# Pair production seeded by electrons in noble gases
as a method for the laser intensity diagnostics

I. A. Aleksandrov<EMAIL_ADDRESS>Department of Physics, St. Petersburg
State University, Universitetskaya Naberezhnaya 7/9, Saint Petersburg 199034,
Russia  Ioffe Institute, Politekhnicheskaya str. 26, Saint Petersburg 194021,
Russia  A. A. Andreev  Department of Physics, St. Petersburg State University,
Universitetskaya Naberezhnaya 7/9, Saint Petersburg 199034, Russia  ELI-ALPS,
ELI-HU NKft. Dugonics tér 13, Szeged 6720, Hungary  Max-Born Institut, Max-
Born Str. 2A, Berlin 12489, Germany

###### Abstract

In this study we explore the possibility of using the process of electron-
positron pair creation in strong laser fields as a tool for measuring the
intensity of the corresponding laser radiation. In the initial state we
consider either free electron gas or gas of neutral xenon, the electrons of
which get ionized. Once these seed electrons gain sufficient energy in the
external laser field, they can emit high-energy photons which subsequently
decay producing electron-positron pairs via the Breit-Wheeler mechanism. By
detecting the resulting positrons, one can recover the value of the laser
intensity by means of the one-to-one correspondences deduced in the present
investigation. We analyze two different configurations of the external field:
the setup involving an individual focused laser pulse and the combination of
two counterpropagating laser pulses. Performing numerical calculations and
analyzing their accuracy, we demonstrate that based on our estimates, the
laser intensity can be determined within the range
$10^{23}$–$10^{26}~{}\text{W}/\text{cm}^{2}$ with a relative uncertainty of
$10$–$50\%$.

##  I Introduction

The current and forthcoming laser facilities should allow one to achieve a
laser intensity of $10^{22}~{}\text{W}/\text{cm}^{2}$ or even higher paving
the way for probing various strong-field QED phenomena (see, e.g., Refs.
vulkan_2010 ; bashinov_epj_2014 ; zou_2015 ; kawanaka_2016 ; papadopoulos_2016
; weber_mre_2017 ; gales_2018 ; guo_2018 ; danson_2019 ; yoon_2019 ).
Nevertheless, the determination of the peak intensity of such strong laser
pulses represents a formidable task. Recently, a number of various techniques
have been extensively discussed in the literature: measuring yields of highly
charged ions due to atomic ionization ciappina_pra_2019 ; ciappina_lpl_2020 ;
ciappina_mra_2020 , detecting the light scattering or additional radiation due
to the interaction between electrons and the laser field har-shemesh_2012 ;
Vais_2016 ; yan_2017 ; harvey_2018 ; he_2019 ; blackburn_2020 , or the
analysis of photoionization or direct acceleration of charged particles
kalashnikov_2015 ; vais_2017 ; vais_2018 ; ivanov_2018 ; krajewska_2019 ;
mackenroth_2019 ; vais_2020 ; vais_2021  (see also references therein). In the
present paper we discuss how the laser intensity diagnostics can be carried
out using the strong-field QED mechanism of pair production due to the
interaction of free electrons or free xenon atoms with an intense laser field.
As this scenario reflects a threshold dependence on the field amplitude, it
can allow one to accurately resolve the laser intensity by measuring the
positron yield.

We examine the following setup. A high-intensity linearly polarized laser
pulse (or two counterpropagating pulses) travels through a gas chamber with
neutral xenon whose electrons after ionization serve as seed particles for the
subsequent process of nonlinear Compton scattering providing high-energy
photons. These photons can then decay via the Breit-Wheeler mechanism yielding
electron-positron pairs which can cascade further. As this process of positron
production possesses a quite sharp threshold, measuring the positrons produced
provides an accurate method for the determination of the laser intensity. We
will show that the intensity can be extracted within the domain
$10^{23}~{}\text{W}/\text{cm}^{2}\lesssim I\lesssim
10^{26}~{}\text{W}/\text{cm}^{2}$ depending on the laser field configuration.
Besides, we also consider free electrons instead of xenon atoms in order to
demonstrate the advantage of the latter scenario due to more efficient
acceleration of the inner-shell electrons maltsev_prl_2003 ; hu_pre_2006 ;
karmakar_2007 ; artemenko_pra_2017 . We also note that positron production via
the Breit-Wheeler process was considered as a method for the intensity
diagnostics in Refs. lecz_andreev_2019 ; lecz_andreev_2020 , where the laser
field was focused on the surface of a thin foil instead of traveling through a
gas chamber.

In this study we use a relatively simple method for estimating the total
number of positrons created without performing Monte-Carlo simulations, which
have been used in numerous studies (see, e.g., Refs. artemenko_pra_2017 ;
lecz_andreev_2019 ; lecz_andreev_2020 ; nerush_prl_2011 ; elkina_2011 ;
king_pra_2013 ; tamburini_srep_2017 ; vranic_sci_rep_2018 ;
baumann_sci_rep_2019 ; blackburn_njp_2019 ; gu_2019 ; yakimenko_2019 ). In
what follows, we will successively take into account ionization, acceleration
in the laser field, photon emission, and Breit-Wheeler pair production
neglecting further stages of the cascade where the particles created emit new
photons. To make sure that we receive significantly accurate predictions for
the total number of positrons, we will not consider too intense laser pulses
which initiate an avalanche-like reaction fedotov_prl_2010 ; 2015_ufn_narozh .
It turns out that even if one disregards the further cascading, our scheme can
be utilized in a quite large intensity interval.

The paper has the following structure. In Sec. II we discuss how the
ionization process is described within our numerical simulations. In Sec. III
we briefly present the external field configuration and the main points of how
the electron dynamics is computed. In Sec. IV we describe how the QED
mechanisms of photon emission and Breit-Wheeler pair production are
incorporated in our calculations. In Sec. V we discuss and estimate other
effects which are not taken into account in our numerical procedures. In Sec.
VI we present and discuss the results obtained. Finally, we conclude in Sec.
VII.

We will use atomic units: Planck constant $\hbar=1$, electron mass $m=1$,
electron charge $e=-1$. In these units the speed of light in vacuum is
$1/\alpha\approx 137.036$, where $\alpha$ is the fine-structure constant.

##  II Ionization model

Atoms interacting with a strong laser background get ionized providing seed
electrons for subsequent emission of high-energy photons which afterwards
yield positrons via the Breit-Wheeler mechanism. Here we discuss how one can
describe the ionization stage. Consider an atom at a given position
$\bm{r}_{0}=(x_{0},y_{0},z_{0})$. Although it interacts with the laser pulse,
we assume it to be motionless since in what follows we will sum the results
over $\bm{r}_{0}$ within the interaction region, where the atoms are
distributed randomly. For this reason we also assume that after ionization the
free electrons appear at rest at the same point $\bm{r}_{0}$. First, we evolve
in time the ionization probabilities $P_{j}$ for each electron level $j$ (in
Xe there are 54 of them) via

$dP_{j}(t)=[1-P_{j}(t)]W_{j}(t)dt,\quad j=1,2,...,N_{\text{lev}},$ (1)

where $W_{j}(t)$ is the ionization rate depending on $j$, the effective charge
$Z_{j}(t)$, and the electric field strength $E(t)\equiv E(\bm{r}_{0},t)$. At
the time instant $t=t_{\text{in}}$, when the atom starts to interact with the
laser field, we assume that $Z_{j}=j$ and at each time step, we recalculate
these charges according to

$Z_{j}(t)=j+\sum_{k=j+1}^{N_{\text{lev}}}P_{k}(t),$ (2)

i.e., we take into account that the screening of the nuclear charge reduces as
the electrons get ionized. The main problem is to accurately evaluate the
ionization rate $W(t)$. Hereinafter we omit $j$ considering a given energy
level. Let $I_{p}$ and $l$ denote the corresponding ionization potential and
orbital quantum number, respectively (they are taken from the NIST database
nist_ie ). We also introduce the following notations:

$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle\sqrt{2I_{p}},$ (3)
$\displaystyle\nu$ $\displaystyle=$ $\displaystyle Z/\kappa,$ (4)
$\displaystyle\mathcal{F}(t)$ $\displaystyle=$
$\displaystyle|E(t)|/\kappa^{3}.$ (5)

Since the external laser field is extremely strong, the corresponding Keldysh
parameter $\gamma_{\text{K}}=\sqrt{2I_{p}}\omega/|E|$ is very small,
$\gamma_{\text{K}}\lesssim 10^{-3}$. It means that one can employ the closed-
form expressions derived for tunnel ionization.

The simplest model is the so-called Ammosov-Delone-Krainov (ADK) model
ammosov_1986 :

$W_{\text{ADK}}(t)=\frac{|E(t)|}{8\pi
Z}\,\sqrt{\frac{3\mathcal{F}(t)}{\pi}}\bigg{(}\frac{4\mathrm{e}}{\nu\mathcal{F}(t)}\bigg{)}^{2\nu}\mathrm{exp}\bigg{(}-\frac{2}{3\mathcal{F}(t)}\bigg{)}.$
(6)

The WKB approach used in the derivation of Eq. (6) is not justified if the
ionization potential is sufficiently small, which leads to the so-called
barrier-suppression (BS) effects extensively discussed in the literature (see,
e.g., Refs. ciappina_lasphyslett_2020 ; artemenko_pra_2017 ; tong_jpb_2005 ;
krainov_josab_1997 ; zhang_pra_2014 ). The characteristic BS field reads
tong_jpb_2005

$\mathcal{F}_{\text{BS}}=\frac{1}{16\nu}.$ (7)

To take into account the BS effects, one can use an empirical formula proposed
in Ref. tong_jpb_2005  multiplying Eq. (6) by the factor
$\mathrm{exp}[-2\overline{\alpha}\nu^{2}\mathcal{F}(t)]$ for
$\mathcal{F}(t)\gtrsim\mathcal{F}_{\text{BS}}$. (for xenon
$\overline{\alpha}=-9.0$). However, this approximate correction yields
relatively accurate results only when $\mathcal{F}\lesssim
2\mathcal{F}_{\text{BS}}$ zhang_pra_2014 . In Ref. zhang_pra_2014  the authors
proposed a more sophisticated correction which is valid up to
$\mathcal{F}\approx 4.5\mathcal{F}_{\text{BS}}$. Nevertheless, in our
computations it is still a very limited domain. To go beyond this region,
i.e., consider $\mathcal{F}\gg\mathcal{F}_{\text{BS}}$, we will employ the
approach presented in Ref. artemenko_pra_2017 . Using the ADK model, we
compute the ionization rates according to

$W(t)=\begin{dcases}W_{\text{ADK}}(t)&\text{if}~{}\mathcal{F}<\mathcal{F}_{\text{BS}},\\\
W_{\text{ADK}}\,\mathrm{exp}[-2\overline{\alpha}\nu^{2}\mathcal{F}(t)](t)&\text{if}~{}\mathcal{F}_{\text{BS}}\leq\mathcal{F}(t)\leq\mathcal{F}_{\text{L}},\\\
W_{\text{L}}(t)&\text{if}~{}\mathcal{F}>\mathcal{F}_{\text{L}},\end{dcases}$
(8)

where $W_{\text{L}}(t)=2I_{p}\mathcal{F}(t)$ and $\mathcal{F}_{\text{L}}$ is
chosen, so that the function (8) is continuous.

Having evaluated the functions $P_{j}(t)$, we then calculate the total number
of electrons ionized $N^{(\text{el})}(t)=\sum_{j}P_{j}(t)$ and split the $t$
axis into small intervals $[t_{i},t_{i}+\Delta t]$. Each “portion of
electrons” $\Delta N^{(\text{el})}_{i}=N^{(\text{el})}(t_{i}+\Delta
t)-N^{(\text{el})}(t_{i})$ propagates then according to the relativistic
equations of motion. In other words, these classical portions
(“macroparticles”) are treated within the particle-in-cell (PIC) approach
hockney_1988 ; birdsall_1991 . They give rise to photon emission with the
corresponding weights $\Delta N^{(\text{el})}_{i}$. In the case of free
electrons, we deal with only one electron portion of unit weight, so the
ionization stage is omitted.

Finally, we note that we also employed the Perelomov-Popov-Terent’ev (PPT)
model ppt_1966 ; ppt_1967  instead of the ADK model and obtained similar
results. The corresponding slight discrepancies were used in estimating the
final uncertainties presented in Sec. VI.

##  III Electron dynamics and the laser field configuration

Assuming the electron to be a classical relativistic particle traveling in
arbitrary electric and magnetic fields $\bm{E}=\bm{E}(t,\bm{r})$ and
$\bm{H}=\bm{H}(t,\bm{r})$, one has to solve the following system of equations:

$\displaystyle\frac{d\bm{p}}{dt}$ $\displaystyle=$
$\displaystyle-\bm{E}-\frac{1}{c}\,\bm{v}\times\bm{H}+\bm{F}_{\text{rec}},$
(9) $\displaystyle\frac{d\bm{r}}{dt}$ $\displaystyle=$ $\displaystyle\bm{v},$
(10) $\displaystyle\bm{v}$ $\displaystyle=$
$\displaystyle\frac{\bm{p}}{\sqrt{1+\bm{p}^{2}/c^{2}}}.$ (11)

Here $\bm{F}_{\text{rec}}$ is the recoil force which appears due to photon
emission and will be specified in the next section. The initial conditions
have the form $\bm{p}(t_{i})=0$, $\bm{r}(t_{i})=\bm{r}_{0}$, where $t_{i}$ is
the time instant when the electron portion under consideration gets ionized
($t_{i}>t_{\text{in}}$). Once the Lorentz parameter
$\gamma(t)=\sqrt{1+\bm{p}^{2}(t)/c^{2}}$ becomes large ($\gamma\gg 1$), the
electron can emit a high-energy photon via nonlinear Compton scattering.

Let us now present the explicit form of the external background in the case of
an individual laser pulse propagating along the $z$ axis. The external field
is assumed to be a focused Gaussian beam multiplied by a spatial envelope
function within the paraxial approximation (see, e.g., Ref. Vais_2016 ):

$\displaystyle H_{x}(t,\bm{r})$ $\displaystyle=$ $\displaystyle 0,$ (12)
$\displaystyle H_{y}(t,\bm{r})$ $\displaystyle=$
$\displaystyle\frac{E_{0}\rho_{0}}{\rho(z)}\,f(\omega t-k_{0}z)$ (13)
$\displaystyle\times$
$\displaystyle\mathrm{exp}\Big{[}-\frac{x^{2}+y^{2}}{2\rho_{0}^{2}}\Big{]}\sin\phi,$
$\displaystyle H_{z}(t,\bm{r})$ $\displaystyle=$
$\displaystyle\frac{E_{0}\rho_{0}y}{k_{0}\rho_{\text{F}}\rho^{2}(z)}\,f(\omega
t-k_{0}z)$ (14) $\displaystyle\times$
$\displaystyle\mathrm{exp}\Big{[}-\frac{x^{2}+y^{2}}{2\rho_{0}^{2}}\Big{]}\cos\tilde{\phi},$
$\displaystyle E_{x}(t,\bm{r})$ $\displaystyle=$ $\displaystyle
H_{y}(t,\bm{r}),$ (15) $\displaystyle E_{y}(t,\bm{r})$ $\displaystyle=$
$\displaystyle 0,$ (16) $\displaystyle E_{z}(t,\bm{r})$ $\displaystyle=$
$\displaystyle\frac{E_{0}\rho_{0}x}{k_{0}\rho_{\text{F}}\rho^{2}(z)}\,f(\omega
t-k_{0}z)$ (17) $\displaystyle\times$
$\displaystyle\mathrm{exp}\Big{[}-\frac{x^{2}+y^{2}}{2\rho_{0}^{2}}\Big{]}\cos\tilde{\phi},$

where $k_{0}\equiv 2\pi/\lambda$,
$\rho(z)=\rho_{0}\sqrt{(1-z/\mathfrak{F})^{2}+(z/z_{*})^{2}}$,

$\displaystyle\phi(t,\bm{r})$ $\displaystyle=$ $\displaystyle\omega
t-k_{0}z+\arctan\frac{z\mathfrak{F}}{z_{*}(\mathfrak{F}-z)}+\pi\theta(z-\mathfrak{F})$
(18) $\displaystyle-$
$\displaystyle\frac{(x^{2}+y^{2})[z+(z_{*}/\mathfrak{F})^{2}(z-\mathfrak{F})]}{zz_{*}\rho^{2}(z)}-\varphi,$
$\displaystyle\tilde{\phi}(t,\bm{r})$ $\displaystyle=$
$\displaystyle\phi(t,\bm{r})+\arctan\frac{z\mathfrak{F}}{z_{*}(\mathfrak{F}-z)}+\pi\theta(z-\mathfrak{F})$
(19) $\displaystyle-$ $\displaystyle\arctan\frac{z_{*}}{\mathfrak{F}},$

and $f(\xi)$ is a smooth envelope function containing $N_{\text{c}}$ carrier
cycles, $f(\xi)=\sin^{2}[\xi/(2N_{\text{c}})]\theta(\pi N_{\text{c}}-|\xi-\pi
N_{\text{c}}|)$. To specify the external field configuration, one has to
define two more parameters besides the field amplitude $E_{0}$, wavelength
$\lambda$, number of the carrier cycles $N_{\text{c}}$, and the CEP parameter
$\varphi$. These can be, for example, focal spot radius $\rho_{\text{F}}$ and
waist position $z_{\text{F}}$. Using these quantities, one derives the rest
parameters:

$\displaystyle\rho_{0}$ $\displaystyle=$
$\displaystyle\rho_{\text{F}}\sqrt{1+[z_{\text{F}}/(k_{0}\rho^{2}_{\text{F}})]^{2}},$
(20) $\displaystyle\mathfrak{F}$ $\displaystyle=$ $\displaystyle
z_{\text{F}}\bigg{(}1+\frac{k_{0}^{2}\rho^{4}_{\text{F}}}{z^{2}_{\text{F}}}\bigg{)},$
(21) $\displaystyle z_{*}$ $\displaystyle=$ $\displaystyle k_{0}\rho_{0}^{2}.$
(22)

In what follows, we assume $\lambda=1.0~{}\mu\text{m}$,
$\rho_{\text{F}}=2.0~{}\mu\text{m}$, $z_{\text{F}}=20.3~{}\mu\text{m}$, and
$\varphi=0$, which leads to $\mathfrak{F}=51.4~{}\mu\text{m}$ and
$f_{\\#}\equiv\mathfrak{F}/(2\rho_{0})=10.0$. The beam divergence
$\Delta=\arctan~{}(k_{0}\rho_{0})^{-1}\approx 0.06$. The number of the carrier
cycles is $N_{\text{c}}=10$, so the pulse duration is $\tau=\lambda
N_{\text{c}}/c=33.3~{}\text{fs}$.

Besides a single laser pulse, we will also analyze the field configuration
consisting of two focused pulses crossing at some angle $\theta$. In
principle, our numerical simulations can be carried out for arbitrary values
of $\theta$, but we will focus on the extreme case $\theta=\pi$ of two
counterpropagating pulses. By means of both the single-pulse case and this
setup, one covers the largest domain of the laser intensities that can be
measured by detecting the resulting positrons. Using intermediate values of
$\theta$ may be more profitable to work within various subintervals as it
allows one to tune the field configuration to make the sharp threshold
behavior of the pair-production process coincide with the intensity region of
interest thus reducing the corresponding uncertainties. We also note that
introducing a high-energy electron beam instead of almost restless electron
(xenon) gas would have made the geometry of the setup substantially more
complex. For instance, various relative angles between the electron beam and
laser pulse are considered in Refs. vranic_sci_rep_2018 ; blackburn_njp_2019 .

##  IV Photon emission and pair production

In order to describe the QED processes leading to the production of positrons,
we will employ the corresponding QED rates within the locally-constant field
approximation (LCFA). Let us introduce the Lorentz-invariant quantum
parameters $\eta=|F_{\mu\nu}p^{\nu}|/c^{4}$ and
$\chi=|F_{\mu\nu}k^{\nu}|/(2c^{4})$ characterizing the electron and photon
dynamics in the external field, respectively. The field itself can be
described by two Lorentz invariants
$\mathcal{F}=(\bm{E}^{2}-\bm{B}^{2})/E_{\text{c}}^{2}$ and
$\mathcal{G}=\bm{E}\cdot\bm{B}/E_{\text{c}}^{2}$, where
$E_{\text{c}}=c^{3}\approx 1.3\times 10^{16}~{}\text{V}/\text{cm}^{2}$ is the
Schwinger critical field strength. The LCFA approach is based on the fact that
if $\mathcal{F}$, $\mathcal{G}\ll 1$ and $\mathcal{F}$,
$\mathcal{G}\ll\eta^{2}$, $\chi^{2}$, then the QED rates are determined only
by $\eta$ and $\chi$ given that $\mathcal{F}$ and $\mathcal{G}$ are small no
matter what specific field configuration is examined nikishov_ritus_1964_1 .
This means that one can describe the necessary QED processes choosing the most
convenient scenario. Basically, one considers either a constant crossed field
nikishov_ritus_1964_1 ; ritus_1985 , or a static magnetic field erber_1966 .
In the present study, we choose the latter configuration following the
emission and pair production model employed in Refs. kirk_2009 ; duclous_2011
; ridgers_jcp_2014 ; arber_ppcf_2015 . Since for the laser intensities
considered in our study, the field amplitude $E_{0}$ is always much smaller
than the Schwinger limit, $E_{0}\ll E_{\text{c}}$, and the dimensionless
parameter $a_{0}=E_{0}/(c\omega_{0})$ obeys $a_{0}\gg 1$, the LCFA is well
justified within our simulations. This issue will be discussed in more detail
in Sec. VI.3.

To incorporate the QED processes in PIC methods, one basically performs Monte-
Carlo simulations governed by the relevant QED rates (see, e.g., Refs.
ridgers_jcp_2014 ; arber_ppcf_2015 ; pukhov_1999 ; fonseca_2002 ;
nerush_prl_2011 ; elkina_2011 ; bastrakov_comp_2012 ; gonoskov_pre_2015 ;
derouillat_2018 ). However, in the present study, we compute the number of
positrons produced according to the following scheme: (i) we evolve the
momentum and coordinates of a given electron portion taking into account the
Lorentz force and the recoil force due to quantum radiation reaction; (ii) at
each time step, we also calculate the number density of photons emitted with a
given quantum parameter $\chi$ which propagates then along a straight line;
(iii) traveling in the external field, this “photon portion” can decay via the
nonlinear Breit-Wheeler process, so we sum the corresponding contributions to
the positron number. The main drawbacks of this approach are the following:
(a) it does not take into account the further stages of the cascading process,
i.e., we consider only photons emitted by the primary (seed) particles; (b) we
neglect the internal plasma field induced by the charged particles. These two
aspects will be discussed in Sec. V. Moreover, this kind of a PIC simulation
does not provide any information on the angular distribution of the positrons
created. Given the threshold behavior of the scenario under consideration, the
total number of particles already represents a relevant quantity which can be
used for the intensity diagnostics. Besides, the advantage of our approach is
that the first stage of the truncated cascade process is described without
Monte-Carlo methods, so we perform only one run of our simulations for given
laser field parameters. Let us turn to the detailed description of our
approach.

The $i$th electron portion, which has the weight $\Delta N^{(\text{el})}_{i}$,
follows the classical equations of motion (9) for $t\geq t_{i}$, i.e., we
evaluate its momentum components and coordinates at each time step $t=t_{j}$,
where $j\geq i$. At a given time instant $t=t_{j}$, the electron can emit
photon with 4-momentum $k^{\mu}_{\gamma}$. In terms of the quantum parameters
$\eta$ and $\chi$, the emission rate reads erber_1966 ; sokolov_ternov :

$\frac{dN^{(\gamma)}}{d\chi
dt}=\frac{\sqrt{3}c}{2\pi}\frac{\eta}{\gamma}\frac{F(\eta,\chi)}{\chi},$ (23)

where $F(\eta,\chi)=F_{x}(\eta,2\chi/\eta)$ is the quantum synchrotron
function,

$F_{x}(\eta,x)=x^{2}yK_{2/3}(y)+(1-x)y\int\limits_{y}^{\infty}dzK_{5/3}(z).$
(24)

Here $y\equiv 2x/[3\eta(1-x)]$ and $K_{n}(z)$ are modified Bessel functions of
the second kind. One also assumes that $F_{x}(\eta,x)=0$ for $x\geq 1$, i.e.,
$\chi$ does not exceed $\eta/2$. Note that a focused external field described
in Sec. III is more favorable for pair production than a simple plane-wave
background since the latter does not allow the electron to reach large values
of the quantum parameter $\eta$. By solving the equations of motion in the
case of a plane wave, one can indeed explicitly demonstrate that
$\eta=|\bm{E}|/c^{3}$ throughout the interaction process no matter what energy
the particle gains.

To evaluate the recoil force, which should be plugged into Eq. (9), we assume
that the photon momentum $\bm{k}$ is parallel to that of the electron:
$k^{\mu}_{\gamma}=(\omega_{\gamma}/c)\,(1,\bm{n})$ and
$\bm{p}=|\bm{p}|\bm{n}$. Using the momentum conservation, we obtain

$\bm{F}_{\text{rec}}=-\bm{n}\int\limits_{0}^{\eta/2}d\chi\,\frac{dN^{(\gamma)}}{d\chi
dt}\,\frac{\omega_{\gamma}(\chi)}{c}.$ (25)

The quantum parameters $\eta$ and $\chi$ can be evaluated via

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{1}{c^{4}}\sqrt{(\bm{E}p_{0}+\bm{p}\times\bm{H})^{2}-(\bm{E}\cdot\bm{p})^{2}},$
(26) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle\frac{\omega_{\gamma}}{2c^{5}}\sqrt{(\bm{E}+\bm{n}\times\bm{H})^{2}-(\bm{E}\cdot\bm{n})^{2}},$
(27)

where $p_{0}=\gamma c$. The recoil force then reads

$\bm{F}_{\text{rec}}=-\bm{n}\frac{\sqrt{3}c}{2\pi}\frac{\eta}{\gamma}\bigg{[}\frac{(\bm{E}p_{0}+\bm{p}\times\bm{H})^{2}-(\bm{E}\cdot\bm{p})^{2}}{(\bm{E}+\bm{n}\times\bm{H})^{2}-(\bm{E}\cdot\bm{n})^{2}}\bigg{]}^{1/2}\int\limits_{0}^{1}dx\,F_{x}(\eta,x).$
(28)

This force enters Eq. (9) together with the classical Lorentz force. At each
time step $t_{j}$, we also evaluate the number density of photons depending on
$\chi$:

$\frac{dN_{ij}^{(\gamma)}}{d\chi}=\frac{dN^{(\gamma)}(t_{j})}{d\chi
dt}\,\Delta t\Delta N^{(\text{el})}_{i},$ (29)

where $i$ is the number of the electron portion and $\Delta t$ is the temporal
grid step. These “photon portions” travel according to
$\bm{r}(t)=\bm{r}_{j}+\bm{n}_{j}c(t-t_{j})$ independently of $\chi$, where
$\bm{r}_{j}$ and $\bm{n}_{j}$ are the electron’s position and direction at
$t=t_{j}$. Finally, we use the same temporal grid to evaluate the total number
of positrons produced by the $j$th photon portions,

$\Delta
N^{(\text{pos})}_{ij}=\int\limits_{0}^{\eta/2}\\!d\chi\\!\int\limits_{t_{j}}^{+\infty}\\!dt^{\prime}\,\frac{dN_{ij}^{(\gamma)}}{d\chi}\,\frac{dN^{(\pm)}(t^{\prime})}{dt^{\prime}}.$
(30)

Here $\eta$ is the quantum parameter of the electron at $t=t_{j}$. The upper
limit of the $t^{\prime}$ integration is practically the time instant when the
photons escape from the external field. The QED rate of pair production can be
represented as erber_1966 :

$\frac{dN^{(\pm)}(t^{\prime})}{dt^{\prime}}=\frac{c^{3}}{\omega_{\gamma}}\,\chi^{\prime}T_{\pm}(\chi^{\prime}).$
(31)

Here $\omega_{\gamma}$ is the photon frequency determined by $\chi$, i.e. the
quantum parameter at $t=t_{j}$, while $\chi^{\prime}$ corresponds to the
photon state at the time instant $t^{\prime}$, i.e. $\chi^{\prime}\equiv
R^{1/2}(t^{\prime})\chi$, where

$R(t^{\prime})=\frac{[(\bm{E}p_{0}+\bm{p}\times\bm{H})^{2}-(\bm{E}\cdot\bm{p})^{2}]~{}\text{at}~{}t^{\prime}}{[(\bm{E}p_{0}+\bm{p}\times\bm{H})^{2}-(\bm{E}\cdot\bm{p})^{2}]~{}\text{at}~{}t_{j}}.$
(32)

The function $T_{\pm}$ has the form

$T_{\pm}(\chi)=\frac{0.16}{\chi}\,K_{1/3}^{2}[2/(3\chi)],$ (33)

so we arrive at

$\Delta N^{(\text{pos})}_{ij}=\Delta t\Delta
N^{(\text{el})}_{i}\frac{0.09\sqrt{3}}{2\pi
c}\frac{\eta^{2}}{\gamma}\sqrt{(\bm{E}p_{0}+\bm{p}\times\bm{H})^{2}-(\bm{E}\cdot\bm{p})^{2}}\,\int\limits_{0}^{1}dx\,F_{x}(\eta,x)\\!\\!\int\limits_{t_{j}}^{+\infty}\\!dt^{\prime}\,R(t^{\prime})\varkappa^{2}\bigg{(}\frac{4}{3R^{1/2}(t^{\prime})\eta
x}\bigg{)},$ (34)

where $\varkappa(z)\equiv zK_{1/3}(z)$. The square root containing the
external fields in Eq. (34) is evaluated at $t=t_{j}$. Note that in this
derivation we have not taken into account the fact that the number of
positrons produced cannot exceed the initial weight of the photon. Since the
pair production mechanism reduces the photon weight, one has to modify Eq.
(30) according to

$\Delta
N^{(\text{pos})}_{ij}=\int\limits_{0}^{\eta/2}\\!d\chi\,\frac{dN_{ij}^{(\gamma)}}{d\chi}\Bigg{[}1-\mathrm{exp}\Bigg{\\{}-\\!\\!\int\limits_{t_{j}}^{+\infty}\\!dt^{\prime}\,\frac{dN^{(\pm)}(t^{\prime})}{dt^{\prime}}\Bigg{\\}}\Bigg{]}.$
(35)

This brings us to

$\Delta N^{(\text{pos})}_{ij}=\Delta t\Delta
N^{(\text{el})}_{i}\frac{\sqrt{3}c}{2\pi}\frac{\eta}{\gamma}\,\int\limits_{0}^{1}dx\,\frac{F_{x}(\eta,x)}{x}\,\Big{[}1-\mathrm{exp}\Big{\\{}-\\!\\!\mathcal{P}(\eta,t_{j},x)\Big{\\}}\Big{]},$
(36)

where

$\mathcal{P}(\eta,t_{j},x)=\frac{0.09\eta
x}{c^{2}}\,\sqrt{(\bm{E}p_{0}+\bm{p}\times\bm{H})^{2}-(\bm{E}\cdot\bm{p})^{2}}\\!\int\limits_{t_{j}}^{+\infty}\\!dt^{\prime}R(t^{\prime})\varkappa^{2}\bigg{(}\frac{4}{3R^{1/2}(t^{\prime})\eta
x}\bigg{)}.$ (37)

The expression (36) is the leading-order contribution of Eq. (36) if the
function (37) is sufficiently small. The final expression (36) should be
summed over the electron ($i$) and photon portions ($j$):

$\Delta N^{(\text{pos})}=\sum_{i}\sum_{j>i}\Delta N^{(\text{pos})}_{ij}.$ (38)

The sum over $j$ involves all $t_{j}>t_{i}$. The quantity (38) represents the
number of positrons for a given initial position $\bm{r}_{0}$ of a Xe atom or
free electron. It should then be summed over the spatial coordinates taking
into account the number density of the atoms or free electrons depending on
the scenario under consideration:

$N^{(\text{pos})}=n\\!\int\\!d\bm{r}_{0}\,\Delta N^{(\text{pos})},$ (39)

where either $n=n_{\text{el}}$ or $n=n_{\text{Xe}}$. We always assume that the
number density of the Xe atoms is $n_{\text{Xe}}=10^{14}~{}\text{cm}^{-3}$.
Within the intensity domain under consideration, the external field usually
ionizes 52 electrons of each atom, so we choose
$n_{\text{el}}=52n_{\text{el}}$ to keep the number of the “active” electrons
constant and focus on the enhancement due to more efficient dynamics of the
particles in the case of xenon.

##  V Other effects and corrections

###  V.1 Nuclear field versus laser field and the initial position of the
electron ionized

Let us estimate the ratio $E_{\text{nucl}}/E$ at the time instant when the
electron gets ionized and becomes a free particle according to our model. If
the ionization energy is sufficiently large, the particle tunnels through the
barrier and appears at some position $x_{0}$. To find this position, we can
simply equate the potential energy of the particle in the combined field of
the nucleus and laser with the ionization energy $I_{p}$. It brings us to

$x_{0}=\frac{I_{p}+\sqrt{I_{p}^{2}-4EZ}}{2E}$ (40)

We see that for $E>E_{\text{BS}}\equiv I_{p}^{2}/(4Z)$ the electron state is
no longer classically bound [this value of $E$ exactly corresponds to the
expression (7)]. It is also convenient to introduce the maximal value of the
particle’s potential energy $-V_{0}=-2\sqrt{ZE}$, so the condition
$E=E_{\text{BS}}$ is equivalent to $I_{p}=V_{0}$.

The barrier-suppression regime relates to $\beta\equiv I_{p}/V_{0}\leq 1$ and
obviously appears when one considers the outer-shell electrons. However, as
the major part of $e^{+}e^{-}$ pairs is produced when the inner-shell
electrons emit high-energy photons, one has to evaluate first $\beta$ for
these states. It turns out that even for the 52th electron of the Xe atom
($Z\approx 51$, $I_{p}=9810.37$ eV), $\beta\approx 1.09$ if
$I=10^{22}~{}\text{W}/\text{cm}^{2}$ and $\beta\approx 0.61$ if
$I=10^{23}~{}\text{W}/\text{cm}^{2}$. This means that for such intense laser
fields, the concept of $x_{0}$ [Eq. (40)] is not well defined. To demonstrate
that the ion field can be neglected once the particle is free, it suffices to
compare the laser field to that of the nucleus for the initial (bound)
electron state, i.e., instead of Eq. (40), we use the corresponding Bohr-orbit
radius $r=\nu^{2}/Z$, so
$E_{\text{nucl}}=Z/r^{2}=Z^{3}/\nu^{4}=\kappa^{4}/Z=16E_{\text{BS}}$.
Accordingly, $E_{\text{nucl}}/E=16E_{\text{BS}}/E=16\beta^{2}$
artemenko_pra_2017 . For the $2s^{2}$ shell of Xe, this ratio obeys
$E_{\text{nucl}}/E\lesssim 1$ if $I\gtrsim 10^{24}~{}\text{W}/\text{cm}^{2}$.
Introducing a second pulse, one reduces the ratio $\beta^{2}$ by a factor of
$2$, so we can safely neglect the nuclear field within the whole range
$10^{23}~{}\text{W}/\text{cm}^{2}\lesssim I\lesssim
10^{26}~{}\text{W}/\text{cm}^{2}$ for our order-of-magnitude estimates.

Next, we will estimate the role of the Coulomb field of the nuclei within the
acceleration stage. In Eq. (9), we totally neglect the ion field $Z/r^{2}$.
Let us first evaluate the distance where the field of the nucleus becomes
small compared to the laser field:

$\frac{Z}{r_{0}^{2}}=\frac{E}{100}.$ (41)

For $I=10^{22}~{}\text{W}/\text{cm}^{2}$ and $Z=54$ it leads to $r_{0}\approx
3.2$ a.u. (for larger $I$ and smaller $Z$, $r_{0}$ is even smaller). We can
estimate the total volume where the ion field is non-negligible according to
$V_{\text{ion}}=(4/3)\pi r_{0}^{3}N_{\text{ion}}$. For the relative volume
factor, it brings us to

$\frac{V_{\text{ion}}}{V}=\frac{4}{3}\pi r_{0}^{3}n\approx 2\times
10^{-9}\,(n\times 10^{-14}~{}\text{cm}^{3}),$ (42)

which amounts to $2\times 10^{-7}$ even for $n=10^{16}~{}\text{cm}^{-3}$. Here
we do not take into account that the attractive force which the nucleus exerts
on the electron can make the latter get closer to the field center, where the
field is larger than $Z/r_{0}^{2}$. Nevertheless, since (a) the laser field
direction is independent of the ion position, (b) we assumed that at $r=r_{0}$
the ion field is 100 times weaker than the laser field, (c) the electron is
unlikely to be captured by the nucleus due to the small capture cross section,
the ion field does not have any significant impact on the particle’s
trajectory. These rough estimates were also confirmed by numerical
simulations.

Finally, we note that the distance $r$ (or $x_{0}$ for $\beta\geq 1$) is
always much smaller than the characteristic length scale of the external
field, which is $\lambda\sim 1~{}\mu\text{m}$. For example, for the $2s^{2}$
electrons $r\sim 10^{-6}~{}\mu\text{m}$, and for the outer-shell electrons
$r\sim 10^{-5}~{}\mu\text{m}$. For weaker fields, e.g.
$I=10^{22}~{}\text{W}/\text{cm}^{2}$, we receive $x_{0}\sim
10^{-5}~{}\mu\text{m}$. Therefore, we always assume that after ionization free
electrons appear at the position of the nucleus.

###  V.2 Ion motion within the ionization process

Now we roughly estimate the effects of the ion motion. The maximal momentum
appearing due to the ponderomotive forces is $p_{\text{max}}\sim EZ/\omega$.
The relativistic parameter $\gamma$ has then the form
$\gamma\approx(1+[EZ/(Mc\omega)]^{2})^{1/2}$. For Xe $Z\lesssim 50$, $M\approx
2.4\times 10^{5}$ a.u., so even for $I=10^{26}~{}\text{W}/\text{cm}^{2}$ we
have $\gamma\lesssim 2.0$, which leads to $v_{\text{max}}/c\lesssim 0.87$ (for
$I=10^{24}~{}\text{W}/\text{cm}^{2}$ it yields $v_{\text{max}}/c\lesssim
0.17$). The maximal displacement of the ion can be estimated as $(\delta
x)_{\text{max}}/\lambda\lesssim v_{\text{max}}/(4c)$, which never exceeds
$0.22$ even for $I=10^{26}~{}\text{W}/\text{cm}^{2}$. However, it is only a
rough upper-bound estimate, so the ion motion is likely to be totally
negligible even for such high intensities, provided one sums the results over
the whole interaction region.

###  V.3 Ion induced QED processes

Although we are mainly interested in the two-stage process of pair production
by a high-energy electron as discussed above, there are also several ion-
induced phenomena that take place during the interaction. Namely, so far we
have not addressed (a) Bethe-Heitler pair production by a photon in the
presence of the Coulomb field of the high-$Z$ nucleus, (b) bremsstrahlung.
These “collisional” processes are in fact negligible according to the
following estimates.

Bethe-Heitler process. Assuming that $\omega_{\gamma}\gg c^{2}$, $\alpha Z\ll
1$, and the energy of the electron/positron is much larger than $c^{2}$, one
can show that the total cross section of the process in our units has the form
blp :
$\sigma_{\text{BH}}\approx(28/9)(1/c^{5})Z^{2}[\ln(2\omega_{\gamma}/c^{2})-109/42]$.
The pair production rate then reads $dN_{\text{BH}}/dt=\sigma_{\text{BH}}cn$.
This quantity should be multiplied by the number of high-energy photons in the
interaction volume and by the characteristic photon lifetime $t_{\gamma}$:
$N_{\text{BH}}=N_{\gamma}\sigma_{\text{BH}}cnt_{\gamma}$. For
$\omega_{\gamma}=1$ GeV, it yields $N_{\text{BH}}\approx 7.6\times
10^{-19}\,N_{\gamma}Z^{2}t_{\gamma}(n\times 10^{-14}~{}\text{cm}^{3})$. The
number of high energy photons is very unlikely to exceed
$10^{3}\,N^{(\text{pos})}$, so for $Z=54$ one finds
$N_{\text{BH}}/N^{(\text{pos})}<2.2\times 10^{-12}\,t_{\gamma}(n\times
10^{-14}~{}\text{cm}^{3})$. Assuming then $t_{\gamma}<100\lambda/c$, we arrive
at $N_{\text{BH}}/N^{(\text{pos})}<10^{-5}$ even for
$n=10^{16}~{}\text{cm}^{-3}$, so the Bethe-Heitler process provides a
negligible contribution.

Bremsstrahlung. If both the initial and final energy of the electron are much
larger than $c^{2}$, the cross section has the form
$\sigma_{\text{Br}}\approx(2^{5/2}/3)(1/c^{5})Z^{2}\gamma_{e}^{1/2}$
artemenko_pra_2017 , where $\gamma_{e}$ is the corresponding Lorentz factor of
the electron. The total number of photons can be estimated as
$N_{\text{Br}}\approx N_{e}\sigma_{\text{Br}}cnt_{e}$, where $N_{e}$ is the
characteristic number of high-energy electrons corresponding to $\gamma_{e}$
and $t_{e}$ is the characteristic interaction time of a high-energy electron.
Accordingly, for $1$ GeV electrons ($\gamma_{e}\approx 2000$),
$\sigma_{\text{Br}}\approx 1.7\times 10^{-9}\,Z^{2}$ and $N_{\text{Br}}\approx
3.5\times 10^{-18}\,N_{e}Z^{2}t_{e}(n\times 10^{-14}~{}\text{cm}^{3})$. For
the upper-bound estimation, we employ $Z=54$, $t_{e}=100\lambda/c$, and
$N_{e}=(10\lambda)^{3}n\sim 10^{7}$. Therefore, $N_{\text{Br}}<1.4\times
10^{-5}\,(n\times 10^{-14}~{}\text{cm}^{3})^{2}$, which amounts to $0.14$ even
for $n=10^{16}~{}\text{cm}^{-3}$. This means that bremsstrahlung can also be
neglected.

Besides, we do not take into account further stages of the cascade process,
when the electron/positron created emits one more photon leading to subsequent
pair production. Such a scenario may give rise to an avalanche-like reaction,
so one should either estimate the additional contributions or demonstrate that
they are small compared to what we incorporate in Sec. IV. This issue will be
discussed in the following subsection.

Figure 1: (left) Final and maximal energy of the electron and (right) maximal
value of $\eta$ as a function of $z_{0}$ for $I=2\times
10^{22}~{}\text{W}/\text{cm}^{2}$ in the case of an individual focused laser
pulse ($x_{0}=y_{0}=0$). The horizontal lines represent the corresponding
values for the plane-wave case.

###  V.4 Further cascading

Unlike the so-called S-type (“shower”) cascades where the $e^{+}e^{-}\gamma$
production processes basically consume energy from the seed high-energy
particles, the phenomenon under consideration may correspond to A-type
(“avalanche”) cascades (see, e.g., Refs. fedotov_prl_2010 ; 2015_ufn_narozh ).
The external laser field not only serves as a background for the Compton and
Breit-Wheeler mechanisms but also accelerates particles, so that they regain
the energy lost. If the external field is sufficiently strong, so the
acceleration time $t_{\text{acc}}$ and the electron/photon lifetime
$t_{e,\gamma}$ are much smaller than the interaction time $t_{\text{esc}}$,
then a continuous avalanche-like reaction occurs being limited only by the
laser pulse duration and ponderomotive expulsion of the particles.

As was demonstrated in Refs. fedotov_prl_2010 ; 2015_ufn_narozh , in the case
of two laser pulses, the hierarchy $t_{\text{acc}}\ll t_{e,\gamma}\ll
t_{\text{esc}}$ takes place when $\mu\equiv E_{0}/(\alpha E_{\text{c}})\gtrsim
1$ and $\mu^{1/4}\gg(1/\alpha)\sqrt{\omega/c^{2}}$. For this scenario, we
analyze the pulses with intensity up to $10^{24}~{}\text{W}/\text{cm}^{2}$,
which yields $\mu=0.28$ and also satisfies the second condition. It means that
the enhancement of the positron number due to the further stages of cascading
does not take place in our case. In the case of an individual focused laser
pulse, an avalanche-like cascade can occur only when $\mu\gtrsim 1/\Delta$ for
sufficiently large divergence of the beam, $\Delta>0.03$ mironov_talk . Since
in our case $\Delta\approx 0.06$, we extent the intensity interval up to
$10^{26}~{}\text{W}/\text{cm}^{2}$. We refrain from using higher intensities
in our simulations as they could make the further stages of cascading non-
negligible.

##  VI Numerical results

In this section, we will discuss the process of acceleration of electrons in
the laser field and pair creation seeded by free electron gas and neutral Xe
atoms. The external field will be chosen in the form of a single focused laser
pulse described by Eqs. (12)–(19) and a combination of two counterpropagating
laser pulses ($\theta=\pi$). The external field parameters are
$\lambda=1.0~{}\mu\text{m}$, $\rho_{\text{F}}=2.0~{}\mu\text{m}$,
$z_{\text{F}}=20.3~{}\mu\text{m}$, $\mathfrak{F}=51.4~{}\mu\text{m}$,
$f_{\\#}\equiv\mathfrak{F}/(2\rho_{0})=10.0$, $\varphi=0$, and
$\tau=33.3~{}\text{fs}$. The number of the carrier cycles is
$N_{\text{c}}=10$.

###  VI.1 Acceleration of particles

First, we evolve the free electron’s trajectory taking into account the recoil
force (28) in order to find out what energies can be achieved when using (a)
focused laser pulse, (b) simple plane-wave field. In Fig. 1 we depict the
$z_{0}$ dependence of the final and maximal energy of the electron and the
parameter $\eta$ for $I=2\times 10^{22}~{}\text{W}/\text{cm}^{2}$ and
$x_{0}=y_{0}=0$. The electron is assumed to be free, i.e., no Coulomb forces
are present. The horizontal lines correspond to the case of a finite plane-
wave pulse (the results are $z_{0}$-independent). Note that in the case of a
plane wave, the parameter $\eta$ is always equal to $|\bm{E}|/c^{3}$, i.e., it
is determined by the local value of the electric field strength, which does
not exceed the amplitude $E_{0}$. For instance, in Fig. 1 (right) the maximal
value of $\eta$ is $1.7\times 10^{-5}$, while $E_{0}/c^{3}=2.9\times 10^{-4}$.
The difference appears due to the fact that the electron’s trajectory does not
cross any of the field maxima. On the other hand, the value of $\eta$ is
crucial since positron production can only be efficient if the electrons emit
high-energy photons. To further illustrate that the focused background is much
more efficient in terms of reaching large $\eta$, we display the maximal (over
$z_{0}$) values of $\eta$ as a function of $I$ together with the upper-bound
estimate $\eta_{\text{max}}=E_{0}/c^{3}$ for the case of a plane-wave pulse
(see Fig. 2).

Figure 2: Maximal value of $\eta$ over time and $z_{0}$ as a function of $I$
in the case of a single focused pulse with the recoil force (28) (blue solid
line) and without it (dashed line). The orange solid line displays the upper-
bound estimate $\eta_{\text{max}}=E_{0}/c^{3}$ for the case of a plane-wave
pulse ($x_{0}=y_{0}=0$).

Figure 3: $z_{0}$ dependences of the maximal energy and $\eta$ in the case of
free electrons traveling in the combination of two counterpropagating laser
pulses with $I=2\times 10^{22}~{}\text{W}/\text{cm}^{2}$ ($x_{0}=y_{0}=0$).

We observe that the results for a focused pulse are always much larger.
Moreover, it may well be that the actual values of $\eta$ in the case of a
plane-wave pulse are even smaller than $E_{0}/c^{3}$. For example, to make the
plane-wave background more realistic, one can introduce a certain spatial cut-
off function allowing the particles to escape from the interaction region.
However, the results strongly depend on this profile, so we do not display
them in Fig. 2. Since the field of a focused pulse is naturally finite in
space, we will study this more realistic setup refraining from further
discussions of the plane-wave background. In Fig. 2 we also present the
results obtained without taking into account the radiation reaction force (28)
(dashed line). This force plays a notable role only when the laser intensity
reaches $10^{24}$–$10^{25}$ $\text{W}/\text{cm}^{2}$. As will be demonstrated
in what follows, the setup under consideration provides a non-negligible
amount of pairs only when $I\gtrsim 10^{25}$ $\text{W}/\text{cm}^{2}$.
Although the focused laser pulse indeed allows one to obtain large $\eta$,
i.e. this scenario seems to be quite promising, the pair creation process is
expected to have a quite high threshold.

To make the setup more efficient, we will introduce a second pulse propagating
in the opposite direction. In Fig. 3 we present the analogous $z_{0}$
dependences for the case of two focused counterpropagating laser pulses. Each
of the pulses has the intensity $I=2\times 10^{22}~{}\text{W}/\text{cm}^{2}$.
The presence of two pulses make the field configuration much more
inhomogeneous and considerably more powerful in terms of the field strength.
As a result, both the energy of the particles and parameter $\eta$ reach much
larger values. Since this scenario is very far from the simplest plane-wave
background not solely due to the focusing, in Fig. 3 (right) we observe the
values which are about two orders of magnitude larger than those given in Fig.
1 (right).

Finally, we take one more step in pursuit of reaching higher kinetic energies
of the electrons by considering Xe atoms instead of free electrons. The
advantage of this scheme consists in very efficient acceleration of the inner-
shell electrons, which are being ionized only in a space-time region where the
external laser field is sufficiently strong maltsev_prl_2003 ; hu_pre_2006 ;
karmakar_2007 . The Coulomb field of the nucleus preserve these electrons from
premature ionization, so they gain relatively higher energies before they
escape from the interaction region of the focused pulse(s). To illustrate this
point, we present the histogram showing the maximal values of $\eta$ reached
by different electrons of a Xe atom in the field of two counterpropagating
laser pulses of the intensity $I=2\times 10^{22}~{}\text{W}/\text{cm}^{2}$
(see Fig. 4). The two deepest inner-shell electrons (53rd and 54th) are not
ionized at all, while the other 52 electrons are fully ionized. This plot
reveals a typical situation where several electrons corresponding to a large
ionization potential reach substantially larger values of $\eta$. Since this
parameter plays then a crucial role in the two-stage process of pair
production, Xe atoms represent a very efficient setup as will also be
demonstrated in what follows.

Figure 4: Maximal value of $\eta$ for various electrons of a Xe atom in the
combination of two counterpropagating laser pulses ($I=2\times
10^{22}~{}\text{W}/\text{cm}^{2}$, $x_{0}=y_{0}=0$, $z_{0}=5~{}\mu\text{m}$).

Figure 5: Total number of positrons produced in (left) an individual laser
pulse and (right) the combinations of two counterpropagating laser pulses as a
function of the laser intensity $I$. The seed particles are either free
electrons (solid lines) or neutral Xe atoms (dashed lines).

###  VI.2 Positron production

In this section, we perform the calculations of the total number of positrons
according to Sec. IV as a function of the laser intensity $I$. We analyze the
process in the case of free Xe atoms randomly distributed over space with the
number density $n_{\text{Xe}}=10^{14}~{}\text{cm}^{-3}$ and free electrons
with the density $n_{\text{el}}=52n_{\text{Xe}}$. As was shown in, e.g., Figs.
3, one has to employ a very high spatial resolution in terms of the initial
position $\bm{r}_{0}$ to accurately describe the electron dynamics, i.e.,
accurately integrate in Eq. (39). Since the pair-production mechanism is
highly nonlinear, analyzing our data , we decided to associate a quite large
uncertainty with the spatial integration assuming that our numerical
simulations provide only order-of-magnitude estimates. On the other hand, as
will be shown further, this relatively low precision already allows one to
recover the field intensity quite accurately. This will be discussed in more
detail in the next subsection.

Before integrating over the initial spatial coordinates in Eq. (39), one has
to sum over the electron and photon portions in Eq. (38). Unfortunately, this
stage is very time consuming in the case of Xe atoms, where there are many
electron portions, whereas a free electron is considered as only one
macroparticle. To save the computational time, we estimate the positron yield
in the case of Xe by using the maximal values of the $\eta$ parameter omitting
the positron production stage described by Eqs. (29)–(38). Namely, we
construct the dependence $\Delta N^{(\text{pos})}(\eta)$ analyzing the data
obtained in the case of free electrons. Then we use this correspondence to
estimate $\Delta N^{(\text{pos})}$ in the case of Xe atoms and integrate the
results over $\bm{r}_{0}$ for each of the Xe electrons. Accordingly, we obtain
54 contributions which should then be summed up and multiplied with the number
density $n_{\text{Xe}}$ (basically, there are 52 nonzero contributions).
Taking into account that there is a significant ambiguity in $\Delta
N^{(\text{pos})}(\eta)$, we estimated the additional uncertainty due to the
simplified treatment of the Xe electrons and found out that the relative error
in the positron yield is always considerably smaller than the order-of-
magnitude error bars that we had introduced before.

The results are summarized in Fig. 5. First, we observe that the pair-
production threshold is indeed substantially lower in the case of two
counterpropagating laser pulses, which can be used to detect positrons already
for $I\gtrsim 10^{23}~{}\text{W}/\text{cm}^{2}$, while an individual focused
pulse should be two orders of magnitude stronger ($I\gtrsim
10^{25}~{}\text{W}/\text{cm}^{2}$). Second, the two configurations considered
in our study do not allow one to determine the laser intensity in the range
$10^{24}~{}\text{W}/\text{cm}^{2}\lesssim I\lesssim
10^{25}~{}\text{W}/\text{cm}^{2}$ for three reasons: (a) an individual pulse
does not yet produce pairs, (b) we do not have reliable data in the case of
two pulses as we neglect QED cascading, (c) even if the further stages of
cascading were taken into account, the uncertainty in the laser intensity
would be quite large because for $I\gtrsim 10^{24}~{}\text{W}/\text{cm}^{2}$
the positron number does not increase that dramatically with increasing $I$.
Nevertheless, one can properly adjust the field configuration by changing the
angle $\theta$ between two laser pulses allowing one to efficiently cover the
whole interval $10^{23}~{}\text{W}/\text{cm}^{2}\lesssim I\lesssim
10^{26}~{}\text{W}/\text{cm}^{2}$ and also minimize the uncertainties. This is
basically the main statement of our study.

Third, in Fig. 5 it is demonstrated that using neutral Xe atoms instead of
free electrons even with a lower number density notably enhances the effect of
positron production. Namely, the results are several times larger than those
obtained for free particles. For larger values of the laser intensity, the
enhancement becomes not that evident since the external field immediately
ionizes a major part of electrons making this setup quite similar to the
initial state containing free electrons. Let us compare our results with those
of Artemenko and Kostyukov artemenko_pra_2017  for the case of a focused
standing wave. First, we employed the data given in Ref. artemenko_pra_2017
for He atoms dividing it by $2$ since the He atom contains two electrons which
get ionized almost immediately due to the small ionization potential. The
results were also normalized by taking into account the different values of
the electron/ion density. Second, we compared the results for Xe atoms. In
both cases our predictions were about one order of magnitude smaller
indicating almost the same enhancement when using atoms instead of free
electrons. Since the positron yield strongly depends on the field amplitude
and the field configurations considered here and in Ref. artemenko_pra_2017
are quantitatively different, we find a good agreement between our approach
and that used in Ref artemenko_pra_2017 .

In the present study we compute only the total number of positrons produced
without specifying their angular distribution. In principle, this information
can also be obtained by using more involved QED expressions. For instance, the
energy distribution between the electron and positron produced via the Breit-
Wheeler process was described in Ref. daugherty_1983 ). Such improvements
require more time-consuming simulations and can be a subject of our future
investigations.

###  VI.3 Intensity diagnostics and uncertainties

Let us now discuss how the uncertainties of the positron yield can be
estimated. To determine the accuracy of the ionization model employed, we
perform the full calculations of the positron number replacing the ADK
expressions with the PPT one. To take into account the possible inadequacy of
the tunnel picture itself, we vary the parameter $\overline{\alpha}$ in Eq.
(8) and the function $W_{\text{L}}(t)$ by $\pm 10\%$. The resulting changes
were found to be small compared to the ADK–PPT discrepancy.

Next, we analyze the accuracy of the external-field model. First, Eqs.
(12)–(19) are valid only when the paraxial approximation is well justified,
i.e. $k_{0}\rho_{\text{F}}\gg 1$. In our case $k_{0}\rho_{\text{F}}\approx
12$. Second, although we propose a method to measure the laser peak intensity,
the other parameters of the external field are not a priori known in the real
experimental setups. For instance, the focal spot radius and the tightness of
laser focusing are difficult to control. However, we do not assign any
uncertainty to these parameters keeping in mind that it should be estimated
later when the actual experimental setup is fully specified. Our numerical
procedures can be applied within a wide domain of the field parameters, i.e.,
it is not limited just to the illustrative configuration analyzed in the
present study.

Another important source of systematic errors is the Coulomb interaction among
the particles. Here we assume that the main contribution appears due to the
interaction between a given electron and the nuclei. To estimate the
corresponding discrepancy, we perform the calculations assuming that the
nuclei are randomly distributed over space with given average density and
taking into account the Coulomb force. The ion motion in the laser field and
the electron tunneling distance can be then completely neglected (see Sec. V).

We note then that the QED expressions (23) and (31) are valid in our case to a
high accuracy because (a) the external field is much weaker than the Schwinger
limit $E_{\text{c}}\approx 1.3\times 10^{16}~{}\text{V}/\text{cm}$, (b) the
parameters $\eta^{2}$ and $\chi^{2}$ are much larger than the Lorentz
invariants of the external field, (c) the dimensionless field strength
$a_{0}\gtrsim 200$ is much larger than unity ridgers_jcp_2014 ;
arber_ppcf_2015 . We also note that in the case of the nonlinear Compton
scattering, the locally constant field approximation underlying the expression
(23) usually either is very accurate or fail in the region of low photon
energy di_piazza_pra_2019 ; ilderton_pra_2019 . Nevertheless, the subsequent
Breit-Wheeler process is likely to occur only when the photon energy is high.
A detailed analysis of the validity of the LCFA in the context of various QED
processes can be found in recent papers meuren_prd_2016 ; di_piazza_pra_2019 ;
blackburn_2018 ; ilderton_pra_2019 ; aleksandrov_prd_2019_1 ; sevostyanov  and
references therein.

Besides the systematic errors, we should also take into account the possible
numerical errors which may arise if the results do not fully converge. We make
sure that the various nonphysical grid steps are sufficiently small, so that
the corresponding inaccuracy is completely negligible. On the other hand, a
large discrepancy is associated with the procedure of summing over the initial
coordinate $\bm{r}_{0}$, which is quite complicated in three dimensions. By
analyzing the data, we decided to assume that the integration provides only
order-of-magnitude estimates. As a result, all of the other discrepancies
discussed above become inessential.

Figure 6: Relative uncertainty in the laser intensity $I$ as a function of $I$
in the case of one laser pulse (solid line) and the combination of two pulses
(dashed line). For each of the two scenarios, the left edge of the intensity
domain corresponds to a small positron yield $N^{\text{(pos)}}<1$, while the
upper boundary appears since we disregard the cascading processes.

To determine the laser intensity, we make use of the sharp threshold behavior
of the pair production process revealed in Fig. 5. Indeed, if the number of
positrons created is of the order $1$–$10^{4}$, even a notable uncertainty in
$N^{\text{(pos)}}$ does not affect much the intensity. To present a
transparent quantitative analysis of the corresponding uncertainty in the
laser intensity, we calculate the relative error according to

$\frac{\delta I}{I}=\frac{dI}{dN^{\text{(pos)}}}\frac{\delta
N^{\text{(pos)}}(I)}{I},$ (43)

where the derivative is computed using the numerical data and $\delta N(I)$ is
chosen as indicated above. For a given scenario, increasing $I$, one receives
larger values of $dI/dN$ as was mentioned above. We also underline that our
simulations are performed only up to $10^{24}~{}\text{W}/\text{cm}^{2}$ in the
case of two laser pulses and $10^{26}~{}\text{W}/\text{cm}^{2}$ in the case of
a single pulse since we do not take into account the cascading process. On the
other hand, for smaller $I$ the uncertainty $\delta N(I)$ becomes substantial
and the number of positrons itself becomes tiny, so the domain where one can
carry out the intensity diagnostics is, of course, limited from both sides. We
summarize the uncertainties in Fig. 6 where the relative error in the laser
intensity $I$ is depicted as a function of $I$. It turned out that the
uncertainties do not depend on the choice of seed particles (free electrons of
xenon). Choosing the appropriate scenario, one can perform a more accurate
diagnostics. Using several setups in combination, one can obtain more reliable
and precise values of the laser intensity.

##  VII Conclusion

In the present study, we provided quantitative estimates for the total
positron yield in several experimental scenarios, which can be used as a tool
for the laser intensity diagnostics. Namely, we examined the setups involving
an individual laser pulse and a combination of two counterpropagating pulses.
As seed particles we considered free electrons and Xe neutral atoms randomly
distributed over space. The laser field accelerates the seed electrons which
can subsequently emit high-energy photons. These photons can, in turn, decay
via the Breit-Wheeler mechanism producing $e^{+}e^{-}$ pairs. Since this
process has a sharp threshold dependence on the laser intensity, measuring the
positron yield allows one to accurately extract the intensity when working in
the vicinity of the pair production onset.

In order to calculate the number of positrons produced, we evolved the
electron trajectories taking into account possible photon emission via the
nonlinear Compton process and the Breit-Wheeler mechanism described by local
expressions derived within QED. Neglecting the further cascading stages
involving the secondary particles, we computed the total number of pairs
summing them over the initial coordinates of the electrons or Xe atoms.

Different scenarios provide different working regimes, i.e. ranges of the
laser intensity where it can be accurately determined. The corresponding
domains cover a wide interval $10^{23}$–$10^{26}~{}\text{W}/\text{cm}^{2}$.
According to our numerical results, choosing the appropriate setup ensures
that the relative uncertainty in the laser intensity does not exceed
$10$–$50\%$.

Although our estimates have already indicated that the laser intensity can be
extracted with a high accuracy, there are several improvements that we aim to
carry out within our future studies. First, one can examine some other
experimental setups involving other field configurations, e.g., two laser
pulses crossing at some angle $\theta\in(0,\pi)$, or other atoms as a source
of seed electrons. Second, it is also desirable to predict the angular
distribution of the positrons produced. Finally, we point out that in order to
reduce the overall uncertainty, one can also take into account the further
cascading stages, which become important at higher intensities, and refine the
ionization model, so that it yields more accurate predictions for the
ionization probabilities.

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