,

# Modeling of the driver transverse profile for laser wakefield electron
acceleration at APOLLON Research Facility

I. Moulanier1, L.T. Dickson1, C. Ballage1, O. Vasilovici1, A. Gremaud1, S.
Dobosz Dufrénoy2, N. Delerue3, L. Bernardi4, A. Mahjoub4, A. Cauchois4, A.
Specka4, F. Massimo1, G. Maynard1, B. Cros1 1 LPGP, CNRS, Université Paris
Saclay, 91405 Orsay, France  2 CEA, CNRS, LIDYL, Université Paris Saclay,
91190 Gif sur Yvette, France  3 IJCLAB, CNRS, Université Paris Saclay, 91405
Orsay, France  4 LLR, CNRS/IN2P3, Ecole Polytechnique, Institut Polytechnique
de Paris, 91120 Palaiseau, France  *Corresponding authors : I. Moulanier,
<EMAIL_ADDRESS>B. Cros,
<EMAIL_ADDRESS>
###### Abstract

The quality of electron bunches accelerated by laser wakefields is highly
dependant on the temporal and spatial features of the laser driver. Analysis
of experiments performed at APOLLON PW-class laser facility shows that spatial
instabilities of the focal spot, such as shot-to-shot pointing fluctuations or
asymmetry of the transverse fluence, lead to charge and energy degradation of
the accelerated electron bunch. It is shown that PIC simulations can reproduce
experimental results with a significantly higher accuracy when the measured
laser asymmetries are included in the simulated laser’s transverse profile,
compared to simulations with ideal, symmetric laser profile. A method based on
a modified Gerchberg-Saxton iterative algorithm is used to retrieve the laser
electric field from fluence measurements in vacuum in the focal volume, and
accurately reproduce experimental results using PIC simulations, leading to
simulated electron spectra in close agreement with experimental results, for
the accelerated charge, energy distribution and pointing of the electron beam
at the exit of the plasma.

##  I Introduction

In the process of Laser WakeField Acceleration (LWFA) [1, 2], an ultra-high
intensity laser is focused inside a gas target, ionizes the medium and creates
a trailing perturbation in its wake in an underdense plasma. The generated
plasma cavity sustains intense longitudinal and transverse electric fields
that can trap, accelerate and focus bunches of electrons to the GeV range [3]
within a few centimeters. Injection of plasma electrons makes LWFA a compact
option for the generation of relativistic electron sources. However, despite
numerous studies, the use of electron beams from LWFA for applications is
impeded by insufficient beam quality and stability. Therefore, detailed
diagnostics, realistic modeling and analysis are needed to achieve a precise
understanding of the key mechanisms controlling the laser plasma interaction
in experiments.

The main schemes for the injection of plasma electrons into the plasma wave
are self-injection [4] and ionization injection [5, 6, 7, 8]. In LWFA
experiments with PW-class laser drivers, both injection schemes can occur in
the so-called bubble regime [9], in which the ponderomotive force of the laser
repels plasma electrons from its propagation axis, generating an electron-free
cavity behind the laser pulse. In the process of self-injection, a portion of
the expelled plasma electrons travel around the cavity before getting trapped
in the wakefield [10]. In ionization injection, the gas target is a mixture of
light atomic species, typically hydrogen, ionized early before the peak of the
laser pulse, and a dopant species, e.g. nitrogen, presenting an energy-gap in
its ionization potential structure [7], leading to ionization of some
electrons close to the peak of the laser pulse.

With peak intensities above $I_{0}=10^{18}$ W/cm2, the pulse temporal front
ionizes hydrogen and nitrogen up to $N^{5+}$. Remaining nitrogen L-shell
electrons are primarily born around laser peak intensities, inside the bubble
[11]. Ionization injection has several properties of interest for tuning
electron injection and trapping, and favors highly charged electron beams. It
operates at an intensity below self-injection [8] and the two mechanisms can
be optimized in different parameter areas. As ionization injection depends on
the local intensity of the laser pulse, it can be particularly sensitive to
laser beam quality and its evolution during propagation in the evolving plasma
density. These properties can be used to control the injection process in
electron density tailored profile or diagnose laser beam quality.

The electron beam charge can be increased by increasing the driving laser
power, providing a large range of parameters to explore for optimizing the
properties of laser driven electron sources with PW class short pulse laser
facilities.

This complex nonlinear physics is described using particle in cell (PIC)
simulations [12], using as input parameters the laser temporal and spatial
shapes, and the gas density profile. Experimentally achieved laser beams often
differ from perfectly symmetrical distributions. Hence, in order to understand
the role of laser imperfections on the quality of the produced electron
bunches, and compare to experimental results, refined PIC simulations
describing realistically the injection and acceleration physics occurring at
the ps scale were performed. These realistic simulations require a proper
description of the gas density profile, as well as of the driver (laser
profile transverse asymmetries) to accurately reproduce laser-plasma
interactions affecting the electron beams characteristics. For instance, laser
asymmetries have been shown to lead to asymmetric wakefields affecting the
output accelerated electron beam, with characteristics directly correlated
with the laser stability and quality [13, 14, 15, 16, 17].

In this paper the method used to analyze characteristic results obtained
during commissioning experiments at APOLLON PW facility [18] is presented.
Focusing the $\mathrm{F_{2}}$ laser beam at 0.4 PW in the long focal area
inside a gas cell [19], experiments were performed to characterize laser beam
quality and evaluate its impact on electron properties.

It is shown that PIC simulations can reproduce experimental results with a
significantly higher accuracy when the measured laser asymmetries are included
in the simulated laser’s transverse profile. This enhanced agreement is meant
in comparison with simulations with an ideal, axisymmetric laser profile,
which is often used in the design stage of LWFA experiments and in preliminary
experimental analyses. The results described in this work thus show the
importance of more realistic initial conditions in numerical modeling used for
these studies. The simulation results shown in this work were obtained with
the quasi-3D PIC code FBPIC [20], but the same method can be applied with
other PIC codes in quasi-3D [21] or full 3D geometry.

In comparison to previous investigations made with realistic PIC simulations
in [14], this work uses an alternative fast Gerchberg-Saxton algorithm [22]
based on mode decomposition to reconstruct the laser field, which allows the
accurate simulation of an experimental electron bunch spectrum in the energy-
angle plane. This method has already been used to present experimental results
in [15], in a regime with a lower peak laser intensity and characterized by a
more stable transverse laser profile from shot-to-shot. In addition, in this
work, the physical effects of the realistic laser driver (in particular its
asymmetries) on the electron injection in the bubble are described. An
agreement between realistic numerical modeling and experiment is obtained also
in the electron beam spectra in the energy-angle plane. Furthermore, the
realistic simulations in [14] have been performed in 3D, while the realistic
simulations of this work were performed in quasi-3D geometry [21],
highlighting that high-fidelity simulations can be obtained also with this
less computationally-demanding technique for preliminary analyses.

The remainder of this paper is organized as follows. Characteristic
experimental results are presented in section II. The method used to retrieve
the laser electric field from experimentally recorded fluence images is
described in section III, followed by the description of the method to
generate data to initialize FBPIC simulations. A comparison of experimental
and numerical electron spectra is discussed in section IV.

##  II Experimental results

An experiment was performed in April 2021 during the commissioning phase of
the long focal area of APOLLON facility to characterize laser beam quality
inside the experimental area and evaluate its impact on electron beam quality.

After compression, the $\mathrm{F_{2}}$ laser beam was transported into the
experimental area and focused in vacuum using an on-axis spherical mirror $3$
m focal length, after reflection from a turning mirror with a hole, as
illustrated in Fig. 1.

Figure 1: Schematic of experimental set-up : the driver laser beam (in red) is
focused through the turning mirror (I) at the entrance of the gas cell (II),
generating a diverging electron bunch (in green) at the exit. The electron
bunch is then deflected by a magnetic dipole (III) and sent onto a LANEX
screen (IV) to measure its Energy-Angle distribution. A small percentage of
the laser driver is used to probe the plasma density transversely. Vacuum
imaging of the focal volume is completed with a CCD (V).

The central part of the laser beam incident on the turning mirror was
collected and used partly to monitor the laser beam energy from shot-to-shot
using a leak through a wedge and a calorimeter. The remainder of the laser
beam was used as a probe laser to diagnose plasma density transversely to the
main pulse. The relative probing time was controlled by an in vacuum delay
stage.

For this experiment, the APOLLON F2 $\mathrm{Ti:Sa}$ laser with a central
wavelength $\lambda=0.8$, $\mu$m was measured to deliver a pulse with a FWHM
intensity duration $\tau_{FWHM}=25$ fs, a post-compression energy $E_{l}=5$ to
$10$ J, with a repetition rate of 1 shot per minute. The peak intensity
estimated in the ideal Gaussian transverse profile approximation in vacuum is
$5\times 10^{19}$ W/cm2. The laser beam was focused inside a $6$ mm long gas
cell [19] filled with a mixture of $99\%\mathrm{~{}}{H_{2}}$ and
$1\%\mathrm{~{}}{N_{2}}$. Electrons were trapped through ionization and
accelerated in the wakefield. After plasma exit, their energy was measured
using a dipole magnet and $\mathrm{LANEX}$ screen imaged onto a $\mathrm{CCD}$
camera. The spectrum was recorded in the $300-900$ MeV energy range within a
$\pm 20$ mrad viewing angle.

The laser beam was characterized in detail in the focal volume every day prior
to shots on the gas-filled target. Using a movable CCD camera in vacuum, the
laser energy distribution was measured inside the interaction chamber before,
at and after focus. The Rayleigh length in vacuum, defined as $z_{R}=(\pi
w_{0}^{2})/\lambda$, with $w_{0}$ being the $1/e^{2}$ radius of the focal
plane intensity, was $z_{R}=1$ mm. The waist of a Gaussian best fit in the
focal plane is $w_{0}=16.6\pm 0.3$ $\mu$m. Fig. 2 shows 2 sets of data taken
on two different days, illustrating instabilities both in laser pointing and
in spatial fluence symmetry between consecutive shots and for different days.
Fig. 2.(a) and (b) show the fluence distribution in the focal plane $z=z_{f}$.

Figure 2: Measured laser beam stability during 2 sequences of multiple vacuum
shots taken on 2 separate days with laser settings 1 and 2: (a) and (b) laser
fluence of 4 consecutive shots measured in the focal plane; (c) and (d)
$\overline{x}-\overline{x}_{f}$, laser beam relative centroid in the $xOz$
plane, with respect to the focal plane centroid, for three positions $z-z_{f}$
along the laser propagation axis; the data for $z=z_{f}$ is from the 4 fluence
images of (a) and (b), while the other points are data collected from 2
separate shots at each position; (e) and (f) fraction of energy
$\alpha_{gauss}$ inside a Gaussian fit of waist $w_{0}$. Error bars along $z$
come from the determination of the focal plane $z_{f}$.

Fig. 2.(c) and (d) show the shot-to-shot transverse centroid displacement
fluctuations $\overline{x}-\overline{x}_{f}$ in the focal volume (vertical
error bars). The horizontal error bars come from the determination of the
focal plane $z_{f}$. The $xOz$ plane is the plane where electrons centroid
fluctuations have been measured, perpendicularly to the laser polarization
plane $yOz$. Fig. 2.(c) and (d) underline instabilities of the laser centroid
in both cases. For Set 1 data the centroid is moving linearly, going from $+3$
$\mu$m at $z-z_{f}=-0.6$ mm to $-6$ $\mu$m at $z-z_{f}=+0.6$ mm. Set 2 data
show larger fluctuations $\overline{x}-\overline{x}_{f}=-7$ $\mu$m at
$z-z_{f}=-0.6$ mm and $\overline{x}-\overline{x}_{f}=-18$ $\mu$m at
$z-z_{f}=+0.6$ mm. Multi-directional fluctuations of the centroid around the
focal plane are the signature of a non zero temporal phase and the asymmetry
of the laser fluence.

In Fig. 2.(e) and (f), $\alpha_{gauss}$, defined as the fraction of total
energy inside a Gaussian fit with waist $w_{0}$, is plotted as a function of
position in the focal volume. Values of $\alpha_{gauss}$ at $z\neq z_{f}$ are
calculated using a waist $w(z)=w_{0}[1+\left(z/z_{R}\right)^{2}]^{1/2}$ for
the Gaussian fit. At $z=z_{f}$, $\alpha_{gauss}$ is averaging $46\,\%$ for Set
1 and $40\,\%$ for Set 2. This demonstrates that the experimental fluence
departs significantly from a perfect Gaussian approximation in both cases.

Figure 3 shows the average of 10 consecutive electron bunch spectra measured
on the same day as Fig. 2 Set 1, in $yOx$, also defined as the
$Energy-\theta_{x}$ angle plane.

Figure 3: Average electron spectrum of 10 consecutive shots acquired the same
day as Fig. 2 Set 1 laser measurements for an average plasma electron density
$\overline{n}_{0}=2.2\times\mathrm{10^{18}}$ cm-3. The color scale shows the
spectral charge density in the $Energy-\theta_{x}$ angle plane. The full green
line along the $Energy$ axis is the average spectral charge density, $dQ/dE$,
profile (scale on left vertical axis) calculated within $\pm 3$ mrad around
each individual maximum $dQ/d\theta_{x}$. The full green line along the
$\theta_{x}$ axis (right-handside vertical axis) represents the average
$dQ/d\theta_{x}$ profile (scale on the top horizontal axis) integrated over
energy. The green areas are the confidence intervals bounded by the standard
deviation extremes (dashed curves). $\theta_{x}=0$ corresponds to the laser
axis alignment position in vacuum.

The average bunch charge measured between $300$ and $900$ MeV and within $\pm
20$ mrad is $\overline{Q}_{tot}=95\pm 46$ pC. For this sequence, the mean
measured laser energy is $\overline{E}_{l}=4.8\pm 0.2$ J, and average plasma
electron density $\overline{n}_{0}=2.2\pm 0.2\times\mathrm{10^{18}}$ cm-3. The
measured charge fluctuations are
$\delta\overline{Q}_{tot}/\overline{Q}_{tot}=47\,\%$, which shows a clear
sensitivity to input parameters.

##  III Modeling of the laser beam

The analysis and implementation of the laser experimental data into PIC
simulations are performed in three steps. First, the laser electric field is
retrieved from the measured fluence data. Then, the reconstructed laser
electric field is represented as a mode sum of fields that are used as
parameters at the start of PIC simulations. Finally, the calculated electron
parameters at the end of each simulation are compared with measured results.

###  III.1 Fit of the laser electric field

A modified version of the reconstructive Gerchberg-Saxton algorithm (GSA)
[22], described in Appendix A, was used to retrieve the laser electric field
corresponding to the laser fluence measured at 3 positions :
$z-z_{f}=0,\,+1.2,\,-1.8$ mm.

The algorithm was used in particular to retrieve an unknown phase map,
$\psi(x,y)$, associated to a set of fluence images measured at different
positions $\left(z_{0},\,z_{1}...\,z_{k_{max}}\right)$.

We selected laser data from the same day as Fig. 2 Set 1, which exhibit better
focal spot stability than Set 2, to reconstruct a realistic laser electric
field distribution. For each position, the corresponding fluence distribution
has been re-centered around the origin in order to reduce in advance the shot-
to-shot fluctuations error on the fit. The selected experimental distributions
(upper row) and results of the fit algorithm (middle row) are plotted in Fig.
4.

Figure 4: Experimental laser fluence measured in the focal volume (upper row),
corresponding Hermite-Gaussian reconstruction (middle row), Laguerre-Gauss fit
with $L=3$ and $M=40$ used in simulations (lower row): (a), (b) and (c)
$z-z_{f}=-1.8$ mm \- (d), (e) and (f) $z-z_{f}=+0$ mm \- (g), (h) and (i)
$z-z_{f}=+1.2$ mm. Each distribution has been normalized by its peak fluence.

Using a combination of low order fits to establish an educated guess, then
refining the optimization with a higher order Hermite-Gaussian modes
projection, a realistic reconstructed distribution is calculated as
represented in Fig.4, (middle row).

The laser fluence reconstructed with Laguerre-Gauss modes is plotted in Fig. 4
(lower row) and shows a relatively good agreement between the reconstructed
Laguerre-Gauss distribution and the experimental laser data.

###  III.2 PIC simulation set-up

Due to the cylindrical representation used in FBPIC, the reconstructed
Hermite-Gauss laser electric field $\mathrm{E}_{\mathrm{HG}}(r,\theta,z_{f})$
is projected on Laguerre-Gauss modes [23] in the focal plane $z=z_{f}$ :

$\begin{split}C_{l,m}=\int_{r=0}^{r_{max}}&\int_{\theta=0}^{2\pi}\mathop{r}\mathop{dr}\mathop{d\theta}~{}\mathrm{E}_{\mathrm{HG}}(r,\theta,z_{f})\\\
&\times\mathrm{LG}_{l,m}^{*}(r,\theta,z_{f})(r,\theta,z_{f},r_{0,opt},\theta_{0,opt},w_{0r})\end{split}$
(1)

with $C_{l,m}$ being the complex amplitudes of the $\mathrm{LG}_{l,m}$
Laguerre-Gauss modes, $l$ the azimuthal order, $m$ the radial order, ∗ the
complex conjugate operator, $w_{0r}=(w_{0x}^{2}+w_{0y}^{2})^{1/2}$ the
projection waist, $\left(r_{0,opt},\theta_{0,opt}\right)$ the GSA cycle
optimized origins in cylindrical coordinates at $z=z_{f}$ and
$r_{max}=\min\left(\Delta X/2,\Delta Y/2\right)$. The rectangular grid length
along each axis is denoted by $\left(\Delta X,\Delta Y\right)$.

For a given number $M$ of radial modes, the quality of the Laguerre-Gauss
projection as a function of the number of azimuthal modes taken into account
(azimuthal order $L$) is evaluated by calculating $\epsilon_{fit}$, the
integral error, defined as :

$\epsilon_{fit}=\dfrac{\int\displaylimits_{r=0}^{r_{max}}\int\displaylimits_{\theta=0}^{2\pi}\mathop{r}\mathop{dr}\mathop{d\theta}\left|F_{\mathrm{HG}}(r,\theta,z_{f})-F_{\mathrm{LG}}(r,\theta,z_{f})\right|}{\int\displaylimits_{r=0}^{r_{max}}\int\displaylimits_{\theta=0}^{2\pi}\mathop{r}\mathop{dr}\mathop{d\theta}F_{\mathrm{HG}}(r,\theta,z_{f})}$
(2)

with $F_{\mathrm{LG}}(r,\theta,z_{f})$ being the normalized fluence of the
Laguerre-Gauss modes, $F_{\mathrm{HG}}(r,\theta,z_{f})$ the normalized fluence
of the Hermite-Gauss modes from the laser electric field fit, both in the
focal position $z=z_{f}$.

The integral error $\epsilon_{fit}$ decreases with the integration radius
$r_{max}$ for a fixed set of ${\mathrm{LG}}$ modes. $\epsilon_{fit}$ was
calculated for an effective interaction radius, $r_{eff}$, to evaluate the
quality of the Laguerre-Gauss fit near the peak fluence of the reconstructed
profile. As injection and acceleration of electrons both occur within a
characteristic radius around the laser centroid, on a scale of the order of
the bubble radius $R_{b}\propto w_{0}$ [24], we set $r_{eff}=R_{b}$ and
evaluate the error for $r\leq r_{eff}$ to measure the accuracy of the fit. In
PIC simulations performed with a perfect Gaussian laser, the plasma cavity has
values of $R_{b}$ ranging from $15$ to $20$ $\mu$m. Therefore, we set
$r_{eff}=20$ $\mu$m as the value corresponding to the maximum observed bubble
radius.

Figure 5 shows the evolution of $\epsilon_{fit}$ for $M=40$ and $L$ ranging
from $0$ to $10$.

Figure 5: Integral error $\epsilon_{fit}$ for $M=40$ radial modes, as a
function of the maximum number of azimuthal modes $L$. Red squares:
$\epsilon_{fit}$ calculated in a disk of radius $r_{eff}=20$ $\mu$m; black
circles: $\epsilon_{fit}$ calculated in a disk of radius $r_{max}=200$ $\mu$m;
blue diamonds (vertical axis on the right): cumulative energy fraction
included in each configuration.

The error of the fit converges toward $\epsilon_{fit}=0$ both within $r_{eff}$
and $r_{max}=200$ $\mu$m as the number of azimuthal modes used for projection
increases. A share of $43\,\%$ of the total energy is contained within the
first mode $L=0$ within a radius $r_{eff}$, which shows that the remainder is
contained within non-symmetrical azimuthal modes.

The number of azimuthal modes for the Laguerre-Gauss harmonics needs $N=5$
azimuthal modes to be simulated with the azimuthal harmonics of type
$\exp{-in\theta}$ used in quasi-3D geometry [21]. In addition, the number of
macro-particles per cell in the simulations in this work has been increased
with the number of azimuthal modes to maintain a constant signal to noise
ratio (whose value in quasi-3D simulations is discussed in [21]). As a result,
the computational time required for a simulation with a Laguerre-Gauss laser
field is increased due to the increased number of modes and of macro-particles
per cell. Every simulation in this work with a Laguerre-Gauss sum laser
profile used $L=3$ and $M=40$, which reproduces 75% of the total energy as
shown in Fig. 5. For $L>$3, the fit error $\varepsilon_{fit}$ decreases slowly
compared to the increase in computational time associated with the
corresponding number of modes.

All numerical results presented in this paper were obtained with input
parameters in the same range as those of the data shown in Fig. 3. The
simulated laser beam has a FWHM duration $\tau_{FWHM,sim}=25$ fs (Gaussian
temporal profile) and the focal plane fluence is fitted with a Gaussian waist
$w_{0,sim}=16$ $\mu$m. The laser is propagated over $10$ mm in a gas cell
filled with $99\,\%\mathrm{H_{2}}-1\,\%\mathrm{N_{2}}$ gas, including a
starting $1.4$ mm up-ramp from $n=0$ to a density plateau of $n=n_{0,sim}$, as
well as a down-ramp to $n=0$ from $ct=7.4$ mm to $ct=10$ mm. The plasma
density longitudinal profile profile is inferred from OpenFOAM simulations
[19]. For both Gaussian and realistic simulations, the laser driver is focused
onto the start of the density plateau at $z_{f}=1.4$ mm.

For simulations using the Gaussian profile, the energy was fixed at
$E_{l,sim}=\alpha_{gauss}\times E_{l}$, with $\alpha_{gauss}=0.46$ the value
calculated in Fig. 2.(e) at $z=z_{f}$. For simulations using the Laguerre-
Gauss profile, simulated with $L=3$ and $M=40$, $E_{l,sim}=0.75\times E_{l}$.

The simulation grid is represented along $z$ by $N_{z}=3000$ points with an
increment $\Delta z=0.025$ $\mu$m, and along $r$ by $N_{r}=1100$ points with
an increment $\Delta r=0.2$ $\mu$m.

In the simulations with the reconstructed laser field profile, each population
of $\mathrm{H_{2}}$ and $\mathrm{N_{2}}$ was simulated with
$[P_{z},P_{r},P_{\theta}]=[2,2,16]$ macro-particles per cell along $z$, $r$
and $\theta$ respectively. In the following, these simulations will be
referred to as ”realistic simulations”.

Instead, every simulation using a perfect Gaussian laser profile was performed
with $N_{m}=2$ azimuthal modes. For these simulations, each population of
$\mathrm{H_{2}}$ and $\mathrm{N_{2}}$ was simulated with
$[P_{z},P_{r},P_{\theta}]=[2,2,4]$ macro-particles per cell.

##  IV Realistic PIC simulations results

###  IV.1 Influence of laser asymmetry on electron beam spectra

To understand the physical impact of laser asymmetry on the electron bunch
quality, realistic simulations were performed with FBPIC using the
reconstructed laser driver retrieved from fluence distributions shown in Fig.
4. Input parameters were set as described in the previous section, and the
electron density was fixed to $n_{0,sim}=2.1\times\mathrm{10^{18}}$ cm-3.

For the sake of comparison between simulation and experimental results, 5
electron spectra are shown in Fig. 6: (a) simulation with a Gaussian laser
driver, (b) simulation with a Laguerre-Gauss laser driver using distribution
described by Figs. 4 and 5, both with electron density
$n_{0,sim}=2.1\times\mathrm{10^{18}}$ cm-3, and (c) to (e) experimental
spectra (single instances of the data shown in Fig. 3) measured with electron
densities $n_{0}=2.1\times\mathrm{10^{18}}$ cm-3 for (c) and (e),
$n_{0}=2\times\mathrm{10^{18}}$ cm-3 for (d) and laser energy $E_{l}=4.7$ J
for (c) to (e). The total charge of the average of the experimental spectra
from (c) to (e) is 111 pC, and their average central divergence is $-6.6$
mrad.

Figure 6: Electron bunch $dQ/dEd\theta_{x}$ spectral density in the energy $\theta_{x}$ divergence plane for 3 different cases : (a) simulated Gaussian spectrum - (b) simulated Laguerre-Gauss spectrum - (c) to (e) single experimental spectra from Fig. 3.(a) sequence. White (green) lines represent spectral charge, $dQ/dE$, (divergence charge density $dQ/d\theta_{x}$) integrated over the divergence (energy) respectively. The shaded area under the green $dQ/d\theta_{x}$ line shows the range $\pm 3$ mrad around the peak of $dQ/d\theta_{x}$, used to produce the $dQ/dE$ curve and calculate the spectrum peaks properties. The label 1 and 2 reference specific peaks within the $dQ/dE$ profiles corresponding to Table 1. Spectrum | $Q_{tot}$ | $\theta_{x,max}$ | Peak Label | $E_{peak}$ | $\Delta E_{peak}/E_{peak}$ | $Q_{peak}$ | $dQ/dE\,max.$
---|---|---|---|---|---|---|---
Units | pC | mrad |  | MeV | $\mathrm{\%}$ | pC | pC/MeV
Simulation Gauss |  174 |  -0.3 | 1 | 556 | 3.2 | 7 | 0.38
2 | 629 | 3.3 | 8 | 0.41
Simulation Laguerre-Gauss |  109 |  -4.7 | 1 | 466 | 23.2 | 18 | 0.23
2 | 632 | 8.6 | 5 | 0.11
Experimental Data |  96 |  -3.8 | 1 | 424 | 20.3 | 17 | 0.21
2 | 653 | 9.8 | 5 | 0.10
Table 1: Comparison of electron properties retrieved from simulated (with
ideal Gaussian and reconstructed laser profile) and experimental
$\theta_{x}-E$ spectra shown in Fig. 6. $Q_{tot}$ is the total charge on each
Fig. 6 spectrum, $\theta_{x,max}$ the central divergence of the angular
distribution, while the ”peak” labelled quantities refer to the $dQ/dE$
profiles that result from selection of $dQ/dEd\theta_{x}$ within a $\pm 3$
mrad range around the maximum $dQ/d\theta_{x}$. Boldface denotes specific peak
labels identified in Fig. 6 (a)-(c) $dQ/dE$ profiles. $\Delta
E_{peak}/E_{peak}$ is the energy spread FWHM for each peak, $Q_{peak}$ is the
FWHM charge contained within each peak and $dQ/dE\,max.$ is the maximum
$dQ/dE$ value reached within each peak.

These results show that a Gaussian driver Fig. 6.(a) gives rise to a wide,
high-energy, high-charge electron spectrum, peaked spatially on the laser
axis. Using a realistic laser driver generates an off-axis electron beam with
lower energy, lower charge, a structure in agreement with experimental results
as shown in Fig. 6.(b) and (c) to (e), where all these spectra exhibit an off-
axis $dQ/d\theta_{x}$ profile centered toward negative values. The final
spectrum contains 20 % (0 %) of self-injected electrons for the Gaussian case
(realistic cases).

Table 1 provides quantitative data for comparison of electron beam properties
for the 3 cases shown in Fig. 6.(a) to (c). The total charge $Q_{tot}$ is
summed between $\pm 20$ mrad and $300-900$ MeV. We define the exit angle
$\theta_{x,max}$ as the angle at which the maximum $dQ/d\theta_{x}$ is
reached. Within $\pm 3$ mrad, peaks with center energy $E_{peak}$, FWHM width
$\Delta E_{peak}/E_{peak}$, and FWHM charge $Q_{peak}$, are identified based
on minimum peak prominence $\delta_{dQ/dE}=0.05$ pC/MeV and minimum base to
base width $\delta_{E}=45$ MeV. Values of Table 1 show that the results of the
simulation with reconstructed laser field profile are in good agreement with
the detailed electron beam structure measured in experiment, while the
Gaussian simulation results in electron beam characteristics significantly
different from the measured ones. Two major differences are $Q_{tot}$ value,
which is $45\,\%$ higher than the experimental data for the Gaussian case, and
the overall $dQ/dE$ structure which has a highest peak energy of $875$ MeV
against $653$ MeV in the experimental data. The charge difference stems from
the quality of the laser angular intensity profile as it is the only input
difference between Gaussian and Laguerre-Gauss simulations.

The analysis of the evolution of laser symmetry during the propagation coupled
to electron injection is discussed in the next section.

###  IV.2 Evolution of the laser asymmetry and effects on electron dynamics

Figure 7 shows the simulated evolution of the laser beam and electron bunch
characteristics in the $xOz$ plane for the case with ideal Gaussian laser
profile (left-hand column) and the case with the reconstructed laser field
(right-hand column). It can be inferred from Fig. 7 that the evolution of
laser asymmetry and maximum field amplitude define the conditions for electron
injection and the dynamics of electrons during the acceleration process in the
plasma cavity.

Figure 7: Simulated evolution of laser pulse and electron bunch
characteristics for the ideal Gaussian laser profile (columns on the left) and
the realistic laser field profile [columns on the right]. Panels (a) and (b):
laser asymmetry function $\sigma_{l}$ (red dots) and spline approximated
asymmetry function curve (red curve) as functions of position on the
propagation axis; peak normalized laser potential $a_{0}$ (black dashed-line),
and normalized longitudinal unperturbed plasma density profile (grey area).
Panels (c) and (d): injected charge $Q_{inj}$ (vertical axis on the right) as
function of position (all electrons plotted as a white line, and angularly
selected $\pm 3$ mrad electrons as green line); The panels (c) and (d) also
show the spectral charge density $dQ/dE_{end}dz$ of electrons (black-red-
yellow colormap) with final energy $E_{end}$ for the angular selection as a
function of the injection position $z$. Panels (e) and (g) [(f) and (h) for
the simulation with realistic laser field]: transverse slice of the normalized
charge density perturbation $\rho/en_{0}$ on the $z-x$ plane at $ct_{1}=1800$
$\mu$m and resp. $ct_{2}=4000$ $\mu$m; in these panels the electron macro-
particles positions are shown as small black dots (with their size pondered by
their individual charge). Superposed in these panels is the laser driver
intensity’s normalized envelope (orange colormap); in the same panels, the
black and red lines correspond respectively to the density and the
longitudinal electric field $E_{z}$ on the axis of maximum laser intensity.

Figures 7.(a) and (b) show the evolution of the laser driver in the plasma
density profile. As the realistic simulated transverse distribution is
asymmetric with respect to the focal plane and changes before and after
$z=z_{f}$ [Fig. 4], different spatial distortions occur during non-linear
self-focusing inside the plasma.

To quantify the deviation from cylindrical symmetry of the transverse fluence
$F(r,\theta)$ and track its evolution throughout the simulated propagation, we
define an asymmetry coefficient $\sigma_{l}$ [15] as :

$\sigma_{l}=\int_{r=0}^{r_{max}}\mathop{r}\mathop{dr}\sqrt{\int_{\theta=0}^{2\pi}\left(\mathop{d\theta}\left(f(r,\theta)-\overline{f}(r)\right)\right)^{2}}\,,$
(3)

where the normalized laser fluence $f(r,\theta)$ is defined as

$f(r,\theta)=\dfrac{F(r,\theta)}{\int_{r=0}^{r_{max}}\int_{\theta=0}^{2\pi}\mathop{r}\mathop{dr}\mathop{d\theta}F(r,\theta)},$
(4)

and $\overline{f}(r)$ is the mean normalized fluence over $\theta$. The
integral origin $r=0$ is defined as the position of the fluence maximum. By
definition, $\sigma_{l}$ converges toward $0$ for a cylindrically symmetric
fluence profile.

This asymmetry coefficient $\sigma_{l}$ is plotted for the Gaussian laser
profile and realistic laser profile in Fig. 7.(a) and (b) respectively as a
function of position along propagation axis. Figure 7.(a) confirms that the
Gaussian profile is angularly symmetric. In this ideal case $\sigma_{l}$
undergoes variations between $0.5e{-5}$ and $2.5e{-5}$. In comparison, its
variations in the simulation with realistic laser field profile are on a scale
$\mathrm{10^{2}}$ times larger. In the realistic case (Fig. 7.(b)), the
amplitude of the symmetry coefficient drops by $67\,\%$ between $600$ and
$2000$ $\mu$m, the focal plane in the plasma being in the middle of this area.
This reduction of $\sigma_{l}$ is simultaneous with the increase in $a_{0}$,
showing that tight focusing of the laser reduces its imperfections. In the
realistic case, $a_{0}$ reaches values similar to the Gaussian $a_{0}$ through
the first self-focusing due to a near Gaussian shape in its focal plane (Fig.
4.(d)). However, the asymmetry of the intensity around the focal plane and
relatively short typical variation length ($z_{R}=1$ mm) are responsible for a
$1$ mm shift of the maximum in comparison to the Gaussian symmetric pulse
[Fig. 7.(a) and (b)].

Figures 7.(c) and (d) show line profiles of injected charge $Q_{inj}$ (right
hand-side vertical axis), and spectral charge density $dQ/dE_{end}dz$ of
electrons (black-red-yellow histogram) with final energy $E_{end}$, as
functions of injection position $z$. In other words, this spectrum describes
distribution of the initial $z$ positions of electrons for given final
energies $E_{end}$.

The simulation with ideal Gaussian laser profile [Fig. 7.(c)] shows a
correlation between final energy and injection position as electrons trapped
earlier gain more energy over the interaction distance. The injection of
electrons in $\pm 3$ mrad around axis (green line) occurs inside a $400$
$\mu$m window centered around the maximum laser amplitude $a_{max}$ at
$ct_{1}=1800$ $\mu$m. The green injected charge in this region (Fig. 7.(c)) is
maximal because the process occurs over $0.4z_{R}$ within the density plateau,
in the portion where $a_{0}\simeq a_{max}$. Electrons injected at the start of
this region see a longer accelerating length than the ones at the end, which
results in decreasing final energy depending on the injection position. The
resulting spectrum is evenly spread between $300$ and $900$ MeV as observed in
Fig. 6.(a).

For a realistic laser driver, the injected charge spectrum (Fig 7.(d))
exhibits multiple injection positions. The higher energy electrons are
injected $1$ mm after $ct_{1}$, when $a_{0}$ has dropped to $2.5$, and
contribute to the high energy peak centered on $632$ MeV in Fig. 6.(b). The
lower energy peak at $466$ MeV in Fig. 6.(b) is composed of electrons trapped
at different positions: $1$ mm after $ct_{1}$ and at $ct_{2}$, which explains
why this peak (Index 1 of Figs. 6.(b) and (c)) has a relatively higher energy
spread than peak 1 for Gaussian driver (Table 1).

The dynamics of electron injection results from the evolution of the laser and
plasma cavity. Snapshots of the first plasma cavity behind the laser pulse are
shown for the Gaussian driver in Fig. 7.(e) and (g), and for a realistic
driver in Fig. 7.(f) and (h), at $ct_{1}=1800$ $\mu$m and $ct_{2}=4000$ $\mu$m
respectively. At $ct_{1}$, for the ideal Gaussian laser simulation, injection
of electrons on axis has just begun in a perfectly symmetric bubble, while for
the realistic case most of the already injected electrons are greatly
defocused and spread from $-30$ to $30$ $\mu$m. The accelerating fields are
similar for Fig. 7.(e) and (f). However, in the realistic case, electrons
ionized early are desynchronized with the trapping portion of the bubble due
to off axis laser fluence fluctuations resulting in a $3$ $\mu$m shorter
bubble compared to the Gaussian case. This prevents continuous injection and
the generation of electrons with energy above $700$ MeV [Fig. 6.(a) and (b)].

Comparison of the two cavity structures at $ct_{2}$ (Fig. 7 (g) and (h)),
provides insight on the effects of transverse asymmetry evolution on the
acceleration process.

In the ideal Gaussian laser case, most of the initially injected charge is
accelerated, and the injection process remains continuous.

The transverse centroid variations and stronger defocusing with the realistic
laser profile induce important losses throughout the interaction process,
which enables trapping of up to $10$ pC around $ct_{2}$, in an off-axis bubble
with half of the accelerating maximum compared to a Gaussian cylindrically
symmetric driver.

Laser and electron beam positions in the $xOz$ transverse plane are plotted in
Fig. 8 as functions of the position along the propagation axis in vacuum. The
ideal Gaussian laser trajectory is centered on the laser axis in vacuum,
closely followed by the electron beam trajectory. The electron beam transverse
size remains relatively constant in the plasma and grows symmetrically around
$x=0$ after plasma exit. For a realistic laser driver, the laser trajectory
[Fig. 8.(b) ]in vacuum oscillates around the $x=0$ axis. Self-focusing in the
plasma lowers the amplitude of the displacement off-axis (compare red line and
black dashed line).

After electron trapping, $z>2000$ $\mu$m, the electron bunch trajectory is
centered on the laser centroid, and its standard deviation reaches up to $10$
$\mu$m. This behavior clearly demonstrates the impact of the asymmetry of the
transverse laser driver on the pointing of the electron beam at the exit of
the plasma.

Figure 8:  Laser beam (solid red line) and electron bunch (solid blue line)
positions in the xOz plane for (a) Gaussian case and (b) Laguerre-Gauss case.
The blue dashed lines are the $\pm$ RMS $\sigma_{x}$ size of the selected
bunch. The laser position is defined as the centroid of the fluence
distribution in an area of radius $r_{eff}$ around the fluence maximum. For
reference, the dash-dotted black line represents the position of the laser
beam in vacuum and the grey area is the longitudinal density profile (right-
end side vertical axis scale).

###  IV.3 Charge fluctuations due to laser intensity distribution

Simulations were performed to analyze the effects of variations of the
intensity profile on the resulting charge. The laser electric field was
reconstructed for 3 reference profiles with the modified GSA algorithm
described in Appendix A and the remaining simulation input parameters are the
same as in Fig. 6.(b) (reported in section IV.1). The density was varied
between $1.7\times\mathrm{10^{18}}$ and $2.3\times\mathrm{10^{18}}$ cm-3.

To characterize the laser energy distribution for each profile, we define the
following effective energy ratio :

$E_{ratio}=\dfrac{1}{N_{z}}\sum_{z=-z_{R}}^{+z_{R}}\dfrac{\int\displaylimits_{r=0}^{r_{eff}}\int\displaylimits_{\theta=0}^{2\pi}\mathop{r}\mathop{dr}\mathop{d\theta}F(r,\theta,z)}{\int\displaylimits_{r=0}^{r_{max}}\int\displaylimits_{\theta=0}^{2\pi}\mathop{r}\mathop{dr}\mathop{d\theta}F(r,\theta,z)}\,,$
(5)

with $z$ being the propagation position of the laser with respect to $z_{f}$,
$N_{z}$ the number of evenly spaced positions $z$ used to perform the
summation in the Rayleigh range $[-z_{R},...,0,...,z_{R}]$ and $F(r,\theta,z)$
the fluence at each propagation position in vacuum. The integral origin is
defined as the position of the fluence maximum. This ratio quantifies the
average effective energy in the characteristic divergence boundaries
$[-z_{R},\,z_{R}]$ of the laser propagation axis, and within an area of radius
$r_{eff}$. Improving this ratio increases the portion of energy usable for the
injection of electrons.

Figure 9: Total injected bunch charge $Q$ for 3 different simulated laser
profiles as a function of $n_{0}$. For each profile, the corresponding
$E_{ratio}$ is calculated by eq. (5), setting $N_{z}=21$: black squares
profile with $E_{ratio}=0.47$ \- red diamonds profile with $E_{ratio}=0.48$ \-
green circles profile with $E_{ratio}=0.52$. For each profile, the dashed
curve is the linear regression of the data points.

The total accelerated charge $Q$ is calculated within the whole $E-\theta_{x}$
space and plotted in Fig. 9 as a function of $n_{0}$ for the 3 laser profiles
and densities ranging from $1.7$ to $2.3\times 10^{18}$ cm-3. In this regime,
within the chosen set of parameters, the total charge $Q$ remains lower than
the theoretical value calculated for a matched laser for most of the input
densities $n_{0}$. This is a consequence of the non-Gaussian laser transverse
distributions, and of the fact that the laser spot size is not matched with
the bubble radius. Comparing the 3 values simulated for $n_{0}=2.1\times
10^{18}$ cm-3, while $E_{ratio}$ fluctuates between $0.47$ and $0.52$, the
total charge $Q$ varies from $198$ to $434$ pC. A relative effective energy
fluctuation $\delta E_{ratio}/E_{ratio}=5\,\%$ leads to a relative charge
fluctuation $\delta Q/Q=38\,\%$. For the laser profile resulting in the
highest total charge, i.e. $E_{ratio}=0.48$, increasing the density $n_{0}$
from $1.9\times 10^{18}$ to $2.1\times 10^{18}$ cm-3 increases $Q$ from $295$
to $434$ pC. A relative density fluctuation of $5\,\%$ leads to a relative
charge fluctuation $\delta Q/Q=19\,\%$. This shows that taking into account
small errors either on the gas density value or on the calculated effective
energy ratio leads to significant charge fluctuations, of the same order of
magnitude as the one measured experimentally (Fig. 3).

The averaged calculated charge measured within the same boundaries as
experimental data is $113$ pC ($\overline{Q}_{tot}=95$ pC in experiment), with
an RMS relative charge fluctuation of 56$\%$
($\delta\overline{Q}_{tot}/\overline{Q}_{tot}=47\,\%$ in experiment). There is
a good agreement between the average simulated and experimental charge for the
same central density $\overline{n}_{0}=2.2\times 10^{18}$ cm-3. In conclusion,
this analysis shows that improving the quality and the stability of the laser
energy distribution and the stability of the plasma density are crucial to
achieve stable and high-energy, high charge electron spectra.

##  V Conclusions

Experiments were carried out during the commissioning phase in the long focal
area of the APOLLON laser facility, to study the influence of the laser beam
properties on the quality of electron beams generated by ionization injection
and laser wakefield acceleration in a gas cell. A detailed analysis of the
laser beam in the transverse plane was performed using fluence measurements in
the focal volume. Particle in Cell simulations were performed with a
reconstruction of the laser field obtained from the measured laser fluence map
as input, leading to electron spectra in close agreement with experimental
results, for the accelerated charge, energy distribution and pointing of the
electron beam at the exit of the plasma. The presented results also show that
this degree of quantitative agreement can be found without using
computationally demanding full 3D simulations.

These high fidelity simulations rely on the calculation of the laser electric
field from experimental data providing the fluence at different positions
along the propagation in vacuum.

An iterative method based on a modified version of the Gerchberg-Saxton
algorithm [22], which allows the reconstruction of a realistic laser electric
field based on a collection of fluence images in vacuum (Fig. 4), was used.
The implementation of asymmetric laser drivers leads to better agreement of
the simulation output bunch characteristics to measured experimental data in
comparison to simulations using a perfect Gaussian driver (Fig. 6 and Table
1).

These realistic simulations highlight the effects of laser field spatial
characteristics (centroid fluctuations and asymmetry quantified by
$\sigma_{l}$) on the injection and acceleration of the electrons (Fig. 7 and
Fig. 8). The symmetry degradation from a Gaussian laser driver leads to a loss
of both peak energy and total accelerated bunch charge (Table 1), which could
be mitigated through optimization of the gas cell density characteristics.
The impact of shot-to-shot fluctuations of the laser transverse distribution
and of plasma density on the accelerated bunch charge, have been quantified
and the source of charge fluctuations in experiments identified. The
stabilization of these fluctuations would lead to an improved stability of the
produced electron spectra.

## Acknowledgments

We acknowledge the resources and assistance of the computing center MesoLUM
managed by ISMO (UMR8214) and LPGP (UMR8578), University Paris-Saclay
(France). Experimental results were achieved using APOLLON Research
Infrastructure and we gratefully acknowledge the work of the Laboratoire pour
l’Utilisation des Lasers Intenses (UMR7605) staff during planning and
execution of the experiment. APOLLON facility was partially funded by Equipex
Cilex (Centre interdisciplinaire lumière extrême) grant N∘ ANR-10-EQPX-25-01,
and by region Ile-de-France.

##  Appendix A Modified Gerchberg-Saxton algorithm (GSA) to model the laser
driver

In the experimental campaign described in this work, only the fluence maps
$F(x,y,z)$ of the laser pulse at specific distances $z_{k}$, with
$k=0,1,...,k_{\mathrm{max}}$ from the focal spot were measured.

A Gaussian temporal profile was assumed for the laser pulse, with the measured
FWHM duration in intensity $\tau_{FWHM}$. Under this hypothesis, the linear
relation between the peak fluence $F_{0}$ and peak intensity $I_{0}$ is :

$F_{0}=\dfrac{\tau_{FWHM}}{2}\sqrt{\dfrac{\pi}{\log 2}}I_{0}.$ (6)

Given this linear relation between fluence $F(x,y,z)$ and intensity
$I(x,y,z)$, an electric field at position $z$ can be defined from a phase map
$\psi(x,y)$ and a fluence $F(x,y,z)$:

$\displaystyle\mathrm{E}(x,y,z)=\mathrm{E}[F(x,y,z),\psi(x,y)]=$
$\displaystyle=\sqrt{I(x,y,z_{0})}\exp{i\psi(x,y)}.$ (7)

To initialize the realistic PIC simulations of this work, a reconstruction of
the laser electric field at a given plane was necessary. Since only the
fluence (and thus intensity) maps at multiple planes were known
experimentally, to reconstruct the laser electric field using Eq. A the field
phase map had to be reconstructed.

For this purpose, a modified implementation of the Gerchberg-Saxton algorithm
(GSA) [22] was used to find the field phase $\psi(x,y)$ at $z_{0}$ and thus
the laser electric field
$\mathrm{E}_{GSA,z_{0}}(x,y,z_{0})=\mathrm{E}[F(x,y,z_{0}),\psi(x,y)]$, from
the available data on the the fluence $F(x,y,z_{0})$ at $z_{0}$ and other
planes, each referred to as $z_{k}$.

In the following a simplified description of the field reconstruction
algorithm used in this work is reported.

This version of the GSA aims at finding a reconstruction of the laser field at
$z_{0}$ through an expansion in Hermite-Gauss (HG) modes [23], using the
measured $F(x,y,z_{k})$ fluence maps. Since the propagation of the HG modes
$\mathrm{HG_{n,p}}$ at $z_{k}$ is analytically known, this reconstruction is
equivalent to finding the estimated HG expansion coefficients $D_{n,p}$ (and
thus the corresponding field phase map). The indices $n$,$p$ denote the order
of the HG mode along the $x$, $y$ axis respectively.

At each iteration $iter$ of the algorithm, the estimated expansion in HG modes
is propagated from the position $z_{0}$ to $z_{k_{\mathrm{max}}}$, with an
improvement of the estimate at each of the intermediate positions $z_{k}$.
This update of the estimated coefficients for the HG expansion uses the
estimated coefficients from the previous measurement plane at $z_{k-1}$ and
the measured fluences $F(x,y,z_{k})$. As in the original GSA formulation [22],
the estimated propagated phase is combined with the fluence at the measurement
planes in the calculations.

After one iteration ends, the procedure is repeated starting from $z_{0}$,
using the coefficients (and thus the phase map) estimated from the previous
iteration.

The implementation of the modified GSA used for this work can be described by
the following pseudocode:

  * •

Find an initial estimate of the coefficients $D_{n,p}$ projecting the
intensity corresponding to the fluence $F(x,y,z_{0})$ on the
$\mathrm{HG_{n,p}}(x,y,z_{0})$ modes at $z_{0}$. In the following we denote
the projection of a function $f(x,y,z)$ on the HG modes with the notation
$\mathrm{Proj}$:

$D_{n,p}=\mathrm{Proj}[f(x,y,z),\mathrm{HG_{n,p}}(x,y,z)].\\\ $ (8)

  * •

For $iter=0\rightarrow N_{\mathrm{iter}}$ and for $k=0\rightarrow
k_{\mathrm{max}}$:

    * –

define the propagated field as

$\mathrm{E}_{GSA,z_{k}}(x,y,z_{k})=\sum_{n,p}D_{n,p}\cdot\mathrm{HG_{n,p}}(x,y,z_{k});$
(9)

    * –

find the phase map $\psi$ as:

$\psi(x,y)=\arg{(\mathrm{E}_{GSA,z_{k}}(x,y,z_{k}))};$ (10)

    * –

combine the measured fluence $F(x,y,z_{k})$ with the phase $\psi(x,y)$ to find
the function $\mathrm{E^{\prime}}_{GSA,z_{k}}$ using Eq. A:

$\mathrm{E^{\prime}}_{GSA,z_{k}}(x,y,z_{k})=\mathrm{E}[F(x,y,z_{k}),\psi(x,y)];$
(11)

    * –

combine the previous estimate of the HG coefficients with those obtained from
the projection of $\mathrm{E^{\prime}}_{GSA,z_{k}}$ on the HG modes at
$z_{k}$:

$\displaystyle D_{n,p}=(1-\alpha)\cdot D_{n,p}+$
$\displaystyle+\alpha\cdot\mathrm{Proj}[\mathrm{E^{\prime}}_{GSA,z_{k}},\mathrm{HG_{n,p}}(x,y,z_{k})];$
(12)

For this work the number of iterations was chosen as $N_{\mathrm{iter}}=10$.
The coefficient $\alpha$ for the weighted sum of the previous and new HG
expansion coefficients are chosen in order to obtain convergence.

Once the algorithm has performed the iterations passing through the
measurements planes, using Eq. A the estimated field phase $\psi(x,y)$ can be
easily found and combined with the measured fluence $F(x,y,z_{0})$ of the
laser to reconstruct its field
$\mathrm{E}_{GSA,z_{0}}(x,y,z_{0})=\mathrm{E}[F(x,y,z_{0}),\psi(x,y)]$.

In this algorithm, the mentioned projection of a function $f(x,y,z_{k})$ on
the HG modes at $z_{k}$ was defined as:

$\displaystyle\mathrm{Proj}[f(x,y,z_{k}),\mathrm{HG_{n,p}}(x,y,z_{k})]=$
$\displaystyle=\int_{-\Delta X/2}^{\Delta X/2}\int_{-\Delta Y/2}^{\Delta
Y/2}\mathop{dx}\mathop{dy}~{}f(x,y,z_{k})\times$
$\displaystyle\times\mathrm{HG}_{n,p}^{*}(x,y,z_{k}),$ (13)

where $\left(\Delta X,\Delta Y\right)$ are the data rectangular grid length
along each axis. The HG modes at $z_{k}$ are defined also using the origins
$\left(x_{0,k},y_{0,k}\right)$ and the projection waists $w_{0x,y}$. The
latter are set sufficiently small to make the Hermite-Gauss modes decay within
the projection integral boundaries, and sufficiently large to fit the part of
the transverse intensity map further away from the origin. The values of
$\left(x_{0,k},y_{0,k}\right)$ are optimized to improve the quality of the
laser field reconstruction, as will be described in a future work.

The field reconstruction obtained with described phase retrieval algorithm
would be sufficient to initialize the laser pulse in a realistic PIC
simulation in 3D Cartesian geometry. However, for the simulations in quasi-3D
geometry [21] of this work, which use a cylindrical grid, a further
decomposition in Laguerre-Gauss modes is necessary, as described in Section
III.2.

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