Effect of laser temporal intensity skew on enhancing pair production in laser—electron-beam collisions

Strong field quantum electrodynamics processes can be investigated with current high power petawatt (PW) lasers, reaching intensities ∼10²¹ W cm⁻². Two important processes are quantum radiation reaction (RR) and multiphoton Breit–Wheeler (BW) pair production. RR describes the self-force on a charged particle as it emits radiation. A classical description of this force is not sufficient in the interaction of energetic electrons with strong laser fields as the quantum parameter χ_e = |E|_RF/E_crit, which compares the ratio of the electric field in the rest frame of the electron E_RF to the critical (Schwinger) field [4] E_crit = 1.38 × 10¹⁸ V m⁻¹ approaches unity. As χ_e approaches unity, 0.44 of the electron's energy is taken by the emitted photons [5, 6]. A complete quantum description of RR is impractical due to the large number of interactions the electrons undergo with the laser photons and instead a practical model has been developed [7–10]. These models require experimental validation with recent evidence showing signs of quantum effects on RR [1, 2]. These experiments used an all-optical setup where electrons were externally accelerated to >100 MeV and then collided with a counter-propagating laser pulse. The resulting Doppler upshift of the laser electric field results in χ_e ≈ 0.02–0.2. Similar experiments have been performed using crystals to provide the strong field [11].

Multiphoton BW pair production [12] occurs when a photon, emitted by an electron in the laser field, interacts with these fields and is converted to an electron-positron pair, γ' + nγ → e⁺ + e⁻ [13]. This was observed at the SLAC facility using a moderate intensity of I ∼ 10¹⁸ W cm⁻² colliding with a 46.6 GeV beam (reached χ_e = 0.36) [14]. However, at the moderate laser intensity the multiplicity of the interactions was not large and they may be described with a full QED calculation, which is not the case in current experiments at higher laser intensity.

As the electromagnetic fields produced with current PW laser systems are E_L ∼ 10⁻³ E_crit, much lower electron energies are required to obtain the same χ_e as the SLAC experiment. Energies accessible to laser wakefield acceleration (LWFA > 100 MeV) are sufficient and we may use an all-optical analogue to the SLAC experiment where a second laser pulse accelerates the electrons [15]. Recent electron energies from LWFA have been demonstrated at BELLA reaching a new record ∼7.8 GeV [16] although most LWFA experiments produce electron beams at ∼1 GeV so there is interest in how we may enhance the quantum parameter in all-optical collider experiments, for example employing an asymmetrical temporal envelope on the colliding pulse.

Modifying the laser pulse with a skew has been used in various simulations to show control of betatron oscillations [17] in a laser wakefield, enhanced proton acceleration [18] and increase electron–positron pair yields [3]. Most notably, in the Leemans experiments [19], it was shown that the asymmetry created by modifying the shape of a short 76 fs pulse improved electron yield in a plasma wakefield. Studies also report on the influence of a pulse asymmetry on the wake behaviour in LWFA simulations [20] and stability in the pointing angle of the electron-beam demonstrated experimentally [21].

Previous studies find enhancement in the yield of multi-photon BW pairs reported in simulations of an all-optical collider where the colliding pulse has a flat-top [22] and supergaussian temporal envelope [23]. In experiments, however, the finite bandwidth limitation of PW laser systems makes producing supergaussian temporal envelopes unattainable [24]. For enhancements to pair yields, this difference is associated to the electrons losing more or less energy to RR as they traverse the initial rise of the pulse [25]. There has not been an investigation of how temporal profiles which may be realistically produced and how the drop in peak power changes the pair yield. In this article we will perform such a study using the particle-in-cell (PIC) code EPOCH [26]. Our study uses parameters available with current PW laser systems with some results still relevant in experiments involving pair-plasma creation expected at multi-PW laser facilities [27, 28]. We will make comparison with BW pair production results from simulations in [3], in contrast using our realistic energy conserved laser intensity temporal envelopes. Pair production from colliding an electron bunch with a laser wakefield produced temporal envelope is also considered.

Control over the temporal envelope using traditional optics can be achieved using an acousto-optic programmable dispersive filter [29] (Dazzler) by changing the spectral phase of the pulse. Other examples of control using optical methods include changing the configuration of a pulse stretcher, shifting the grating distance of a chirp-pulse amplification compressor away from an optimal value or, produce high order phase components [30] in which a grating rotation gives a 3rd order phase component.

Plasma optics offers an attractive alternative to changing the laser temporal envelope. Experimental characterisation of pulses from a laser wakefield has been performed [31], relevant to above a₀ = 1, where asymmetric self-phase modulation has shown to shorten the pulse length with blueshift altering the rear and a redshift giving rise to steepening at the front of the pulse. Ellipsoidal plasma mirrors offer promising features by enhancing the laser contrast [32] and intensity [33] (expected increase ∼8 at the PERL laser facility [34]), although with minimal effect on shaping the temporal envelope. One previous study has used plasma mirrors to study pair production enhancement [35] as well as other target and geometry parameters [36]. Two of the envelopes we simulate in our study can be produced using traditional optics via the methods aforementioned while the third envelope uses only plasma.

The first envelope is a short 50 fs optimally compressed Gaussian in (a) of figure 2. The skewed temporal envelope in (b) has been produced by a spectral phase off-setting and has a sharper rising edge and a slower trailing edge, in contrast to (c) which shows an envelope produced by driving the laser pulse through a 20 mm underdense plasma with electron density n_e = 2 × 10¹⁸ cm⁻³. Figure 1 shows the peak reduction to the power spectrum by changing higher order phase components of a theoretical 53 fs Gaussian pulse. The plasma generated pulse has been produced with the Fourier–Bessel PIC (FBPIC) simulation code [37] and then simulated in our QED-PIC using EPOCH, described in section 4. Each of the three envelopes have been simulated respectively changing only the temporal envelope.

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The envelopes in figure 2 are plotted in terms of the envelopes normalised vector potential a₀, where ${a}_{0}\approx \sqrt{I{\lambda }_{\text{L}}^{2}/1{0}^{18}\enspace {\mathrm{W}\enspace \mathrm{c}\mathrm{m}}^{-2}\enspace \mu {\text{m}}^{2}}$, for a given laser intensity I in units of W cm⁻² and wavelength λ_L in microns. The magnitude of the peak a₀ is given in table 1 and differs considerably between the respective laser envelopes with the lowest a₀ for the skewed envelope. Another important distinction is that, although the LWFA generated pulse has the highest peak a₀, the pulse length is a lot shorter than the optimally compressed 50 fs Gaussian envelope or the longer skewed envelope. Electron bunches traversing these three envelopes will therefore experience different amounts of energy loss due to RR causing χ_e to change and the pair yields to be different in the three cases.

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These envelopes were then simulated using the pair production QED-PIC code described in [7]. The electron–positron pair yields have been calculated, taking the ratio of the number of pairs N_± to the total number of electrons in the electron bunch, after the collision with a 1.5 GeV (γ₀ = 2931) electron-beam. We calculate that the pair yields are maximum after interaction with the laser wakefield generated pulse, where N_±/N_e = 7.90 × 10⁻², followed by the short 50 fs compressed Gaussian N_±/N_e = 1.90 × 10⁻² and giving lowest pairs is the skewed envelope with ∼10² magnitude lower, N_±/N_e = 8.90 × 10⁻⁵. This drop found to the number of pairs produced per electron are consistent with the following results, as we go on to describe the effect of skewing the temporal envelope of theoretical Gaussian laser pulses.

Consider the temporal envelope to be Gaussian and written as the piecewise defined spectrum

$$\begin{equation}I(t)=\begin{cases}_{0}\enspace \mathrm{exp}(-{t}^{2}/{\tau }_{\text{r}}^{2})\quad & \quad {\tau }_{\text{r}}{< }{t}_{0}\\ {I}_{0}\enspace \mathrm{exp}(-{t}^{2}/{\tau }_{\text{f}}^{2})\quad & \quad {\tau }_{\text{f}}{ >}{t}_{0},\end{cases}\end{equation} \tag{ 1 }$$

where the peak intensity is related to the electric field by ${I}_{0}=0.5c{{\epsilon}}_{0}{E}_{0}^{2}$ and E₀ is the associated peak electric field. Modified envelopes may be produced by varying the two parameters τ_r or τ_f corresponding to the rise and fall times respectively. A symmetric Gaussian is recovered by setting τ_r = τ_f. The total pulse duration τ_p (FWHM) is then related to the rise and fall times by τ_p = 0.5(τ_r + τ_f). In order to keep the energy in the pulse constant we multiply the peak intensity of the skewed pulse relative to that of an unskewed Gaussian with e-folding time τ₀ by 2τ₀/τ_p. In order to keep the skewed pulse energy conserved we also derive the reduction factor $\mathcal{R}$ used to scale the intensity envelope, (full derivation in appendix A.1)

$$\begin{equation}\mathcal{R}=\frac{2{\tau }_{\text{p}}}{{\tau }_{\text{r}}+{\tau }_{\text{f}}}.\end{equation} \tag{ 2 }$$

The skewed pulse in figure 1 and the inset of figure 3 gives indication of the affect pulse skewing has on the peak temporal intensity in our simulations. This is important as it ensures equal comparison so that the pulse energy is always conserved (identical energy to the Gaussian pulse) by lowering the peak intensity of the skewed envelope. We also impose a condition in the choice of values for the rise τ_r and fall times τ_f such that 2τ_p < τ_r + τ_f as this is the case for energy conservation to be satisfied, so $\mathcal{R}{< }1$ and I₁ < I₀. Consider the case, pertinent to the all-optical collider, of a relativistic electron-beam counter-propagating relative to a skewed laser pulse. Assume the electron-beam is monoenergetic with initial energy ${\mathcal{E}}_{0}={\gamma }_{0}{m}_{\text{e}}{c}^{2}$. If the hard photon (energies E_γ ∼ MeV) emission from these electrons is synchrotron-like (valid for I₀ ≫ 10¹⁸ W cm⁻²), the energy evolves according to

$$\begin{equation}{m}_{\text{e}}{c}^{2}\frac{\mathrm{d}}{\mathrm{d}t}\langle \gamma \rangle =-\mathcal{P}({\chi }_{\text{e}}),\end{equation} \tag{ 3 }$$

where ⟨γ⟩ is the average electron Lorentz factor and $\mathcal{P}({\chi }_{\text{e}})$ is the instantaneous radiated power [38]

$$\begin{equation}\mathcal{P}({\chi }_{\text{e}})=\frac{2}{3}\frac{{\alpha }_{\text{f}}c}{{\bar\lambda}_{\text{c}}}{m}_{\text{e}}{c}^{2}{\chi }_{\text{e}}^{2}g({\chi }_{\text{e}}),\end{equation} \tag{ 4 }$$

where α_f ≈ 0.0073 is the fine structure constant and ƛ ≈ 3.86 × 10⁻¹³ m is the reduced Compton wavelength. Here the Gaunt factor g(χ_e) [39] accounts for the reduction in the emitted power and is a correction to the synchrotron emission spectrum, due to quantum effects [40]. An approximate fit to this function is given by (full definition in appendix A.2)

$$\begin{equation}g({\chi }_{\text{e}})={[1+4.8(1+{\chi }_{\text{e}})\mathrm{ln}(1+1.7{\chi }_{\text{e}})+2.44{\chi }_{\text{e}}^{2}]}^{-2/3}.\end{equation} \tag{ 5 }$$

Since the electrons propagate counter to the laser pulse, χ_e ≈ γ/E_crit|E_⊥ + v × B| (E_⊥ is the electric field perpendicular to the electron's motion) is

$$\begin{equation}{\chi }_{\text{e}}=2\gamma (t)\frac{{E}_{\text{L}}(t)}{{E}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}},\end{equation} \tag{ 6 }$$

where the laser electric field temporal envelope has the form

$$\begin{equation*}{E}_{\text{L}}(t)={E}_{0}\enspace \mathrm{exp}(-{t}^{2}/2{\tau }_{\text{p}}^{2}).\end{equation*}$$

Note a conversion factor is introduced here in the definition of E_L(t) from the relation I ∝ E². Now we determine the solution of equation (3) for the asymmetric temporal envelope by substituting equation (6) into the instantaneous power radiated (4) and solving the differential equation in (3) to find the average behaviour of γ

$$\begin{equation}\langle \gamma (t)\rangle =\begin{cases}_{0}/\left(\right. 1+\delta (1+\text{erf}(t/{\tau }_{\text{r}}))\quad & \quad {\tau }_{\text{r}}{< }{t}_{0}\\ {\gamma }_{0}/\left(\right. 1+\delta (1+\text{erf}(t/{\tau }_{\text{f}}))\quad & \quad {\tau }_{\text{f}}{ >}{t}_{0},\end{cases}\end{equation} \tag{ 7 }$$

where the term $\delta =2\sqrt{\pi }{\tau }_{\text{r,}\;\text{f}}{\gamma }_{0}{\alpha }_{\text{f}}c{E}_{0}^{2}/(3{\lambda }_{\text{c}}{E}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{2})$ which depends on the peak laser electric field E₀, duration τ_r,f = 0.5τ_p and initial Lorentz factor γ₀. We can now simply derive the classical average quantum parameter of the electrons by multiplying (7) by the relation of χ_e in (6) to find

$$\begin{equation}\langle {\chi }_{\text{e}}(t)\mathcal{R}\rangle =\begin{cases}_{0}{\xi }^{-1}\mathcal{R}\enspace \mathrm{exp}(-{t}^{2}/2{\tau }_{r}^{2})\quad & \quad {\tau }_{\text{r}}{< }{t}_{0}\\ {\chi }_{0}{\xi }^{-1}\mathcal{R}\enspace \mathrm{exp}(-{t}^{2}/2{\tau }_{\text{f}}^{2})\quad & \quad {\tau }_{\text{f}}{ >}{t}_{0},\end{cases}\end{equation} \tag{ 8 }$$

where χ₀ is the initial quantum parameter before entering the pulse and we define the term ξ = 1 + δ(1 + erf(t/τ_r,f)). The validity of this equation agrees with equation (7) found in [41] which has been solved as a function of the phase χ_e(ϕ). Taking the derivative of the average χ_e gives the time at which the average electron χ_e(t) is maximised in the envelope (i.e. when d⟨χ_e(t)⟩/dt = 0)

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}t}\langle {\chi }_{\text{e}}(t)\rangle =\frac{{\chi }_{0}t}{\xi {\tau }_{p}^{2}}\enspace \mathrm{exp}(-{t}^{2}/2{\tau }_{\text{r,}\;\text{f}}^{2})+\frac{{\chi }_{0}{\gamma }_{0}}{{\xi }^{2}{\tau }_{\text{R}}}\enspace \mathrm{exp}(-3{t}^{2}/2{\tau }_{\text{r,}\;\text{f}}^{2}),\end{equation} \tag{ 9 }$$

where

$$\begin{equation*}{\tau }_{\text{R}}=\frac{3{\bar\lambda}_{\text{c}}}{8{\alpha }_{\text{f}}c}{\left(\frac{{E}_{\text{crit}}}{{E}_{0}}\right)}^{2}.\end{equation*}$$

These equations describe a classical RR on the electrons and does not account for quantum effects. A modified-classical model in which equation (8) includes g(χ_e) to give ⟨χ_e(t)g(χ_e)⟩ has been solved numerically and is plotted in figure 3 simulated with three increasing amounts of skew controlled by the rise time τ_r. Here we have shown two important observations; the first is that the electrons rate of energy loss is less than with the skewed envelope and therefore changing the shape of the synchrotron emission spectra, verified by PIC simulation in figure 7. Secondly, the time in which the average χ_e is maximised is extended in the Gaussian case and slowly decreases whereas the peak χ_e is shorter and decreases much more rapidly for the asymmetric envelope. We find peak ⟨χ_e⟩ ≈ 0.15 in colliding a Gaussian pulse and a factor 2 increase ⟨χ_e⟩ ≈ 0.3 if the envelope has a significant skew while the intensity is scaled by 0.52.

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EPOCH PIC code was used to simulate the collision of the laser pulse with the electron-beam. The photon emission is calculated in the QED-PIC by using a Monte Carlo algorithm [7] capturing quantum stochasticity in the emission and resulting RR. These assumptions use the weak field (E_L ⩽ 10⁻³ E_crit) and the quasi-static approximations. This is to ensure that the formation length of the emitted hard photons are smaller than the laser wavelength so that the emission can be treated as synchrotron-like. In simulating BW pair production, we use a pair creation multiplier, multiplying the rate by a large factor and then dividing the weight of the produced pairs by the same fraction.

Our investigation uses the following simulation parameters. A domain which is −100 μm ⩽ x ⩽ 100 μm with 10³ global grid points. The laser enters the simulation box from the left-hand boundary and has a 1 micron wavelength, 5 × 10²¹ W cm⁻² peak intensity and is circularly polarised with a pulse duration (FWHM) of τ_p = 40 fs. The electron bunch, propagating from the right-hand boundary, consists of 10⁵ macroparticles. Convergence tests have been performed on the number of pairs produced per electron as a function of both macroparticles to represent the bunch and convergence on the simulation grid points. These tests were performed in order to determine that our baseline parameters have converged on the number of pairs produced. The electron-beam is monoenergetic with an initial beam energy centred at 1.5 GeV and initial density n_e = 1.8 × 10¹⁸ m⁻³. The electrons travel in the −x direction (initial position at x₀ = 99 μm) colliding with the laser pulse at time t.

4.1. Skew effect on maximising the electron quantum parameter

Measuring the energy spread of the electron energy distribution after the interaction with the laser provides a clear indication of which RR model is observed [42, 43]. In a real RR experiment, both the electron energy spread as well as the photon energy spectra is measured. Our main focus is on the influence that skew has to the electron quantum parameter and how it is enhanced to maximise χ_e. This parameter depends on the collision angle [44], the initial electron beam energy and the shape of the laser envelope. Here we only consider the latter as we assume the electrons are counter-propagating relative to the laser pulse (i.e. the collision is head-on). By changing the laser pulse duration and amount of asymmetry in the pulse, we may determine the effect that a skewed laser envelope has on maximising the bunches peak ⟨χ_e⟩. These results are shown in figure 4 giving the average Lorentz factor in (a), average quantum parameter in (b) and its derivative in (c). Observe that as ⟨χ_e⟩ for the skewed envelope is maximum for the interaction in (b), the Gaussian pulse gives a peak ⟨χ_e⟩ over a longer interaction time ∼3fs. In (c) where the derivative passes zero, we can see the temporal displacement of peak χ_e in which the Gaussian pulse has a maximum ⟨χ_e⟩ ≈ 0.25 at around ≈40 fs while this is slightly later for the negatively skewed envelope at ≈44 fs, where peak ⟨χ_e⟩ > 0.3.

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Figure 5 shows the maximum electron average χ_e as a function of the rise time for both a constant Gaussian intensity along with a scaled intensity ${I}_{1}=\mathcal{R}{I}_{0}$. This linear dependence found on the rise time of the intensity envelope in this figure is expected in a non-deterministic emission model as the photon spectrum becomes hardened [45]. The rapid increase of the black line shows that with the omission of $\mathcal{R}$ in equation (1), the increase in ⟨χ_e⟩ is almost double as the rise time, becoming considerably shorter as the pulse is further skewed. This resulting difference is because of the competition between peak intensity and the time in which the electrons pass through the envelope and experience energy loss by RR. In the case of the Gaussian envelope, the electron bunch could have already produced high-energy photons. These PIC code results, while agreeing with the analytical result of figure 3, shows the model predicts that the initial energy of the electron bunch is critical in maximising the electron's χ_e and secondly that shorter pulses encourage maximum average χ_e because of the shorter interaction time as the electrons pass through the laser envelope. Therefore, the time in which χ_e is maximum in the pulse is some time before reaching the most intense part of the pulse, namely at the peak intensity I₀. This shows a skew on the colliding envelope is therefore beneficial in reaching maximum peak ⟨χ_e⟩.

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4.2. Effect that the temporal envelope has on electron-positron pair yields

We are now in a position to determine the effect that skewing the envelope has on electron–positron pair yields. Four simulations have been ran with varying degree of skew to the envelope, giving the number of pairs per electron N_±/N_e provided in table 2. Furthering our results alongside the results for realistic envelopes in section 2, we have considered the pair yields via the BW process recording the pair yields as skew is added, by decreasing the rise time in the range 5 fs ⩽ τ_r ⩽ 40 fs to resolve a Gaussian while the intensity is accordingly scaled ($\mathcal{R}{< }1$). This result is shown in figure 6, finding that as the peak temporal intensity drops from optimal 5 × 10²¹ W cm⁻² down to 2.4 × 10²¹ W cm⁻², the pair yield is reduced by an order of magnitude. We find pairs per electron for a short 40 fs optimally compressed envelope is 1.96 × 10⁻⁵, while with the extreme skewed envelope, pair yields are reduced to 1.50 × 10⁻⁶. These simulations are also used to see the effect skew has in the electron χ_e distribution and photon energy spectrum for this corresponding number of pairs per electron.

Table 2. Results from four simulation runs with a Gaussian and skewed Gaussian laser envelope giving the maximum quantum parameter and number of electron–positron pairs produced per electron at the end of the collision.

Simulation	I0,1(1021 W cm−2)	${\mathcal{E}}_{0}$	max⟨χe⟩	N±/Ne(10−6)
1	5	1.5	0.255	19.60
2	3.7	1.5	0.272	6.20
3	2.9	1.5	0.291	2.80
4	2.4	1.5	0.313	1.50

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These are provided in figure 7 once the electrons are at the peak of the Gaussian and skewed Gaussian envelopes (a) and (c). Also plotted are the distributions at some later time (8 fs) after the electron bunch is at the peak of the pulse in (b) and (d). RR has clearly changed the electron χ_e distribution for the Gaussian envelope where ⟨χ_e⟩ is lower, while finding that the skewed envelope has more electrons with a significantly higher average χ_e. In fact, at the peak of the pulse, the average electron χ_e for the skewed Gaussian is 0.23 and with the optimally compressed Gaussian is 0.16. At 8 fs after the peak, these averages are 0.23 and 0.22 respectively. This difference becomes less clear after the peak of the pulse with both skewed or unskewed distributions converging. This behaviour is also identified in the photon energy spectrums $\mathrm{d}{N}_{\gamma }/\mathrm{d}{\mathcal{E}}_{\gamma }$ which shows that at the peak there is a shape distinction between the soft lower energy photons, suggesting greater energy loss with the Gaussian pulse. This also becomes less observable in comparison after the electrons have traversed beyond the peak of the pulse. Finally, simulations in a higher intensity regime with parameters relevant to multi-PW laser facilities were performed. The reduction to the number of pairs for the skewed envelope in this regime is now the same order as with the Gaussian pulse. A 5.0 GeV electron bunch with a maximum quantum parameter of χ_e ≈ 9, produced electron–positron pairs per electron of N_±/N_e = 0.19 after colliding with a 40 fs Gaussian I₀ = 1 × 10²³ W cm⁻² peak intensity laser pulse and N_±/N_e = 0.11 with the comparable skewed envelope.

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This simulation study has attempted to show the effect that the laser intensity temporal envelope has on enhancing electron–positron pair yields in a variety of envelope cases. The maximum pairs per electron found in section 2 for the pulse generated in an LWFA scenario shows that a plasma-based method, employing skew with plasma optics, gives enhanced pair production via BW and is the ideal method. Further investigation may consider how different plasma densities influence the laser envelope to control the rise and fall times and find parameters that give the maximum intensity ${a}_{0}^{2}$ over short durations. In section 3, we gave equations describing the average Lorentz factor γ and considered how the average electron χ_e evolves in time in the laser envelope for symmetric and asymmetric pulses. In consistency with simulation results in [46], these results show that the pulse duration and initial energy of the electron-beam is critical to maximising χ_e throughout the interaction and therefore presenting the most significant change to the synchrotron emission spectrum.

The results of our study have been made in contrast to the pair production yields in figure 3 of [3]. In figure 3(a) nonlinear BW pairs are shown to decrease for S = −0.7 but increase by a small factor for a skew of S = 0.7 (S = 0 is no skew). We show in figure 6 that BW pairs decrease by an order of magnitude from 2 × 10⁻⁵ to 1.5 × 10⁻⁶ pairs per electron once a significant skew is applied to the envelope, namely τ_r = 10 fs and τ_f = 160 fs. This is because in our simulation methodology, a combination of maximum peak intensity and a fast rising leading edge at the front of the pulse is unattainable in a realistic laser pulse. As a result of ignoring energy conservation, the laser intensity spectrum of skewed pulses as in [3], when compressed are unphysical with a sharp peak power.

Figure 5 shows the importance of keeping the energy in the skewed and non-skewed envelopes constant. Ignoring this and omitting the scaling by $\mathcal{R}=2{\tau }_{\text{p}}/({\tau }_{\text{r}}+{\tau }_{\text{f}})$ to the peak intensity while skewing the envelope presents unphysical results in which the average χ_e is overestimated and appears high. We have also shown similar effects occurring in figure 5 (inset) as the initial electron beam energy increases, a skewed envelope gives a maximum ⟨χ_e⟩ of 1.0 while it is only 0.4 for the optimally compressed Gaussian pulse. This suggests that while χ_e is maximised by a temporal skew, this is at the detriment of producing higher energy photons over longer times that may go on to produce pairs.

This naturally leads to consider the effect that electron straggling [47] has on the electrons in an envelope with a faster rise time. As a result of the probabilistic nature of the quantum emission process for χ_e ⩾ 0.1, some electrons may reach the peak of the temporal envelope without emitting any or very few high-energy photons. If one finds straggling in a sufficient number of electrons, it would be beneficial in that electrons would reach the pulse peak with high values of χ_e, as the rate of energy loss of these electrons is lower and significant energy remains for possible pair production [48], reports that straggling increases the number of pairs and that a lower intensity can be compensated by straggling electrons. In realistic RR experiments as the laser is focused, the electrons experience varying intensities along longitudinal slices of the bunch, as reported in the Gemini RR experiment [1]. Therefore, in theory a pulse with a faster rising leading edge could improve the number of straggling electrons as they reach the peak of the pulse on a shorter time duration. This effect can be seen in figure 4(b) in the PIC simulations, where straggling is taken into account, the electrons have a higher maximum average χ_e after traversing the skewed envelope.

As we mainly focus on showers of electron–positron pairs at lower values of χ_e ∼ 0.3 and not avalanches in which further pairs are produced after the initial pair creation [49]. It was necessary to consider our results more broadly in a higher intensity regime. Figure 1 in [6] shows the scaling of χ_e as a function of laser intensity. Here it is noted that at intensities >10²³ W cm⁻², the quantum parameter begins to saturate and therefore so does the total number of pairs. Therefore, in this intensity regime, a reduction to peak intensity may not change the final pair yields substantially and so the Gaussian and skewed Gaussian collisions would generate similar numbers of pairs. At the end of section 4.2, simulations were ran to determine whether the reduced pairs seen in figure 6 for the skewed envelope still holds for χ_γ > 1. Equal orders of magnitude in pair yields for both Gaussian and skewed Gaussian pulses were obtained. This suggests that above an intensity threshold, the order of magnitude increase to pair yields with the optimally compressed pulse is lost.

Employing a skew to the laser temporal envelope and changing the peak field strength changes the pair production probability rate [50]. In the trident pair production process, in which an electron produces a pair in a strong field via a virtual photon, the production rate depends on the electron energy and strength of the field. For BW pairs, this creation is by real photons interacting with the external field and depends more on the shape of the synchrotron emission spectrum. A change in the shape of the spectrum of figure 7 can be seen at the peak of the pulse and explains the higher number of pairs for a short 40 fs compressed pulse. Further investigation could compare these two pair production processes as was studied in [3], not ignoring the trident pair production process as we have in this study. This has been ignored due to the complex PIC modelling of this process and overall validity concerns of QED processes in PIC codes [51–54].

In conclusion, we have shown that skewing the laser intensity temporal envelope, lowering the peak intensity, gives a lower electron–positron pair yield per electron 1.5 × 10⁻⁶ compared to optimally compressed 1.9 × 10⁻², or in colliding an electron-beam with a pulse generated using plasma optics 7.9 × 10⁻². We have considered how the electron quantum parameter χ_e evolves in a pulse with a temporal skew, showing χ_e is maximised not at the most intense part of the pulse but some time before significant energy loss to RR. While the electrons that reach the peak in a shorter interaction time of a skewed envelope have a higher maximum average χ_e, this advantage is compensated by a lower pair yield as the energy in the laser is conserved by reducing the peak intensity as expected with current PW class laser systems. Our results provide evidence that colliding electrons into a laser envelope with a skew at the front of the pulse yields lower BW pairs, leading to enhanced pair production at current or future multi-PW facilities with either a short optimally compressed Gaussian or pulses generated by driving a laser wakefield.

Computing resources were provided by STFC Scientific Computing Department's SCARF cluster.

The data that support the findings of this study are openly available at the following URL/DOI: https://etheses.whiterose.ac.uk/25601/1/Bradley_thesis.pdf.

A.1. Conserving the energy in an asymmetric temporal envelope

Let the total energy in an optimally compressed Gaussian pulse be

$$\begin{equation}{\mathcal{E}}_{\text{C}}={\int }_{-\infty }^{\infty }{I}_{0}\enspace \mathrm{exp}(-{(t/{\tau }_{\text{p}})}^{2})\mathrm{d}t={I}_{0}{\tau }_{\text{p}}\sqrt{\pi }\end{equation} \tag{ 10 }$$

given the relation that

$$\begin{equation}{\int }_{-\infty }^{\infty }\mathrm{exp}(-{x}^{2})\mathrm{d}x=\sqrt{\pi }.\end{equation} \tag{ 11 }$$

Consider splitting the integrals for the asymmetric envelope as follows

$$\begin{align}{\mathcal{E}}_{S}& ={\int }_{-\infty }^{0}{I}_{1}\enspace \mathrm{exp}(-{(t/{\tau }_{\text{r}})}^{2})\mathrm{d}t+{\int }_{0}^{\infty }{I}_{1}\enspace \mathrm{exp}(-{(t/{\tau }_{\text{f}})}^{2})\mathrm{d}t\\ & =\left({\int }_{-\infty }^{0}{I}_{1}\enspace \mathrm{exp}(-{(t/{\tau }_{\text{r}})}^{2})\mathrm{d}t+{\int }_{0}^{\infty }{I}_{1}\enspace \mathrm{exp}\left(\right. -{(t/{\tau }_{\text{f}})}^{2}\enspace \mathrm{d}t\right)\\ & =\frac{1}{2}\sqrt{\pi }{I}_{1}({\tau }_{\text{r}}+{\tau }_{\text{f}}).\end{align} \tag{ 12 }$$

By setting the energy in the compressed Gaussian equal to the energy in the asymmetric Gaussian, we may find the ratio of intensities

$$\begin{equation*}{I}_{0}{\tau }_{\text{p}}\sqrt{\pi }=\frac{1}{2}\sqrt{\pi }{I}_{1}({\tau }_{\text{r}}+{\tau }_{\text{f}}).\end{equation*}$$

Solving for $\mathcal{R}$ factor

$$\begin{equation}\mathcal{R}=\frac{{I}_{1}}{{I}_{0}}=\frac{2{\tau }_{\text{p}}}{({\tau }_{\text{r}}+{\tau }_{\text{f}})}.\end{equation} \tag{ 13 }$$

A.2. Quantum synchrotron emissivity

The Gaunt factor g(χ_e) (figure 8) is responsible for modifying the photon spectrum leading to a reduction to the radiated power in equation (4). It is given by integrating the photon synchrotron function

$$\begin{equation*}g({\chi }_{\text{e}})=\frac{{\int }_{0}^{{\chi }_{\text{e}}/2}F({\chi }_{\text{e}},{\chi }_{\gamma })\mathrm{d}{\chi }_{\gamma }}{{\int }_{0}^{\infty }{F}_{\text{c}}\left(\frac{4{\chi }_{\gamma }}{3{\chi }_{\text{e}}^{2}}\right)\mathrm{d}{\chi }_{\gamma }}=\frac{3\sqrt{3}}{2\pi {\chi }_{\text{e}}^{2}}{\int }_{0}^{{\chi }_{\text{e}}/2}F({\chi }_{\text{e}},{\chi }_{\gamma })\mathrm{d}{\chi }_{\gamma },\end{equation*}$$

where the function F(χ_e, χ_γ ) is the quantum synchrotron function provided by Sokolov and Ternov [55]

$$\begin{equation*}F({\chi }_{\text{e}},{\chi }_{\gamma })=\frac{4{\chi }_{\gamma }^{2}}{{\chi }_{\text{e}}^{2}}y{K}_{2/3}(y)+\left(1-\frac{2{\chi }_{\gamma }}{{\chi }_{\text{e}}}\right)y{\int }_{y}^{\infty }{K}_{5/3}(t)\mathrm{d}t,\end{equation*}$$

where y = 4χ_γ /(3χ_e(χ_e − 2χ_γ )) and K_n are modified Bessel functions of the second kind. In the classical limit as g(χ_e) is exactly unity, F(χ_e, χ_γ ) reduces to the classical synchrotron function F_c(χ_e, χ_γ )

$$\begin{equation*}{F}_{\text{c}}({\chi }_{\text{e}},{\chi }_{\gamma })={y}_{\text{c}}{\int }_{{y}_{\text{c}}}^{\infty }{K}_{5/3}(u)\mathrm{d}u,\end{equation*}$$

where the term y_c is $4{\chi }_{\gamma }/3{\chi }_{\text{e}}^{2}$ including χ_γ and χ_e which are the usual quantum parameters for the photon and electron respectively.

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