Theoretical Study of the Pre-Plasma Density Scale Length’s Influence on the Absorption Efficiency in Laser–Solid Interaction at Relativistic Laser Intensities for PW-Class Lasers

Abstract

This work studied the pre-plasma that builds up in interactions of focused high-power PW-class lasers with solid targets at the target surface facing the laser beam, and its impact on the global laser absorption efficiency as well as on the spectral cut-off energy of laser-generated proton beams. Our practical heuristic estimates were derived from the example of the VEGA-3 laser at CLPU. Our modeling results for the pre-plasma expansion due to the laser pedestal of VEGA-3 were benchmarked by hydrodynamic simulations, revealing good agreement for the evolution before the arrival of the main Gaussian laser intensity peak. Our detailed numerical two-dimensional Particle-in-Cell simulations showed the impact of different pre-plasma scale lengths on the absorption efficiency of laser energy into electrons, relevant for the seeding of other types of radiation. It was shown that the absorption can increase manyfold when increasing the pre-plasma scale length. This effect can be beneficial for the spectral cut-off energy of accelerated protons, where a trade-off between absorption and electron dynamics yields an optimum pre-plasma scale length. The findings can be applied to other PW-class laser facilities.

1. Introduction

The interaction of a high-intensity laser pulse with a solid target can yield bright sources of secondary radiation, ranging from gamma rays [1,2] and other forms of ionizing radiation [3,4,5] to XUV and THz pulses [6,7,8,9,10], as well as antenna emissions based on pulsed currents [11,12]. Solid-density metal targets have been used to create ion sources [13,14,15], flashes of high energetic X-rays [16,17] and extreme-ultraviolet light sources [18]. Pulsed bright ion beams are generated from solid-density targets by well-known mechanisms, such as Target Normal Sheath Acceleration (TNSA), Radiation Pressure Acceleration (RPA) and others [4]. Many applications in science and technology benefit from laser-driven ion beams, such as isotope production [19], positron emission tomography [20], ion beam microscopy [21], Particle-Induced X-ray Emission (PIXE) [22] and inertial confinement fusion [23]. Optimization of laser-to-target energy coupling is fundamental to rendering these laser-driven secondary sources efficient. One avenue to this end opens up via the mechanical tailoring of the target properties, e.g., morphology [24]. We will discuss, hereinafter, all-optical means that allow the use of simple flat-foil targets without the need of advanced methods of target fabrication.

Laser light arriving on the target before the main laser pulse is commonly referred to as pre-pulse or pedestal, comprising Amplified Spontaneous Emission (ASE) and the ps-pedestal of the main pulse (PED). The pre-pulse of high-power lasers is intense enough to trigger plasma formation (hereafter referred to as pre-plasma) through collisional and multi-photon ionization above the respective thresholds [25], at ≈$1\times {10}^{10} \mathrm{W} {\mathrm{cm}}^{-2}$ and ≈$1\times {10}^{12} \mathrm{W} {\mathrm{cm}}^{-2}$. Such pre-plasma can be characterized by an exponential ramp with scale length typically ranging from below 100 nm to several $\mathsf{\mu }$m in typical high-intensity laser systems [26,27,28,29].

When the main pulse arrives at the target, it will interact with the electron population within the pre-plasma. The laser pulse is able to penetrate the pre-plasma up to the critical density $n_{\mathrm{c}}$, which is much smaller than the solid density of a cold solid-density material $n_{\mathrm{e}0}$ (typically for metals $100\times n_{\mathrm{c}}\ll n_{\mathrm{e}0}$). Consequently, the density profile of the preformed plasma will change the interaction dynamics of the main pulse with the target, when compared to the interaction with a solid-density surface. Without pre-plasma, the laser pulse encounters a sharp density rim from which it triggers plasma expansion. With a pre-plasma, the laser pulse first crawls up a density ramp and then reflects from a less steep density gradient. Depending on the size of the plasma density scale length, the laser pulse travels a shorter or a longer distance until it reaches the critical target density. This has important consequences for the laser–solid interaction, as detailed below, i.e., for absorption efficiency, the acceleration mechanisms at play, and the characteristics of the emitted radiation. Therefore, the plasma density scale length directly plays an important role in the suitability of laser-generated secondary sources from laser–solid interaction in various applications.

Experimental studies have measured a total absorption coefficient of the laser energy in the plasma constituents of up to 80–90% for laser intensities above 1 × 10²⁰ W/c²m interacting with solid targets. It has been demonstrated through two-dimensional (2D) Particle-in-Cell (PIC) simulations that these high absorption coefficients are linked to the presence of pre-plasma on a 6 $\mathsf{\mu }$m scale length (for, initially, $\approx 10\mathsf{\mu }\mathrm{m}$ thick targets of $n_{\mathrm{e}0}=30\times n_{\mathrm{c}}$), and hole boring inside the target by the laser pulse [30]. Moreover, self-focusing in the pre-pulse can increase the laser intensity manyfold [31].

The formation of a pre-plasma can lead to an increase in the energy of the accelerated electrons [32,33,34] as well as the temperature of the accelerated electron distribution [35]. Further theoretical and experimental studies have also shown that the divergence angle of electron beams increases with the increase in the plasma density scale length [36,37,38].

Recent studies on the generation of bright laser-generated photon sources have revealed that a large pre-plasma can lead to the enhancement of high-harmonic generation [39] and to the enhancement and the collimation of $\gamma$ ray emissions [40].

Ion sources based on thick $\mathsf{\mu }$m-scale targets can benefit from the presence of pre-plasma, i.e., studies have shown an increase in the maximum proton energy in laser–solid interaction [41] and the tunability of the ion beam direction and spectrum [42,43,44].

Pre-plasma effects can be beneficial for specific aspects of an application and detrimental to others, depending on the target configuration. In the field of fusion science, it has been shown [45] that a pre-plasma can benefit fast ignition by laser self-focusing and production of energetic electrons, while a large pre-plasma can reduce the quality of the electron beam needed for fast ignition for a laser-cone target configuration [46].

In this work, we studied, numerically, laser absorption efficiency in high-power laser facilities, for cases in which a small $\mathsf{\mu }$m-scale pre-plasma is present, i.e., for parameters relevant to the VEGA-3 laser system in the Centro de Láseres Pulsados, Salamanca, Spain (CLPU). The first part focuses on a heuristic study of laser–target interaction, aiming at a model for pre-plasma generation; and the second part details our numerical studies of absorption efficiency and proton cut-off energy with non-negligible pre-plasma scale lengths.

2. Materials and Methods

The VEGA-3 laser at CLPU [47,48] is a Ti:Sa system able to deliver pulses of up to 30 J energy compressed down to ${\tau }_{\mathrm{L}}=30\mathrm{fs}$ pulse duration with a grating compressor. The laser pulse is transported in a high vacuum of 1 × ${10}^{-6}$ mbar via a $f=2.5\mathrm{m}$ Off-Axis Parabola (OAP) onto the target, irradiating it under an incidence angle of ${12}^{\circ }$. Up to $E_{\mathrm{L}}=(6.9\pm 0.3) \mathrm{J}$ are on target within the first Airy disk of $r_{\mathrm{L}}=(12.8\pm 1.9) \mathsf{\mu }\mathrm{m}$. Therefore, the relevant peak intensity throughout this work was chosen to be ≈$1\times {10}^{20} \mathrm{W} {\mathrm{cm}}^{-2}$.

2.1. Experimental Measurements of Laser Parameters

Full Width at Half Maximum (FWHM) focal spot size and pulse energy were extrapolated from measurements at low power, with a high-resolution imaging system and potentiometer, respectively—the laser pulse duration was measured for every shot with a second-order autocorrelator [48]. A commercially available acousto-optic programmable dispersive filter [49] (the Dazzler device of Fastlite, Paris, France) was used to modify the pulse duration by adjusting the Group Delay Dispersion (GDD) relative to the optimum position (at best compression on target). The main compressor, a heavy grating compressor, was not moved, for better reproducibility of working points. The Dazzler was installed after the last stretcher of the double CPA system and before the amplification stages. Time-resolved intensity profiles of VEGA-3 for different GDDs were measured with a single-shot third-order autocorrelator [50] (the Sequoia device of Amplitude, Paris, France). The latter was installed after the grating compressor, and it had to be removed for laser shots on target. In addition, the spectrum of the pulse was tested, to ensure that it was preserved in the scanned range covered by the Dazzler.

2.2. Numerical Simulations

2.2.1. Laser–Matter Interaction Through Hydrodynamic Simulations

The influence of the laser profile on the generation of pre-plasma was investigated using 1-D Lagrangian radiation–hydrodynamics simulations performed with the code HELIOS-CR [51]. For simulation, we used the PROPACEOS equation of state/multigroup opacity data tables for a titanium or carbon target with a thickness of 6 $\mathsf{\mu }$m and 50 $\mathsf{\mu }$m in the planar geometry, respectively. As in the experimental situation, the laser beam with a wavelength of 800 nm irradiated the target at an angle of 12 degrees to the target’s normal (see Figure 1). The laser contrast in Figure 2 was mimicked by an approximation curve in the same way as was done in Perez et al. [48]. The simulation box was 120 ps and contained the information about the laser pre-pulse up to 105 ps in advance. The main laser pulse had a peak laser intensity of about ${10}^{18}$ ${\mathrm{W/cm}}^2$, due to simulation restrictions and reliability. However, this did not affect the results, because of the slower hydrodynamic expansion at pre-pulse-related intensities. Our consideration was, rather, focused on the measurement of the 1/e scale length for the peak intensity in the laser continuum given in Figure 2, with $I={10}^{16}$ ${\mathrm{W/cm}}^2$.

2.2.2. Laser–Plasma Interaction Through PIC Simulations

We performed 2D PIC simulations, to study the effect of the plasma density scale length on the laser energy absorption into hot electrons and on the proton cut-off energy. The simulations were performed with Simulating Matter Irradiated by Light at Extreme Intensities (SMILEI) [52] on the cluster Supercomputación Castilla y León (SCAYLE) [53].

The simulation setup is shown in Figure 1: a laser pulse of peak intensity $I_0$, normalized field amplitude $a_0$ and pulse duration $\tau$ irradiated a target with the initial density $n_{e0}$ and thickness L. Prior to the arrival of the main peak of the laser pulse, the laser pedestal interacted with the target, which, in turn, expanded and exhibited a modified density profile, with lower- and higher-density regions. We considered an exponential expansion of the target towards the laser direction, with the characteristic plasma density scale length $L_{preplasma}$, where $L_{preplasma}$ was the length for which the density decreased by $1/e$.

For good comparability of our numerical studies to future experimental work, we considered the characteristics of the VEGA-3 laser with linear polarization, a wavelength $\lambda$ of $800\mathrm{nm}$, a peak intensity of $1.955·{10}^{19}\mathrm{W}/{\mathrm{cm}}^2$ (corresponding to a normalized field amplitude $a_0=3$), a pulse duration of $100\mathrm{fs}$ FWHM and a transverse waist of 12 $\mathsf{\mu }$m. The interaction with the target occurred at an incidence angle of ${12}^{\circ }$, as this was the operating angle of VEGA-3 in the experimental chamber, to avoid back reflections towards the parabola. A longer pulse duration was chosen instead of maximum compression, as this has been proven in experiments to be beneficial to proton acceleration [48].

To investigate the influence of the plasma density scale length on laser absorption and the spectral cut-off energy of a TNSA spectrum of protons, we considered two cases for the target characteristics: (i) a thin solid target and (ii) a thick solid target.

(i) The thin solid target consisted of a fully ionized titanium plasma, having a density of $100\times n_{\mathrm{c}}$ (where $n_{\mathrm{c}}=1.71\times {10}^{21} {\mathrm{cm}}^{-3}$ was the critical density for 800 nm lasers) and a thickness of 6 $\mathsf{\mu }$m. The plasma density scale length was varying, from 60 nm to 4 $\mathsf{\mu }$m, having a decaying exponential profile over a length of 20 $\mathsf{\mu }$m. The transverse width of the target was 40 $\mathsf{\mu }$m. At the rear side of the target, we considered a thin layer of neutral hydrogen plasma of 60 nm thickness and $10\times n_{\mathrm{c}}$ density to simulate target contaminants available for acceleration. The simulation box had 80 $\mathsf{\mu }$m in the target normal direction (the x coordinate) and 40 $\mathsf{\mu }$m in the transverse direction (the y coordinate).

(ii) The thick solid target consisted of a fully ionized plastic plasma (CH material), having a density $50\times n_{\mathrm{c}}$ and a thickness of 50 $\mathsf{\mu }$m. The plasma density scale length was varying, from 1 $\mathsf{\mu }$m to 12 $\mathsf{\mu }$m, having a decaying exponential profile over a length of 20 $\mathsf{\mu }$m. When the initial density of the ramp profile was much higher than $1{\mathrm{n}}_{\mathrm{c}}$, the pre-plasma profile was prolonged over a distance higher than 20 $\mathsf{\mu }$m to ensure an initial density of $0.1{\mathrm{n}}_{\mathrm{c}}$. The transverse width of the target was 40 $\mathsf{\mu }$m. At the rear side of the target, we considered a thin layer of neutral hydrogen plasma of 50 nm thickness and $5\times n_{\mathrm{c}}$ density to simulate target contaminants. The simulation box was 128 $\mathsf{\mu }$m in the x direction (being adjusted accordingly when the pre-plasma profile was prolonged, to ensure the laser entry in the box before the interaction began) and 40 $\mathsf{\mu }$m in the y direction.

Note that the corresponding density for both materials was reduced to a lower density in the PIC simulations, due to computational limits, keeping their values well over the critical density. In both cases, the cell length was $\mathrm{d}x=\mathrm{d}y=12.5 \mathrm{nm}$ and the number of particles per cell was 20 for each species. The particles were deleted while crossing the domain boundaries and the fields were absorbed. Collisions between electrons and ions were taken into account in all the simulations.

3. Results and Discussion

After a first estimation of the typically small plasma scale length on PW-class high-power lasers, we will discuss the influence of such pre-plasma profiles on laser–plasma interaction, notably the absorption efficiency and the spectral cut-off energy of laser-driven proton beams.

3.1. Estimation of the Pre-Plasma Density Scale Length

This section is dedicated to the heuristic estimation of the plasma density scale length for the high-power laser VEGA-3 [47,48]. The time-resolved intensity profile of VEGA-3 is plotted in Figure 2 for different durations of the main pulse as a variation of GDD relative to the optimum position for best compression. The measurements date from 2023 and may differ from earlier and later states of the laser system, as its components’ quality can be slightly affected by its extensive use at high repetition rates [12,15].

The laser profile can be fitted as a combination of exponential functions modeling the pre-pulse, and a Gaussian representing the main pulse as

$$I\left(t\right)=max\left(\begin{array}{c}{I}_{\mathrm{ASE}}·exp\left[\left(t-t_{\mathrm{ASE}}\right)/{\tau }_{\mathrm{ASE}}\right],\hfill \\ I_{\mathrm{PED}}·exp\left[\left(t-t_{\mathrm{PED}}\right)/{\tau }_{\mathrm{PED}}\right],\hfill \\ {\displaystyle \frac{F_{\mathrm{main}}}{{\tau }_{\mathrm{main}}\sqrt{2\pi }}}·exp\left[{\displaystyle \frac{-{\left(t-t_{\mathrm{main}}\right)}^2}{2{\tau }_{\mathrm{main}}^2}}\right]\hfill \end{array}\right),$$

(1)

with the resulting best-fitting parameters summarized in

Table 1. Fit parameters (with their standard deviation) for the time-resolved intensity profile from individual Sequoia scans done at different GDDs induced by a Dazzler. The underlying laser contrast is measured after the compressor, and intensities are extrapolated to the best focus position. Values for $t_{\mathrm{ASE}}=-8\mathrm{ps}$ and $t_{\mathrm{PED}}=-1.4\mathrm{ps}$ are forced.

GDD	ASE Pre-Pulse		ps-Pedestal		Main Pulse
$\ddot{\mathsf{\Phi }}$/${\mathbf{fs}}^{\mathbf{2}}$	${\mathit{I}}_{\mathbf{ASE}}$/$\mathbf{W} {\mathbf{cm}}^{\mathbf{-}\mathbf{2}}$	${\mathit{\tau }}_{\mathbf{ASE}}$/fs	${\mathit{I}}_{\mathbf{PED}}$/$\mathbf{W} {\mathbf{cm}}^{\mathbf{-}\mathbf{2}}$	${\mathit{\tau }}_{\mathbf{PED}}$/fs	${\mathit{F}}_{\mathbf{main}}$/$\mathbf{J} {\mathbf{cm}}^{\mathbf{-}\mathbf{2}}$	${\mathit{t}}_{\mathbf{main}}$/fs	${\mathit{\tau }}_{\mathbf{main}}$/fs
$-2000$	(3.6 ± 0.6) × ${10}^{11}$	781 ± 23	(2.6 ± 0.2) × ${10}^{15}$	336 ± 25	(3.1 ± 0.2) × ${10}^6$	12 ± 8	196 ± 8
−1000	(2.9 ± 0.4) × ${10}^{11}$	833 ± 25	(6.8 ± 0.7) × ${10}^{14}$	267 ± 14	(3.1 ± 0.2) × ${10}^6$	31 ± 1	170 ± 16
−200	(0.8 ± 0.2) × ${10}^{11}$	681 ± 26	(1.4 ± 0.2) × ${10}^{15}$	351 ± 26	(8.2 ± 0.4) × ${10}^6$	27 ± 6	79 ± 7
200	(3.5 ± 0.8) × ${10}^{10}$	593 ± 17	(2.0 ± 0.2) × ${10}^{15}$	344 ± 19	(7.9 ± 1.9) × ${10}^6$	−77 ± 34	99 ± 28
600	(7.9 ± 1.5) × ${10}^{10}$	648 ± 18	(2.0 ± 0.2) × ${10}^{15}$	401 ± 18	(3.8 ± 0.4) × ${10}^6$	−11 ± 8	97 ± 9
1000	(5.5 ± 0.9) × ${10}^{11}$	862 ± 27	(4.1 ± 0.3) × ${10}^{15}$	506 ± 33	(1.4 ± 0.3) × ${10}^6$	−82 ± 10	117 ± 7
2000	(5.4 ± 0.6) × ${10}^{11}$	830 ± 18	(2.6 ± 0.1) × ${10}^{15}$	465 ± 29	(1.4 ± 0.3) × ${10}^6$	−96 ± 8	156 ± 5

. Considering the uncertainties of the fit, which reflect the noisy nature of the data, the pre-pulse does not vary significantly when changing the GDD. Averaging over multiple shot sequences would reduce the discrepancies between the values for ${\tau }_{\mathrm{ASE}}$ and ${\tau }_{\mathrm{PED}}$.

The plasma density scale-length $L_{preplasma}$ can be estimated [54] based on a laser-sided target expansion with the ion speed of sound $c_{\mathrm{s}}$:

$$L_{preplasma}\left(t\right)={\int }_{-\infty }^t{c}_{\mathrm{s}}\left(t\right)\mathrm{d}t={\int }_{-\infty }^t\sqrt{{\mathfrak{Y}}_c·T_{\mathrm{e}}\left(t\right)·Z_{\mathrm{i}}^{*}\left(t\right)/m_{\mathrm{i}}}\mathrm{d}t,$$

(2)

with ${\mathfrak{Y}}_c$ being the heat capacity ratio (adiabatic parameter or also index), $m_{\mathrm{i}}$ the ion mass of the target bulk, $Z_{\mathrm{i}}^{*}$ the spatially uniform charge state of the ion population within the pre-plasma and $T_{\mathrm{e}}$ the uniform electron temperature from a Maxwell–Boltzmann (i.e., non-relativistic) or Maxwell–Jüttner (i.e., relativistic) electron distribution function in units of energy. The estimate requires the target to be thick enough, such that the ablation front does not reach the target rear during the laser interaction.

In cases where enough energy is supplied by the laser in order to allow a sustained heating of the expanding pre-plasma volume, the electron temperature depends on the laser intensity and wavelength. Heating efficiency varies in the regimes of ponderomotive acceleration, resonance absorption, vacuum heating and collisional absorption—and their respective transitions. For near-normal laser incidence, the electron temperature follows the scaling laws proposed by Forslund et al. [55], Beg et al. [9] and Wilks et al. [56]; also supported by data of Brunel [57], Malka & Miquel [58] and Wilks & Kruer [59]:

$$T_{\mathrm{e}}\left(a_0\right)=511\mathrm{k}\mathrm{e}\mathrm{V}·\left\{\begin{array}{cc}\sqrt{1+a_0^2}-1\hfill & \mathrm{if}{a}_0\ge 1.1266,\\ 0.4678·a_0^{2/3}\hfill & \mathrm{if}1.1266>a_0\ge 0.1477,\\ 0.1013·a_0^{1/2}\hfill & \mathrm{if}0.1477>a_0\ge 0.0311,\\ 1.8240·a_0^{4/3}\hfill & \mathrm{if}0.0311>a_0\ge 0.8531\times {10}^{-4},\end{array}\right.$$

(3)

with the normalized vector potential of the laser $a_0=0.853\sqrt{I_{\mathrm{L}}{\lambda }_{\mathrm{L}}^2/1 \times {10}^{18} \mathsf{\mu }{\mathrm{m}}^2\mathrm{W} {\mathrm{cm}}^{-2}}$. Owing to the simplicity of this approach, one discards the regime of collisional absorption below 1 × 10¹² $\mathsf{\mu }$m²W cm⁻².

The adiabatic parameter $1<{\mathfrak{Y}}_c<5/3$ of hot and dense plasmas is a function of electron temperature, ionization state and ion density [60]. Owing to the simplicity of our model approach, and given that this parameter does not change the order of magnitude of the following estimations, we assume ${\mathfrak{Y}}_c\approx 1$.

In a first approximation, the charge state can be estimated based on the Thomas–Fermi-Model [61], which evaluates from the electron temperature and material properties:

$$\begin{array}{cc}\hfill R& ={\rho }_{\mathrm{i}}/{\mathfrak{Z}}_{\mathrm{i}}/m_{\mathrm{i}}^3,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill T_0& =T_{\mathrm{e}}\left(t\right)/{\mathfrak{Z}}_{\mathrm{i}}^{4/3},\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill T_f& =T_0/(1+T_0),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill a_1=0.003,a_2=0.9718,& a_3=9.26148e-5,a_4=3.10165,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill A& =a_1·T_0^{a_2}+a_3·T_0^{a_4},\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill b_0=-1.7630,b_1& =1.43175,b_2=0.31546,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill B& =-exp\left[b_0+b_1·T_f+b_2·T_f^7\right],\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill c_1=-0.366667,& c_2=0.983333,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill C& =c_1·T_f+c_2,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill Q_1& =A·R^B,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill Q& ={\left(R^C+Q_1^C\right)}^{1/C},\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \alpha =14.3139,& \beta =0.6624,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill x& =\alpha ·Q^{\beta }\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill Z_{\mathrm{i}}^{*}\left(t\right)& ={\mathfrak{Z}}_{\mathrm{i}}·x/(1+x+\sqrt{1+2·x}),\hfill \end{array}$$

(17)

where ${\rho }_{\mathrm{i}}$ is the mass density of the target bulk and ${\mathfrak{Z}}_{\mathrm{i}}$ the atomic charge. Here, all superscripts are exponents.

Solving the integral Equation (2) in a range from −8 ps to 0 ps is now straightforward when calculating the hot electron temperature via Equation (3) using the averaged fitted laser intensity profile following Equation (1). For aluminum, the average plasma density scale-length $L_{\mathrm{full}\mathrm{pulse}}\approx (2961\pm 977)\mathrm{nm}$. Only taking into account the ASE and ps-pre-pulse leads to $L_{\mathrm{pre}-\mathrm{pulse}}\approx (1985\pm 387)\mathrm{nm}$. The short duration of relativistic heating yields a comparably large additional expansion of 50%. Checking the energy balance, one notes that the energy stored in ionized atoms and heated electrons exceeds the energy of the laser pulse by a factor of ≈10. For the intensity profile of VEGA-3, the deposited laser energy is not high enough for maximum heating of free electrons, and one needs to adjust the model.

The limited amount of available energy motivates a more detailed calculation of electron temperature and ionization state. As a first refinement, the time-dependent electron energy distribution function is estimated as a Maxwell–Boltzmann distribution $f_{\mathrm{MB}}\left(E\right)=2N_{\mathrm{e}}·\mathfrak{NA}\sqrt{E/\pi }·T_{\mathrm{MB},\mathrm{e}}^{-3/2}·exp\left[-E/T_{\mathrm{MB},\mathrm{e}}\right]$ that contains all the laser energy that is not consumed for ionization. Here, $\mathfrak{NA}$ is the Avogadro constant and $N_{\mathrm{e}}=Z_{\mathrm{EB},\mathrm{i}}^{*}·N_{\mathrm{i}}$ is the number of electrons within the pre-plasma (in units of mol), with $N_{\mathrm{i}}={\int }_{S}L·{\rho }_0/m_{\mathrm{i}}\mathrm{d}S$ being the respective ion number (in units of mol). The integral ${\int }_S\mathrm{d}S$ describes an integration over the full irradiated surface. The total kinetic energy of the electrons computes as $E_{\mathrm{kin},\mathrm{e}}={\int }_0^{\infty }E·f_{\mathrm{MB}}\left(E\right)\mathrm{d}E=N_{\mathrm{e}}·\mathfrak{NA}·T_{\mathrm{MB},\mathrm{e}}·3/4$. To find the electron temperature, one consequently evaluates the average kinetic energy available to one electron, $E_{\mathrm{kin},\mathrm{e}}/N_{\mathrm{e}}/\mathfrak{NA}$, and one derives $T_{\mathrm{MB},\mathrm{e}}$. Moreover, Equation (3) is used to estimate an upper bound, above which laser energy is not further absorbed by electrons. Therewith, the electron temperature is given by

$$\begin{array}{cc}\hfill T_{\mathrm{MB},\mathrm{e}}\left(t\right)& =min\left[T_{\mathrm{e}}\left(a_0\left(t\right)\right);{\displaystyle \frac{4}{3}}·E_{\mathrm{kin},\mathrm{e}}\left(t\right)/N_{\mathrm{e}}/\mathfrak{NA}\right],\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill \mathrm{with}{E}_{\mathrm{kin},\mathrm{e}}\left(t\right)& ={\int }_S\left({\int }_{-\infty }^t{I}_{\mathrm{L}}\left(t\right)\mathrm{d}t\right)\mathrm{d}S-N_{\mathrm{i}}\sum _{j=0}^{Z_{\mathrm{EB},\mathrm{i}}^{*}\left(t\right)}{\xi }_j,\hfill \end{array}$$

(19)

where ${\xi }_j$ is the ionization energy per mole that is required to ionize matter to the respective next-higher ionization state (with ${\xi }_0=0\mathrm{kJ}{\mathrm{mol}}^{-1}$).

The time-dependent charge state $Z_{\mathrm{EB},\mathrm{i}}^{*}\left(t\right)$ has to be estimated in iterations. The moment $t_0$ where the atoms attain the first ionization level is found with an energy-balance equation, which postulates that the energy needed for ionization must be smaller than the laser energy transferred to the target:

$${\displaystyle \frac{{\rho }_{\mathrm{i}}}{m_{\mathrm{i}}}}·{\delta }_{\omega }·\sum _{j=0}^{Z_{\mathrm{EB},\mathrm{i}}^{*}\left(t_0\right)=1}{\xi }_j<{\int }_{-\infty }^{t_0}{I}_{\mathrm{L}}\left(t\right)\mathrm{d}t,$$

(20)

where ${\delta }_{\omega }=\sqrt{2/(\sigma ·\mu ·{\omega }_{\mathrm{L}})}$ is the skin depth for the central frequency of the laser pulse ${\omega }_{\mathrm{L}}\approx 800\mathrm{nm}$, with $\sigma$ being the conductivity of the target bulk and $\mu ={\mu }_r{\mu }_0$ its permeability.

The ionization level for $t>t_0$ is calculated based on the energy transfer from electrons to ions. One assumes that ions can be promoted to the next-higher ionization level if the most likely electron energy (the maximum of the distribution function) attains $5\times$ the required ionization energy, which is a typical estimate for the maximum ionization cross-section [62]. Therefore, the charge state is elevated by one if

$${\displaystyle \frac{1}{2}}·T_{\mathrm{MB},\mathrm{e}}\left(t\right)\ge 5·{\xi }_{Z_{\mathrm{EB},\mathrm{i}}^{*}\left(t\right)+1}/\mathfrak{NA}.$$

(21)

The plasma density scale length for $t>t_0$ can now be calculated from Equation (2), using, first, Equation (18) to evaluate the hot electron temperature and successively checking for possible ionization with Equation (21). Overall, this approach neglects the energy that is transferred to ions and laser plasma instabilities or removed from the system via radiation, and it also neglects the absorption efficiency of the laser pulse by assuming perfect coupling to the plasma (i.e., 100% absorption).

The results for an aluminum target are shown in Figure 3. The average scale-length $L_{\mathrm{full}\mathrm{pulse}}\approx (982\pm 369)\mathrm{nm}$, and taking into account only the ASE and ps-pre-pulse led to $L_{\mathrm{pre}}-\mathrm{pulse}\approx (585\pm 154)\mathrm{nm}$. In the margin of uncertainty, the scale-length $L_{\mathrm{preplasma}}$ attained similar values for different GDDs. This was expected, due to the similar intensity profiles. Unexpectedly, $L_{\mathrm{full}\mathrm{pulse}}$ varied also on smaller scales than the magnitude of its uncertainty, which can be explained by the few data points that captured the main intensity peak (the resolution of the Sequoia scan was 16.8 fs).

The use of different target materials led to variations of the scale length, as the ionization energies were characteristic for each element, the number of ions per mole scaled with ${\rho }_{\mathrm{i}}/m_{\mathrm{i}}$ and the speed of sound depended on $\sqrt{Z_{\mathrm{i}}^{*}/m_{\mathrm{i}}}$. The results for typical target materials are summarized in

Table 2. Averaged plasma density scale length for typical target materials, taking into account the full laser pulse for calculating $L_{\mathrm{full}-\mathrm{pulse}}$ and the pre-pulse only for $L_{\mathrm{preplasma}}$. The calculation followed Equation (2), using, first, Equation (18) to evaluate the hot electron temperature and successively checking for possible ionization with Equation (21).

Target Material	C	Al	Si	Ti	Ni	Cu	Au
$L_{\mathrm{full}-\mathrm{pulse}}$/nm	1081 ± 362	982 ± 369	1081 ± 357	827 ± 310	660 ± 249	659 ± 249	511 ± 193
$L_{\mathrm{preplasma}}$/nm	658 ± 130	585 ± 154	599 ± 166	493 ± 130	393 ± 104	393 ± 104	303 ± 80

. The presented values resulted from averaging the outcome of the individual calculations over all GDD-dependent intensity profiles. One notes that the scale length that was caused by ASE and ps-pre-pulse was smaller than the laser wavelength.

Note that as the absorption efficiency was assumed to be $100\%$, these results may have led to an overestimation of the pre-plasma scale length. The estimates disregarded, particularly, energy losses due to radiation or plasma instabilities. A further possible refinement of this method would take into account the absorption efficiency of the laser energy to the energy of the hot electron population in the low-intensity regime, which was well beyond the scope of this paper. Interested readers are referred to a recent review [63].

3.2. Hydrodynamic Simulations

We simulated two temporal points at −8 ps and −0.5 ps, to describe a pre-plasma expansion. The first time point with the maximum laser intensity of 1 × ${10}^{12}$ ${\mathrm{W/cm}}^2$ shows the effect of a long-range laser contrast, while the second time point mimics the moment corresponding to the maximum expansion of the pre-plasma just before the arrival of the fs laser. The mass density profiles simulated at the two different times are shown in Figure 4. The 1/e scale length in these cases was in the range of (0.11–0.66) $\mathsf{\mu }$m for titanium and (0.15–1.1) $\mathsf{\mu }$m for carbon.

3.3. Pre-Plasma Influence on the Laser–Plasma Interaction

In this section, we will investigate the influence of the plasma density scale length on the laser energy absorption coefficient and on the maximum proton energy achieved, using the results from our 2D PIC simulations.

Figure 5 shows the dependence of the absorption coefficient on the plasma density scale length for both targets: the thick CH films and the thin titanium foils. The absorption coefficient was defined as the total energy retrieved in the plasma, including the kinetic energy of the particles and electromagnetic fields, divided by the initial laser energy. It ranged from ≈20% to ≈85% when varying the pre-plasma scale length from 60 nm to 4 $\mathsf{\mu }$m for the thin solid case. A similar behavior was reported by Ping et al. [30], where the experimentally measured absorption coefficient in laser–solid interaction was explained by the presence of a pre-plasma. Note that most of the laser energy was absorbed by the hot electrons, which, in turn, lost their energy for the creation of the fields and ion acceleration [64]. The reflection coefficient R was defined as the electromagnetic energy measured at the left side of the simulation box, after the laser pulse was reflected by the target. In a similar way, the transmission coefficient was defined by the electromagnetic energy measured at the right side of the target. The reflection and the transmission coefficients were calculated using the instantaneous Poynting vector at the left and right boundaries of the simulation box, as used in [64]. In all cases, the transmission coefficient was negligible, as the laser energy was either transferred to the particles or reflected. Note that the plateau value of the absorption coefficient could not increase if more pre-plasma was added, as it coincided with a constant reflection coefficient. Moreover, the sum of the reflection and absorption coefficient was not $100\%$, due to the losses of electromagnetic energy on the Y axes, which could go from $1\%$ up to $15\%$.

For plasma density scale lengths below 1 $\mathsf{\mu }$m, the laser propagated almost unperturbed through a longer region of underdense plasma before reaching the critical density in the vicinity of the solid-density plateau, where a part of its energy was absorbed (about 20%) and the greater part of it was reflected. When the plasma density scale length increased, the laser propagated through a longer region of density that was, in magnitude, closer to the critical density. In this regime, most of the laser energy would have been absorbed before reaching the critical target density (about 85%). Consequently, increasing the scale length led to a rise in absorption efficiency, an effect seen for both the thick and the thin targets.

It becomes clear that the absorption efficiency was similar between the two studied materials, with a small variation in the order of 5%. This was a consequence of the laser propagation in equally underdense or near-critical pre-plasma until reaching the critical density, in both cases. The absorption of the laser energy is mostly to the electrons in areal density, which depends on the thickness and on the target density.

Figure 6 shows the variation of the spectral proton cut-off energy with the plasma density scale length. The cut-off energies were probed with a screen positioned 25 $\mathsf{\mu }$m behind the the target. As can be observed, the presence of a pre-plasma can be beneficial to the acceleration of protons: for the titanium (the thin) target in this study, the proton cut-off energy increased from a value of $3\mathrm{MeV}$ when there was no pre-plasma up to $21\mathrm{MeV}$ when the pre-plasma scale length was 1 $\mathsf{\mu }$m. For the thick CH target, the proton cut-off energy increased from a value of $9\mathrm{MeV}$ with a small pre-plasma of 1 $\mathsf{\mu }$m scale length up to $13\mathrm{MeV}$ when the pre-plasma scale length was 3 $\mathsf{\mu }$m. Also, the cut-off presented an optimum of 1 $\mathsf{\mu }$m for Ti and 3 $\mathsf{\mu }$m for the CH target, even though the absorption rose monotonically over the full interval. Previous numerical studies, involving targets without pre-plasma, have shown that the optimum thickness for proton acceleration and the optimum thickness for maximizing absorption are different [64,65,66]. This occurs due to the dilution of hot electrons in a larger volume, which leads to the creation of a less strong electrostatic field on the rear side of the target, consequently leading to less-energetic protons [66]. Moreover, the building time of the electrostatic field, which can be longer for the thicker targets, also plays an important role, as already shown in [67,68]. The pre-plasma effect on the electrostatic field and on the proton energy cut-off is discussed further in Perez et al. [48].

4. Conclusions

The pre-plasma build-up typical for PW-class laser facilities was modeled, and the benchmarked modeling results were applied to delimit the parametric range for a study of laser absorption efficiency and related effects on accelerated proton beams.

This work proposes a heuristic model to estimate the pre-plasma scale length, which is based on the spatio-temporal laser profile and target material constants. The model disregards energy losses due to radiation or plasma instabilities that can lead to over-estimations for the pre-plasma scale lengths. In the typical example of the VEGA-3 laser system of CLPU, one obtains an ASE pre-plasma scale length that is smaller than the laser wavelength ${\lambda }_{\mathrm{L}}\approx 800\mathrm{nm}$, ranging from (303 ± 80) nm for gold targets to (658 ± 130) nm for carbon targets. The heuristic model was benchmarked with hydrodynamic simulations that predicted comparable scale lengths. Particularly, the 1/e scale length in the two compared cases was in the range of (0.11–0.66) $\mathsf{\mu }$m for titanium and (0.15–1.1) $\mathsf{\mu }$m for carbon targets. These results can help to give an initial estimate for the pre-plasma scale length when hydrodynamic simulations are not available, and they can be used for the comparison of the experimental results obtained at VEGA-3 with predictions made by numerical simulations.

Our subsequently performed 2D PIC simulations investigated the effect of different pre-plasma scale lengths on the absorption of laser energy and the spectral proton energy cut-off. The scale lengths were chosen according to the results of the pre-plasma build-up, from 0 nm to 4 $\mathsf{\mu }$m for the titanium and from 1 $\mathsf{\mu }$m to 12 $\mathsf{\mu }$m for the CH targets.

In the performed PIC simulations, we observed that the presence of a pre-plasma can be beneficial to the absorption of laser energy in hot electrons and proton acceleration. The absorption efficiency ranged from ≈20% to ≈90% when varying the pre-plasma scale length from 60 nm to 12 $\mathsf{\mu }$m, summarizing the results for both the thin titanium foils and the thick CH films, while the proton cut-off energy could increase from $3\mathrm{MeV}$ when there was no pre-plasma up to $21\mathrm{MeV}$ when the pre-plasma scale length was 1 $\mathsf{\mu }$m (for the titanium target). While the absorption monotonically increased for growing pre-plasma scale length, the cut-off energy of the proton spectra showed an optimum, which was related to other effects, such as the dilution of the electron cloud.

This study could be directly used as the basis for an experimental work deploying the VEGA-3 laser at various operation points (laser energy, pulse duration or focusing), as well as for other laser facilities in the PW class with fs-pulses. Simulation results could be benchmarked via measurements of proton spectra and the reflected laser light.

Future PIC simulations could focus on the minimum pre-plasma required to infer the predicted trends from theoretical or experimental results. For simplicity, this work considered full ionization from the start of each simulation; a further study could investigate the effect of a lower ionization level.