Multiplicity of electron- and photon-seeded electromagnetic showers
at multi-petawatt laser facilities

We present a reduced model for electromagnetic showers developing in the head-on collision of a high-energy electron or photon beam with an ultra-intense laser pulse, see Fig. 1 (a) and (b).

The initial (seed) beam is considered to be a point-like (zero spatial extension and angular aperture) beam of initially monoenergetic particles with energy $\mathcal{E}_{0}=\gamma_{0}\,mc^{2}$ for photons and $\mathcal{E}_{0}=(\gamma_{0}-1)\,mc^{2}$ for charged particles. In this work, we will focus on ultra-high energy seed particles, $\gamma_{0}\gg 1$, focusing in particular on the range $\gamma_{0}\in[100-40000]$, i.e. energies in the range 50 MeV - 20 GeV. Furthermore, we will consider optical laser systems with a micrometric wavelength $\lambda$, and large relativistic field strength parameter $a_{0}=eE_{0}/(mc\omega)\gg 1$ (with $E_{0}$ the maximum laser electric field and $\omega=2\pi c/\lambda$ the laser angular frequency). As our interest lies in future experiments on multi-petawatt laser systems, we focus on $a_{0}\in[10,1000]$, corresponding to $I_{0}\lambda^{2}\sim 10^{20}-10^{24}~{}{\rm W/cm^{2}\mu m^{2}}$ (with $I_{0}=c\,\varepsilon_{0}E_{0}^{2}/2$ the peak laser intensity). Furthermore, predictions of the number of produced pairs will be discussed for two types of laser systems: ultra-short (20 fs) Ti:Sapphire ($\lambda=0.8~{}{\rm\mu m}$) laser pulses as delivered at Apollon, CoReLS, or ZEUS, and longer (150 fs) Nd:Glass ($\lambda=1.057~{}{\rm\mu m}$) laser pulses as delivered by the L4 ATON laser beam at ELI Beamlines.

The range of parameters discussed in this work is reported in Fig. 1(c) considering $\lambda=0.8~{}{\rm\mu m}$ and a full-width-at-half-maximum-intensity laser pulse duration $\tau_{p}=20~{}{\rm fs}$. Superimposed is a rectangular region that corresponds to seed particle energies in between 1 and 10 GeV and field strengths $a_{0}$ in between $95$ and $300$ computed considering a 1PW or 10 PW Gaussian laser pulse focused with an angular aperture $f/3$. It is relevant to most forthcoming experiments on multi-petawatt laser systems.

We also report the value of the maximum quantum parameter $\chi_{0}$ a particle (either photon, electron or positron) with energy $\gamma_{0}\,mc^{2}$ can achieve colliding head-on with an electromagnetic wave of relativistic field strength $a_{0}$:

In this representation, contours of iso-$\chi_{0}$ are oblique lines and iso-contours $\chi_{0}=0.1,1$ and $10$ are reported as grey lines.

We further report two regions of interest. The first, blue one, corresponds to the region of validity of the model developed by Blackburn et al. [42] for electron-seeded showers. It is limited to low $\chi_{0}$ values which is not of direct interest for multi-petawatt laser facilities, but rather corresponds to regimes explored coupling standard accelerators with near-100 TW laser systems, as planned for experiments such as LUXE or FACET-II. The second, in red, corresponds to the domain of validity of the model proposed by Mercuri-Baron et al. [43] for photon-seeded showers. More precisely, it corresponds to the limit of the so-called soft-shower regime in which pair production is dominated by the decay of the seed photons only. This region covers part, but not all of the (green-hatched) region of interest for multi-petawatt laser systems.

Our objective is to extend the region of validity of both models to cover the whole region of interest for multi-petawatt laser systems. To do so, we will go beyond some of the hypotheses made in the previous models. To extend Blackburn’s model to higher $\chi_{0}$, we (i) make the distinction between rates and probabilities, and (ii) include the contribution of intermediate-energy electrons to the shower. To extend Mercuri-Baron’s model  [43], we go beyond the soft-shower regime and account for the secondary pairs considering that primary pairs can radiate away high-energy photons that can, in turn, decay into new electron-positron pairs [see Fig. 1(b)].

By doing so, we derive a set of equations that can be integrated numerically. To make this model tractable and gain insights into the development of the shower, we retain some hypotheses which we now detail.

First, we split all species (photons, electrons, and positrons) into successive generations which we treat separately, each species generation being described by its own distribution function (for a treatment of the full distribution functions see Appendix A). When considering electron-seeded showers [see Fig. 1(a)], the seed electrons will emit a first generation of (primary) high-energy photons that may then decay in a first generation of (primary) electron-positron pairs. Our model of electron-seeded showers considers that these primary pairs account for most of the produced pairs and we neglect secondary photons that can be emitted by those primary pairs. Note that the number of pairs is not, here, limited to the number $N_{0}$ of seed electrons as those electrons may emit more than a single photon. This is different from the case of photon-seeded showers [see Fig. 1(b)] for which going beyond the soft-shower limit is needed if one wants to enter regimes where the number of produced pairs exceeds that of seed photons. In the photon-seeded model, one thus allows the primary pairs²²2As further discussed in Sec. II.3, we actually consider that only one particle (electron or positron) per primary pair can further contribute to the cascade. to radiate high-energy photons that can decay into a second generation of pairs. Further emission of high-energy photons by this second generation of pairs will however be neglected. We will show in Sec. III.2 that considering only the first two generations provides an excellent approximation in many regimes of interest, and we will give an intuitive justification of why this is so.

Second, we assume that all particles participating in high-energy photon emission and pair production are ultra-relativistic ($\gamma\gg 1$) at all times. This allows us to approximate electron and positron velocity as that of light and to consider that high-energy photons and electron-positron pairs are emitted/created with a momentum aligned with that of the particle they originate from. We further assume that all particles propagate in the same direction, $-{\bf\hat{z}}$, and will in particular neglect the change of momentum of the charged particles in the laser electromagnetic field. This is a good approximation assuming $\gamma_{0}\gg a_{0}$ (satisfied for parameters of interest for this study). As a consequence, all particles experience at any given time $t$ the same electric and magnetic fields ${\bf E}({\bf z}_{b}(t),t)$ and ${\bf B}({\bf z}_{b}(t),t)$, with ${\bf z}_{b}(t)=-ct\,{\bf{\hat{z}}}$ denoting the particles’ position at time $t$. While the model developed in Secs. II.2 and II.3 can be applied to arbitrary polarizations, within this work, we will focus on linearly-polarized electromagnetic pulses with an electric field:

and magnetic field ${\bf B}(z,t)=-{\bf\hat{z}}\times{\bf E}/c$.

Here, we have considered a sin² intensity profile with $\tau_{p}$ its full-width-at-half-maximum-intensity. It follows that knowing a particle’s energy at any time $t$ uniquely defines its time-dependent quantum parameter³³3Valid for photons as well as electrons or positrons.

Third, the rate of photon emission by electrons and positrons is computed using the average charged particle energy and the corresponding quantum parameter. This approximation, which we have found to provide a good prediction for the spectrum of emitted photons, is different from the one considered by Blackburn in his model of electron-seeded showers. Indeed, in Ref. [42], the authors compute the rate of photon emission using the initial (seed) electron energy $\gamma_{0}$ (even though they do account for energy loss when computing the electron quantum parameter). This approach can be justified in the low $\chi_{0}$ regime where pair production is dominated by those photons that have been emitted by the highest energy electrons. It does not hold however in the regime of interest for his work and radiation losses have to be accounted for.

Last, as we are mostly interested in the range of parameters that will be explored by multi-petawatt laser facilities ($a_{0}\gg 1$), the processes of high-energy photon emission (nonlinear inverse Compton scattering) and electron-positron pair production (Breit-Wheeler process) are described by their (energy) differential rates computed in the locally constant cross-field approximation (LCFA):

where $w_{\chi}(\gamma,\gamma_{\gamma})\,dt\,d\gamma_{\gamma}$ denotes the probability for an electron (equivalently positron) of energy $\gamma\,mc^{2}$ and quantum parameter $\chi$ to emit a high-energy photon with energy in between $\gamma_{\gamma}mc^{2}$ and $(\gamma_{\gamma}+d\gamma_{\gamma})mc^{2}$ between times $t$ and $t+dt$, while $\overline{w}_{\chi_{\gamma}}(\gamma_{\gamma},\gamma)\,dt\,d\gamma$ denotes that of a photon to decay into a pair with the electron carrying an energy in between $\gamma mc^{2}$ and $(\gamma+d\gamma)mc^{2}$. The functions $F(\chi,\xi)$ and $G(\chi_{\gamma},\zeta)$ are reported in Appendix B. Integrating these differential rates over all possible daughter particle energy [photons for Eq. (4) and electrons/positrons for Eq. (5)] leads to the photon emission and pair production rates

where the functions $a(\chi)$ and $b(\chi_{\gamma})$ and their asymptotic limits are reported in Appendix B. In Fig.1(c), black and red dotted lines report the values of ($a_{0}$,$\gamma_{0}$) for which $W_{\rm\scriptscriptstyle{CS}}(\gamma_{0},\chi_{0})=\omega$ and $W_{\rm\scriptscriptstyle{BW}}(\gamma_{0},\chi_{0})=\omega$, respectively. The values of ($a_{0}$,$\gamma_{0}$) for which $W_{\rm\scriptscriptstyle{BW}}(\gamma_{0},\chi_{0})=1/\tau_{p}$ (with $\tau_{p}=20~{}{\rm fs}$) are also reported as a blue solid line defining the regime of validity of the model proposed by Blackburn et al. [42]. It is clear that, in the regime of interest for future experiments at multi-petawatt laser facilities, abundant high-energy photon emission is expected on timescales of the order of or shorter than the optical cycle ($W_{\rm\scriptscriptstyle{CS}}>\omega$). In contrast, the characteristic time for a photon to decay into an electron-positron pair will be larger than the optical cycle, yet shorter than the characteristic pulse duration (considering $\tau_{p}\geq 20~{}{\rm fs}$).

In what follows, we describe the models that have been developed for electron-seeded and photon-seeded cascades. In both cases, we first give the governing equations for the distribution functions of the successive generation of species (electrons, positrons and photons) participating in the shower. We then derive our model and discuss our assumptions. Last we briefly summarise how the model is used to compute the final number of pairs produced per incident (seed) particle, henceforth referred to as the shower multiplicity.

When the shower is seeded by an incident electron beam, we need to follow the dynamics of the seeding electrons through their distribution function $f^{(0)}(\gamma,t)$, the first generation of gamma photons, i.e. those emitted by the seeding electron, though their distribution function $f_{\gamma}^{(1)}(\gamma_{\gamma},t)$, and the first generation of created pairs through their distribution function $f_{\pm}^{(1)}(\gamma,t)$. The equations of evolution of these distribution functions read⁴⁴4As stressed in Sec. II.1, we neglect the work of the laser electric field on all charged particles.:

Note that all differential rates are time-dependent through their dependency in $\chi$.

Equation (8) describes the cooling of the seeding electron beam as it collides with the laser pulse and electrons emit high-energy gamma photons (see, e.g., Ref. [57]). As will be demonstrated in Sec. III.2, in the parameter range of interest for this study, it is not mandatory to capture the details of the seed electron distribution function $f^{(0)}$. It is sufficient to compute the time-dependent seed electron average energy:

which temporal evolution is well approximated by  [57]:

with $W_{0}=2\alpha/(3\tau_{c})$ and $\langle\gamma\rangle_{{}_{\!0}}(t=0)=\gamma_{0}$. Here $\langle\chi\rangle_{{}_{\!0}}(t)$ denotes the quantum parameter⁵⁵5Note that here $\langle\chi\rangle_{{}_{\!0}}(t=0)=0\neq\chi_{0}$ with $\chi_{0}$ defined by Eq. (1). of a charged particle (either electron or positron) with energy $\langle\gamma\rangle_{{}_{\!0}}mc^{2}$ computed at time $t$ [using Eq. (3)], and

is the quantum correction (Gaunt factor) on the power radiated away by an ultra-relativistic electron in a strong field⁶⁶6In our model, we use the approximate form proposed by Baier et al. [58]: $g(\chi)\simeq[1+4.8(1+\chi)\ln(1+1.7\chi)+2.44\chi^{2}]^{-2/3}$..

Let us note that Eq. (11) is the same as Eq. (6) in [42]. It was solved analytically by Blackburn et al. in the limit of small quantum parameters and pair production probability, and considering a laser pulse with a Gaussian temporal profile. This solution is however not valid in the regime of interest for this work and Eq. (11) will be solved numerically.

Equation (9) describes the evolution of the first generation of high-energy photons. Those photons are emitted by the seed electrons as described by the first term in the right-hand-side (rhs) and may decay into electron-positron pairs as described by the second term. We now rewrite this equation introducing the pair production rate $W_{\rm\scriptscriptstyle{BW}}(\gamma_{\gamma},\chi_{\gamma})$ [Eq. (7)] and considering that the emission of high-energy photons is dominated by seed electrons at the average energy $\langle\gamma\rangle_{{}_{\!0}}$, i.e. using $f^{(0)}\simeq N_{0}\,\delta\left(\gamma-\langle\gamma\rangle_{{}_{\!0}}(t)\right)$ in the first term:

with $\chi_{\gamma}(t)$ the quantum parameter of a photon with energy $\gamma_{\gamma}$ computed at time $t$ using Eq. (3). This leads to:

with

Finally, Eq. (10) describes the evolution of the first generation of pairs produced from the decay of the (first generation) high-energy photons as described by the first term in the equation’s rhs. The last two terms account for the modification of the pair energy spectrum due to high-energy photon emission (radiation reaction). The temporal evolution of the number of pairs $N_{\pm}^{(1)}$ of the first generation is computed integrating⁷⁷7Note that the last two terms in Eq. (10) cancel upon integration in $\gamma$ as high-energy photon emission conserves the number of charges. over all $\gamma$, which, after time integration leads to:

Using Eq. (12) for $f_{\gamma}^{(1)}$ and integrating by part finally leads:

The final number of produced pairs, approximated as $N_{\pm}^{(1)}$, can then be obtained integrating Eq. (13) numerically.

When the shower is seeded by a high-energy photon beam, we need to follow the dynamics of this photon beam through its distribution function $f^{(0)}(\gamma_{\gamma},t)$, and at least the first generation of pairs it can produce. However, rather than following the full pair distribution function $f_{\pm}^{(1)}(\gamma,t)$ [as described by Eq.(10)], we focus our attention only on those particles, either electrons or positrons, that carry the most energy at their moment of creation. Indeed, simulations presented in Sec. III show that those particles, hereinafter described by their distribution function $\hat{f}_{\pm}^{(1)}(\gamma,t)$, are the ones that can radiate secondary photons, described by $f_{\gamma}^{(1)}(\gamma_{\gamma},t)$, that in turn will create more electron-positron pairs, described by their distribution function $f_{\pm}^{(2)}(\gamma,t)$. The equations of evolution of these distribution functions read:

where $H(x)$ denotes the Heaviside function.

Equation (14) describes the decay of the incident photons into electron-positron pairs as they collide with the laser pulse. Considering an initially mono-energectic photon beam, the solution of Eq. (14) can be written in the form:

where $\chi_{\gamma_{0}}(t)$ is the quantum parameter of a seed photon with energy $\gamma_{0}mc^{2}$, computed at time $t$ using Eq. (3).

Equation (15) describes the evolution of the resulting (first) generation of pairs as they are created (first term in the equation’s rhs) and then emit high-energy radiation (the last two terms in the equation’s rhs). As stressed earlier, only those particles (either electrons or positrons) that are created with the most energy are accounted for here. Simulations⁸⁸8Let us add that considering the full distribution function of primary pairs $f_{\pm}^{(1)}$ (rather than $\hat{f}_{\pm}^{(1)}$) and then considering that pairs at the average energy dominate the emission of (secondary) high-energy photons does not lead a good agreement with PIC simulations. indeed show that, of the primary pair particles, those (either electron or positron) which have been created with the highest energy contribute most to the following development of the shower. This can be understood as follows. In the small-$\chi$ limit, both electron and positron are statistically created with a similar yet different energy, equivalently quantum parameter. Because of exponential suppression of the pair production rate in the small-$\chi$ limit, the particle with the highest energy/quantum parameter is strongly advantaged compared to the other one in further participating in the shower. In the large-$\chi$ limit, the pair production rate is such that, at the moment the electron-positron pair is created, most of the decayed photon energy is carried away by one or the other particle.

Consistently with Eq. (14), Eq. (15) provides the time evolution of the number of pairs of the first generation:

An analytical expression of the integral has been proposed in [43] which is reproduced in Appendix C for the case of Eq. (2). In what follows we will however integrate this equation numerically.

Proceeding as in Sec. II.2, we will focus our attention on those particle mean energy. Multiplying Eq. (15) by $\gamma$ and integrating over all $\gamma$ provides us with the equation for the average energy $\langle\gamma\rangle_{\!{}_{1}}(t)$ for $t>0$⁹⁹9Formally, the term $\left[\gamma_{0}-\langle\gamma\rangle_{\!{}_{1}}(t)\right]$ in the rhs of Eq. (19) should have been written $\left[\gamma_{\rm cr}(t)-\langle\gamma\rangle_{\!{}_{1}}(t)\right]$ with $\gamma_{\rm cr}(t)=\int_{\gamma_{0}/2}^{\gamma_{0}}\!d\gamma\,\gamma\,\overline{w}(\gamma_{\gamma},\gamma)\,\big{/}\int_{\gamma_{0}/2}^{\gamma_{0}}\!d\gamma\,\overline{w}(\gamma_{\gamma},\gamma)$. Extensive tests have shown that, while using $\gamma_{\rm cr}(t)$ provides fair predictions for the final number of pairs, a better agreement is found using $\gamma_{\rm cr}(t)=\gamma_{0}$ at all times.:

with $\langle\chi\rangle_{1}(t)$ the quantum parameter of an electron with the energy $\langle\gamma\rangle_{\!{}_{1}}(t)$ computed at time $t$ using Eq. (3) and¹⁰¹⁰10At the very beginning of the interaction ($t\sim 0$), electrons experience a very small electromagnetic field, hence $\chi_{\gamma_{0}}\rightarrow 0$. In the unlikely case a pair is created, it is thus very probable that the photon energy will be equally split between the electron and positron. $\langle\gamma\rangle_{\!{}_{1}}(t=0)=\gamma_{0}/2$.

Equation (16) then describes the evolution of the energy distribution of secondary photons that, on the one hand, are created by those (highest energy) first generation of electron-positron pairs [first term in the rhs of Eq. (16)] and, on the other hand, decay into secondary electron-positron pairs [second term in the rhs of Eq. (16)]. This equation can be solved considering that photon emission is dominated by electrons and positrons at the average mean energy, i.e. using $\hat{f}_{\pm}^{(1)}(\gamma,t)\simeq N_{\pm}^{(1)}(t)\,\delta\left(\gamma-\langle\gamma\rangle_{\!{}_{1}}(t)\right)$, leading to:

with

Finally, Eq. (17) describes the evolution of the secondary pairs. Using Eq. (20), it gives, after integration over $\gamma$, the numbers of secondary pairs:

The total number of pairs is finally computed summing the contribution of primary and secondary pairs. While an analytical approximation for the number of primary pairs $N_{\pm}^{(1)}$ was proposed in Ref. [43] (see also Appendix C, Eq. (2C)), we here integrate Eq. (18) numerically. We proceed similarly with Eq. (19) (for $\langle\gamma\rangle_{\!{}_{1}}$) and Eq. (21) (for $N_{\pm}^{(2)}$).

In this Section, we discuss the results of a series of one-dimensional (written 1D3V since velocities are three-dimensional) PIC simulations that follow the head-on collision of a thin (two-cell wide) monoenergetic beam of seed particles (either electrons or photons) against an ultra-intense laser pulse. Seed particles are initialized as a mono-energetic beam, all particles of the beam propagating in the same direction and having the same initial energy $\mathcal{E}_{0}=(\gamma_{0}-1)mc^{2}$ for electrons and $\mathcal{E}_{0}=\gamma_{0}mc^{2}$ for photons. The intense laser pulse is a linearly polarized plane wave with maximum relativistic field strength parameter $a_{0}=eE_{0}/(mc\omega)$ ($E_{0}$ the maximum electric field), and a sin² intensity profile with full-width-at-half-maximum duration $\tau_{p}$, as described by Eq. (2). In general, and unless specified otherwise, two types of laser systems are discussed: ultra-short (20 fs) Ti:Sapphire ($\lambda=0.8{\rm\mu m}$) pulses, and longer (150 fs) Nd-glass ($\lambda=1.057{\rm\mu m}$) pulses.

All simulations were performed with the open-source code Smilei [51] that embarks Monte-Carlo modules¹¹¹¹11For better accuracy, precomputed tables with 1024 points were used as described in https://smileipic.github.io/Smilei/Use/tables.html. to account for high-energy photon emission (nonlinear inverse Compton scattering) and pair production (nonlinear Breit-Wheeler). For those simulations, the spatial and temporal resolution were $\Delta x=~{}\lambda/64$ and $c\Delta t=~{}\Delta x/2$, respectively. All $2\times 10^{4}$ (if $\chi_{0}>2$) or $2\times 10^{6}$ (if $\chi_{0}\leq 2$) seed macro-particles were initialized in two cells. The simulations were run long enough that no more particles (either photons or pairs) were created by the end of each simulation.

In Fig. 2, we report, as a function of $\gamma_{0}$ and $a_{0}$, the multiplicity ($N_{\pm}/N_{0}$) extracted from a set of 1D3V PIC simulations of electron-seeded (left panels) and photon-seeded (right panels) showers, for ultra-short ($20~{}{\rm fs}$, $\lambda=0.8~{}{\rm\mu m}$) pulses (top panels) and longer ($150~{}{\rm fs}$, $\lambda=1.057~{}{\rm\mu m}$) pulses (bottom panels). Each panel summarizes the results of $256$ simulations considering $16$ values of $a_{0}$ from $10$ to $1000$ and $16$ values of $\gamma_{0}$ from $100$ to $4\times 10^{4}$ (both axes discretized in logarithmic scale).

The white region in the bottom-left part of each panel corresponds to a low multiplicity regime ($N_{\pm}/N_{0}\lesssim{10^{-3}}$), that will not be much discussed here for two reasons. First, it is of marginal interest for forthcoming experiments on multi-petawatt laser systems, see Fig. 1. Second, it corresponds to a regime where the standard Monte-Carlo approach implemented in Smilei is not optimal (at least for systematic studies performed without consuming too much computer resources). Indeed, Monte-Carlo methods for the description of photon disintegration require a particle number greater than $N$ to ensure precision at $1/\sqrt{N}$ level. As our simulations were performed with $2\times 10^{6}$ initial particles, the low multiplicity regime ($\lesssim 10^{-3}$) is not well described.

The color-scaled region thus reports the number of produced pairs per incident particle, or multiplicity, as small as $10^{-3}$ at moderate $\chi_{0}\lesssim 1$, and up to near a hundred at very large $\chi_{0}$ (top-right corner). Iso-contours of multiplicity $N_{\pm}/N_{0}=10^{-2}$, $10^{-1}$, $1$, $5$ and $15$ are also reported: in black when extracted from our PIC simulations, and in white for the model predictions. For electron-seeded showers, a very good agreement is observed up to multiplicities of $N_{\pm}/N_{0}=2$ (not shown) and fair agreement for multiplicities of $5$. For photon-seeded showers, a very good agreement is observed up to multiplicities of $N_{\pm}/N_{0}=5$.

Before discussing the differences between electron- and photon seeding, as well as ultra-short (20 fs) and longer (150 fs) pulses, let us note that the multiplicity iso-contours are not, in general, straight lines parallel to iso-$\chi_{0}$ (gray dotted) contours. This means that $\chi_{0}$ is not the only parameter determining the total number of pairs, and that by extension $\gamma_{0}$ and $a_{0}$ do not play a symmetrical role in the shower development. This was already discussed for the case of photon-seeded showers in [43], and could have been expected from Eqs. (6) and (7) where the rates depend on the incident particle energy as well as on its quantum parameter.

We are now in a position to discuss the model assumptions, namely the fact that we limit ourselves to a few generations and that to describe the energy evolution of the leptons we don’t need the full distribution function but we can limit ourselves to the first moment i.e. the average energy. In Fig. 3 we present (with the same hierarchy defined in Fig. 2) the final number of pairs divided by the number of pairs produced from the first (only for electron seeding) and the second (for photon seeding) generations extracted from PIC simulations. This plot acts as a definition of the regime of validity of our model where the black solid line (or the color map) corresponds to the iso-contour equal to $1.1,2,5$ and represents the error factor induced by the consideration of the first generations only. It is clear that for future experimental parameters, only a few generations of creation: first for electron seeding and first and second for photon seeding are enough to describe the number of pairs with at most an error of a factor $2$.

Even if $\chi$ is not the only relevant parameter and because our results have a weak dependence on the pulse duration, at zero order, pair creation will not happen below a threshold of $\chi=1$. Moreover as discussed in Fig. 3 our model is very well satisfied for $\chi_{0}$ less than $\approx 20$. We can thus consider a simple argument to estimate the number of relevant generations [59]. If $\chi\gg 1$ on average an initial electron with an initial energy of $\gamma_{0}$ emits a photon with an energy of $\approx\gamma_{0}/4$. This assumption holds for a large range of $\chi_{0}$ and is still approximately correct down to $\chi=1$ (the average energy varying from $\approx\gamma_{0}/4$ at $\chi\gg 1$ to $\approx\gamma_{0}/7$ at $\chi=1$). The produced photon will create the first pair so that all its energy goes to one particle that will again emit a photon of energy $\approx\gamma_{0}/16$. The reaction chain stopping for $\chi=1$, we can infer that the initial value of $\chi_{0}$ consistent with one generation will be of the order of $16$. For an initial photon at $\gamma_{0}$, the reasoning is similar. This photon decay to create one electron with the same energy $\gamma_{0}$ and no one to the positron. Using the same intuitive result, the third generation starts for $\chi_{0}$ larger or order of $16$. Even if all the parameters are not considered here, this simple estimation is in agreement with the full numerical analysis presented in Fig. 3 and allows us to get a rough estimate of the range of validity of the model.

The comparison between the model and PIC simulations shown in Fig. 2 proves that the average energy approximation gives a good estimation of the number of pairs. The initial particles are mono-energetic so at the beginning of the interaction the average energy is enough to describe the photon emission. After a certain cooling characteristic time, the spectrum also tends to a sharp Gaussian confirming the average approximation. During the transition regime, the spectrum is particularly broad but the average approximation still well describes the lower energetic photons emission. By using this hypothesis, we always underestimate the number of emitted photons with the energy of the order of their parent electron. This is a problem only in low multiplicity regimes ($\lessapprox 0.1$) where the only photons capable of decay are those that have an energy near to their parent electron. Even if the model well predicts the scaling of the number we get an underestimation at low multiplicity. While in multiplicity order of $1$, these rare photons do not influence the number of pairs because the decay process is mainly due to a high number of photons with lower energy. The average energy approximation allows first to take into account the energy conservation in the photon emission and second to describe these low energetic photons with a good precision giving an estimation in excellent agreement with simulations for moderate multiplicity (order of $1$) regime.

Over the whole range of parameters studied here, we find that photon seeding always provides multiplicities larger or equal to those obtained using electron seeding. However, forthcoming experiments will likely produce a significantly larger number of GeV-level electrons than photons of similar energy. Apart from this practical consideration, it is interesting to explain the difference between electron and photon seeding. For clarity, we plot in Fig. 4(a) the multiplicity of pairs as a function of $\gamma_{0}$ for $a_{0}=200$, $\tau_{p}=20$fs and $\lambda=0.8\mu$m in the case of electron seeding (blue circles) and photon seeding (red squares). The black lines report our theoretical predictions obtained from Eq. (13) (for electron seeding) and Eqs. (18) and (21) (for photon seeding). The same quantity is shown as a function of $a_{0}$ for $\gamma_{0}=10^{4}$, $\tau_{p}=20$fs and $\lambda=0.8\mu$m in panel (b).

The largest difference between the two seeding strategies appears in the low multiplicity regime [$\gamma_{0}<10^{3}$ in Fig. 4(a) or $a_{0}<100$ in Fig. 4(b)] where the number of produced pairs per incident electron is significantly smaller than that obtained seeding the shower with photons. This can be easily explained as, in this regime, the seed electrons will emit many photons but with an energy significantly smaller than $\gamma_{0}$. Their quantum parameter will be much less than the quantum parameter $\chi_{0}$ associated to $\gamma_{0}$. The probability of these low-energy photons to decay will thus be significantly (exponentially) suppressed compared to the initial photon seeding case.

In contrast, in the large multiplicity regime, there is virtually no difference between seeding the shower with an electron or a photon beam at the same initial energy. A seed photon will quickly turn into a pair with one of the particles (either the electron or the positron) carrying most of its energy, so the situation is very similar to seeding the shower with an electron beam.

To illustrate the discussion, we show in Fig. 5 (top panel) the evolution of the quantum parameter of the incident electrons (in blue) or primary positrons (for photon seeding in red). Dashed red line represent $N_{\gamma}^{(0)}(t)\times\chi(t)$ in photon seeding case. On the bottom panel, we report the evolution of the multiplicity as a function of time in red for photons seeding and in blue for electrons seeding. In the case of a long pulse duration, the incident electrons dissipate their energy before reaching maximum intensity. As a result, the maximum average quantum parameters become less pronounced compared to a short-pulse case, and low-energy photon emission is favored. The competition between pair production rates over long and short periods can be succinctly characterized as follows: a set of low-energy photons interacting over a long duration versus a few high-energy photons interacting briefly. The nonlinearities of the proposed equations prevent a direct resolution of this question. Simulations reveal growth in the number of pairs with pulse duration throughout the $a_{0},\gamma_{0}$ parameter space explored see Fig. 4(c). However, for some excessive parameters not shown here, we saw that the number of pairs can decrease with the pulse duration. From this analysis, it is important to understand that at the same intensity increasing the total pulse duration (so the total energy) is not always favorable to pair production. Also, it is important to note that for a longer pulse, pairs will also lose their energy by radiation and then have more chance to bounce back. The detectors have to be set with this consideration and an explicit study of the bounce effect is studied in [60].

Calculations of the multiplicity in a similar configuration to the one considered here have been performed before by Blackburn et al. [42] for the electron seeding case and by Mercuri-Baron et al. [43] for the photon seeding one. In this section, we compare our results with these previous models and further discuss the different hypotheses in use and their validity.

In Fig. 6(a,b) we report the final number of pairs produced per incident particle as a function of the initial energy $\gamma_{0}$ and $a_{0}=200$ [panels (a) and (b) for electron- and photon seeding, respectively]. In Fig. 6(c,d) we report the same quantity as a function of $a_{0}$ and fixed $\gamma_{0}=2\times 10^{4}$ [panels (c) and (d) for electron- and photon seeding, respectively]. In all cases, the pulse parameters are $\tau_{p}=20$ fs and $\lambda=0.8$ $\mu$m. Red squares (photon seeding) and blue dots (electron seeding) are extracted from 1D3V PIC simulations. Our model predictions, Eq. (13) (electron seeding) or Eqs. (18) and (21) (photon seeding), are plotted as solid lines.

We first focus on electron seeding [panels (a) and (c)] and compare our predictions with those obtained using the model of Blackburn et al. [42], shown as black dotted lines. In that paper, to propose a completely analytical model, the authors performed several approximations: i) The expression of the rate and not the real probability is used for pair creation. This induces a systematic error when the multiplicity $\gtrapprox 0.1$ (or more accurately when $W_{\rm\scriptscriptstyle{BW}}\simeq 1/\tau_{p}$), leading to an overestimate of the number of produced pairs that, at high probability, can even overcome the number of available photons. ii) The nonlinear Compton Scattering rate is asymptotically expanded around $\gamma-\gamma_{\gamma}\ll 1$. This is a very good assumption, as verified in our simulations, when the highest-energy photons are the ones responsible for pair creation, which is the case in the low-$\chi_{0}$, low-multiplicity regime. However, as $\chi_{0}$ and the shower multiplicity increase, this approach fails in capturing the lower-energy photons that can contribute further to the pair production, and this assumption leads to underestimating the final number of final pairs. iii) The nonlinear Compton Scattering rate is evaluated at the initial electron energy $\gamma_{0}$ and yet considering the time-dependent quantum parameter $\langle\chi\rangle$. This contrasts with our model (see Sec. II) we consistently evaluate the rate of photon emission using the time-dependent average energy and quantum parameter. iv) Finally, the number of pairs is calculated heuristically integrating (over energies) a function built as the product of the nonlinear Breit-Wheeler rate by the emitted photon energy distribution. This approach leads to another systematic overestimate of the number of produced pairs. In our work, we rather use the exact form given by Eq. (13) of this work.

Thanks to these approximations, Blackburn et al. obtain a fully analytical expression for the number of pairs produced in a shower driven by a linearly polarised Gaussian laser. Their prediction appears in fair agreement with simulations shown in Figs. 6 (a) and (c).

Our approach extends Blackburn’s by alleviating certain hypotheses that are not well justified in the regime of intermediate quantum parameters $\chi_{0}$ and potentially large shower multiplicities relevant to multi-petawatt laser facilities. Doing so however requires a numerical integration of the main equations derived in Sec. II.2, but also allows for a more general treatment, not limited to the case of linearly polarized Gaussian beam.

Let us now turn our attention to the case of photon seeding. In Figs. 6(b) and (d) we show the prediction of the shower multiplicity obtained from the model of Mercuri-Baron et al. [43] (black dotted lines). In this case, the multiplicity is estimated by considering exactly one generation, i.e. assuming the soft-shower regime. The model is equivalent to Eq. eq18 and as we can see gives a very good agreement with simulations until the multiplicity becomes larger than $1$. Note that in the work of [43], a fully analytical expression of Eq. 18 is given for arbitrary geometry (see also Appendix C).

In this work, we improve on the model of Mercuri-Baron et al. by computing the number of pairs of the second generation. Doing so extends the model of photon-seeded shower to multiplicities larger than 1, see Figs. 6 (a)-(d), and allows to cover the whole range of parameters relevant to forthcoming facilities.

To conclude, the models developed in Sec. II improve on previous works by Blackburn, Mercuri-Baron, and co-workers, and extend our prediction capabilities to multiplicities of the order of or larger than unity, relevant for multi-petawatt laser facilities. At very large values of $\chi_{0}$, our models however underestimate the final number of pairs. This happens for multiplicities above typically 5, for which PIC simulations show that later generations of pairs and high-energy photons need to be accounted for. Such regimes of abundant pair production may not be easily accessible on near-feature facilities.

To construct our three-dimensional model, we consider the head-on collision between a linearly polarized Gaussian laser pulse with beam waist $w_{0}$ (radius at $1/e^{2}$ in intensity) propagating in the $+z$-direction (henceforth referred to as the longitudinal direction) and an electron or gamma-photon beam propagating in the $-z$-direction. The latter (seed particle) beam has a characteristic longitudinal length $L_{b}$ and transverse size $w_{b}$ and is described, right before the collision, by its energy distribution function $dn_{b}/d\gamma$ which we parameterize as follows:

Here $(x_{0},y_{0},z_{0})$ denotes the position of the seed particles before the collision, $n_{b}$ stands for the seed particle beam density, $f_{b}(x_{0},y_{0})$ and $g_{b}(z_{0})$ the transverse and longitudinal beam density profile normalized such that¹²¹²12For a rectangular beam profile with transverse widths $w_{bx}$ and $w_{by}$ one thus has $w_{b}^{2}=w_{bx}\,w_{by}$. For a cylindrical beam with transverse radius $r_{b}$, one has $w_{b}^{2}=\pi r_{b}^{2}$. $\int\!dx_{0}\,dy_{0}\,f_{b}=w_{b}^{2}$and $\int\!dz_{0}\,g_{b}=L_{b}$, and $h_{b}(\gamma)$ the beam energy distribution normalized such that $\int\!d\gamma\,h_{b}=1$. The total number of particles in the beam then reads $N_{b}=n_{b}\,L_{b}\,w_{b}^{2}$.

The total number of pairs resulting from the collision can then be written in the form:

where $M_{\rm traj}(\gamma;x_{0},y_{0},z_{0})$ measures the number of pairs produced by a single seed particle with energy $\gamma mc^{2}$ initially located at $(x_{0},y_{0},z_{0})$ and considering its full trajectory across the laser pulse.

To compute Eq. (23), we now introduce some assumptions on both the laser pulse and seed particle beam. As done for Sec. II, we first assume that all (seed and produced) particles move at ultra-relativistic speed along straight lines (see Sec. II.1 for details) so that the trajectories over which $M_{\rm traj}$ is computed are simple. We further assume that the laser and seed particle beams are perfectly synchronized and aligned¹³¹³13That is, an ideal particle located at the center of the seeding beam and moving at the speed of light will reach the peak of the laser pulse at its focal plane. and that the characteristic (longitudinal) length $L_{b}\equiv\int\!dz_{0}\,g_{b}(z_{0})$ of the seed particle beam is smaller than or at most of the order of the laser beam Rayleigh length $z_{R}=\pi w_{0}^{2}/\lambda$ and pulse length $c\tau_{p}$. This ensures that all seed particles within the focal spot can experience the laser field at its maximum and makes the integration over $z_{0}$ in Eq. (23) straightforward ($M_{\rm traj}$ can then be taken independent of $z_{0}$). Under these assumptions, Eq. (23) can be rewritten:

Considering that pair production occurs mainly within the focal volume, the laser electromagnetic field is well approximated by simply multiplying Eq. (2) by a Gaussian dependency $\exp\left(-(x_{0}^{2}+y_{0}^{2})/w_{0}^{2}\right)$. As a result, $M_{\rm traj}=N_{\pm}^{1D}/N_{0}$ is easily computed from the one-dimensional developed in Sec. II, using $N_{\pm}^{1D}=N_{\pm}^{(1)}$ as given by Eq. (13) for electron-seeded showers, and $N_{\pm}^{1D}=(N_{\pm}^{(1)}+N_{\pm}^{(2)})$ with $N_{\pm}^{(1)}$ and $N_{\pm}^{(2)}$ given by Eqs. (18) and (21), respectively, for photon-seeded showers. Here, $N_{\pm}^{(1)}$ and $N_{\pm}^{(2)}$ are computed for all values of $x_{0}$ and $y_{0}$ using the laser field strength evaluated at $(x_{0},y_{0})$.

In the limit of a thin seed-particle beam, $w_{b}\ll w_{0}$, one can make the approximation $f_{b}(x_{0},y_{0})\simeq w_{b}^{2}\,\delta(x_{0})\,\delta(y_{0})$ in Eq. (24). The resulting number of produced pairs can then be simply extracted from the one-dimensional predictions of Sec. II:

In the limit of a broad (pancake-like) seed-particle beam, $w_{b}\gg w_{0}$, one can take $f_{b}(x_{0},y_{0})\simeq 1$ in Eq. (24) and the number of produced pairs can be computed as:

For this broad (pancake-like) seed-particle beam, we see that the electron beam is fully defined by its areal density $n_{b}\,L_{b}$ and the total number of produced pairs is proportional to an effective, total cross-section $\sigma_{\rm tot}$ (as introduced in Ref. [43]). It is interesting to rewrite Eq. (26) as $N_{\pm}^{3D}=n_{b}\,L_{b}\,w_{0}^{2}\,M_{\rm eff}$, where $n_{b}\,L_{b}\,w_{0}^{2}$ is the number of seed particles in the focal volume and $M_{\rm eff}\equiv\sigma_{\rm tot}/w_{0}^{2}$ thus measures an effective multiplicity. For a Gaussian laser pulse, this effective multiplicity is independent of $w_{0}$.

In what follows, we benchmark this three-dimensional model against self-consistent particle-in-cell simulations. As the case of a thin beam is a direct extrapolation of one-dimensional results, we rather focus our attention on the case of a pancake beam ($w_{b}\gg w_{0}$). In addition, we now focus our attention on the case of a monochromatic seed-particle beam [$h_{b}(\gamma)=\delta(\gamma-\gamma_{0})$].

In what follows we present 3D3V PIC simulations performed with Smilei. Each simulation reproduces a volume of $14\lambda\times 14\lambda\times(c\tau_{p}/\lambda+1)\lambda$ (in the $x$, $y$ and $z$ directions respectively) with spatial resolution $\Delta x=\Delta y=~{}\lambda/16$ and $\Delta z=~{}\lambda/32$. Time resolution is set to $c\Delta t=~{}\Delta z/2$. The laser propagates in the $z$-direction and collides head-on with a counter-propagating beam of electrons or photons with finite longitudinal extension $L_{b}=~{}\lambda/8$ and an infinite transverse extension (periodic boundary conditions are used in the transverse direction). Each cell of the beam contains $4$ macro particles for a total of $802816$ (seed) particles. The initial position of the particle beam is set so that if a particle at the center of the particle beam was always moving at $c$, it would reach the peak intensity at the beam focus. The duration of the simulation is longer than the interaction time of the laser and the particles. All results presented hereafter have been extracted at the end of the simulations.

A set of simulations was performed considering the two types of laser facilities previously discussed, i.e. short Ti:Sapphire laser systems such as Apollon ($\lambda=0.8$ $\mu$m and $\tau_{p}=20$ fs), and long Ne:Glass laser systems such as the ELI Beamlines A4 ATON ($\lambda=1.057$ $\mu$m and $\tau_{p}=150$ fs). To compare the results on both types of facilities, results are presented as a function of the laser intensity (not field strength $a_{0}$). In addition, simulations have been performed using a fixed focusing aperture $f/3$.

In Fig. 7, the resulting total cross sections (dots), in units of $w_{0}^{2}$, are compared to those from the numerical integration of Eq. (26) (lines), as a function of $I_{0}$ and $\gamma_{0}$. Excellent agreement is observed for the range of parameters considered. Since the effect of finite transverse size is that many particles will interact with an intensity lower than the peak value $a_{0}$, the approximation of considering only the first generation holds over a wider range of parameters in the three-dimensional case than in the one-dimensional limit. If we compare pulses with different duration and same field peak amplitude (here $I_{0}=4.9\times 10^{22}$ W/cm² i.e. $a_{0}=200$ at $\lambda=1.057$ $\mu$m) it is interesting to notice that in the case of electron seeding, for the lowest electron energies we obtain a slightly larger cross-section for $\tau_{p}=20$ fs than $\tau_{p}=150$ fs. As discussed before, electrons lose more energy before reaching the peak of intensity in a long pulse, resulting in nonlinear Compton Scattering emitted photons with lower energy. This means that the number of pairs is not always a growing function of the pulse duration.