# Impact of the laser spatio-temporal shape on Breit-Wheeler pair production

A. Mercuri-Baron1, M. Grech2, F. Niel1, A. Grassi1, M.Lobet3, A. Di Piazza4
and C. Riconda1 1 LULI, Sorbonne Université, CNRS, CEA, École Polytechnique,
Institut Polytechnique de Paris, F-75255 Paris, France  2 LULI, CNRS, CEA,
Sorbonne Université, École Polytechnique, Institut Polytechnique de Paris,
F-91128 Palaiseau, France  3 Maison de la Simulation, CEA, CNRS, Université
Paris-Sud, UVSQ, Université Paris-Saclay, F-91191 Gif-sur-Yvette,France  4 Max
Planck Institute for Nuclear Physics, Saupfercheckweg 1, D-69117, Heidelberg,
Germany<EMAIL_ADDRESS>
###### Abstract

The forthcoming generation of multi-petawatt lasers opens the way to abundant
pair production by the nonlinear Breit-Wheeler process, i.e., the decay of a
photon into an electron-positron pair inside an intense laser field. In this
paper we explore the optimal conditions for Breit-Wheeler pair production in
the head-on collision of a laser pulse with gamma photons. The role of the
laser peak intensity versus the focal spot size and shape is examined keeping
a constant laser energy to match experimental constraints. A simple model for
the soft-shower case, where most pairs originate from the decay of the initial
gamma photons, is derived. This approach provides us with a semi-analytical
model for more complex situations involving either Gaussian or Laguerre-Gauss
(LG) laser beams. We then explore the influence of the order of the LG beams
on pair creation. Finally we obtain the result that, above a given threshold,
a larger spot size (or a higher order in the case of LG laser beams) is more
favorable than a higher peak intensity. Our results match very well with
three-dimensional particle-in-cell simulations and can be used to guide
upcoming experimental campaigns.

##  1 Introduction

Abundant electron-positron pair production is among the most exotic and
striking processes of extremely high-intensity laser interaction with light
and/or matter that will be made possible by the emerging class of multi-
petawatt (multi-PW) laser systems [1]. Under conditions envisioned for
forthcoming facilities such as Apollon [2], CoReLS [3], ELI [4], OMEGA-EP OPAL
[5], XCELS [6] and zeus [7], the dominant process leading to pair production
is nonlinear Breit-Wheeler , i.e. the decay of a high-energy photon, as it
interacts with the strong laser field, in an electron-positron pair [8, 9].

In the last decade, various simulation campaigns have been conducted to help
design the future experiments to efficiently drive this process, an effort
that was largely made possible and supported by particle-in-cell (PIC)
simulations accounting for quantum electrodynamics (QED) processes [10, 11,
12, 13, 14, 15]. Several configurations have been considered. Electron-seeded
electromagnetic/QED cascades in the collision of two counter-propagating laser
pulses, a setup originally proposed by Bell & Kirk [16], have attracted
particular attention [11, 14, 17, 18, 19, 20] as such cascades were identified
as a possible limitation on the attainable intensity of high power lasers
[21]. This setup was further built upon considering either the use of plasma
channels [22] or the collision of multiple laser pulses [23, 24, 25]. All
these works considered multiple laser beams, but electromagnetic cascades were
also predicted considering the direct irradiation of a solid target by a
single, extremely intense laser pulse, a situation that leads to the
production of dense pair plasmas [26] (see also [27] for an application to
laboratory astrophysics). In these cases, the solid target, an overdense
plasma acts as a mirror and pair production is efficiently achieved in the
field of the incident and reflected laser light at the target front [28].

All the previously discussed configurations considered cascades seeded by
electrons initially at rest in the laboratory frame. The seed electrons,
strongly accelerated in the electromagnetic fields of laser pulses with
extreme intensity (typically $\sim 10^{24}\,{\rm W/cm^{2}}$), gain energies at
the GeV-level (as expected, at such intensities, from the ponderomotive
scaling) and are the source (via nonlinear Compton scattering) of the high-
energy photons triggering pair production. Another possible path toward pair
production in the laboratory relies on an already existing high-energy
electron or photon source. The two latter schemes are related, as photons with
energy of the order of the electron one are produced via nonlinear Compton
scattering of electrons in the intense laser fields [29, 30, 31]. Indeed, this
configuration was exploited in the seminal E-144 experiment at SLAC [32],
where the head-on collision of 46.6 GeV electron beams with TW laser pulses
gave the first direct demonstration of nonlinear Breit-Wheeler pair
production. However, the laser intensity in this experiment was only
marginally relativistic, and very few positrons were produced ($\sim 100$
positrons over nearly $22000$ electron-laser collisions).

Combining GeV-electron beams with PW and multi-PW lasers will allow to extend
this study deep into the relativistic regime, as investigated in [33] that
numerically demonstrated the possibility of producing extremely bright high-
energy photon sources as well as positrons bunches for laser intensities
$\gtrsim 10^{22}{\rm W/cm^{2}}$, in conditions relevant to the Apollon
facility. This configuration was further studied theoretically and via PIC
simulations in [34, 35, 36], while the possibility to produce pairs via
nonlinear Breit-Wheeler with high-energy photons from a Bremsstrahlung source
was presented in [37].

In this paper, we focus on the study of nonlinear Breit-Wheeler pair
production following from the head-on collision of an extremely intense laser
(intensities in the range $10^{21}-10^{25}{\rm W/cm^{2}}$) with a burst of
gamma photons (with energies ranging from 100 MeV to few 10s of GeV). In
contrast with [37], where the authors focused on the optimization of the
photon source, we consider here the high-energy photon burst as given (only
its energy will vary), and do not discuss its origin (various sources of high-
energy photons have been proposed [38, 39, 40, 41, 42, 43]). Rather, we aim to
optimize the conditions of interaction with the colliding high-intensity
laser. Motivated by experimental constraints, we investigate the optimal
conditions for pair production varying the laser polarisation, focusing,
spatial or temporal profiles, always considering a fixed laser energy.

In particular, special attention will be paid to the use of Laguerre-Gauss
(LG) beams, which have recently attracted the interest of the ultra-high
intensity laser-plasma community [36, 44, 45, 46, 47] for their ability to
carry orbital angular momentum (OAM) [48]. These beams, that emerge as eigen-
modes of the paraxial equation, have unique properties, such as a ring-shaped
intensity distribution and the large OAM, that could have an impact on pair
production. It was for instance demonstrated in [36] that the collision of an
electron beam with a LG beam can lead to efficient production of gamma rays
carrying large OAM, to enhanced secondary radiation emission and pair
production. However, that study was performed at constant maximum intensity,
so it is hard to distinguish the impact of the increased energy (up to
$3\times$ higher for the LG beam) from the role of the laser spatial profile
itself. We will discuss this aspect in detail in this paper.

Because of the remarkable number of free parameters to study (polarisation,
focusing, spatial and temporal profile) and the complex spatio-temporal
dependence of the electromagnetic fields of LG beams, it is useful to develop
and validate a reduced analytical model for pair creation (see Ref. [49] for
an analytical study of the Breit-Wheeler pair production process in a Gussian
beam). Starting from the probability for a given high-energy photon to decay
in an electron-positron pair while it crosses an extremely intense laser
pulse, we propose a simple, yet accurate, model to compute the number of pairs
created during this interaction. The computation relies on three assumptions.
(i) The electron-positron pair production rates are obtained in the locally
constant cross-field approximation [50]. (ii) The newly created charged
particles do not further contribute to pair production. (iii) The potentially
complex electromagnetic fields explored by the high-energy photons as they
cross the laser pulse are approximated, assuming that each photon sees a
succession of single half-periods of monochromatic plane waves, the field
amplitude of each half-period being computed so that both the temporal and
spatial profiles of the laser pulse at focus are accounted for. As will be
further discussed in this work, these assumptions are well satisfied for a
wide range of parameters of interest for this study and currently accessible
laser parameters. Our model goes further than previous studies [34, 50] as it
accounts for arbitrary polarisation and spatio-temporal profile of the laser
beam, and it is not limited to small pair production probability and/or rate.

Three-dimensional PIC simulations performed with the code Smilei [51] and
embarking the relevant QED modules [15] are presented and validate our model.
Assumptions (ii) and (iii) are shown to hold under conditions relevant to
ultra-short laser pulses (up to few tens laser cycles) typical of the
Ti:Sapphire technology. Assumption (ii) holding is of particular importance as
it demonstrates that, under conditions relevant to forthcoming ultra-short
laser facilities, the high-energy-photon-laser interaction proceeds
essentially in a regime which we refer to as the soft-shower regime [52].
Furthermore, our PIC simulations allow to test our model against complex laser
pulse geometries, e.g. considering LG beams, showing that assumption (iii) -
which is at the core of the simplification of our model - provides accurate
predictions even for non trivial electromagnetic field configurations.

The proposed model and performed study make it possible to clearly identify
the laser and high-energy photon parameters that optimize pair production for
a given laser energy. This work also highlights the conditions in which either
stronger focusing or increased gamma photon energy does not improve the
production of primary pairs.

The paper is structured as follows. In Sec. 2, we introduce the key properties
of the nonlinear Breit-Wheeler process, and we propose a reduced model for
pair creation in the head-on collision between gamma photons and a plane wave
with arbitrary polarization. The possibility to account for a time envelope of
the laser field is also considered and the model is benchmarked against 1D
particle-in-cell (PIC) simulations. In Sec. 3 we consider the effect of the
spatial dependence of the laser field. Both Laguerre-Gauss and Gaussian beams
are considered and we introduce the notion of a total cross-section for which
we propose a reduced model, shown to be in very good agreement with 3D PIC
simulations. The optimal conditions for pair creation with respect to the
laser geometry and intensity are also discussed. In Sec. 4 we then use our
model to discuss pair production on upcoming Ti:Sapphire laser facilities,
before giving our conclusions in Sec. 5.

##  2 Reduced model for pair production in a time-varying background field

This section provides the theoretical framework for describing nonlinear
Breit-Wheeler pair production, and aims to establish a simple model for pair
production in a time-varying background field. Throughout this section, we
consider the head-on collision of high-energy gamma photons with a strong,
optical, laser pulse (also referred to as the background field). In addition
to exact expressions, we provide practical formulae relevant to forthcoming
high-power laser facilities, focusing on optical laser systems with
micrometric wavelength $\lambda$, with relativistic field strength
$a_{0}=eE_{0}/(m_{e}c\omega)\gg 1$ (with $\omega=2\pi c/\lambda$ the
background field angular frequency), corresponding to $I_{0}\lambda^{2}$ well
beyond $10^{18}{\rm W/cm^{2}\,\mu m^{2}}$, with
$I_{0}=\epsilon_{0}cE_{0}^{2}/2$ the laser intensity. Note that SI units are
used throughout this work: $e$, $m_{e}$ and $c$ denote the elementary charge,
electron mass and speed of light in vacuum, respectively, $\epsilon_{0}$ the
vacuum permittivity, $\tau_{e}=e^{2}/(4\pi\epsilon_{0}m_{e}c^{3})$ the time
for light to cross the classical radius of the electron, $\hbar$ the reduced
Planck constant and $\alpha=e^{2}/(4\pi\epsilon_{0}\hbar c)$ the fine-
structure constant.

###  2.1 Pair production instantaneous rate and probability

Nonlinear Breit-Wheeler is the dominant QED process leading to electron-
positron pair production from the collision of a high-energy photon with a
strong (optical) laser pulse [53, 54]. As discussed in Ref. [50], for a
relativistically intense background field, $a_{0}\gg 1$, the local constant
field approximation (LCFA) holds and the pair production probability can be
computed by integrating the instantaneous pair production rate. For a high-
energy photon propagating with velocity ${\bf c}$, normalized energy
$\gamma_{\gamma}=\hbar\omega_{\gamma}/(m_{e}c^{2})$ and instantaneous quantum
parameter

$\displaystyle\chi_{\gamma}=\frac{\gamma_{\gamma}}{E_{S}}\,\sqrt{\big{(}{\bf
E}+{\bf c}\times{\bf B}\big{)}^{2}-\big{(}{\bf c}\cdot{\bf
E}\big{)}^{2}\\!/c^{2}}\,,$ (1)

where $E_{S}=m_{e}^{2}c^{3}/(e\hbar)\simeq 1.32\times 10^{18}{\rm V/m}$ is the
Schwinger field and ${\bf E}$ and ${\bf B}$ denote the electric and magnetic
fields at the electron position, the instantaneous pair production rate is
given by (see, e.g., Refs. [55, 56]):

$\displaystyle
W_{\rm\scriptscriptstyle{BW}}=W_{0}\,\frac{b_{0}(\chi_{\gamma})}{\gamma_{\gamma}}\,,$
(2)

where $W_{0}=2\alpha^{2}/(3\tau_{e})$. For a fixed value of the quantum
parameter $\chi_{\gamma}$, the rate is inversely proportional to the gamma
photon energy $\gamma_{\gamma}$, and depends non-trivially on $\chi_{\gamma}$
through the function

$\displaystyle b_{0}\left(\chi_{\gamma}\right)$ $\displaystyle=$
$\displaystyle\dfrac{\sqrt{3}}{2\pi}\int_{0}^{1}\dfrac{\rm{d}\xi}{\xi\left(1-\xi\right)}\left[\dfrac{2}{3\chi_{\gamma}}\dfrac{1-2\xi}{\left(1-\xi\right)}\rm{K}_{5/3}\left(\mu\right)+\rm{K}_{2/3}\left(\mu\right)\right]\,,$
(3)

with $\mu=2/[3\chi_{\gamma}\xi(1-\xi)]$ and ${\rm K}_{n}(x)$ the modified
Bessel function of the second kind. This function, shown in Fig. 1(a), can be
well approximated (within less than 1% from the numerically computed values
for $\chi_{\gamma}\in[10^{-2},10^{3}]$) by111Note that $0.242\sim 0.16\times
3/2$, and $b_{0}(\chi)=3\chi\,T(\chi)/2$, with $T(\chi)$ the function used by
Erber [57] and Blackburn [34].

$\displaystyle b_{0}(\chi_{\gamma})\simeq 0.242\,\frac{{\rm
K}_{1/3}^{2}\big{(}4/(3\chi_{\gamma})\big{)}}{1-0.172/(1+0.295\,\chi_{\gamma}^{2/3})}\,.$
(4)

The denominator in Eq. (4) provides a simple improvement of the well known
Erber approximation [57], which also allows to recover the correct asymptotic
behavior [dashed grey lines in Fig. 1(a)] of $b_{0}(\chi)$:

$\displaystyle b_{0}(\chi_{\gamma})\rightarrow
c_{1}\,\chi_{\gamma}\,e^{-8/(3\chi_{\gamma})}\quad$ $\displaystyle{\rm for}$
$\displaystyle\quad\chi_{\gamma}\ll 1\,,$ (5) $\displaystyle
b_{0}(\chi_{\gamma})\rightarrow c_{2}\,\chi_{\gamma}^{2/3}\quad$
$\displaystyle{\rm for}$ $\displaystyle\quad\chi_{\gamma}\gg 1\,,$ (6)

with $c_{1}=(3/16)\left(3/2\right)^{3/2}\simeq 0.344$ and
$c_{2}=3^{2/3}45\,\Gamma^{4}(2/3)/(56\pi^{2})\simeq 0.569$ [$\Gamma(x)$
denoting the gamma function].

Figure 1:  (a) Dependence of $b_{0}(\chi_{\gamma})$ on the photon quantum
parameter $\chi_{\gamma}$ as defined by Eq. (3) (black line) and asymptotic
behaviors given by Eqs. (5) and (6) (grey dashed lines). (b) Dependence of
$\mathcal{I}_{\varepsilon}(\chi_{0})$ [Eq. (19)] on the maximum photon quantum
parameter $\chi_{0}$ [as defined by Eq. (13)], for a linearly polarized (LP)
background field ($\varepsilon=0$, black line) and a circularly polarized (CP)
background field ($\varepsilon=\pm 1$, blue line). Dashed lines show the
asymptotic behavior given by Eqs. (26) and (27) for the LP case. The inset
highlights the values of $\chi_{0}$ for which LP ($\chi_{0}<2.52$) or CP
($\chi_{0}>2.52$) gives higher pair production probability and the relative
difference of $\mathcal{I}_{\varepsilon}$ (in absolute value) between the two
cases.

The quantity $W_{\rm\scriptscriptstyle{BW}}$ defined by Eq. (2) is the
instantaneous rate of pair production, so that considering, at a time $t_{0}$,
a given number $N_{0}$ of identical high-energy photons entering a strong
background field, the number of photons $N_{\gamma}$ remaining at later times
$t>t_{0}$ satisfies the rate equation:

$\displaystyle\frac{d}{dt}N_{\gamma}=-W_{\rm\scriptscriptstyle{BW}}N_{\gamma}\,,$
(7)

where both $N_{\gamma}$ and $W_{\rm\scriptscriptstyle{BW}}$ depend on time
(the latter when the high-energy photons explore electromagnetic fields with
spatial and/or temporal variations). Note that Eq. (7) does not account for
subsequent radiation from the produced pairs, which will be verified to be
accurate for the investigated parameter range in Sec. 3. In the following
$t_{0}$ will refer to the time at which the gamma flash first encounters the
pulse. The solution to this equation reads

$\displaystyle
N_{\gamma}\left(t\right)=N_{0}\,\exp\left[-\int_{t_{0}}^{t}W_{\rm\scriptscriptstyle{BW}}\\!\left(t^{\prime}\right)dt^{\prime}\right]\,,$
(8)

from which one can derive the probability $P(t_{0},\Delta
t)=1-N_{\gamma}(t_{0}+\Delta t)/N_{0}$ for a given high-energy photon to decay
into an electron-positron pair between time $t_{0}$ and $t_{0}+\Delta t$:

$\displaystyle P(t_{0},\Delta t)=1-\exp\left[-\int_{t_{0}}^{t_{0}+\Delta
t}W_{\rm\scriptscriptstyle{BW}}\\!\left(t^{\prime}\right)dt^{\prime}\right]\,.$
(9)

When the time-integrated rate [i.e. the integral term in Eq. (8)] is small, it
can be assimilated to the probability $P(t_{0},\Delta t)$, as $P(t_{0},\Delta
t)\rightarrow\int_{t_{0}}^{t_{0}+\Delta
t}W_{\rm\scriptscriptstyle{BW}}\\!\left(t^{\prime}\right)dt^{\prime}\ll 1$.
This leads some authors to refer to $W_{\rm\scriptscriptstyle{BW}}$ as the
pair production probability per unit time. While this is correct in the limit
${P(t_{0},\Delta t)\ll 1}$, it leads to inconsistencies as the probability
increases and the time-integrated rate assumes values greater than unity (see,
also, Refs. [58, 50, 59, 60]). Hence, throughout this work, we will
distinguish pair production rate [as defined by Eq. (2)] and probability [as
defined by Eq. (9)].

###  2.2 Pair production probability in an arbitrarily polarized plane wave

Let us now focus on the case of a high-energy photon colliding head-on with an
ultra-intense laser pulse described by a plane wave (propagating in the
$\hat{\bf z}$-direction), with electric field:

$\displaystyle{\bf E}(z,t)$ $\displaystyle=$
$\displaystyle\dfrac{E_{0}}{\sqrt{1+\varepsilon^{2}}}\left[\sin{\left(\omega
t-kz\right)}\,{\bf\hat{x}}+\varepsilon\cos{\left(\omega
t-kz\right)}\,{\bf\hat{y}}\right]\,,$ (10)

magnetic field ${\bf B}=\hat{\bf z}\times{\bf E}/c$, and polarization
ellipticity $\varepsilon\in[-1,1]$. It takes the particular values
$\varepsilon=\pm 1$ for a circularly polarized (CP) wave ($\pm$ standing for
opposite helicities), and $\varepsilon=0$ for a linearly polarized (LP) wave
($\hat{\bf x}$ denoting by convention the direction of polarization). It is
important to stress that our choice of parametrisation [Eq. (10)] ensures that
the laser energy is the same for any value of $\varepsilon$. On the contrary,
$E_{0}/\sqrt{1+\varepsilon^{2}}$ measures the maximum laser electric field
amplitude and varies with the ellipticity.

When a high-energy photon collides with such a laser field (we define
$t_{0}=0$ as the time at which the photon enters the background field), the
value of its quantum parameter

$\displaystyle\chi_{\gamma}(t)=\frac{\gamma_{\gamma}}{E_{S}}\,\Big{|}{\bf
E}\big{(}z_{\gamma}(t),t)+{\bf c}\times{\bf
B}(z_{\gamma}(t),t\big{)}\Big{|}\,$ (11)

evolves in time as the photon explores regions with different electric and
magnetic field amplitude [here $z_{\gamma}(t)$ denotes the photon time-
dependent position]. In the case of head-on collisions (${\bf c}=-c\,\hat{\bf
z}$), the time-dependent photon quantum parameter reduces to

$\displaystyle\chi_{\gamma}(t)=\chi_{0}\,\Psi_{\\!\varepsilon}(2\omega
t)\quad\text{with}\quad\Psi_{\\!\varepsilon}(\varphi)=\sqrt{\frac{\sin^{2}\\!\varphi+\varepsilon^{2}\cos^{2}\\!\varphi}{1+\varepsilon^{2}}}\,,$
(12)

and

$\displaystyle\chi_{0}=2\,\gamma_{\gamma}\,\frac{E_{0}}{E_{S}}\simeq
0.801\,\left(\frac{\hbar\omega_{\gamma}}{1\,{\rm
GeV}}\right)\,\sqrt{\frac{I_{0}}{10^{22}~{}{\rm W/cm^{2}}}}\,.$ (13)

Note that, with our convention on the background field parametrization [Eq.
(10)], $\chi_{0}$ denotes the maximum quantum parameter for the LP case only.
For arbitrary $\varepsilon$, the maximum achievable quantum parameter
$\chi_{m}$ is a decreasing function of $|\varepsilon|$

$\displaystyle\chi_{m}=\frac{\chi_{0}}{\sqrt{1+\varepsilon^{2}}}.$ (14)

Note also that Eq. (12) takes a simple form when considering either a CP or a
LP background field:

$\displaystyle\chi_{\gamma}(t)=\chi_{0}/\sqrt{2}\quad$ for
$\displaystyle\quad\varepsilon=\pm 1\,,$ (15)
$\displaystyle\chi_{\gamma}(t)=\chi_{0}\,\big{|}\sin(2\omega t)\big{|}\quad$
for $\displaystyle\quad\varepsilon=0\,.$ (16)

####  2.2.1 Decay probability after propagating through half a period of the
background field

For arbitrary values of $\varepsilon$, $\chi_{\gamma}(t)$ is either a constant
(for $\varepsilon=\pm 1$) or a periodic function of time with period $\tau/4$
(we recall $\tau=2\pi/\omega$) corresponding to the time for the high-energy
photon to cross half a wavelength of the background field. It is thus
convenient to compute the probability $P_{m}$ of the high-energy photon to
decay into a pair during this interval of time. Let us then consider a time
$t_{m}$ at which the high-energy photon sees a maximum of the background
electric field, and compute the probability $P_{m}$ for the high-energy photon
to decay into an electron-positron pair in between the times
$t_{0}=t_{m}-\tau/8$ and $t_{0}+\tau/4=t_{m}+\tau/8$:

$\displaystyle
P_{m}=P(t_{0},\tau/4)=1-\exp\left(-R_{m}\frac{\tau}{4}\right)\quad\text{with}\quad
R_{m}\frac{\tau}{4}=\frac{W_{0}}{2\omega\gamma_{\gamma}}\int_{0}^{\pi}\\!b_{0}\big{(}\chi_{0}\Psi_{\\!\varepsilon}(\varphi)\big{)}\,d\varphi\,$
(17)

where $R_{m}\tau/4$ is the time-integrate rate, while $R_{m}$ denotes the
average rate.

Let us stress that the dimensionless quantity $W_{0}/(2\omega\gamma_{\gamma})$
in Eq. (17) is of order 1 for optical background fields (with micrometric
wavelength $\lambda$) and GeV-level high-energy photons:

$\displaystyle\frac{W_{0}}{2\omega\gamma_{\gamma}}=\frac{\alpha}{3}\,\frac{m_{e}c^{2}}{\hbar\omega}\,\frac{m_{e}c^{2}}{\hbar\omega_{\gamma}}\simeq
0.512\,\left(\frac{\lambda}{1\,{\rm\mu m}}\right)\,\left(\frac{1\,{\rm
GeV}}{\hbar\omega_{\gamma}}\right)\,.$ (18)

The probability $P_{m}$, that measures the contribution of a single field
maximum, depends on the integral quantity

$\displaystyle\mathcal{I}_{\varepsilon}(\chi_{0})=\int_{0}^{\pi}\\!b_{0}\big{(}\chi_{0}\Psi_{\\!\varepsilon}(\varphi)\big{)}\,d\varphi\,,$
(19)

which, for a given polarization ellipticity $\varepsilon$, is a function of
$\chi_{0}$ only as shown in Fig. 1(b) for LP (black line) and CP (blue line).

This integral is the key object to compute the pair creation probability and
can be calculated either numerically or by means of an approximated formula as
described in the following. In A, we generalize the approach proposed in [34]
to arbitrary $\varepsilon$ and $\chi$, and show that
$\mathcal{I}_{\varepsilon}(\chi_{0})$ can be written in the approximate form:

$\displaystyle\mathcal{I}_{\varepsilon}(\chi_{0})\simeq\pi\,b_{0}\\!\left(\chi_{m}\right)\,{\rm
min}\\!\left\\{F\big{(}s_{\varepsilon}(\chi_{m})\big{)},f(\varepsilon)\right\\}\,\text{with}\,\left\\{\begin{array}[]{ll}F\\!\left(s\right)=\sqrt{2/\pi}\,s\,{\rm
erf}\\!\left(\pi\sqrt{2}/(4s)\right)\\\
f(\varepsilon)=\frac{1}{\pi}\int_{0}^{\pi}\\!\left[\sin^{2}\varphi+\varepsilon^{2}\cos^{2}\varphi\right]^{1/3}\\!d\varphi\end{array}\right.$
(22)

where $\chi_{m}$ is defined in Eq. (14) and ${\rm
erf}(x)=\tfrac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}dt$ is the error function.
The function $F(s)$ emerges from the saddle point approximation used to
compute the integral when the main contribution comes from the vicinity of the
field maximum. It varies slowly with $s$, as
$F(s)\simeq\tfrac{2}{\sqrt{\pi}}s$ for $s\ll 1$ and $F(s)\rightarrow 1$ for
$s\rightarrow+\infty$. In Eq. (22), $F(s)$ takes for argument
$s_{\varepsilon}(\chi_{m})$, with

$\displaystyle
s_{\varepsilon}(\chi)=\sqrt{\frac{3}{2}}\,\frac{c\\!\left(\chi\right)}{\sqrt{1-\varepsilon^{2}}}\quad\text{and}\quad
c(\chi)=\sqrt{\frac{2\,b_{0}(\chi)}{3\chi\,b_{0}^{\prime}(\chi)}}\,,$ (23)

where $b_{0}^{\prime}(\chi)$ denotes the derivative of $b_{0}(\chi)$ and
$c(\chi)$ is a slowly varying function of $\chi$, as
$c(\chi)\simeq\sqrt{\chi}/2$ for $\chi\ll 1$ and $c(\chi)\rightarrow 1$ for
$\chi\rightarrow+\infty$.

Equation (22) is exact for CP background fields ($\varepsilon=\pm 1$) for
which ${s_{\pm 1}(\chi_{m})\rightarrow+\infty}$, so that $\mathcal{I}_{\pm
1}(\chi_{0})=\pi\,b_{0}\big{(}\chi_{0}/\sqrt{2}\big{)}$. Moreover it allows to
recover the asymptotic behavior of Eq. (19) in the limiting cases of small and
large quantum parameter $\chi_{0}$:

$\displaystyle\mathcal{I}_{\varepsilon}(\chi_{0})$
$\displaystyle\xrightarrow[]{\chi_{0}\ll 1}$
$\displaystyle\left\\{\begin{array}[]{ll}c_{3}\,(1-\varepsilon^{2})^{-1/2}(1+\varepsilon^{2})^{-1/4}\,\chi_{0}^{3/2}\,\exp\left(-\frac{8\sqrt{1+\varepsilon^{2}}}{3\chi_{0}}\right)\quad{\rm
for}\quad\chi_{0}\ll 1-\varepsilon^{2}\,,\\\
c_{4}\,\chi_{0}\,\exp(-\frac{8\sqrt{2}}{3\chi_{0}})\quad{\rm
otherwise}\,,\end{array}\right.$ (26)
$\displaystyle\mathcal{I}_{\varepsilon}(\chi_{0})$
$\displaystyle\xrightarrow[]{\chi_{0}\gg 1}$ $\displaystyle
c_{5}\,\frac{f(\varepsilon)}{(1+\varepsilon^{2})^{1/3}}\,\chi_{0}^{2/3}\,,$
(27)

where we used Eqs. (5) and (6), and $c_{3}=27\sqrt{\pi/2}/64\simeq 0.529$,
$c_{4}=9\sqrt{3}\pi/64\simeq 0.765$ and
${c_{5}=3^{2/3}45}\Gamma^{4}(2/3)/(56\pi)$ ${\simeq 1.789}$.

####  2.2.2 Total decay probability

As will be further discussed in Sec. 2.3, considering parameters typical of
the upcoming generation of laser facilities, the probability for a high-energy
photon to decay into a pair after crossing a single half-wavelength of the
background field will in general be small compared to 1. This will not however
be the case when considering the cumulative effect of crossing several
wavelengths.

The probability for a high-energy photon to decay into a pair after
$t=n\tau/4$ of interaction with the background field (or, equivalently, after
crossing $n$ local maxima of the field amplitude) $P_{\rm tot}(t=n\tau/4)$,
can be derived from the number $N_{\gamma}$ of photons surviving after a time
$t$:

$\displaystyle N_{\gamma}(t=n\tau/4)=N_{0}\,\prod_{m=1}^{n}\vbox{\hrule
height=0.5pt\kern
0.6458pt\hbox{\kern-1.99997pt$P$\kern-1.00006pt}}_{m}\quad\text{with}\quad\vbox{\hrule
height=0.5pt\kern
0.6458pt\hbox{\kern-1.99997pt$P$\kern-1.00006pt}}_{m}=1-P_{m}=\exp(-R_{m}\tau/4)\,.$
(28)

Here we have used $m$ as a running index for compactness, and $P_{m}$ [given
by Eq. (17)] and $\vbox{\hrule height=0.5pt\kern
0.6458pt\hbox{\kern-1.99997pt$P$\kern-1.00006pt}}_{m}$ denote the
probabilities for the photon to decay and not to decay, respectively, during
the $m^{\rm th}$ interval. We can then write the total probability of
producing a pair as

$\displaystyle P_{\rm tot}(t=n\tau/4)=1-\prod_{m=1}^{n}\vbox{\hrule
height=0.5pt\kern
0.6458pt\hbox{\kern-1.99997pt$P$\kern-1.00006pt}}_{m}=1-\exp\left(-R\,t\right)\quad\text{with}\quad
R=\frac{1}{n}\sum_{m=1}^{n}R_{m}\,,$ (29)

$R$ denoting the average pair production rate.

Let us note that this formula is exact at $t=n\tau/4$ with $n\in\mathbb{N}$,
and – as will be shown later – can also be used with good approximation at all
times $t\gg\tau/4$. Moreover, in the case of a monochromatic plane wave (no
temporal envelop), the average pair production rate simply reduces to
$R=R_{m}$. When dealing with a pulse with a finite temporal envelop, however,
Eq. (29) needs to be computed combining the contribution of successive
$(\tau/4)$-long intervals, using the local maximum of the background field
strength for each time interval. This approach gives very good results even
considering ultra-short, few cycles laser pulses. To confirm this, Figure 2
shows the temporal evolution of the decay probability for a high-energy photon
with $\gamma_{\gamma}=10^{3}$ colliding head-on with different background
fields. In panels (a) and (b), the high-energy photon interacts with a
background field with constant amplitude $a_{0}\simeq 82$ [$\chi_{0}=0.5$,
Fig. 2(a)] and $a_{0}\simeq 660$ [$\chi_{0}=4$, Fig. 2(b)]. The interaction
lasts for a time $\tau$, i.e. the photon explores two wavelengths of the
background field. In panels (c) and (d), the high-energy photon interacts with
a background field with a sin2 temporal profile in intensity, a full-width-
half-maximum (fwhm) of $5\tau$, and maximum field strength $a_{0}\simeq 82$
and $a_{0}\simeq 660$, respectively.

In all panels, the solid lines denote the exact probability computed by
numerically integrating Eq. (9), considering either LP (black lines) or CP
(blue lines) background field. We can see that, for $\chi_{0}=0.5$, CP
produces less pairs than LP, while for $\chi_{0}=4$ CP is slightly more
efficient [the difference between the two polarizations will be discussed in
details in Sec. 2.3.1]. These probabilities are compared against the
prediction of Eq. (29), computed using either the numerically evaluated
$\mathcal{I}_{\varepsilon}(\chi_{0})$ [dashed lines in panels (a) and (b),
dots in panels (c) and (d)] or the approximation given by Eq. (22) to compute
$R_{m}$ (red dot-dashed lines, only computed for LP as it is exact for CP).
Note that here and in the following, whenever Eq. (22) is used, the improved
Erber approximation Eq. (4) is exploited.

Panels (a) and (b) show an excellent agreement between the exact computations
(solid lines) and Eq. (29) (dashed lines), both using the numerically
integrated values of $\mathcal{I}_{\varepsilon}(\chi_{0})$. A remarkably good
agreement is also found when considering the fully analytical approximation of
Eq. (22) for $\mathcal{I}_{\varepsilon}(\chi_{0})$ (red dot-dashed lines). The
probability is slightly overestimated in this case, as expected for
$\chi_{0}\sim 1$ for which Eq. (22) shows the greatest departure from the
exact expression of $\mathcal{I}_{\varepsilon}(\chi_{0})$.

Similarly, an excellent agreement between all three approaches is found in
panels (c) and (d) considering a short ($5\tau$ fwhm in intensity) background
field. In this case the value of $R_{m}$ in each $\tau/4$ interval is computed
by taking the value of $\mathcal{I}_{\varepsilon}$ given by the local maximum
of the field, i.e. the time variation of the envelope is considered slow
during $\tau/4$. The validity of this assumption is confirmed by PIC
simulations discussed in the following. In particular, Figure 2 shows that the
approach leading to the derivation of Eq. (29), which consists in treating the
cumulative contributions of successive maxima of the background field, remains
a very good approach even for ultra-short, few cycle, background fields.

Last, we note that the results of 1D PIC simulations performed with Smilei
(see B for details) are not distinguishable from the exact computations (solid
lines) for all reported cases, and are therefore not shown in Fig. 2.

Figure 2:  Temporal evolution of the probability for a high-energy photon with
normalized energy $\gamma_{\gamma}=\hbar\omega_{\gamma}/(m_{e}c^{2})=10^{3}$
to decay into an electron-positron pair while interacting with a plane wave
background field in head-on collision. Considering a constant background field
envelop with (a) $a_{0}\simeq 82$ corresponding to $\chi_{0}=0.5$ and (b)
$a_{0}\simeq 660$ corresponding to $\chi_{0}=4$. Considering a background
field with sin2 temporal profile with maximum field amplitude (c) $a_{0}\simeq
82$ corresponding to $\chi_{0}=0.5$, (d) $a_{0}\simeq 660$ corresponding to
$\chi_{0}=4$. Solid lines correspond to the exact probability obtained by
integrating numerically Eq. (9), dashed lines and dots to the theoretical
prediction obtained from Eq. (29), for LP ($\varepsilon=0$, black) and CP
($\varepsilon=\pm 1$, blue) cases. The red dash-dotted lines correspond to the
theoretical prediction from Eq. (29) using the approximation of Eq. (22) for
LP.

###  2.3 Guidelines for pair production optimizations

####  2.3.1 Influence of the background field polarization

Interesting insights into the role of the background field polarization on
pair production can be gained from the asymptotic behavior provided by Eqs.
(26) and (27).

We recall that, motivated by experimental constraints, we compare LP and CP at
fixed energy: this condition implies that the CP beam has a lower maximum
amplitude. Because of this, for $\chi_{0}\ll 1$, Eq. (26) shows that any
departure from the LP case leads to a tremendous decrease of
$\mathcal{I}_{\varepsilon}(\chi_{0})$. Indeed the exponential cut-off of the
pair production rate is strongly affected by the decrease of $\chi_{m}$ with
$|\varepsilon|$ as shown in Eq. (14). In contrast, for $\chi_{0}\gg 1$,
$f(\varepsilon)/(1+\varepsilon^{2})^{1/3}$ is an increasing function of
$|\varepsilon|$: the reduction of the field peak amplitude is compensated by
the fact that the absolute value of the amplitude is not time dependent [see
Eq.(15)]. Hence, in this range, increasing the background field ellipticity
increases the pair production rate. However, the rate increase from the LP to
the CP case is small (less than $12\%$), and it is thus of marginal use to
consider CP for boosting pair production222In particular as, in practice,
implementing CP on multi-petawatt facilities would lead to a decrease of the
delivered laser pulse energy..

The differences between the LP and CP cases are clearly shown in the inset in
Fig. 1(b), where $\mathcal{I}_{0}(\chi_{0})$ is found to be orders of
magnitude larger than $\mathcal{I}_{\pm 1}(\chi_{0})$ for $\chi_{0}\ll 1$
while $\mathcal{I}_{\pm 1}(\chi_{0})$ is only about 10% larger than
$\mathcal{I}_{0}(\chi_{0})$ for $\chi_{0}\gg 1$. Similar conclusions can be
drawn from Fig. 2. For small values of the photon parameter, $\chi_{0}=0.5$
(left panels), the pair production probability is significantly larger
considering a LP background field instead of a CP one. For the higher value of
$\chi_{0}=4$ (right panels), CP increases, but only marginally, pair
production. For this reason, we will from now on focus on LP background
fields, unless otherwise specified.

####  2.3.2 Influence of the background field amplitude and gamma-photon
energy

Figure 3:  Probability $P_{m}$ for a high-energy photon to decay into an
electron-positron pair after crossing half-a-wavelength of a LP background
field, as a function of the photon energy
$\gamma_{\gamma}=\hbar\omega_{\gamma}/(m_{e}c^{2})$ and background field
amplitude $a_{0}$: (a) integrating numerically Eq. (17), (b) computed using
the approximate expression Eq. (22), (c) extracted from one-dimensional PIC
simulations. In all three panels, the solid white lines report the isocontours
of the first panel. In panel b), the isocontours obtained from Eq. (22) are
shown in red dot-dashed lines. White dotted lines correspond to constant
values of $\chi_{0}=0.1,1,16.5$ from bottom left to top right corners.

The relative influence on pair production of the photon energy
$\gamma_{\gamma}$ and the background field amplitude $a_{0}$ (or equivalently
intensity $I_{0}$, as we consider here a given background field wavelength
$\lambda=0.8\,{\rm\mu m}$), is summarized in Fig. 3. It shows the probability
$P_{m}$ for a high-energy photon to decay in a pair after crossing half a
wavelength of a LP background field: (a) integrating numerically Eq. (17)
which provides an exact value of $P_{m}$; (b) using the approximate but fully
analytical expression given by Eq. (22); (c) extracted from 1D PIC simulations
of the interaction of a flash of high-energy photons colliding head-on with a
plane wave (details are given in B).

Let us first discuss panel (a) and note that, in addition to the probability
isocontours (solid white lines), the dotted white lines represent the contours
of constant quantum parameter, $\chi_{0}=0.1$, $1$ and $16.5$. As
$\chi_{0}\propto\gamma_{\gamma}\,a_{0}$, these are straight lines with a
$-45^{\circ}$ inclination.

In the limit $\chi_{0}\ll 1$, i.e. the bottom-left of Fig. 3, $P_{m}$ assumes
very small values. In this limit, the time-integrated rate $R_{m}\tau/4$
scales as
$\gamma_{\gamma}^{-1}\chi_{0}^{3/2}\exp\\!\left[-8/(3\chi_{0})\right]$ [Eq.
(26)] and the exponential term, depending only on $\chi_{0}$, gives the
dominating contribution to $P_{m}$. As a result, the isocontours of $P_{m}$
(solid white lines) behave as nearly straight lines, roughly parallel to the
contours of constant $\chi_{0}$. We wish to stress however that the dependence
on $\gamma_{\gamma}^{-1}$ cannot be fully ignored: the isocontours of $P_{m}$
are less steep than the isocontours of $\chi_{0}$ which indicates that $P_{m}$
increases faster with $a_{0}$ than with $\gamma_{\gamma}$. In this range of
small $\chi_{0}$, for which probability and time-integrated rate are
equivalent, the importance of increasing $a_{0}$ to improve pair production
was already pointed out in [34]. However, the probability variation with both
$a_{0}$ and $\gamma_{\gamma}$ needs to be examined in more details at higher
$\chi_{0}$.

Indeed, the dependence with $\gamma_{\gamma}^{-1}$ plays a more important role
as $\chi_{0}$ increases so that $\chi_{0}$ can not be considered as the only
relevant parameter in order to optimize the pair production rate. This is
clearly seen by considering the isocontours of $P_{m}$ in
($a_{0}$,$\gamma_{\gamma}$)-plane, Fig. 3(a): the isocontour slope becomes
shallow and eventually changes sign. Thus, for any value of $P_{m}$, a minimum
field strength $a_{0}$ is needed in order to obtain the desired level of
probability. A corollary is that, at constant $a_{0}$, increasing
$\gamma_{\gamma}$ increases the probability $P_{m}$ up to a maximum value,
beyond which a further increase of $\gamma_{\gamma}$ would only decrease
$P_{m}$. The minimum of each isocontour can thus be found by solving $\partial
P_{m}/\partial\gamma_{\gamma}=0$, which one can recast in the form

$\displaystyle\dfrac{\rm{d}\mathcal{I}_{\varepsilon}}{\rm{d}\chi_{0}}-\dfrac{\mathcal{I}_{\varepsilon}(\chi_{0})}{\chi_{0}}=0\,.$
(30)

Interestingly, this equation involves only $\chi_{0}$ so that the minima of
the probability isocontours lie on a straight line of constant $\chi_{0}$.
This a priori surprising result can be better understood noting that both
$a_{0}$ and $P_{m}$ are Lorentz invariants, the minimum value of $a_{0}$ to
reach a given probability $P_{m}$ can depend only on the Lorentz invariant
involving the photon energy, i.e. $\chi_{0}$. Solving numerically Eq. (30) for
$\varepsilon=0$ (LP case), we obtain333Solving Eq. (30) for arbitrary values
of $\varepsilon\in[-1,1]$ leads to $\chi_{0}$ in between $16.5$ and $17.2$,
and corresponding values of $I_{\varepsilon}(\chi_{0})$ in between $5.3$ and
$6.0$. $\chi_{0}\simeq 16.5$, which is reported by a dotted line in Fig. 3.

To conclude with Fig. 3, let us now turn to panels (b) and (c). Panel (b)
reports the probability $P_{m}$ computed from Eq. (22) and the corresponding
isocontours (red dashed lines). Superimposing the isocontours (white lines) of
the probability from panel (a), we can see an excellent agreement between the
two approaches. This validates the approximate form, Eq. (22), which has the
advantage to be completely analytical and can now be used to get quick yet
precise estimates of the probability $P_{m}$. Last, panel (c) reports the
probability extracted from 1D PIC simulations of the head-on collision of a
flash of high-energy photons with a laser beam. The probability was extracted
from the depletion of high-energy photons after crossing half a laser
wavelength. Here again, a very good agreement is observed with the isocontours
(white lines) obtained from the exact integration shown in panel (a). This
last panel thus provides a cross-benchmark of our model and the Monte-Carlo
module for Breit-Wheeler pair production implemented in our PIC code Smilei .

Figure 4:  Position in the ($\gamma_{\gamma},a_{0}$)-plane [equivalently in
the ($\mathcal{E}_{\gamma},I_{0}$)-plane] at which $R\Delta t=1$ [$P_{\rm
tot}(\Delta t)\simeq 0.63$] for the interaction with a single maximum of the
incoming pulse (half-wavelength, i.e. pulse duration equal to $\tau/2$) for
which the interaction lasts for $\Delta t=\tau/4$ (yellow line) and for the
interaction with pulses with a step-like time profile (solid lines) of
duration $5\tau$ (green) and $100\tau$ (blue), and pulses with a $\sin^{2}$
time envelope with fwhm of $5\tau$ (green dashed line) and $100\tau$ (blue
dashed line). The color-shaded areas above the plain lines correspond to
$R\Delta t>1$ (considering the step-like time profiles). The dotted black
lines correspond to constant $\chi_{0}=0.1\,,1\,,4\,,16.5\,$. The symbols
highlight the parameters of the simulations discussed in Sec. 3.4 and 3.5.

Our model can be used to predict the effectiveness of pair production for a
given set of laser and high-energy photon parameters444In the case of large
secondary pair production, our prediction will underestimate the number of
pairs as our analysis is limited to the soft-shower regime.. Most of the
photons crossing the laser will be converted in pairs over a time $\Delta t$
if the quantity $R\Delta t$ in Eq. (29) becomes of order 1, for which555Note
that in this limit, the probability cannot be assimilated to the time-
integrated rate, as in the case of very weak pair production. $P_{\rm
tot}(\Delta t)\geq 0.63$. In Fig. 4, we highlight in yellow the region in the
$(a_{0},\gamma_{\gamma})$-plane where the probability for a high-energy photon
to decay into a pair after interacting with a single maximum (half-wavelength)
of the laser pulse $P_{\rm tot}(\tau/4)$ is equal or larger than $0.63$ (the
pulse duration is here $\tau/2$, but the time required for the photon to
explore half the laser wavelength is $\Delta t=\tau/4$). The condition $P_{\rm
tot}(\Delta t)=0.63$ is also shown for pulses with a step-like time profile
(solid lines) and duration $5\,\tau$ (green, $\Delta t=2.5\tau$) or
$100\,\tau$ (blue, $\Delta t=50\tau$), and pulses with a $\sin^{2}$ intensity
time profile with fwhm $5\tau$ (green dashed line, $\Delta t=5\tau$) and
$100\,\tau$ (blue dashed line, $\Delta t=100\tau$). Note that both step-like
and $\sin^{2}$ time profiles correspond to laser pulses with the same energy
and peak power, and lead to similar requirements to reach $P_{\rm tot}\simeq
0.63$. Comparing the blue and green lines, we see that at constant
$\gamma_{\gamma}$ a longer pulse satisfies the condition of efficient
conversion for a lower value of $a_{0}$. Hence, even though the condition
$R\,\tau/4=1$ (yellow solid line) is quite stringent and achieving a high
probability level over a time interval $\tau/4$ can be difficult, the
condition $R\,\Delta t\gtrsim 1$ is significantly relaxed considering longer
interaction time. Indeed, significant (order 1) pair production probability is
expected on forthcoming multi-PW laser facilities (see Sec. 4 for details).

Let us finally note that, when $R\,\tau/4\geq 1$ (yellow area), more than
$63\%$ of conversion is achieved after crossing a single wavelength of the
background field, and more than $99\%$ of the incident gamma photons are
converted into pairs in less than 3 periods. This gives a very robust
condition for abundant pair creation, and it determines a limit above which is
not useful to further increase the laser amplitude to increase the number of
primary pairs. As we will show in the following, this condition
($R\,\tau/4\geq 1$) can also be invoked to find the optimal laser transverse
shape and focal spot for a given laser energy. Indeed 3D PIC simulations,
shown by the symbols in Fig. 4 and discussed in detail in the next Sec. 3,
show a different behaviour depending on whether their initial parameters lie
in the yellow or green region of Fig. 4. In the case of the two simulations
reported as squares in the figure, we will see that these differences are
observed even though the simulation parameters correspond to the same value of
$\chi_{0}$.

##  3 Geometrical effects on pair production: the case of LG beams

This Section aims to generalize the model developed in Sec. 2.2.2, to overcome
the plane wave approximation and account for the laser beam transverse
profile. We start by reminding some properties of the LG beams, limiting
ourselves to the critical aspects for the understanding of pair production. We
then compare the predictions of our theoretical model (generalized to account
for the laser transverse profile in Sec. 3.3) with the results of 3D PIC
simulations, whose initialization is discussed in Sec. 3.2. A thorough
comparison of the efficiency of pair production considering beams with the
same energy and different field structures is presented in Secs. 3.4 and 3.5.
We explore first the impact of the beam geometry and amplitude in a regime of
efficient pair production (time-integrated rate $R_{m}\tau/4>1$) and then in a
regime ($R_{m}\tau/4<1$) more relevant for current experimental facilities (as
will be discussed in the following Sec.4). Finally in Sec. 3.6, we discuss
some properties of the produced pairs, such as the energy spectrum and the
propagation direction.

###  3.1 Properties of LG beams

The LG modes are solutions of the paraxial equation with cylindrical symmetry
and a generalization of the Gaussian beam. Each mode is indexed by two
integers: $p\geq 0$ for the radial order and $\ell$ for the azimuthal one.
Hence, the notation ${\rm LG}_{p\ell}$ will be used in the following to
identify a specific mode.

Starting from Eq. (10), the electric field of an LG beam can be expressed
using $E_{0}=a_{0}\,\mathfrak{R}\left(u_{p\ell}\right)$, where the complex
envelope $u_{p\ell}$ in cylindrical coordinates is [61]

$\displaystyle u_{p\ell}\left(\rho,\phi,z\right)$ $\displaystyle=$
$\displaystyle
C_{p\ell}\,\dfrac{w_{0}}{w(z)}\left(\dfrac{\sqrt{2}\rho}{w(z)}\right)^{|\ell|}L_{p}^{|\ell|}\\!\left(\dfrac{2\rho^{2}}{w^{2}(z)}\right)$
(31) $\displaystyle\times$
$\displaystyle\exp{\left[-\dfrac{\rho^{2}}{w^{2}(z)}\right]}\exp{\left[-i\psi_{pl}\left(z\right)+i\ell\phi+i\dfrac{z\rho^{2}}{w^{2}(z)}\right]}\,,$

where $C_{p\ell}=\sqrt{p!/(p+|\ell|)!}$ is a normalization constant,
$L_{p}^{\ell}(x)$ are the generalized Laguerre polynomials,
$w(z)=w_{0}\sqrt{1-z^{2}/z_{R}^{2}}$ is the beam waist with $w_{0}=w(z=0)$ the
waist at focus and $z_{R}=\pi w_{0}^{2}/\lambda$ the Rayleigh length, and
$\psi_{p\ell}\left(z\right)=\left(2p+|\ell|+1\right)\arctan{\left(z/z_{R}\right)}$
is the generalized Gouy phase. Note that the LG modes described by Eq. (31)
have all the same energy and that ${\rm LG}_{00}$ corresponds to a Gaussian
beam with maximum field amplitude $a_{0}$.

Figure 5: Photon quantum parameter $\chi_{\gamma}$ in the head-on collision of
photons of energy $\gamma_{\gamma}=400$ with a LP LG beam at focus [i.e. $z=0$
in Eq. (31)] with $a_{0}=2000$, $p=0$ and a) $\ell=0$, b) $\ell=1$, c)
$\ell=2$, d) $\ell=3$. e) Maximum of the complex envelope $u^{\rm
max}_{p\ell}$ from Eq. (33) (black line) and for $p\in[0,3]$ as a function of
$|\ell|$. For $p\neq 0$, Eq. (32) is solved numerically.

In order to infer how the structure of the LG modes might affect pair
production we show in Fig. 5(a)-(d) the quantum photon parameter
$\chi_{\gamma}$ for the head-on collision of photons of energy
$\gamma_{\gamma}=400$ with a LP LG beam at focus (i.e. $z=0$) with
$a_{0}=2000$, $p=0$ and $\ell\in[0,3]$. As can be seen in Fig. 5(a)-(d), for
$\ell\neq 0$ the maximum of the field, and thus the peak values of the quantum
parameter, are no longer along the beam axis (i.e. $x=y=0$), in contrast with
the Gaussian beam case. The radius at which the field has its maximum
amplitude $\rho^{\rm max}$ can be obtained by solving
$\partial_{\rho}|u_{p\ell}|^{2}=0$, leading to:

$\displaystyle\frac{2\rho^{2}}{w(z)^{2}}+\left(1-\delta_{0p}\right)\frac{4\rho^{2}}{w(z)^{2}}\frac{L_{p-1}^{|\ell|+1}\\!\left(\frac{2\rho^{2}}{w(z)^{2}}\right)}{L_{p}^{|\ell|}\\!\left(\frac{2\rho^{2}}{w(z)^{2}}\right)}-|\ell|=0\,,$
(32)

with $\delta_{ij}$ the Kronecker delta. For $p=0$, Eq. (32) can be solved
analytically and gives $\rho^{\rm max}=w(z)\sqrt{|\ell|/2}$. While the radial
position of the field maximum for $p=0$ increases with $|\ell|$, the amplitude
decreases, as shown in Fig. 5(e), and tends asymptotically to zero for
$|\ell|\gg 1$ as $|2\pi\ell|^{-1/4}$. The maximum amplitude of the envelope
$u_{0\ell}$ can be computed inserting $\rho^{\rm max}$ into Eq. (31), which
gives

$\displaystyle u^{\rm
max}_{0\ell}=|u_{0\ell}(\rho^{\rm{max}},\varphi,z)|=\frac{1}{\sqrt{1+z^{2}/z_{R}^{2}}}\,\frac{|\ell|^{|\ell|/2}\,{\rm
e}^{-|\ell|/2}}{\sqrt{|\ell|!}}\,.$ (33)

Figure 5(e) also shows $u^{\rm max}_{p\ell}$ for $p\neq 0$, obtained by
inserting the numerical solution of Eq. (32) into Eq. (31). The maximum field
amplitude for the LG beams decreases with both $|\ell|$ and $p$, and as a
consequence the same behaviour is observed for the maximum value of
$\chi_{\gamma}$ [see Fig. 5(a)-(d)]. Therefore, in the following we will
consider only $|\ell|\leq 5$ and $p=0$, because of the higher field amplitude
(hence higher values of $\chi_{\gamma}$) and because of the practical
challenges in producing high order modes at relativistic intensity.

Note that in the considered cases with $\ell\neq 0$, $\chi_{\gamma}$ has
$2|\ell|$ maxima in the transverse plane [see Fig. 5(b)-(d)] and, looking at
the evolution in time, each maximum makes a full rotation around the laser
propagation axis in $|\ell|\tau$. This means that, in a fixed transverse
plane, a maximum rotates to the location of the next one in half a period.

If we consider the head-on collision of the LG beam with a gamma photon, the
photon is crossing half a period of the field in $\tau/4$. Hence, the photon
crosses a rotating local maximum in $\tau/4$. This observation is fundamental
for generalization of the reduced model presented in the previous Sec. 2.2.2,
where we consider subsequent intervals of duration $\tau/4$ to infer the total
probability.

As noted in Fig. 5, the LG beams not only have a non-trivial spatial
distribution of the fields in the transverse plane, but also a maximum field
amplitude decreasing with $|\ell|$ [following Eq. (33)]. In order to
distinguish the impact on pair production of these two aspects, we discuss the
interaction with extended Gaussian (EG) beams. For each LG beam (i.e. each
value of $|\ell|$), we define the associated EG beam as the Gaussian beam that
has the same maximum field amplitude and carries the same total energy. As
$u^{\rm max}_{p\ell}$ decreases with $|\ell|$ [Eq. (33)] and the energy is
kept constant, the waist of the EG beams increases with respect to the
fundamental Gaussian (equivalent to $\rm LG_{00}$) as
$w_{\ell}=w_{0}\sqrt{|\ell|!}\,{\rm e}^{|\ell|/2}|\ell|^{-|\ell|/2}\geq
w_{0}\,.$

###  3.2 Simulation set-up

In what follows, we present 3D PIC simulations performed with the open-source
code Smilei [51], considering the setup schematically represented in Fig. 6.
Each simulation reproduces a volume of $40\lambda\times 40\lambda\times
14\lambda$ (in the $x$, $y$ and $z$ directions, respectively) with spatial
resolution $\lambda/24$ and temporal resolution at 95% of the CFL condition
[62].

An intense laser pulse (with wavelength $\lambda=0.8\,\mu\rm m$) is injected
from the left boundary $z=-5\lambda$ using the method presented in Ref. [63]
that allows for the Maxwell-consistent injection of an electromagnetic wave
with an arbitrary spatio-temporal profile. Here, this method is used to inject
the laser pulse by prescribing its spatio-temporal profile in its focal plane
[i.e. the $(x,y)$-plane at $z=0$], as given by the real part of Eq. (31) at
$z=0$, multiplied by a temporal envelope so that the laser pulse has a
$\sin^{2}$ temporal profile in intensity with fwhm
$\tau_{\rm\scriptscriptstyle{FWHM}}=5\,\tau$, and total duration of
$10\,\tau$. The laser beam collides head-on with a counter-propagating flash
of high-energy photons with a finite longitudinal extension
$L_{\gamma}=\lambda/6$, transverse extension equal to the simulation box, and
a density $n_{\gamma}$ equal to the critical density
$n_{c}=\epsilon_{0}m_{e}\omega^{2}/e^{2}$. The duration of the simulation is
long enough for the two light beams to stop interacting, but short enough so
that none of the photons or secondary particles escape from the simulation
box. All results presented hereafter have been extracted at the end of the
simulations.

In Secs. 3.4 and 3.5, we present two series of simulations: the first one with
$a_{0}=2000$ and $\gamma_{\gamma}=400$, the second with $a_{0}=400$ and
$\gamma_{\gamma}=2000$, both series corresponding to a reference quantum
parameter $\chi_{0}\simeq 6\times 10^{-6}\,a_{0}\,\gamma_{\gamma}\sim 4.85$
(for $\lambda=0.8~{}{\rm\mu m}$). For each series, we varied the transverse
spatial profile of the high-intensity laser pulse in order to study the impact
of the spatio-temporal field profile, comparing different LG beams (varying
$\ell$, but keeping $p=0$) and the corresponding EG beams. Note however that
within each series, $a_{0}$ is left unchanged and refers to the maximum field
amplitude of the reference Gaussian beam (for which we consider
$w_{0}=3\lambda$). As a result, within each set of simulations, the laser
pulse energy is unchanged, equal to $\simeq\rm 10\,kJ$ for $a_{0}=2000$ and
$\simeq\rm 410\,J$ for $a_{0}=400$. Conversely, the photon flash surface
energy density $m_{e}c^{2}\,\gamma_{\gamma}n_{\gamma}d$ is $\simeq 5\,\rm
mJ/\lambda^{2}$ for the first series (with $\gamma_{\gamma}=400$) and $\simeq
25\,\rm mJ/\lambda^{2}$ for the second one ($\gamma_{\gamma}=2000$).

As highlighted in Fig. 4 (squares), the reference Gaussian case of the first
series of simulations allows us to explore the physics in the regime of high
pair production probability after a single $\tau/4$ interval (i.e.
$R_{m}\tau/4>1$, yellow region). On the contrary, with the second series we
investigate a regime where efficient pair production is achieved thanks to the
cumulative effects of several wavelength (i.e. $R_{m}\tau/4<1$, but the
reference Gaussian case is still above the green line in Fig. 4).

Figure 6: Typical set-up of the 3D PIC simulations, with $\lambda$ and $\tau$
the laser wavelength and the optical cycle, respectively.

###  3.3 Reduced model for pair creation with arbitrary spatio-temporal laser
beam profiles

In this Section we generalize the model developed in Sec. 2.2.2 to accurately
describe pair production from complex, spatially structured laser beams.

To do so, we exploit the fact that we are here dealing with high-energy
photons colliding head-on with the strong background field. Hence, the photon
momenta are left unchanged as the photon crosses the background field, and
their trajectories are straight lines so that the field seen by a given photon
along its trajectory is known and depends only on the photon initial position
($x_{0},y_{0}$) in the plane transverse to the direction of propagation of
both light beams666For analytical tractability, we consider all photons to be
propagating exactly in the $z$-direction.. It follows that the quantum
parameter of any photon is known throughout the photon interaction with the
background field, and for a given background field, is determined uniquely by
the photon initial position ($x_{0},y_{0}$). As a result, the total
probability $P_{\rm tot}$ for any photon to decay into a pair after
interacting with the laser pulse is also determined by the photon initial
position, and the total number of produced pairs at the end of the interaction
between the flash of high-energy photons (with density $n_{\gamma}$ and
longitudinal width $L_{\gamma}$) simply reads

$\displaystyle N_{pair}=n_{\gamma}L_{\gamma}\,\sigma_{\rm
tot}\quad\text{with}\quad\sigma_{\rm tot}=\int\\!dx_{0}\,dy_{0}\,P_{\rm
tot}(x_{0},y_{0})\,.$ (34)

A key quantity to model pair production is thus the total cross-section
$\sigma_{\rm tot}$, here defined as the integral of the total probability over
the transverse plane.

To compute the total probability at any position ($x_{0},y_{0}$) we use Eq.
(29) and proceed as described in Sec. 2.2.2, reconstructing the total
probability from successive time intervals of duration $\tau/4$ (as discussed
in Sec. 3.1, a photon at any position in the transverse plane explores in
$\tau/4$ a local maximum of the field amplitude whatever value $\ell$
assumes). One then only needs to compute the maximum field amplitude
satisfactorily. For simplicity, we do so measuring the field amplitude at
focus, taking the absolute value of the field envelope from Eq. (31) at $z=0$.
This approach, which neglects the effects of diffraction, is justified
whenever the duration of the interaction
($\sim\tau_{\rm\scriptscriptstyle{FWHM}}$) does not exceed the time $z_{R}/c$
for light to cross the laser pulse Rayleigh length. As shown in the following,
this approximation proves satisfactory considering typical ultra-short
(Ti:Sapphire) light pulses.

Figure 7: Probability map for a photon with energy $\gamma_{\gamma}=400$ to
produce a pair during the interaction with the $\rm LG_{05}$ beam with fwhm of
$5\tau$ and maximum amplitude of the reference Gaussian beam equal to
$a_{0}=2000$, corresponding to $\chi_{0}\simeq 4.85$. a) Analytical prediction
from Eq. (29). b) Numerical results from the 3D PIC simulation.

The result of this procedure is shown in Fig. 7(a) for the case of a LG beam
with $\ell=5$, $p=0$ and $a_{0}=2000$ colliding with a flash of photons with
energy $\gamma_{\gamma}=400$ (corresponding to $\chi_{0}=4.85$; all other
parameters are specified in Sec. 3.2). Panel (a) shows a map, in the
transverse plane, of the probability $P_{\rm tot}$ computed over the whole
interaction time in the transverse plane. Integrating over the transverse
plane provides the measure of the total cross-section $\sigma_{\rm tot}\simeq
91.4\,\lambda^{2}$.

To test the impact of the approximations used in the model (phase and
diffraction effects are neglected), we show in Fig. 7(b) the probability map
extracted from a 3D PIC simulation, where all effects neglected in our model
are taken into account. Here the probability is extracted at the end of the
interaction as the ratio of the remaining photon linear density by the initial
one. An excellent agreement is found with the theoretical prediction of panel
(b), with a measured total cross-section $\sigma_{\rm tot}\simeq
91.0\,\lambda^{2}$, which confirms the predictive capability of our simple
model for the case of spatially structured beams.

###  3.4 Simulations in the regime of high pair production probability
$(R_{m}\tau/4>1)$

Figure 8: a) Number of produced pairs, and corresponding cross section
$\sigma_{\rm tot}/\lambda^{2}$ after the interaction of gamma photons with
energy $\gamma_{\gamma}=400$ and a counter-propagating LG with $p=0$,
$\ell\in[\\![-1,5]\\!]$ and $a_{0}=2000$ [CP (blue), LP (black)], and the
corresponding LP EG (green). Frozen simulations (dots), full simulations
(squares) and theoretical predictions from Eq. (34) (dashed lines). The insets
show the probability map for the $\rm LG_{00}$, $\rm LG_{05}$ and $\rm
EG_{05}$ cases. b) Relative difference in the number of produced pairs between
the frozen and full simulation results.

In order to investigate the dependence on the laser structure of the number of
produced pairs, we present here a study in the regime satisfying
$R_{m}\tau/4>1$ for the reference Gaussian beam, in which we expect efficient
pair production. We remind that we consider here photons with normalized
energy $\gamma_{\gamma}=400$ and a reference Gaussian beam with maximum
amplitude $a_{0}=2000$, leading to $\chi_{0}\simeq 4.85$ (all other parameters
have been specified in Sec. 3.2).

To test our analytical predictions and the assumptions at the base of our
model, we performed simulations (hereafter referred to as frozen cases) in
which the produced pairs are prevented to further radiate. In this condition,
no secondary pairs are produced and the simulations allow to directly test the
proposed analytical model. We also performed simulations including the full
system dynamics (hereafter referred to as full cases), meaning that the
produced pairs interact with the laser fields and potentially radiate. This
allowed us to verify that we are still in the soft-shower regime, and that
only a small departure from predicted number of produced pairs is indeed
observed.

Figure 8(a) shows the number of produced pairs at the end of the interaction
for all the tested cases, LP LG (in black), CP LG (blue) and EG (green), dots
refer to the frozen simulations and squares to the full ones. Considering a LP
LG laser beam, pair production efficiency increases with $\ell$. This means
that for the high value of $a_{0}$ considered here, it is preferable to
increase $\ell$, even if it is decreasing the maximum amplitude [as shown in
Fig. 5(e)], as it increases the characteristic transverse size of the beam,
and so the interaction area where the probability is high [see inset in Fig.
8(a)]. The EG beams (which have by construction the same maximum amplitude of
the LG beams but Gaussian profile) are more efficient than the corresponding
LG, up to to $\ell=4$, as they have a bigger region with substantial
probability in their probability maps. However, a slightly greater number of
pairs are produced in the simulation with $\rm LG_{05}$ than $\rm EG_{05}$
beam. This because the maximum field amplitude has dropped substantially and a
large part of the interaction region in the EG case has low probability of
pair production [see inset in Fig. 8(a)]. Predictions form our model discussed
in Sec. 3.3 and based on Eq. (34), shown in dashed lines, are in very good
agreement with both EG and LG simulations.

Based on the prediction from Fig. 1, for the considered $\chi_{0}\simeq 4.85$
we would expect at first look a higher efficiency in the CP case than with LP,
contrary to what is obtained from PIC simulations [blue symbols in Fig. 8(a)].
However, given the spatial structure of the laser field in the transverse
plane, in a large part of the interaction region the quantum number is
$\chi<2.5$, i.e. the value above which CP should be favoured based on the
analysis shown in Fig. 1(b) obtained within the plane wave approximation.
Hence, the slightly higher number of pairs produced in the CP case with
respect to LP in the region where $\chi>2.5$, cannot compensate the higher
efficiency of LP in the region of $\chi<2.5$. Note that the results using a CP
beam are independent from the sign of $\ell$ (i.e from the total angular
momentum). This suggests that the total angular momentum of the laser has no
effect on the number of pair produced.

For completeness, we compare the results of the frozen cases with simulations
reproducing the full dynamics of the system [squares in Fig. 8(a)]. As
highlighted in Fig. 8(b), the difference in the produced number of pairs is
always below $25\%$ of its value in the frozen case, and it decreases with
$|\ell|$. This confirms that the majority of pairs comes from the conversion
of primary gamma photons and not from subsequent radiation, as expected for a
soft shower. In this regime, our model, which accurately captures the physics
of pair production for the frozen simulations (dashed lines), can be therefore
used to predict pair production in realistic conditions.

Figure 9: a) Number of produced pairs and corresponding cross section
$\sigma_{\rm tot}/\lambda^{2}$, after the interaction of gamma photons of
energy $\gamma_{\gamma}=2000$ with a counter-propagating LG beam with $p=0$,
$\ell\in[\\![-1,5]\\!]$ and $a_{0}=400$ [CP (blue) and LP (black)], and the
corresponding LP EG (green). Full simulations (squares) and theoretical
predictions from Eq. (34) (dashed lines). The insets show the probability map
for the $\rm LG_{00}$, $\rm LG_{05}$ and $\rm EG_{05}$ cases. b) Relative
difference in the number of produced pairs between the frozen and full
simulation results. The frozen cases are not shown in panel (a) as the
difference with the full ones is always below 15%.

###  3.5 Simulations in the regime of moderate pair production probability
$(R_{m}\tau/4<1)$

In this second series of simulations we explore the regime of $R_{m}\tau/4<1$,
maintaining the same maximum $\chi_{0}$ as in the previous section by
exchanging the values of field amplitude (now equal to $a_{0}=400$) and
photons energy (now equal to $\gamma_{\gamma}=2000$). Note that considering a
laser duration ($\tau_{\rm\scriptscriptstyle{FWHM}}=5\tau$), we still expect
efficient pair production (the reference plane wave simulation lies above the
green dashed line in Fig. 4). This can be seen in the inset of Fig. 9(a) for
the Gaussian case $\rm LG_{00}$ where a significant region exhibits a
probability of order $1$.

As Figure 9 shows, for the simulations with LG beams the number of produced
pairs is optimized for $\ell=3$. The increase for $|\ell|\leq 3$ can be
explained as in the previous section by geometrical considerations, i.e. the
area with the field being large enough to reach $P_{\rm tot}$ of order one is
getting wider with $|\ell|$. In the contrary for $|\ell|>3$, the laser field
amplitude becomes too low and having a laser beam with a larger transverse
extension does not improve pair production. This is due to the fact that the
field is becoming too low, so that the effect of decreasing $P_{tot}$ on pair
production is more important, and is not compensated anymore by the increase
in transverse size.

Moreover, in this regime, all LG beams perform better than the corresponding
EG, given that the region with high (even though smaller than in the previous
Sec. 3.4) probability is larger for the LG than EG, as shown in the inset of
Fig. 9 for $\ell=5$. Also, a similar behaviour as discussed in the previous
Sec. 3.4 is observed for the CP simulations and can be explained with the same
reasoning. Last but not least, our model is reproducing with high accuracy the
observed number of pair in this case too.

###  3.6 Energy and angular distribution of the produced pairs

Figure 10: Positron distribution in Lorentz factor $\gamma$ (equiv. normalized
energy) and angle $\theta=\arctan(p_{x}/p_{z})$ for the first simulation
series (left column) and for the second one (right column); LP (CP) in the top
(bottom) row. The insets show the energy spectra for positrons propagating in
the positive (black line) and in the negative (red line) directions.

The energy-angular distribution of the produced pairs in the $z-x$ plane (i.e
the plane formed by the laser propagation direction $\hat{\bf z}$ and the
polarization direction $\hat{\bf x}$ for the LP case) recorded at the end of
the interaction is shown in Fig. 10 considering the reference LP and CP
Gaussian beams. The top row gives the positron distribution in
$\theta=\arctan(p_{x}/p_{z})$ and in energy for the LP beams and the bottom
one for the CP beams. The insets show the energy spectra of particles moving
in the positive (same as the laser, black line) and negative (same as the
gamma flash, red line) $z$-directions.

At high $a_{0}$ (panels a and c) the produced pairs, initially generated with
momentum in the negative $z$-direction, are predominantly moving in the same
direction as the laser (corresponding to $\theta=0^{\circ}$) within a cone of
aperture $40^{\circ}$, meaning that they have been slowed down and then pushed
by the laser. They reach a maximum energy of $\simeq 10^{4}\,m_{e}c^{2}$ for
the LP case and $\simeq 8\times 10^{3}\,m_{e}c^{2}$ for the CP beam,
consistent with the decrease of the peak field amplitude in CP with respect to
LP at constant energy.

The spatial distribution is drastically modified when considering $a_{0}=400$.
Even if there still is a substantial amount of pairs propagating in the laser
direction, the majority of them keep their original direction, along the
initial gamma photons propagation axis and anti-parallel to the laser pulse
one. For these pairs the maximum energy is independent from the polarization,
as shown in Fig. 10(b)-(d), contrarily to the pairs that are slowed down and
pushed by the laser (right side of the quadrants or black line in the insets).

This study, supported by complementary simulations (not shown), suggests that
for $a_{0}\gtrsim\gamma_{\gamma}$ most of the created particles will end up
propagating along the laser direction, while for smaller values of the laser
maximum amplitude a large fraction of pairs will keep their original
direction. A complete characterisation of the pairs spectrum and
directionality is beyond the scope of this work, and will be investigated in
more details in the future, as it can give important information for the
planning of upcoming experimental campaigns.

##  4 Discussion

The model developed in this work allows to predict the pair production
capability of upcoming facilities. In Fig. 11, we discuss the pair production
maximum probability and total cross-section as a function of the photon energy
and laser amplitude. The considered parameter range is relevant to ultra-short
Ti:Sapphire ($\lambda=0.8~{}{\rm\mu m}$) lasers, where studies of high-field
physics and QED are already planned [64, 65, 66, 67]. Two types of facilities
can be distinguished. First, the Apollon777The Apollon, located on the Plateau
de Saclay, south of Paris, France, started operating at the PW-level [2, 68];
see also the Apollon website., CoReLS888CoReLS, in Gwanju, South Korea, is
operating at up to 4PW [3]; see also the CoReLS website. and zeus999A 3PW
laser facility being built at the University of Michigan; see the zeus
website. facilities (shown in black in Fig. 11) are designed to deliver
multiple light beams with duration in between 15 and 30 fs (fwhm) and peak
power from 1 to 10 PW. Second, the facet-ii101010facet-ii is planned at SLAC
National Accelerator Facility [69]; see also the facet-ii website. and
luxe111111luxe is an experiment proposed at the European XFEL [70]; see also
the luxe experiment website. experiments will couple conventional electron
accelerators with high-intensity (100 TW-class) ultra-short laser pulses (in
white in Fig. 11).

In Fig. 11(a,b) the accessible $a_{0}$ and typical $\gamma_{\gamma}$
envisioned with these facilities (following the reference articles) are
reported. For Apollon only, we computed $a_{0}$ considering a 20 fs light
pulse delivering 20, 60 and 150 J on target, corresponding to peak power of 1,
3 and 7.5 PW, respectively. The reported value of $a_{0}$ is then computed for
a LP Gaussian beam with focusing aperture $f/3$ (where the beam waist for a
focusing aperture $f/N$ is $w_{0}\simeq 0.90\,N\lambda\simeq 2.2~{}{\rm\mu
m}$), using

$\displaystyle a_{0}$ $\displaystyle\simeq$
$\displaystyle\dfrac{151}{N}\sqrt{\dfrac{\mathcal{E}_{\rm
laser}}{10\rm{J}}}\sqrt{\dfrac{25\,\rm{fs}}{\tau_{\rm\scriptscriptstyle{FWHM}}}}\simeq\dfrac{239}{N}\sqrt{\dfrac{P_{\rm
laser}}{1\,\rm{PW}}}\,.$ (35)

Note that, as most of these facilities will operate in the $\chi_{0}\gtrsim 1$
regime, we consider photon energy close to that of the electron beams either
expected from laser-driven wakefield acceleration (for Apollon, CoReLS and
zeus) or emerging from the linear accelerators (for facet-ii and luxe).

Figure 11: Model predictions considering a laser pulse with $\sin^{2}$ time-
envelope in intensity ($\tau_{\rm{FWHM}}=25\,\rm{fs}$) and
$\lambda=0.8\,\mu\rm{m}$. (a) Maximum pair production probability from Eq.
(29). (b) Total cross section normalised to $w_{0}^{2}$ using the model
discussed in Sec. 3.3. In both panels (a) and (b) the red lines denote
contours of constant $\chi_{0}=0.1\,,\,1\,,\,16.5$. The regimes accessible
with current and upcoming facilities are also reported. c) Total cross section
normalised to $\lambda^{2}$ as a function of $\gamma_{\gamma}$ computed for
the Apollon facility operating at 1 PW (solid lines), 3 PW (dashed lines) and
7.5 PW (dash-dotted line) and with focusing aperture $f/3$ (black lines) and
$f/1.5$ (red lines). In this last panel, at a given power, the laser energy is
the same for the different focusing aperture $f/N$.

Let us now focus on panel (a), which shows the maximum total decay probability
[from Eq. (29)] for a photon with energy $\gamma_{\gamma}$ colliding head-on
with a LP Gaussian laser pulse with maximum field strength $a_{0}$ and
duration $\tau_{\rm\scriptscriptstyle{FWHM}}=25~{}{\rm fs}$ (typical for the
facilities listed above). This panel suggests the strategy to optimize pair
production depending on the type of facility.

First, despite the extremely high electron beam energies, facilities such as
luxe and facet-ii (in white in Fig. 11) offer the possibility to probe the
regime of moderate quantum parameter ($\chi_{0}\lesssim 1$) for which pair
production may be observed ($P_{\rm tot}\simeq 10^{-2}$) but will not be
abundant. This limit might be overcome by increasing the laser power up to few
100’s of TW (300 TW are envisioned at luxe), which allows to enter the quantum
regime ($\chi_{0}>1$) and increase $P_{\rm tot}$ to values exceeding $0.1$. On
these facilities however, because $P_{\rm tot}$ will not assume large values,
increasing $a_{0}$ (e.g. by focusing the laser pulse as much as technically
possible) is one of the most promising path to achieve abundant pair
production.

In contrast, provided that multi-GeV electron beams can be obtained from laser
wakefield acceleration, abundant pair production ($P_{\rm tot}\gtrsim 0.63$)
is expected on all multi-PW laser facilities with the focalisation technique
already in place (typically an aperture in between $f/3$ and $f/4$ was
considered in Fig. 11).

Similar conclusions can be drawn from panel (b) where we examine in more
details the influence of the laser pulse spatial profile, by looking at the
total cross-section121212Since the Gaussian field profiles depend on the
transverse coordinate through the ratio $\rho/w_{0}$ [Eq. (31)], $\sigma_{\rm
tot}/w_{0}^{2}$ is independent of $w_{0}$. $\sigma_{\rm tot}$ [as defined by
Eq. (34)] normalized to $w_{0}^{2}$, considering a Gaussian laser beam with
the same temporal properties as in panel (a).

For the facet-ii and luxe facilities, the total cross-section assumes very
small values but increases fast as $a_{0}$ is increased: e.g. for luxe,
$\sigma_{\rm tot}$ increases by 3 orders of magnitude increasing $a_{0}$ by a
factor 10. Clearly, operating at large field strength will be a bottleneck for
achieving abundant pair production on these facilities.

In contrast, in the range of parameters covered by multi-PW facilities, the
dependence of $\sigma_{\rm tot}$ with $a_{0}$ is much weaker. As in addition,
$\sigma_{\rm tot}\simeq w_{0}^{2}$ (consistent with $P_{\rm tot}\simeq 1$ at
the center of the beam), it becomes interesting for these facilities to
increase the laser transverse size rather than opt for tight focusing.

Let us finally discuss in more details the impact of focusing for the three
upgrades of the Apollon facility (at 1, 3 and 7.5 PW). In panel (c), we
present the total cross section (in units of $\lambda^{-2}$) as a function of
$\gamma_{\gamma}$, considering either the standard $f/3$ aperture (black
lines) or the more challenging $f/1.5$ aperture (red lines). For each laser
power, a tighter focusing reduces the beam waist but increases the maximum
$a_{0}$.

In all cases, the cross section is rapidly increasing with the photon energy
for $\gamma_{\gamma}\lesssim 2000$ ($\simeq 1$ GeV). It is almost flat as
$\gamma_{\gamma}$ reaches $10^{4}$ ($\simeq 5$ GeV), which suggests that it is
not worth increasing the photon energy above this value in forthcoming
experiments to optimize pair production in the soft shower regime.

As shown in panel (c), for a given laser power the cross section for $f/1.5$
is larger than for $f/3$ only for small values of $\gamma_{\gamma}$, while the
opposite behaviour is observed at large $\gamma_{\gamma}$ (of the order of a
few GeV). This means that such tight focus, which is technologically very
challenging, is not necessary for pair creation in forthcoming experiments on
Apollon.

To test this prediction, we have performed complementary 3D PIC simulations of
the 3 PW case with $f/3$ and $f/1.5$ ($a_{0}=138.8$ and $276.7$, respectively)
interacting with a gamma flash with energy $\gamma_{\gamma}=10^{4}$. For the
$f/3$ case we find $\sigma_{\rm tot}=26.8\,\lambda^{2}$ which is about 30%
higher than the frozen case ($\sigma_{\rm tot}=22.1~{}\lambda^{2}$). This
remains larger than the total cross-section obtained with the $f/1.5$
aperture, for which $\sigma_{\rm tot}=19.5~{}\lambda^{2}$. In that case
however, we note a significant (by a factor $\sim 2$) departure from the
frozen case ($\sigma_{\rm tot}\simeq 9.8~{}\lambda^{2}$) as more secondary
pairs are produced.

##  5 Conclusion

In summary, we have presented a systematic study of nonlinear Breit-Wheeler
pair production in the head-on collision of high-energy gamma photons with an
ultra-high intensity laser beam. Combining analytical modeling and PIC
simulations embarking the relevant QED modules, we have evaluated the impact
on the efficiency of pair production of the laser spatio-temporal profile
[comparing e.g. Gaussian with different focal spots and Laguerre-Gaussian (LG)
pulses] and parameters such as polarisation, intensity and duration. Motivated
by experimental constraints, we have considered fixed laser energies while
changing these parameters.

We have explored laser field strength and photon energy parameter ranges
relevant for currently and/or upcoming high-power laser facilities, focusing
in particular on ultra-short Ti:Sapphire laser facilities. A reduced model was
proposed that allows to describe pair production in the regime of soft-shower,
where secondary pair production can be neglected. This model, that would in
principle allow to predict a minimum number of produced pairs, is found to
agree remarkably well with 3D PIC simulations over a broad range of
parameters, highlighting the importance of the soft-shower regime for
experiments on the forthcoming laser facilities.

The model also allows to distinguish two regimes of interaction depending on
whether the probability for a photon to decay into a pair as it crosses a
single half-wavelength of the laser pulse is large (of order 1) or not. The
two regimes are not however fully determined by the photon maximum quantum
parameter $\chi_{0}$, so that the laser field strength parameter $a_{0}$ and
photon energy $\gamma_{\gamma}$ do not play a symmetric role in the system
dynamics. This was confirmed in 3D PIC simulations where the two regimes were
investigated at $\chi_{0}=4.85$ considering laser beams with complex spatio-
temporal profiles, LG beams in particular, and different polarizations. It was
found that, for a fixed laser energy, using circular polarization or LG does
not improve significantly pair production. As they are also quite difficult to
achieve experimentally, their effect on pair production is too marginal to be
interesting for applications. It was found however that, in the regime of high
pair production probability, it can be preferable not to focus much the laser
beam (keeping the same total energy with lower peak intensity) and maintain a
Gaussian shape in order to maximise the area with high-enough fields for
efficient pair production.

These findings help to draw guidelines for future experiments on ultra-high
intensity facilities. We show in particular that the path to abundant pair
production is different whether one considers $100\,$TW-class or multi-PW
laser systems. In particular, for facilities such as facet-ii or luxe, 10 to
$100\,$TW-class lasers will be coupled to ultra-relativistic 10-20 GeV
electron beams emerging from conventional electron accelerators. On these
facilities, pair production will develop in the low probability regime and
operating in a tightly-focused configuration to access the highest possible
laser field strength will be a major experimental challenge. In contrast,
using multiple laser beam multi-PW facilities will operate in the regime of
high pair production probability. In this regime, the pair production
efficiency (measured in our work in terms of a total cross section) depends
more strongly on the laser pulse transverse size than on its maximal field
strength so that operating with standard (not too tight) focusing aperture
increases the number of produced pairs. Interestingly, producing high-energy
photons (or electrons, e.g. through laser wakefield acceleration) of a few
($\sim\\!5$) GeV shall be enough for these facilities as higher energies do no
help increase significantly the pair production efficiency in this high
probability regime.

To conclude, this work provides a deeper understanding of the optimal
conditions for pair production in upcoming experimental campaigns exploring
nonlinear Breit-Wheeler pair production. It focused on pair production seeded
by high-energy photons, and it will be interesting for future works to
investigate the differences with the case where ultra-relativistic electrons
are used to seed the pair production process. Studying the transition from the
soft-shower to the cascade/avalanche regime in which secondary pairs and
radiations play a dominant role is another interesting perspective.

## Acknowledgements

The authors are grateful to F. Amiranoff, P. Audebert, S. Corde, L. Gremillet,
L. Lancia, R. Nuter and C. Thaury for stimulating discussions. This work used
the open-source PIC code Smilei , the authors are grateful to all Smilei
contributors and to the Smilei -dev team for its support. Simulations were
performed on the Irene-Joliot-Curie machine hosted at TGCC, France, using High
Performance Computing resources from GENCI-TGCC (Grant No. 2020-x2016057678).
Support by Sorbonne Université in the framework of the Initiative Physique des
Infinis (IDEX SUPER) is acknowledged. This publication is also supported by
the Collaborative Research Centre 1225 funded by Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID
273811115 - SFB 1225.

##  Appendix A Approximation of $\mathcal{I}_{\varepsilon}(\chi_{0})$

In this Appendix, we detail how the integral
$\mathcal{I}_{\varepsilon}(\chi_{0})=\int_{0}^{\pi}b_{0}\\!\left(\chi_{0}\Psi_{\varepsilon}(\varphi)\right)d\varphi$
[Eq. (19)] can be efficiently approximated by Eq. (22). To do so, let us start
by stressing that, in general, and in particular at small $\chi$,
$b_{0}(\chi)$ is a steep function of $\chi$. One thus expects the principal
contribution to the integral to come from the phase $\varphi$ around
$\varphi_{m}=\pi/2$, for which
$\chi_{0}\,\Psi_{\varepsilon}(\varphi)\simeq\chi_{m}$ with
$\chi_{m}=\chi_{0}/\sqrt{1+\varepsilon^{2}}$. To compute the contribution of
this maximum analytically, we approximate the integrand in Eq. (19) around
$\phi_{m}$ as

$\displaystyle b_{0}\\!\left(\chi_{0}\Psi_{\varepsilon}(\varphi)\right)\simeq
b_{0}(\chi_{m})\,\exp\left(-\frac{(\varphi-\varphi_{m})^{2}}{2s_{\varepsilon}^{2}(\chi_{m})}\right)\,,$
(36)

where $s_{\varepsilon}(\chi_{m})$ is chosen so that the exact and approximated
integrand have the same second derivative at $\varphi=\varphi_{m}$, which
gives

$\displaystyle
s_{\varepsilon}(\chi_{m})=\sqrt{\frac{3}{2}}\,\frac{c(\chi_{m})}{\sqrt{1-\varepsilon^{2}}}\quad{\rm
with}\quad c(\chi)=\sqrt{\frac{2b_{0}(\chi)}{3\chi b_{0}^{\prime}(\chi)}}\,.$
(37)

With the ansatz Eq. (36), the integral can be performed analytically, leading
to

$\displaystyle\mathcal{I}_{\varepsilon}(\chi_{0})\simeq\pi\,b_{0}\left(\chi_{m}\right)\,F\\!\left(s_{\varepsilon}(\chi_{m})\right)\quad{\rm
with}\quad F(s)=\sqrt{2/\pi}\,s\,{\rm erf}\big{(}\pi\sqrt{2}/(4s)\big{)}\,.$
(38)

The ansatz Eq. (36) gives a good approximation of the integrand for arbitrary
values of $\varepsilon$ as long as $\chi_{m}$ is small enough. It is also
exact (for all $\chi_{0}$) in the cases $\varepsilon=\pm 1$ for which $s_{\pm
1}(\chi_{m})\rightarrow+\infty$. Hence, Eq. (38) provides a very good, fully
analytical approximation of Eq. (19) for a wide range of $\varepsilon$ and
$\chi_{0}$.

The approximation however needs to be corrected for $\chi_{0}\gg 1$ and
$|\varepsilon|<1$. In this limit, the integrand in Eq. (19) can be
approximated using Eq. (6) leading to

$\displaystyle\mathcal{I}_{\varepsilon}(\chi_{0})\simeq\pi\,b_{0}\left(\chi_{m}\right)\,f(\varepsilon)\quad{\rm
with}\quad
f(\varepsilon)=\frac{1}{\pi}\int_{0}^{\pi}\left[\sin^{2}\varphi+\varepsilon^{2}\cos^{2}\varphi\right]^{1/3}d\varphi\,.$
(39)

Equations (38) and (39) can be combined in Eq. (22). This form ensures the
correct asymptotic behaviour of the integral Eq. (19) in both small and large
$\chi_{0}$ limits. It departs from the exact expression (integrated
numerically) by less than 20$\%$, the error being maximum for intermediate
values of $\chi_{0}\sim 1$. This was tested over a broad range of
$\chi_{0}\in[0.1\,,\,1000]$ for various polarization parameter
$\varepsilon\in\\{0,0.25,0.5,0.75,1\\}$.

##  Appendix B Details on 1D PIC simulations

A series of 1D3V (1 dimensional in space and 3 dimensional in velocity) PIC
simulations, considering a plane wave colliding head-on with a flash of gamma
photons, have been performed to produce Fig. 3c. We simulated a box of length
$2.5\,\lambda$ with spatial resolution $\lambda/256$, for a simulation time of
$1.75\,\tau$, with temporal resolution $\tau/512$. This is a long enough time
to study the full interaction of the plane wave with a flash of gamma photons
of extension $\lambda/2$ and extract the probability from the number of
surviving photons. The probability map in Fig. 3c is obtained from the results
of 4096 ($64\times 64$) simulations performed over a the logarithmically
spaced range of $a_{0}\in[10,4000]$ and $\gamma_{\gamma}\in[100,40000]$.

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