# Coupled dynamics of laser pulse and electron distribution from accurate modeling of laser wakefield accelerators

I. Moulanier, L.T. Dickson, Gilles Maynard, Francesco Massimo, Brigitte Cros

► To cite this version:

I. Moulanier, L.T. Dickson, Gilles Maynard, Francesco Massimo, Brigitte Cros. Coupled dynamics of laser pulse and electron distribution from accurate modeling of laser wakefield accelerators. Journal of Physics: Conference Series, 2025, 6th European Advanced Accelerator Concepts Workshop 17/09/2023 - 23/09/2023 Elba, Italy, 3124 (1), pp.012013. 10.1088/1742-6596/3124/1/012013. hal-05441470

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PAPER • OPEN ACCESS

Coupled dynamics of laser pulse and electron
distribution from accurate modeling of laser
wakefield accelerators

To cite this article: I. Moulanier et al 2025 J. Phys.: Conf. Ser. 3124 012013

View the [article online](#) for updates and enhancements.

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# Coupled dynamics of laser pulse and electron distribution from accurate modeling of laser wakefield accelerators

I. Moulanier¹, L.T. Dickson¹, G. Maynard¹, F. Massimo¹, B. Cros¹

¹ Laboratoire de Physique des Gaz et des Plasmas, CNRS, Université Paris Saclay

E-mail:

ioaquin.moulanier@universite-paris-saclay.fr,

brigitte.cros@universite-paris-saclay.fr

**Abstract.** Using a fast and flexible field reconstruction method based on the Gerchberg-Saxton Algorithm with Hermite-Gauss Modes Decomposition (GSA-MD), a precise modeling of the laser driver distribution was performed and used to simulate the creation of a relativistic electron bunch by Laser WakeField Acceleration (LWFA). In the bubble regime, the fluence distribution of the laser driving the plasma waves directly affects plasma cavity formation, as well as the injection and acceleration of electron beams in these waves. A detailed analysis of the transverse dynamics of the laser energy distribution, as well as of its effects on the evolution of the plasma wave shape and of the accelerated electron beam, is provided. This high accuracy analysis technique can be applied to explain the data measured in experiments, often conducted with lasers presenting an asymmetric transverse energy distribution.

## 1. Introduction

Numerical simulations of laser-plasma acceleration in the non-linear regime require an accurate modeling of the laser driving the plasma waves in order to provide insight on experimental results or tools to evaluate tolerance for future facilities. To reconstruct the transverse laser field distribution from fluence measurements, a fast and flexible field reconstruction method based on the Gerchberg-Saxton Algorithm with Hermite-Gauss Modes Decomposition (GSA-MD) has been developed [1]. This algorithm was used to reconstruct the asymmetries of the energy distribution in the focal volume [1, 2, 3] from fluence images collected during two experimental campaigns at the APOLLON and Lund Laser Centre facilities, aiming at generating electron sources using ionisation injection schemes in Laser Wakefield Acceleration (LWFA).

In LWFA [4, 5], an ultra-high intensity laser pulse is focused in an under-dense plasma target. The ionisation of gas and the expulsion of electrons from the laser propagation axis results in a trailing density perturbation in the form of a plasma wave, or plasma cavity. This plasma wake is characterised by intense transverse and longitudinal electric fields, whose amplitude can reach the order of 100 GV/m. These longitudinal electric fields accelerate relativistic electrons trapped in these waves with energy gains reaching the GeV range over a few mm [6]. LWFA has the potential to become a compact alternative for injector stages of large scale accelerators [7, 8] and various applications of relativistic electrons sources [9].

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In the ionisation injection scheme [10, 11, 12], the gas target is composed of a light species (e.g. hydrogen) and a dopant species (e.g. nitrogen), the latter presenting a large gap in its ionisation potential structure. The laser intensity value is chosen such that the light species is fully ionised at the front of the laser, and inner shell electrons of the dopant are only freed near the maximum of the laser intensity. The hydrogen ionisation threshold occurs for a laser intensity of the order of $10^{14}$ W·cm$^{-2}$, while the threshold for the K-shells electrons of nitrogen is at $7.6 \times 10^{18}$ W·cm$^{-2}$ [13].

LWFA driven by a laser beam with asymmetric fluence distribution has been observed in experiments. In previous work, we have shown that the laser transverse asymmetries affect its propagation within the plasma, resulting in a deformation of the accelerating structure. The produced electron bunch typically gains a larger transverse angular spread and reaches lower energy in comparison to the characteristics predicted by simulations with symmetric distribution [2, 3]. Building on this work, this article presents additional results from numerical diagnostics of laser and electron beam distribution in the plane perpendicular to the propagation direction. These results provide detailed information on the electron beam dynamics and its correlation with the laser evolution in the plasma.

## 2. Simulations setting

Simulations presented in the remainder of this article were performed with parameters of an experiment conducted at the APOLLON laser facility during the commissioning phase at the 200 TW level [2]. In that experiment, the average maximum energy gained by the electrons was 700 MeV, while the corresponding distribution displayed a transverse angular asymmetry, with exit angles close to -5 mrad. The average measured charge in the 300-900 MeV energy range and $\pm 20$ mrad transverse exit angle was 95 $\pm$ 46 pC. The values of charge and energy are lower than expected; a detailed analysis [2] showed that measured electron spectra were reproduced more accurately by simulations using as input the reconstructed transverse laser field distribution measured in vacuum.

To further understand the transverse dynamics of the laser propagation leading to this agreement, an ideal symmetrical case with a Gaussian distribution [14] is compared to a case reconstructed from the fluence measured in the experiment. The reconstructed case simulation uses a Laguerre-Gauss decomposition [15] of the reconstructed laser field in vacuum via GSA-MD [1]. The fluence measured in the focal plane was fitted with a Gaussian distribution with waist $w_0 = 16$ µm, and the temporal profile was assumed to be Gaussian with a full width half maximum (FWHM) duration of 25 fs and central wavelength $\lambda_0 = 0.8$ µm. For simulations using the Gaussian distribution, the simulated energy, $E_{l,sim}$ was fixed at $E_{l,sim} = 0.46 \times E_l$ with $E_l = 4.7$ J the measured energy of the laser. The coefficient 0.46 corresponds to the fraction of energy of the measured laser pulse fitted by a Gaussian distribution with waist $w_0$ in the focal plane. The reconstructed field distribution was simulated with $L = 3$ and $M = 40$, with $L$ and $M$ being the maximum azimuthal number and the maximum radial number of Laguerre-Gauss modes. The fraction of laser energy reconstructed with these mode parameters corresponds to $E_{l,sim} = 0.75 \times E_l$ and is therefore an improved fit.

The laser is linearly polarized in a direction transverse to its propagation direction. In the following, $z$ is the variable along the direction of propagation of the laser in vacuum, and $r_\|$ is along the transverse direction parallel to the laser polarization axis and $r_\perp$ along the transverse direction perpendicular to the polarization axis.

The laser pulse is focused in a gas distribution containing a mixture of 99 % H₂ and 1 % N₂, with a peak electron density $n_0 = 2.1 \times 10^{18}$ cm$^{-3}$. The simulated gas distribution is transversely uniform and the longitudinal profile is plotted as the shaded area in Fig. 1. The simulated laser pulses are focused at the start of the density plateau at $z = 1400$ µm. The simulated gas is assumed as partially pre-ionised, with the hydrogen completely ionised and the nitrogen ionised
up to the 5th level.

The reported PIC simulations have been performed in quasi-3D cylindrical geometry using the code FBPIC [16]. The simulation grid is divided in $N_z = 3000$ points in the direction of propagation, with an increment $\Delta z = 0.025$ µm. In the radial direction transverse to the propagation, the grid is divided by $N_r = 1100$ points with an increment $\Delta r = 0.2$ µm. A cylindrical geometry with azimuthal mode decomposition (as described in [17]) is used. For the reconstructed case, the number of azimuthal modes used is $N_m = 4$. Both H₂ and N₂ species were simulated with $[P_z, P_r, P_\theta] = [2, 2, 16]$ macro-particles per cell along $z$, $r$ and $\theta$ respectively. For the Gaussian case, $N_m = 2$ and $[P_z, P_r, P_\theta] = [2, 2, 4]$.

### 3. Evolution of the laser pulse asymmetry and effects on electron dynamics

The evolution of the laser and electron distributions in the plane transverse to propagation has been simulated and is compared in this section for symmetrical and reconstructed laser input.

Fig. 1 shows the evolution of the laser pulse FWHM width in both transverse directions, denoted with $w_{F,\perp}$ and $w_{F,\|}$, and the evolution of the laser peak normalised potential, $a_0$ (green line), as a function of position, $z$, along the propagation axis in vacuum. The peak normalised potential is equal to $a_0 = eE_0/(\omega m_e c)$, where $E_0$ is the peak electric field, $\omega$ the laser pulsation and $m_e$ the electron rest mass. For the Gaussian symmetrical case, shown in Fig. 1.(a), $w_{F,\perp} = w_{F,\|}$ in both directions at every position, showing that the pulse shape preserves its symmetry during the interaction. In comparison, in Fig. 1.(b) the width of the reconstructed pulse size oscillates and has different values in both directions. The best symmetry is achieved in the focal plane of the laser, where $a_0$ is near its peak, as it is in this area that the difference between $w_{F,\perp}$ and $w_{F,\|}$ is minimal.

Five specific positions are indicated by arrows in Fig. 1. and labeled $z_{0,1,2,3,4} = 1400, 3000, 4000, 5000$ and $6000$ µm. The first position, $z_0$, is chosen at the very start of electron injection in the plasma cavity for the Gaussian case. The other positions, from $z_1$ to $z_4$, are located after the injection of all electrons in the reconstructed case. Between $z_0 = 1400$ µm and $z_2 = 4000$ µm, $w_{F,\perp} > w_{F,\|}$. After $z_2$ the laser pulse is subject to more defocusing along $r_\|$ than along $r_\perp$, with $w_{F,\perp} < w_{F,\|}$ at $z_{3,4} = 5000, 6000$ µm.

Fig. 2 illustrates the transverse coupling of laser pulse and electron beam distributions at five selected positions along the propagation for the Gaussian and reconstructed cases. The behavior of electrons trapped in the plasma cavity is analysed by selecting electrons within an interval of $[\pm 10$ mrad] around the final exit angle of the electrons bunch in both transverse directions $r_\perp$ and $r_\|$. The angular selection guarantees that the majority of tracked electrons were trapped within the plasma cavity.

In Fig. 2.(a) and (c), the evolution of the laser pulse shape contour (grey area), the plasma cavity boundaries (black line), and the electron bunch distribution are represented for $z_{0,1,2,3,4}$ in the plane transverse to the propagation direction. In Fig. 2.(b) and (d), for the same positions, the plasma density perturbation is represented in the plane of propagation containing the laser polarisation axis. In Fig. 2.(a) and (b), for the Gaussian case, the laser transverse distribution preserves a cylindrical symmetry centered on the propagation axis at all positions, as was also shown in Fig. 1.(a). In comparison, the reconstructed case [Fig. 2.(c) and (d)] exhibits an evolution of its shape and centroid from $z_0$ to $z_4$. The pulse centroid, $\{r_\perp, r_\|\}$, drifts from $\{0.4$ µm, $0.5$ µm$\}$ at $z_0$ to $\{-1.7$ µm, $5.8$ µm$\}$ at $z_4$. Additionally, the pulse shape contour varies between $z_0$ and $z_1$. In Fig. 2.(c), at $z_0 = 1400$ µm, the laser shape is 7 µm larger in the $r_\perp$ direction than in $r_\|$. At $z_1$ and $z_2$, this asymmetry is enhanced, as the gap between $w_{F,\perp}$ and $w_{F,\|}$ is larger than around the focal plane [Fig. 1.(b)]. After 1 mm of propagation, in $z_3 = 5000$ µm, the pulse is now more elongated in $r_\|$ than in $r_\perp$.

In Fig. 2.(a), the Gaussian case's plasma cavity contour preserves the same cylindrical
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Evolution of $w_{F,\perp}$ (resp. $w_{F,\|}$), i.e. the laser pulse FWHM width in the direction perpendicular (resp. parallel) to the laser polarization axis, along the propagation axis for a) Gaussian case - b) reconstructed case. $w_{F,\perp}$ is represented in blue and $w_{F,\|}$ in red dashed lines. The laser normalised potential $a_0$ is plotted as a green line. The location of the positions $z_0$ to $z_4$ are indicated along the longitudinal axis. The longitudinal plasma density profile is represented by the grey shaded area.</figcaption></figure>

symmetry as the laser during propagation. At $z_1$, the cavity radius at the bunch position is 5 µm in both planes. This value gradually grows to 10 µm at $z_4$ [Fig. 2.(e)], which is an effect of beam loading of the bunch generating its own wake [Fig. 2.(b)]. In Fig. 2.(c), for the reconstructed case, the plasma cavity contour and centroid mirror those of the laser. At $z_0$, the cavity is more elongated along $r_\perp$, while it becomes more elongated along $r_\|$, at $z_2$, which is the same behaviour as the laser fluence. This shows that the asymmetric behavior of the reconstructed laser pulse in the non-linear regime leads to deformation of the plasma cavity during propagation.

The shape variations of the plasma cavity which can be seen in Fig. 2.(c) and (d) affect in return the distribution of the transverse electric field, which is focusing at the position where the accelerated beam is injected. Due to this correlation, the transverse electron distribution's shape within the plasma cavity is affected in return.

In the Gaussian case, the bunch transverse distribution remains symmetric. In Fig. 2.(a), across the all planes subsequent to $z_0$, the bunch preserves its size once all the charge is injected.

In Fig. 2.(c), the bunch from the reconstructed case becomes more elongated along $r_\perp$ at
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Evolution of electron bunch, plasma cavity and laser fluence at propagation positions $z_{0,1,2,3,4} = 1400, 3000, 4000, 5000$ and $6000$ µm. In the plane transverse to the direction of propagation for a) Gaussian case - c) reconstructed case, and at the longitudinal position of the red dashed line in row (b) and (c) respectively, the plasma cavity boundaries are plotted as a black line, along $r_\|$ and $r_\perp$ (perpendicular direction to the laser polarisation). The electron bunch transverse distribution in the plasma cavity is plotted for charge density values above $0.1 \text{ pC.}\mu\text{m}^{-2}$. In grey, the laser shape is represented as a surface with fluence values above $10\%$ the peak of the Gaussian case at $z_0$ (a). In the reconstructed case (c), the centroid of the laser fluence is plotted in gray. The plasma density perturbation (color scale) and laser envelope (orange shade) are represented in the simulation grid containing both the propagation axis (co-propagating coordinates $z-ct$) and laser polarisation axis (parallel coordinates $r_\|$) for b) Gaussian case and d) reconstructed case. The electron distribution is plotted for charge density values above $0.1 \text{ pC.}\mu\text{m}^{-2}$. The red dashed vertical line corresponds at $z = z_0$ to $z-ct = -45$ µm, and at $z \neq z_0$ to the longitudinal center of mass of the electron bunch in the plasma cavity.</figcaption></figure>

$z_1$. Between $z_2$ and $z_4$, the electron distribution gradually compresses in $r_\perp$, following the compression of the plasma cavity sheath. The distribution ends up being more elongated along
the parallel direction at $z_4$. The shape variation of the bunch over 3 mm between $z_1$ and $z_4$ occurs at the same time as fluctuations of the laser pulse FWHM widths $w_{F,⊥,‖}$ observed in Fig. 1.(b). Initially, at $z_1$, the pulse is wider in $r_⊥$ than in $r_‖$, with $\Delta_⊥ = 31$ µm, $\Delta_‖ = 22$ µm. From $z_1$ to $z_2$ this gap is narrowed, until the curves of $w_{F,⊥}$ (blue) and $w_{F,‖}$ (red) cross at 4400 µm. The electron beam transverse distribution is also centered around the laser centroid, which is represented in Fig. 2.(d) The electron distribution drifts according to the evolution of the laser centroid, which is also the centroid of the focusing area of the plasma cavity. This indicates a correlation between the dynamics of the laser and that of the electrons via the deformation of the plasma cavity.

As seen in Fig. 2.(d), between $z_1$ and $z_4$, the fluctuations of the charge distribution take place in a region of the simulation where beam loading [18] is prevalent. Thus, in addition to the deformation induced by the laser, the plasma cavity radius is affected by the beam loading of the asymmetric bunch [19, 20]. As a result, the evolution of the electron beam distribution is not only determined by the effects of the evolution of the laser pulse on the plasma cavity shape, but also by the effects of the beam's electromagnetic fields on the plasma cavity, which in turn influence the distribution of the electron beam. The complexity of these competing effects further highlights the importance of simulations with reconstructed laser fields to obtain an accurate description of LWFA in nonlinear regimes.

## 4. Conclusion

The dynamics of the laser and electrons in the transverse plane of PIC simulations have been analysed. The impact of the asymmetry of the laser field on the electron bunch propagation was discussed by comparing two simulations using an ideal symmetric laser pulse and an asymmetric laser field reconstructed from fluence measurements.

The correlation between the evolution of the shape of the laser and that of the plasma cavity boundaries, as well as the distribution of the produced electrons, was demonstrated. In the transverse plane, numerical diagnostics show that fluctuations of the energy distribution of the laser contribute to the shaping of the plasma cavity sheath. As a result, the accelerated electron bunch experiences asymmetric focusing fields, which shapes its distribution and fields its compactness.

Realistic simulations, performed using data from prior laser measurements in vacuum, allow to predict accurately the characteristics of the produced electron bunch of PW-class laser facilities LWFA experiments. The correlated propagation dynamics of the laser and electrons, which occur in the reconstructed case due to fluctuations of the laser transverse distribution, demonstrate the need for realistic modeling of the laser in the non-linear regime.

## 5. Acknowledgements

We acknowledge the resources and assistance of the computing center MesoLUM managed by ISMO (UMR8214) and LPGP (UMR8578), University Paris Saclay (France).

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