives the BG corrected yield of (4.66 ± 0.17) × 10⁷ counts_n /pC_r. This measured signal is now directly correlated to the simulated number of incident positrons to obtain the SPD calibration of (2.15 ± 0.03) × 10⁶ counts_n  /positron_ams.

A second SPD system relied on two Timepix3-based hybrid detectors [55] 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 × 14 mm² (256 × 256 pixels, each 55 μm × 55 μm). Timepix3 assemblies with silicon sensors were chosen with the front detector having a 300 μm thick sensor and the rear 150 μm thick. Any minimum ionising particles (MIPs) passing through the silicon deposit ∼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 ∼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 γ photon noise. For example, the probability of γ-ray absorption in 150 μm silicon is ∼8 × 10⁻⁴–2 × 10⁻³ [56]. 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 γ 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 figures 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.

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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.

Detector	Charge sum	Num. pixels	Height	Width
Front 300 μm	40 ⩽ X ⩽ 145	$\leqslant 5$	$\leqslant 2$	$\leqslant 3$
Rear 150 μm	10 ⩽X ⩽ 85	$\leqslant 3$	$\leqslant 2$	$\leqslant 2$

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).

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 AMS and low energy diffuse radiation from the γ-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 ± 0.1 × 10⁻³ positrons_ams/pC_r/crystal. This is assumed to originate from diffuse low energy radiation leaking through the γ-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 ± 0.1 × 10⁻³ positrons_ams/pC_r/crystal, almost twice as large compared to the shielded region. The difference, (BG_open − BG_diffuse) = BG_AMS = 1.2 ± 0.1 × 10⁻³ positrons_ams/pC_r/crystal is therefore assumed to originate from high energy positrons that propagate through the AMS. For context, this corresponds to 1 background positron for every 4 pC_r 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 γ-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 γ 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 γ 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 γ beam profile and γ spectrum detectors and the beam dump.

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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 γ 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 γ 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 γ 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 ⟨N_bg⟩ varying linearly with the electron charge, C, i.e. ⟨N_bg⟩ = σ_bg C shows that the signal on the CsI detector corresponds to 11.2 ± 0.6 positron events on a shot with the average charge. The number of events counted on the Timepix3 detector corresponds to 3.8 ± 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.

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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 γ events generated in the γ beam dump. This background subtraction is possible on the CsI detector because part of the detector is shielded from positrons transported by the AMS. 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 γ 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 AMS. This suggests that both detectors should be fairly insensitive to the diffuse γ background caused by the beam dump.

To test the detection capability of the SPDs, a series of commissioning shots were performed. These involved inserting a 25 μm thick Kapton tape target into the γ 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 ≈20 cm of tape length. This meant it covered the entire vertical length of the γ footprint, but some fraction of the footprint in the horizontal plane. By translating the target so that it was only partially intercepting the γ beam profile we could vary the number of positrons produced and transported to the detectors. Only a limited number of shots (≲10) were taken at each position. The γ 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 γ 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 γ 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 γ 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 SPDs 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.

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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(ω, θ, ϕ). 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

$$\begin{equation}P(x)={\lambda }_{\mathrm{d}}^{-1}\enspace {\mathrm{e}}^{-x/{\lambda }_{\mathrm{d}}},\end{equation} \tag{ 1 }$$

where λ_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 [58]

$$\begin{equation}\frac{1}{{\lambda }_{\mathrm{d}}}={\int }_{0}^{2\pi }\mathrm{d}\phi {\int }_{0}^{\pi }\mathrm{d}\theta {\int }_{0}^{\infty }\mathrm{d}\omega \enspace \sigma (s)\enspace n(\omega ,\theta ,\phi )\enspace (1-\mathrm{cos}\enspace \theta ),\end{equation} \tag{ 2 }$$

where the BW cross-section is

$$\begin{equation}\sigma (s)=\frac{\pi {r}_{\mathrm{e}}^{2}(1-{\beta }^{2})}{2}\left[(3-{\beta }^{4})\mathrm{log}\enspace \frac{1+\beta }{1-\beta }-2\beta (2-{\beta }^{2})\right],\end{equation} \tag{ 3 }$$

r_e is the classical electron radius, $\beta =\sqrt{1-4{m}_{\mathrm{e}}^{2}{c}^{4}/s}$, and s = 2(1 − cos θ_s)E₁ E₂ for two photons of energy E₁ and E₂ interacting at angle θ_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_± and polar, θ_± and azimuthal, ϕ_± scattering angles, where the ± 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

$$\begin{equation}\frac{\mathrm{d}\sigma }{\mathrm{d}{\Omega}}=\frac{{r}_{\mathrm{e}}^{2}\beta {m}_{\mathrm{e}}^{2}{c}^{4}}{s}\left[\frac{1+2\beta \enspace {\mathrm{sin}}^{2}\enspace \theta -{\beta }^{4}-{\beta }^{4}\enspace {\mathrm{sin}}^{4}\enspace \theta }{{(1-{\beta }^{2}\enspace {\mathrm{cos}}^{2}\enspace \theta )}^{2}}\right],\end{equation} \tag{ 4 }$$

whereas the positron azimuthal scattering angle is sampled uniformly from 0 to 2π. Due to conservation of momentum, this also defines the scattering angles of the electron, which are given by θ₋ = π − θ₊ and ϕ₋ = ϕ₊ − π. 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(ω, θ, ϕ), 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, the x-ray source is designated as the static field and the γ-ray beam is treated dynamically. Ray tracing simulations of γ-rays propagating through a time varying x-ray field were performed to test the validity of the static x-ray field assumption. These show that the error introduced is between 5% and 15%.

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 nine times more BW signal than the TimePix3.

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 γ 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  pC (corresponding to $\sim 3\times 1{0}^{8}$ primary particles), the mean number of events detected on full and null shots are μ_F = 0.95 + (0.9 × 10⁻⁴) and μ_N = 0.95 respectively. The number of events detected on a shot is a Poisson variable, so after N shots the error on μ_N/F is ${\sigma }_{\mathrm{N}/\mathrm{F}}=\sqrt{{\mu }_{\mathrm{N}/\mathrm{F}}/N}$. If a 2σ significance observation is desired, then it is required that μ_F − 2σ_F > μ_N + 2σ_N. Figure 13(a) shows this condition occurs after $\sim 1{0}^{9}$ shots. This is far more shots than can be achieved over the duration of a Gemini experiment.

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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 reference [59]). These spectra extend to far higher energies than figure 3, with higher beam charges, in excess of 100 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 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 pC (corresponding to 10⁹ primary electrons), figure 13(b) shows $\sim 5000$ shots are required to make a 2σ observation with optimum experimental parameters.

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