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

Recent high-intensity laser experiments (Cole et al 2018 Phys. Rev. X 8 011020; Poder et al 2018 Phys. Rev. X 8 031004) have shown evidence of strong radiation reaction in the quantum regime. Experimental evidence of quantum effects on radiation reaction and electron–positron pair cascades has, however, proven challenging to obtain and crucially depends on maximising the quantum parameter of the electron (defined as the ratio of the electric field it feels in its rest frame to the Schwinger field). The quantum parameter can be suppressed as the electrons lose energy by radiation reaction as they traverse the initial rise in the laser intensity. As a result the shape of the intensity temporal envelope becomes important in enhancing quantum radiation reaction effects and pair cascades. Here we show that a realistic laser pulse with a faster rise time on the leading edge, achieved by skewing the temporal envelope, results in curtailing of pair yields as the peak power is reduced. We find a reduction in pair yields by orders of magnitude in contrast to only small reductions reported previously in large-scale particle-in-cell code simulations (Hojbota et al 2018 Plasma Phys. Control. Fusion 60 064004). Maximum pairs per electron are found in colliding 1.5 GeV electrons with a laser wakefield produced envelope 7.90 × 10−2 followed by a short 50 fs Gaussian envelope, 1.90 × 10−2, while it is reduced to 8.90 × 10−5, a factor of 100, for an asymmetric envelope.


Introduction
Strong field quantum electrodynamics processes can be investigated with current high power petawatt (PW) lasers, reaching intensities ∼10 21 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 × 10 18 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][8][9][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 ∼ 10 18 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.

Modifying the laser temporal envelope using plasma optics
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 a 0 = 1, where Simulating realistic laser temporal envelopes using a 50 fs symmetric Gaussian (blue), an asymmetric Gaussian generated by phase off-setting (red) and a pulse produced from a laser wakefield (orange). Simulation results on number of pairs per electron N ± /N e for the respective envelopes are provided in table 1. 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 × 10 18 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.
The envelopes in figure 2 are plotted in terms of the envelopes normalised vector potential a 0 , where a 0 ≈ Iλ 2 L /10 18 W cm −2 μm 2 , for a given laser intensity I in units of W cm −2 and wavelength λ L in microns. The magnitude of the peak a 0 is given in table 1 and differs considerably between the respective laser envelopes with the lowest a 0 for the skewed envelope. Another important distinction is that, although the LWFA generated pulse has the highest peak a 0 , 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.
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 (γ 0 = 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 ∼ 10 2 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.

Analytical solution for a simplified asymmetric temporal envelope
Consider the temporal envelope to be Gaussian and written as the piecewise defined spectrum where the peak intensity is related to the electric field by I 0 = 0.5c 0 E 2 0 and E 0 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 τ 0 by 2τ 0 /τ p . In order to keep the skewed pulse energy conserved we also derive the reduction factor R used to scale the intensity envelope, (full derivation in appendix A.1) 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 R < 1 and I 1 < I 0 .
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 E 0 = γ 0 m e c 2 . If the hard photon (energies E γ ∼ MeV) emission from these electrons is synchrotron-like (valid for I 0 10 18 W cm −2 ), the energy evolves according to where γ is the average electron Lorentz factor and P(χ e ) is the instantaneous radiated power [38] P(χ e ) = 2 3 where α f ≈ 0.0073 is the fine structure constant and λ − ≈ 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) 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 where the laser electric field temporal envelope has the form Note a conversion factor is introduced here in the definition of E L (t) from the relation I ∝ E 2 . 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 where the term δ = 2 √ πτ r, f γ 0 α f cE 2 0 /(3λ c E 2 crit ) which depends on the peak laser electric field E 0 , duration τ r,f = 0.5τ p and initial Lorentz factor γ 0 . 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 where χ 0 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 where 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.

Simulating asymmetric temporal envelopes in a QED-PIC code
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 10 3 global grid points. The laser enters the simulation box from the left-hand boundary and has a 1 micron wavelength, 5 × 10 21 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 10 5 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 × 10 18 m −3 . The electrons travel in the −x direction (initial position at x 0 = 99 μm) colliding with the laser pulse at time t.  6. PIC code results showing the electron-positron pair yields per electron N ± /N e as the laser envelope is skewed. As the rising edge of a I 0 = 5 × 10 21 W cm −2 Gaussian laser pulse becomes faster, the peak intensity drops by a factor R giving the new peak intensity I 1 = RI 0 , displayed on the right-hand axis.

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. 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 = RI 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 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 I 0 . This shows a skew on the colliding envelope is therefore beneficial in reaching maximum peak χ e .

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 (R < 1). This result is shown in figure 6, finding that as the peak temporal intensity drops from optimal 5 × 10 21 W cm −2 down to 2.4 × 10 21 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.
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 dN γ /dE γ 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 I 0 = 1 × 10 23 W cm −2 peak intensity laser pulse and N ± /N e = 0.11 with the comparable skewed envelope.

Discussion
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 2 0 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 R = 2τ p /(τ r + τ 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 > 10 23 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][52][53][54].

Conclusion
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.

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

Data availability statement
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 given the relation that √ πI 1 (τ r + τ f ).
By setting the energy in the compressed Gaussian equal to the energy in the asymmetric Gaussian, we may find the ratio of intensities I 0 τ p √ π = 1 2 √ πI 1 (τ r + τ f ).
Solving for R factor

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 where the function F(χ e , χ γ ) is the quantum synchrotron function provided by Sokolov and Ternov [55] F(χ e , χ γ ) = 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 ) where the term y c is 4χ γ /3χ 2 e including χ γ and χ e which are the usual quantum parameters for the photon and electron respectively.