# QScatter: Numerical Framework for Fast Prediction of Particle Distributions
in Electron-Laser Scattering

Óscar Amaro , Marija Vranic  0000-0003-0615-0686 0000-0003-3764-0645
GoLP/Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico,
Universidade de Lisboa, 1049-001 Lisbon, Portugal
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The new generation of multi-PetaWatt laser facilities will allow tests of
Strong Field QED, as well as provide an opportunity for novel photon and
lepton sources. The first experiments are planned to study the (nearly) head-
on scattering of intense, focused laser pulses with either relativistic
electron beams or high-energy photon sources. In this work, we present a
numerical framework that can provide fast predictions of the asymptotic
particle and photon distributions after the scattering. The works presented in
this manuscript includes multiple features such as spatial and temporal
misalignment between the laser and the scattering beam, broadband electron
beams, and beam divergence. The expected mean energy, energy spread,
divergence or other observables are calculated by combining an analytical
description and numerical integration. This method can provide results within
minutes on a personal computer, which would otherwise require full-scale 3D
QED-PIC simulations using thousands of cores. The model, which has been
compiled into an open-source code QScatter, may be used to support the
analysis of large-size data sets from high-repetition rate experiments,
leveraging its speed for optimization or reconstruction of experimental
parameters.

##  1 Introduction

Strong-field quantum electrodynamics (SFQED) is a rapidly developing research
field that studies the interaction between matter and intense electromagnetic
fields. In recent years, there has been a growing interest in this area due to
the availability of high-intensity laser sources, which enable the exploration
of novel physical phenomena, with experiments being planned for the near
future: ELI [1], Apollon [2], CoReLS [3], FACET-II [4, 5], LUXE [6, 7], EXCELS
[8], ZEUS[9], Omega Laser Facility [10], HIBEF [11], among others. The
proposed experimental setups consist of scattering of intense, focused laser
pulses with either relativistic electron beams or high-energy photons,
allowing for precision studies of radiation reaction (the recoil on the
charged particles that emit high-energy photons) and electron-positron
production in the lab.

In these studies, different regimes of radiation reaction can be identified
based on the energies of the probe particles and the strength of the
electromagnetic fields involved.

Recent experiments have demonstrated electron energy loss that can be
attributed to radiation reaction [12, 13]. For low enough electron energies
and laser intensities, the effect of radiation on the radiation emitting
electrons can be described through a continuous correction to the equation of
motion, e.g. the Landau-Lifshitz equation [14]. For intermediate values of
energies and intensities, the evolution of the particle distribution can be
modeled by a Fokker-Planck equation, where the energy of each emitted photon
is assumed to be much smaller than the emitting lepton [15, 16]. In this case,
the particle can still lose a significant fraction of its energy through
multiple emissions of low-energy photons. For higher energies and intensities,
a single photon emission can lead to almost complete energy depletion of the
parent lepton, and thus a Boltzmann/transport equation becomes a better
description. A comparison between the different models of radiation reaction
and their range of validity can be found in the recent literature [17, 18,
19].

Besides radiation reaction, a phenomena likely to occur in electron-laser
collisions is Breit-Wheeler [20] electron-positron pair production. The pair
production rates for this process are known for particles in a constant
intense background field, and can be mapped onto a plane wave scenario. There
are also extended models that take into account the fact that the laser is a
wavepacket with a temporal envelope. For example, the authors in Ref. [21]
derive approximate scaling laws for the expected positron yield for both
photon-laser and electron-laser collisions. Further theoretical work was
pursued by [22], where the authors investigated photon-laser scattering for
different Laguerre-Gauss modes of the laser, and in [23] where optimal
focusing conditions were found for maximizing the number of pairs from the
scattering between Gaussian laser pulses and different electron beam profiles.

Since the strong-field regime of QED is characterized by the breakdown of
perturbative approaches, it requires the development of new theoretical and
computational tools. Numerical modeling of experiments with enough
spatiotemporal resolution and accounting for all relevant scattering
parameters becomes too computationally expensive if relying on full-scale 3D
PIC and Monte Carlo simulations, especially if the analysis requires multiple
parameter studies. New methods are currently being evaluated for accelerating
the evaluation of QED rates (e.g. Machine-Learning based [24], and Chebyshev
polynomial fits [25]).

In this paper, we describe the development of a reduced semi-analytical model
for the final energy and angular distribution of particles in electron-laser
scattering (see figure 1). This approach simplifies the calculation for
various beam geometries, focusing and collision synchronization. The results
can be obtained within minutes on a personal computer. We perform benchmarks
with the fully-relativistic particle-in-cell code OSIRIS [26], and demonstrate
that with this model quantitative predictions can be obtained without the need
for the full-scale simulations. This can be particularly useful for real-time
parameter scans during the course of an experiment.

Figure 1: Scattering of a relativistic electron beam (yellow) and a counter-
propagating laser pulse (red and blue). Electron-positron pairs and high
energy photons can be produced in the interaction, and their number, spectra
and divergence will depend on the precise geometry of the collision.

This manuscript is structured as follows. In section 2, we discuss the mapping
between plane wave (PW) models and more realistic “3D” scattering setups. In
section 3, we use this approach to derive the final spectra of photons and
electrons after scattering with a finite, focused laser pulse. In section 4,
we apply an approximate semi-classical numerical model for the photon energy
distribution. In section 5, we investigate the electron and photon angular
distribution as well as positron yield. Finally, the conclusions are presented
in section 6.

##  2 Effective intensity in focused laser scattering

This section introduces the theoretical framework for describing the
distribution of particles in the maximum effective intensity of the laser they
interact with along their ballistic trajectories. This approach aims at
simplifying the mapping between 1D plane wave models to more realistic
3D-focused-laser environments.

To better understand the derivation of the effective intensity distribution,
it is useful to first recall the case of a plane wave pulse. When a beam of
particles collides with this laser pulse (there is no focusing, the wavepacket
has a temporal envelope and no transverse structure), all particles get to
interact with the maximum intensity. For this case, the distribution of
particles $\mathrm{d}N_{b}$ according to the peak laser vector potential they
interact with $a_{0,\mathrm{eff}}$ is of the form
$\mathrm{d}N_{b}/\mathrm{d}a_{0,\mathrm{eff}}\sim\delta(a_{0,\mathrm{eff}}-a_{0})$,
as all particles experience the maximum vector potential $a_{0}$.

However, in the case of a diffraction-limited focused laser, not all beam
particles (which can be either ultra-relativistic electrons or photons)
interact with the peak laser intensity, but rather with a maximum
$a_{0,\mathrm{eff}}$ that depends on the longitudinal and transverse offsets
from the focus at the collision time. If one considers a cylindrical
coordinate system $\vec{r}=(\rho,\phi,z)$, each beam particle is assigned an
effective vector potential $a_{0,\mathrm{eff}}(\rho,\phi,z)$ according to the
distribution function

$\frac{\mathrm{d}N_{b}}{\mathrm{d}a_{0,\mathrm{eff}}}(a_{0,\mathrm{eff}})=\int_{V}2~{}\delta(a(\vec{r})-a_{0,\mathrm{eff}})~{}n_{b}(\vec{r})~{}\mathrm{d}V=\int_{S}\frac{2~{}n_{b}\
\mathrm{d}S}{||\nabla a||}.$ (1)

where $a_{0,\mathrm{eff}}$ is the argument of the distribution function,
$a(\vec{r})$ is the effective vector potential as a function of the
coordinates (to be specified later), $n_{b}$ is the particle number density in
the lab frame, the factor 2 accounts for crossing time (at twice the speed of
light for ultrarelativistic particles), and the surface integral (right-hand
side) is evaluated at $a=a_{0,\mathrm{eff}}$, . The functional expression
above assigns each macro-particle (which is equivalent to a volume element in
the electron distribution) a spatial coordinate corresponding to the position
along its trajectory where it interacts with the maximum laser field. The
volumetric integral then assigns the volume element to the correct bin in the
$a_{0,\mathrm{eff}}$ distribution. As a Gaussian focusing geometry is
cylindrically symmetric around the laser propagation axis, we can convert
equation (1) to a surface integral, where
$\mathrm{d}S=\rho\sqrt{\mathrm{d}\rho^{2}+\mathrm{d}z^{2}}~{}\mathrm{d}\varphi=\rho\sqrt{1+(\partial\rho/\partial
z)^{2}}~{}\mathrm{d}z~{}\mathrm{d}\varphi$ is calculated at the isosurface
that is by definition perpendicular to the gradient of the vector potential
given by $||\nabla a||=\sqrt{\left(\partial
a/\partial\rho\right)^{2}+\left(\partial a/\partial z\right)^{2}}$.

The distribution becomes a function of both the beam density profile and the
laser’s spatial structure (neglecting the wave’s phase, and in particular its
wavefront curvature). For example, the ideal Gaussian laser has a spatial
dependence

$a(\rho,\phi,z)=\frac{a_{0}}{\sqrt{1+(z/z_{R})^{2}}}\exp\left(-\frac{\rho^{2}/W_{0}^{2}}{1+(z/z_{R})^{2}}\right)$
(2)

where $W_{0}$ is the spotsize, $z_{R}\equiv\pi W_{0}^{2}/\lambda$ the Rayleigh
length, and $\lambda$ is the laser central wavelength.

For an arbitrary beam density profile $n_{b}(\vec{r})$ with length $L$ and
radius $R$, it can be challenging to analytically compute the particle
distribution in $a_{0,\mathrm{eff}}$. However, in the limiting cases of a
Short beam ($L\ll z_{R}$, transverse Gaussian density), Wide beam ($R\gg
W_{0}$, longitudinal and transverse flat-top density) and Thin beam ($R\ll
W_{0}$, longitudinal flat-top density) these distributions can be calculated.
In figure 2, we present several instances of these geometries and their
respective distributions. In table 1, we collect the distributions for the
three above-mentioned geometries, generalized to the case of non-synchronized
scattering, i.e., when the particle beam and laser collide outside of the
focus.

Figure 2: Particle distributions for different geometries, where $\lambda=1~{}\mu\mathrm{m}$ and $W_{0}=2~{}\mu\mathrm{m}$. The dashed line corresponds to the limit of either infinite beam radius (Short beam) or infinite length (Thin and Wide beams). Setup | Particle distribution for temporally unsynchronized beams
---|---
$\begin{array}[]{cc}\textrm{Wide}\\\\[-10.0pt] \textrm{beam}\end{array}$ | $\dfrac{\mathrm{d}N_{b}}{\mathrm{d}a}=\dfrac{2\pi~{}n_{b}~{}W_{0}^{2}~{}z_{R}}{a}\left(\dfrac{z_{+}}{z_{R}}\left(1+\dfrac{1}{3}\left(\dfrac{z_{+}}{z_{R}}\right)^{2}\right)\theta(a_{z_{+}}-a)-\dfrac{z_{-}}{z_{R}}\left(1+\dfrac{1}{3}\left(\dfrac{z_{-}}{z_{R}}\right)^{2}\right)\theta(a_{z_{-}}-a)\right.$
|
$\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\dfrac{\sqrt{a_{0}^{2}-a^{2}}}{3~{}a}\left(2+\dfrac{a_{0}^{2}}{a^{2}}\right)\left(\theta(a-a_{z_{+}})\pm\theta(a-a_{z_{-}})~{}\right)\right)$
$\begin{array}[]{cc}\textrm{Thin}\\\\[-10.0pt] \textrm{beam}\end{array}$ | $\dfrac{\mathrm{d}N_{b}}{\mathrm{d}a}=\dfrac{2N_{b}z_{R}}{L}\dfrac{a_{0}^{2}}{a^{2}}\dfrac{1}{\sqrt{a_{0}^{2}-a^{2}}}\left(\theta(a-a_{z_{+}})\pm\theta(a-a_{z_{-}})~{}\right)$
$\begin{array}[]{cc}\textrm{Short}\\\\[-10.0pt] \textrm{beam}\end{array}$ | $\dfrac{\mathrm{d}N_{b}}{\mathrm{d}a}=N_{b}\dfrac{W^{2}}{R^{2}}\dfrac{1}{a}\left(\dfrac{a}{a_{0}}~{}\dfrac{W}{W_{0}}\right)^{W^{2}/R^{2}}I_{0}\left(2\dfrac{\Delta_{\bot}}{R}\dfrac{W}{R}\log^{1/2}\left(\dfrac{a_{0}}{a}~{}\dfrac{W_{0}}{W}\right)\right)\exp\left(-\dfrac{\Delta_{\bot}^{2}}{R^{2}}\right)\theta(a-a_{\parallel})$
Table 1: Particle distributions for arbitrary temporal synchronization. A
shorter notation is used $a\equiv a_{0,\mathrm{eff}}<a_{0}$. Here,
$a_{z}\equiv a_{0}/\sqrt{1+(L/4z_{R})^{2}}$ is the $a_{0,\mathrm{eff}}$
associated with the integration limits imposed by the longitudinal size of the
electron beam, $N_{b}$ represents the total number of particles in the beam,
$n_{b}$ is the beam density, $R$ and $L$ are the beam radius and length
respectively. The laser spot size is $W_{0}$ in the focal plane and
$W=W_{0}\sqrt{1+\Delta_{\parallel}^{2}/z_{R}^{2}}$ outside, $z_{R}\equiv\pi
W_{0}^{2}/\lambda$ is the Rayleigh length, and $\Delta_{\parallel}$,
$\Delta_{\bot}$ are the displacements of the collision center from the laser
propagation axis and focal plane, respectively. The $\pm$ sign corresponds to
situations where $\Delta_{\parallel}<L/4$ or $\Delta_{\parallel}>L/4$
respectively, $\theta(x)$ is the Heaviside Theta function,
$z_{\pm}=\Delta_{\parallel}\pm L/4$,
$a_{z_{\pm}}=a_{0}/\sqrt{1+(z_{\pm}/z_{R})^{2}}$ and
$a_{\parallel}=a_{0}/\sqrt{1+(\Delta_{\parallel}/z_{R})^{2}}$.

We first introduced the concept of the
$\mathrm{d}N_{b}/\mathrm{d}a_{0,\mathrm{eff}}$ distribution in our previous
work [23], where we applied it to the optimization of the number of pairs
produced in electron-laser scattering. We accomplished that by generalizing
the scaling law previously derived for positron production in a plane wave
[21]. This approach is cost-effective and amenable to analytical calculations
pairing with PW models for other observables. In the next section, we apply
our method to estimate the final electron energy spectrum in a focused laser
scattering.

##  3 Electron and photon spectra

In this section, we apply the effective interaction intensity distributions to
derive the asymptotic energy distributions of the electrons after the
scattering. Simulation parameters for the given examples can be found in
appendix Appendix A: PIC simulation parameters.

###  3.1 Classical Radiation Reaction

If the scattering between the laser and the electron beam is in the classical
regime of interaction, one can assign a deterministic trajectory to an
electron within the beam. If there is considerable radiation emission (e.g. a
few percent of the electron energy is radiated) during the interaction, then
we can describe the motion though the Landau-Lifshitz equation, which has an
exact solution in a plane wave [27]. This solution calculates the full space-
time trajectory of the particle as it interacts with the laser field (in
configuration and momentum space). For the case of a pulse with a temporal
envelope, this solution can be condensed into a scaling law for the final
energy of the electron [28]

$\gamma_{f}=\frac{\gamma_{0}}{1+c_{RR}~{}\gamma_{0}~{}a_{0}^{2}}~{},\quad
c_{RR}\equiv(1-\cos\theta)^{2}\frac{\eta}{3}\frac{e^{2}\omega_{0}^{2}}{mc^{3}}\tau_{0}.$
(3)

Here, $\gamma_{0}$ is the initial electron energy, $\tau_{0}$ is the laser
pulse duration, $\omega_{0}$ the fundamental laser frequency, $c_{RR}$ is a
numerical factor quantifying radiation reaction that depends on the laser
shape and scattering angle, $\eta$ is a numerical factor associated with the
temporal shape of the laser pulse, $\theta$ is the interaction angle between
the laser wavefront and the colliding electron, $e$ is the elementary electric
charge, $m$ the electron mass, and $c$ is the speed of light in vacuum.

Equation (3) shows a 1-to-1 relation between the final energy and the $a_{0}$
of the plane wave (fixing all other parameters), which means it is possible to
directly calculate the final energy for each particle, provided we know the
maximum laser intensity and $\gamma_{0}$. Combining the effective intensity
distributions from table 1 and the equation (3), we obtain the full energy
distribution function after the interaction

$\dfrac{\mathrm{d}N_{b}}{\mathrm{d}\gamma_{f}}(\gamma_{f})=\dfrac{\mathrm{d}N_{b}}{\mathrm{d}a_{0,\mathrm{eff}}}~{}\left(\dfrac{\mathrm{d}\gamma_{f}}{\mathrm{d}a_{0,\mathrm{eff}}}\right)^{-1}$
(4)

where the second part is the inverse of the derivative of equation 3 with
$a_{0}=a_{0,\mathrm{eff}}$. In figure 3, we compare the distributions obtained
from the analytical expression (4) against PIC simulations. Here, electrons
radiate energy following the Landau-Lifschitz classical radiation reaction
model, while the quantum radiation reaction/QED module was turned off in the
simulation parameters to demonstrate the applicability of the model. For the
different geometries and aspect ratios $R/W_{0}$, we find good agreement
between the predicted distributions and simulation results. It is worth noting
that the cut-offs in the spectra (specially for the Short and Thin beams) is a
feature of deterministic radiation reaction; however, this will be smoothed
out if the initial electron beam has large enough energy spread or a
stochastic quantum radiation reaction model is used.

Figure 3: Final electron energy distributions for Short, Thin and Wide beam
geometries after classical radiation reaction. Parameters: $\gamma_{0}=5000$,
$a_{0}=12$, $\lambda=0.8~{}\mu\mathrm{m}$, $\tau_{0}=50~{}\omega_{0}^{-1}$.
The exact formulas for the theoretical distribution are presented in Table 2

The summary of formulas for the final energy distributions of electrons is
given in table 2. We show the expression for the same examples as in the
previous table, where for the sake of simplicity we consider the scattering
synchronized and aligned ($\Delta_{\parallel}=0,\Delta_{\bot}=0$). The
distributions can be derived for any case (from table 1 or not) following the
procedure that we just outlined.

Setup | Energy distributions after classical radiation reaction
---|---
Wide beam | $\dfrac{\mathrm{d}N_{b}}{\mathrm{d}\gamma}=4\pi~{}n_{b}~{}W_{0}^{2}~{}z_{R}\dfrac{\gamma_{0}/\gamma}{\gamma_{0}-\gamma}~{}\left(\dfrac{L(L^{2}+16z_{R}^{2})}{128z_{R}^{3}}~{}\theta_{1}\right.$
|
$\left.\quad\quad+\dfrac{1}{6(\gamma_{0}-\gamma)^{3/2}}(c_{RR}\gamma\gamma_{0}a_{0}^{2}+2(\gamma_{0}-\gamma))\sqrt{c_{RR}\gamma\gamma_{0}a_{0}^{2}+\gamma-\gamma_{0}}~{}(1-\theta_{1})\right)$
Thin beam | $\dfrac{\mathrm{d}N_{b}}{\mathrm{d}\gamma}=\dfrac{2N_{b}~{}z_{R}}{L}\dfrac{\gamma_{0}}{(\gamma_{0}-\gamma)^{3/2}}\dfrac{c_{RR}\gamma_{0}a_{0}^{2}}{\sqrt{c_{RR}\gamma\gamma_{0}a_{0}^{2}+\gamma-\gamma_{0}}}~{}\theta_{1}~{}\theta_{2}$
Short beam | $\dfrac{\mathrm{d}N_{b}}{\mathrm{d}\gamma}=N_{b}~{}\dfrac{W_{0}^{2}}{2R^{2}}\dfrac{\gamma_{0}/\gamma}{\gamma_{0}-\gamma}\left(\dfrac{\gamma_{0}-\gamma}{c_{RR}\gamma_{0}\gamma a_{0}^{2}}\right)^{\dfrac{W_{0}^{2}}{2R^{2}}}~{}\theta_{2}$
Table 2: Electron spectra after classical scattering with focused laser pulses
for different beam geometries. Distributions are obtained from table 1 and the
scaling law (3). The minimum electron energy is
$\gamma_{\mathrm{min}}\equiv\gamma_{0}/(1+c_{RR}\gamma_{0}a_{0}^{2})$ and the
energy at branch transition is
$\gamma_{z}\equiv\gamma_{0}/(1+c_{RR}\gamma_{0}a_{z}^{2})$.
$\theta_{1}\equiv\theta(\gamma-\gamma_{z})$,
$\theta_{2}\equiv\theta(\gamma-\gamma_{\mathrm{min}})$

###  3.2 Quantum Radiation Reaction and misaligned beams: semi-analytical
approach

In the previous sub-section, we considered the case of deterministic,
classical radiation reaction. As the laser field and the electron energy
increase, the photon emission becomes stochastic due to its quantum nature and
tends to spread the distribution function of the photon emitting electrons.
There is no general analytical solution for the evolution of the coupled
electron and photon distributions, even in the simplest field configurations.
However, the plane wave case is easy to simulate numerically in one dimension.
In this subsection, we show that we can use specific plane wave solutions from
1D simulations to reconstruct a more general 3D result. Similarly to the
deterministic case, the energy lost to photons is expected to increase as the
laser $a_{0}$ increases and the ratio $R/W_{0}$ decreases.

In figures 4 a) and b) we show simulation results for the final photon and
electron spectra after scattering with a pulsed plane wave. Analogously with
the classical case, there are more electrons in the low energy tail of the
distribution for the high values of $a_{0}$. To map these results to 3D,
similarly to equation (4), we weight the plane wave spectra $dN^{PW}/d\gamma$
by the $a_{0,\mathrm{eff}}$ distribution function

$\dfrac{\mathrm{d}N}{\mathrm{d}\gamma}^{3D}(\gamma;a_{0})\sim\sum_{a_{0,\mathrm{eff}}}\dfrac{\mathrm{d}N}{\mathrm{d}\gamma}^{PW}(\gamma,a_{0,\mathrm{eff}})~{}\dfrac{\mathrm{d}N_{b}}{\mathrm{d}a_{0,\mathrm{eff}}}(a_{0,\mathrm{eff}})$
(5)

where $a_{0,\mathrm{eff}}$ is a uniformly distributed, discrete set of $a_{0}$
values with enough resolution to represent the full effective intensity
distribution.

By running several PW simulations in parallel, this approach can be used to
calculate the asymptotic distributions expected after the interaction with a
focused Gaussian laser pulse (including and all the variations on alignment),
which has orders of magnitude lower computational cost than one full-scale 3D
or quasi-3D simulation. In figures 4 c) and d) we present the results of
quasi-3D simulations in color against the reconstruction method presented
before, showing good agreement. We used PW simulations with $a_{0}$ from $0$
to $12$ in step increments of $0.2$.

Figure 4: Reconstructing particle spectra in focused laser collisions using
pulsed plane wave datasets. a) and b) photon and electron final spectra in
pulsed plane wave collisions, respectively. c) and d) Final spectra from a 3D
simulation of the interaction of a diffraction-limited (Gaussian) laser pulse
against the reconstruction from 1D plane wave samples.

The same PW dataset can be effectively used to reconstruct the electron and
photon distributions from a collision with a transverse offset (presented in
figure 5).

Figure 5: Perpendicular displacement of Short beam. Left: $a_{0,\mathrm{eff}}$
distribution. Middle: photon spectra. Right: electron spectra. Same main
parameters as in figure 4, with maximum $a_{0}=12$.

It is also possible to reconstruct the expected distributions for longitudinal
(temporal) offsets (see figure 6).

Figure 6: Parallel displacement of Short beam. Left: $a_{0,\mathrm{eff}}$
distribution. Middle: photon spectra. Right: electron spectra. Same main
parameters as in figure 4, with maximum $a_{0}=12$.

In later sections, this approach is applied to other observables from laser-
beam scattering.

##  4 The approximate photon spectra

In this section, we discuss the possibility of computing an approximate
asymptotic photon spectrum without resorting to PIC simulations. This can
replace the 1D plane wave calculations of the photon spectra from the previous
section.

As an electron interacts with a wavepacket, it loses energy, on average,
following the semi-classical equation of motion (the Lorentz force is omitted
in the equation, we just display the radiation reaction term)

$\left(\dfrac{\mathrm{d}p}{\mathrm{d}t}\right)_{RR}=-~{}g(\chi_{e})~{}P_{cl}/c,~{}P_{cl}=\dfrac{2~{}\alpha~{}c}{3~{}{\mkern
0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{c}}m_{e}c^{2}\chi_{e}^{2}$ (6)

where $\chi_{e}\sim 2~{}\gamma~{}a/a_{S}$ is the lepton quantum nonlinearity
parameter; $a_{S}$ is the normalized Schwinger vector potential; $P_{cl}$ the
classical synchrotron-like emitted power, $\alpha$ is the fine-structure
constant; ${\mkern 0.75mu\mathchar 22\relax\mkern-9.75mu\lambda}_{c}$ is the
Compton wavelength, and $g(\chi_{e})$ the electron Gaunt-factor (which
corrects the average emitted power to be consistent with a full quantum
approach).

According to equation (6), for each phase of the laser $\phi$ we can assign an
associated average electron energy $\gamma(\phi)$ and combine this with an
instantaneous laser field vector potential $a(\phi)$. For each of these pairs
of values, we compute the quantum synchrotron spectrum (in local constant
field approximation) and integrate them to obtain the final, cumulative photon
spectrum as shown in figure 7.

Figure 7: Left: comparison between final cumulative photon spectra in
electron-laser PW interaction for varying $a_{0}$ in simulations (solid) and
reconstructed (dashed). Right: comparison between final cumulative photon
spectra in electron-laser PW interaction for varying $R/W_{0}$ in simulations
(solid) and reconstructed (dashed).

This results in a very good estimate of the photon spectra. Figure 7 also
shows small deviations between the simulations and the theoretically predicted
reconstructions. These deviations can be attributed to the neglected energy
spread of the photon emitting electrons. This model could be further improved
by applying a Boltzmann equation solver to capture a self-consistent, multi-
particle evolution of all the energy momenta.

##  5 Angular distribution of electrons, photons and positron yield

After having addressed the average energy, energy spread and the evolution of
the electron energy distribution function during the laser-electron
scattering, in this section we discuss the possibility of incorporating
several scaling laws for different scattering observables into the model. In
particular, the highlight is given to the angular distribution and of both
electrons and photons, as this can be directly measured in experiments. For
completeness, we also mention how the same method is applied to calculate the
positron yield, which was addressed in more detail in our previous work [23].

###  5.1 Photon angular distribution

Relativistic electrons emit photons within an angle $\sim 1/\gamma$ around
their propagation direction. When an electron interacts head-on with a
linearly polarized laser (LP), the transverse momentum is of the order of the
laser vector potential, so the expected opening angle of the emission is
$\sim\left<a\right>/\gamma$ in the polarization direction. The third
direction, perpendicular both to the propagation and the polarisation still
has an expected typical divergence of $\sim 1/\gamma$. Here $a$ and $\gamma$
refer to the instantaneous laser amplitude and electron energy. To account for
a laser temporal envelope (in particular, we use a $\sin^{2}$ pulse envelope)
and linear polarization, the average instantaneous value of the laser vector
potential is twice decreased by a factor of $\sqrt{2}$ compared to the peak
field. This gives an overall expected root-mean-squared photon angle of $\sim
0.5~{}a_{0}/\gamma_{0}$.

For moderate to high laser intensity, the number of photons above a certain
energy scales linearly with $a_{0}$ (see figure 8 a). Since the number of
photons emitted by the electrons is a function of $a_{0}$, this has to be
included in the weighted average over the instantaneous emission angle to
obtain the expected final divergence of the signal.

As shown in figure 8, using this information and pairing it with the effective
intensity distributions allows predicting the final photon divergence for
diffraction-limited laser-electron collisions, including spatio-temporal
misalignment.

$\theta_{3D}^{2}\sim\int\theta_{PW}^{2}~{}\dfrac{N_{\gamma}}{N_{e}}~{}\dfrac{\mathrm{d}N_{b}}{\mathrm{d}a_{0,\mathrm{eff}}}~{}\mathrm{d}a_{0,\mathrm{eff}}~{}/\int\dfrac{N_{\gamma}}{N_{e}}~{}\dfrac{\mathrm{d}N_{b}}{\mathrm{d}a_{0,\mathrm{eff}}}~{}\mathrm{d}a_{0,\mathrm{eff}}$
(7)

Figure 8: Left: number of photons per electron (only including photons above 1
MeV energy). Middle: photon angle rms in plane wave. Right: photon angle rms
for setups using focused lasers.

###  5.2 Electron angular distribution

As the photon divergence depends directly on the instantaneous electron
divergence at emission time, estimating the final photon divergence can be
accomplished with a simple integration, as shown in previous subsection. The
asymptotic electron divergence is more challenging to estimate because of the
accumulated diffusion of the distribution function due to the quantum
stochasticity of photon emission. In [16], a scaling law was derived for the
electron transverse momentum divergence in a PW as a function of average final
energy $\gamma_{f}$ and energy spread $\sigma_{f}$, following the quantum
radiation reaction on the emitting electrons. In [29], it was shown that if
the precise values of final average energy and energy spread were used in the
previously mentioned scaling law, it could accurately predict the final
electron divergence. However, the estimate of the final divergence is
sensitive to the accuracy of the estimates for $\sigma_{f}$ and $\gamma_{f}$.
The final electron divergence, measured as root-mean-squared angle in the
electron momentum [16], can be written as

$\theta_{e}^{PW}\sim\sqrt{\dfrac{2}{\pi}}\dfrac{a_{0}~{}\sigma_{f}}{\gamma_{f}^{2}}$
(8)

where the electron energy spread is upper-bounded by

$\sigma_{\mathrm{f}}^{2}\leq 1.455\times
10^{-4}\sqrt{I_{22}}~{}\frac{\gamma_{0}^{3}}{\left(1+6.12\times
10^{-5}~{}\gamma_{0}~{}I_{22}~{}\tau_{0}[\mathrm{fs}]\right)^{3}}.$ (9)

Here, $I_{22}=10^{-22}~{}I[\mathrm{W/cm}^{2}]$ is the laser intensity and
$\gamma_{f}$ is the final electron energy given by equation (3). Similarly to
the photon divergence, the equivalent electron divergence in 3D will be the
root-mean-squared plane wave divergence weighted by the $a_{0,\mathrm{eff}}$
distribution.

In figure 9, we compare this scaling law against PIC simulations, first in the
case of a plane wave for different $a_{0}$ values, and then for focused
Gaussian laser scattering with different collision offsets.

Figure 9: Left: electron angle rms in plane wave. Right: electron angle rms in
focused laser setups.

Equation (9) was originally derived for scattering against a laser pulse with
a long, flat temporal envelope, where sufficient time would have passed for an
equilibrium between radiative cooling and stochastic diffusion to have
occurred. For the laser pulse durations considered in this work ($\ll
100~{}\mathrm{fs}$), this is not always satisfied. Consequently, we correct
the scaling law with a multiplicative pre-factor to match the plane wave
results with a shorter interaction time. The results are presented in figure
9, where there is an overall good correspondence between the model and
simulations, and once again we can incorporate the spatio-temporal
misalignment.

Knowing the asymptotic divergence of the electron beam can be very useful in
experiments. For example, in [30, 31], the authors explore the possibility of
using the induced asymmetry in the electron angular distributions in a
linearly polarized laser as a robust sign of quantum stochasticity broadening.
It may also be used to estimate the peak laser intensity during scattering,
which cannot be directly measured.

###  5.3 Positron yield in electron-laser scattering

In electron-laser scattering for pair production, both nonlinear Compton
Scattering and nonlinear Breit-Wheeler pair production have to be taken into
account. These two stochastic processes have to occur in succession and within
the crossing time of the laser pulse. In the soft-shower regime (only one
generation of pairs produced), the electrons emit photons of broad spectra and
at different phases of the laser. As a consequence, photons are distributed
not only in energy but also in their integrated probability of decaying into
pairs. This makes theoretical modeling of these events somewhat more
challenging than direct photon-laser scattering.

An approximate scaling law for the positron yield (and average energy) in the
scattering of electrons with a pulsed plane wave was derived in [21]. This
expression can then be implemented numerically into this framework (as we have
done in [23]) to estimate the positron yield in focused laser setups.

##  6 Conclusions

We present a semi-analytical model which offers a fast and computationally
inexpensive method of predicting electron yield, photon yield, spectra and
angular divergence in realistic electron scattering setups.

We introduce the model by considering the interaction between a distribution
of electrons interacting with a focused intense laser pulse, assuming the
probe particles perform ballistic trajectories. We then derive the closed-form
electron spectra for different collision geometries after experiencing energy
lost due to classical radiation reaction. Furthermore, we have shown that data
from quasi-1d simulations of scattering against a plane wave with a temporal
envelope can be recombined to yield the equivalent particle spectra in a
focused laser setup, both for classical and quantum radiation reaction models.
In both cases, the simulations and the theory agree very well, demonstrating
that the presented model can be used as a cheap and fast alternative to 3D-PIC
simulations. By employing scaling laws for the final photon and electron
divergence in a plane wave, we have also shown that these observables
scenarios can also be predicted. Besides Gaussian profiles, the model could
also be extended to other beam profiles, such as Laguerre or Hermite-Gauss
modes.

The need for fast computation of observables in high-repetition-rate
facilities requires a robust and efficient framework. The incoherent nature of
the nonlinear Compton Scattering gamma radiation (independent between
different electrons, but not within a single electron trajectory) allows
calculation of the final spectrum as a sum of single-particle spectra
(contrary to Thomson scattering where there will be interference patterns).
Having a sufficiently accurate forward model could enable the reconstruction
of experimental parameters and profiles (beam and laser), and quantify the
associated uncertainties. This will be particularly important in high
repetition-rate facilities, where data needs to be processed quickly and can
also be statistically relevant to train automated models. In the future, we
will investigate this possibility.

Another possible extension of this framework is the positron production at 90º
of incidence for Direct Laser Acceleration (DLA). Contrary to a head-on
collision, here the energy cutoff of the electron-positron pairs is no longer
limited to the initial energy of the interacting electrons. If positrons are
generated at low energies, they can be trapped, and accelerated along the
laser propagation direction [32].

We strongly believe these findings will contribute to further analytical
calculations on classical and quantum radiation reaction and pair production
and help design future experiments to validate current models of Strong-Field
QED.

##  7 Acknowledgments

This article comprises part of the PhD thesis work of Óscar Amaro, which will
be submitted to Instituto Superior Técnico, University of Lisbon.

The authors thank Mr. B. Barbosa, Dr. B. Martinez, and Dr. L. I. Iñigo Gamiz
for fruitful discussions and proofreading the manuscript. This work was
supported by the European Research Council (ERC-2015-AdG Grant No. 695088) and
Portuguese Science Foundation (FCT) Grants No. CEECIND/01906/2018, PTDC/FIS-
PLA/3800/2021, and UI/BD/153735/2022. Simulations were performed at the IST
cluster (Lisbon, Portugal).

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## Appendix A: PIC simulation parameters

QScatter (quick-scattering-toolkit) is an open-source library written in
Python. Examples showing its usage are provided on GitHub at:
https://github.com/OsAmaro/QScatter

In this appendix we specify the parameters of the simulations referred in this
work. Unless specified, common to all simulations are the electron energy
$\gamma_{0}=5000$, a laser wavelength of $\lambda$ = 0.8 micron
($\omega_{0}=2.35\times 10^{15}~{}s^{-1}$), linear polarization, a temporal
envelope of the type $\sin^{2}$, and a corresponding rise time of the pulse of
$50~{}1/\omega_{p}$. Additionally, when applicable, the laser spotsize was
$18.85~{}c/\omega_{p}$ and focused at the center of the simulation box. All
macro-electrons and photons from the simulations are included in the particle
distributions shown in the different plots of this work.

In section 3.1, we consider setups where the classical model of radiation
reaction is applied. For both the Wide and Thin beam geometries, the quasi-3d
simulation box dimensions were $1100~{}c/\omega_{p}$ and $100~{}c/\omega_{p}$
with $5500$ and $500$ grid cells in the longitudinal and transverse
directions, respectively. The time step was dt = $0.02~{}1/\omega_{p}$ and
particle information was retrieved at tmax = $320~{}1/\omega_{p}$ (after
interaction with the laser). The electron beam profile was flat top, with
dimensions $356~{}c/\omega_{p}$ and $80~{}c/\omega_{p}$ in the Wide beam
geometry, and with 20 particles-per-cell (ppc). For the Thin beam, the
transverse dimension was much smaller than the laser spot size. In the case of
the Short beam, the quasi-3d box dimensions were $300~{}c/\omega_{p}$ and
$100~{}c/\omega_{p}$, with grid cells $1500$ and $500$, with a time step of dt
= 0.02 $1/\omega_{p}$, and a total duration of tmax = 120 $1/\omega_{p}$. The
electron beam profile was flat top longitudinally, with length 0.8
$c/\omega_{p}$, and Gaussian in the transverse direction, with radiae
$13.33~{}c/\omega_{p}$ and $6.67~{}c/\omega_{p}$ for the different $R/W_{0}$
ratios, and ppc=320.

In section 3.2 we consider the quantum model of radiation reaction. For the
quasi-1d simulations, the box size was $200~{}c/\omega_{p}$ with 2500 grid
cells, a time step of dt = $0.021~{}1/\omega_{p}$, and a total duration of
tmax = $130~{}1/\omega_{p}$. The electron density was flat top with length
$4~{}c/\omega_{p}$, and ppc=$4000$. In the case of the synchronized Short Beam
setup, the quasi-3d simulation box and temporal parameters were the same as in
the Short beam with classical radiation reaction. The electron density profile
was Gaussian in the transverse direction, radiae $13.33$, $26.66$ and
$39.99~{}c/\omega_{p}$ respectively, and ppc=320. For the Short beam with
parallel offset, the quasi-3d simulation box dimensions were
$1100~{}c/\omega_{p}$ and $100~{}c/\omega_{p}$ with $5500$ and $500$ grid
cells in the longitudinal and transverse directions, respectively. The time
step was dt = $0.02~{}1/\omega_{p}$ and particle information was retrieved at
tmax = $120~{}1/\omega_{p}$. The electron Gaussian density profile had a
radius of $13.33~{}c/\omega_{p}$, with the initial center of the beam varying
in the longitudinal direction in multiples of 89 $c/\omega_{p}$ for different
offsets. In the case of the perpendicular offset, the simulation box had
dimensions $300~{}c/\omega_{p}$ in the longitudinal direction and
$240~{}c/\omega_{p}$ in both transverse directions, with 3000 and 1200 grid
cells, respectively. Temporal resolution was dt = 0.02 $1/\omega_{p}$, with
tmax = 120 $1/\omega_{p}$. The electron density profile was flat top in the
longitudinal direction with length $0.8~{}c/\omega_{p}$, and Gaussian in the
transverse direction with radius $13.33~{}c/\omega_{p}$. The initial center of
the electron beam was varied along the x2 direction in multiples of 13.33
$c/\omega_{p}$ for different offsets, and ppc=8.