The coupling of high-energy (>1 kJ) picosecond-pulse lasers with high-energy (>100 kJ) nanosecond-pulse laser systems opens new perspectives for the study of high-energy-density (HED) physics, including research on laboratory astrophysics1 and inertial confinement fusion (ICF).2,3 High-intensity short-pulse lasers can generate strong bursts of energetic particles (electrons and ions) and photons (x rays and gamma rays) ideally suited for probing matter under extreme conditions, such as those achieved using megajoule-level nanosecond lasers. Among the possible advanced diagnostic techniques enabled by those particle and radiation sources are proton radiography,4 time-resolved x-ray radiography,5 and x-ray phase contrast imaging.6 In addition, the laser-driven electron and ion beams can serve to isochorically heat dense materials, with the goal of creating plasmas under unprecedented HED conditions,7,8 launching blast waves,9 or igniting ICF targets.10,11

Examples of operational laser facilities combining high-energy nanosecond and picosecond laser beams are Omega/Omega EP12 and NIF/ARC13 in the USA, Gekko/LFEX14 in Japan, SG-II/SG-II PW15 in China, and CEA's Laser Mégajoule (LMJ)/PETAL16–18 in France. These laser systems, unlike lower-energy facilities, make it possible to investigate relativistic laser interactions sustained over picosecond time scales and ∼100 μm spatial scales. Such conditions have been predicted to increase the efficiency of electron acceleration and reduce the divergence of the hot-electron distribution, hence leading to more energetic and collimated proton beams.19–23 Increasing experimental evidence supports these claims.24–27

This paper describes the results of the 2018 PETAL experiment at the LMJ on the generation of high-energy particles and radiation from laser–solid interactions. The spatial and spectral distributions of the energetic electrons, ions, and x rays emitted by plastic foil targets of varying thicknesses are characterized by a variety of diagnostics.28,29 A major observation is the acceleration of protons to unexpectedly high energies (between 30 and 50 MeV) given the relatively modest (<10¹⁹ W/cm²) laser intensity achieved on target. Those measurements are interpreted by coupling hydrodynamic and kinetic simulations using as input the measured laser parameters. These simulations reveal that the laser-driven hot-electron generation accounting for the ion acceleration essentially takes place through the extended preplasma formed at the irradiated target surface. The electron energization appears to be significantly enhanced by a combination of stochastic effects due to laser backscatter, combined with local intensifications of the laser field due to filamentation. The simulated electron and proton spectra are found to be in fair agreement with the observations.

The PETAL beamline, located at CEA-CESTA near Bordeaux, France, delivers laser pulses in the kJ-ps range, coupled with the LMJ beams at the target chamber center (TCC). The PETAL architecture is based on the chirped pulse amplification (CPA) technique.30 The first 1.15 PW shot, out of the compression stage, was performed in May 201531 and the commissioning shots on target in October 2017. On-target intensities in the range of 10¹⁸–10¹⁹ W/cm² have been demonstrated.17

The PETAL amplifier can deliver up to 6.4 kJ energy pulses at a wavelength of 1.053 μm. It has the same architecture as the LMJ amplifier section, but the mechanical structure of the 4 × 2 beams LMJ amplifier has been modified to handle a single beam. The amplifier comprises a four-pass system with angular multiplexing and a back-reflector. The main difference with the LMJ power chain is a correction of the longitudinal chromatism defects generated during the beam propagation through the optical components.32 Without such a correction, the on-target intensity would be decreased by a factor of 10.

The PETAL compressor scheme is based on coherent beam combining (instead of the grating mosaic phasing implemented at OMEGA-EP or at LFEX33). The initial beam is divided into four beamlets, which are independently compressed and synchronized within micrometer-range precision. These beamlets are coherently combined using a single segmented mirror with three interferometric displacements for each subaperture. The pulse duration routinely obtained in high-energy shots is less than 700 fs on target.

After compression, the beam is transported under vacuum to the focusing stage. The available energy on target was to be kept below 400 J during the first campaigns to prevent damage to the turning mirrors, a common risk with high-energy picosecond laser pulses.34 The focusing system consists of an off-axis parabola (7.8 m focal length) with a 90° deviation angle, followed by a pointing mirror. The focal spot is measured in each high-energy shot after the compression stage by using the leakage from a mirror, and on target with the Tωist diagnostic (see below).35Figure 1 shows the focal spot measured after the compression stage in a representative shot with 450 J energy and 610 fs pulse duration. The inferred intensity is 7.9 × 10¹⁸ W/cm².

The Tωist (three-ω imaging system) diagnostic serves for shot-to-shot characterization of the laser focal spot in high-energy shots. Tωist images the third-harmonic (3ω) light generated by the interaction of the 1ω (1053 nm) PETAL pulse with a solid target.35 The results are in fair agreement with the measurements performed using the light leaking out of the compressor. Both diagnostics reveal a pattern of multiple spots, each of about 50 μm full width at half maximum (FWHM), likely induced by beam phase distortion in the subaperture compression scheme. The beam phasing is performed mechanically by using tilts and pistons, the imperfections of which readily lead to fragmented focal distributions, as seen in Fig. 2.

The temporal contrast of the PETAL pedestal relative to the main pulse was measured with two diagnostics installed after the compression stage. A silicon integrator measured the long-term contrast to be 10⁻³ in energy and 10⁻⁷ in intensity during a 5 ns period prior to the pulse maximum. A house-developed, single-shot cross-correlator showed that the contrast was ∼10⁻⁵ to 10⁻⁶ about 250 ps before the intensity peak, with two prepulses with 10⁻⁴ contrast.

During PETAL commissioning, 14 PETAL shots were performed on either multilayer targets (Zn/Ag/W/Au), thick W targets, or thin plastic (Parylene) and Au targets. The laser energy ranged from 187 to 450 J and the pulse duration from 610 to 660 fs.

This paper presents the results related to particle emission (electrons, protons, and ions) obtained in three shots conducted under comparable laser and target conditions, namely,•shot #176: 450 J pulse energy, 610 fs pulse duration, 50 μm thick plastic (CH) foil target coated with a 1 μm Al layer on the front side;•shot #177: 409 J pulse energy, 660 fs pulse duration, 10 μm thick Parylene foil target;•shot #178: 187 J pulse energy, 610 fs pulse duration, 10 μm thick Parylene foil target.

The energetic particle distributions were characterized using the following diagnostics:•SEPAGE (Spectromètre Electron Proton à Grande Energie) is composed of two Thomson parabolas: a low- and a high-energy channel, respectively covering the spectral ranges from 0.1 to 20 MeV and from 8 to 200 MeV for protons and from 0.1 to 150 MeV for electrons.^36,37 Particles are detected with imaging plates (IPs), absolutely calibrated over the whole energy range.³⁸ On the front side is placed a cassette containing a stack of radiochromic films (RCFs) for radiographic imaging. SEPAGE is inserted close to the TCC using a system for insertion of diagnostics (SID).•SESAME (Spectromètre Electrons Angulaires Moyenne Energies) consists of two magnetic-dipole electron spectrometers directly mounted on the wall of the interaction chamber at angles of 0° and 45° from the PETAL beam axis.³⁹ They provide electron energy spectra in the 5–150 MeV range, with an IP calibrated for electron detection.^40,41•A diagnostic based on B-dot coils was specifically developed for the commissioning experiments to characterize strong electromagnetic pulses (EMPs) produced in the interaction of the PETAL beam with the target. This diagnostic, placed inside the interaction chamber, probes a broad frequency range from 50 MHz to 6 GHz.

Figure 3 displays the experimental setup and the locations of the diagnostics in the equatorial plane of the target chamber. The SEPAGE diagnostic and the associated RCF cassette were positioned at a distance of 100 mm from the TCC perpendicularly to the rear side of the target. The Tωist diagnostic was located perpendicularly to the front side of the target. The two SESAME modules observed the target rear side from the chamber wall (6 m away from the TCC) at 0° and 45° angles from the PETAL beam axis, i.e., at 13.5° and 58.5° angles from the target normal.

Electrons emitted from the rear side of the target were measured using the two dipole modules of SESAME on the target chamber wall (5.5 m from the target, with a 5 mm × 50 μm slit). Figure 4, related to shot #176 reveals high hot-electron temperatures: 8.3 MeV at 13.5°, and 3 MeV at 58.5° (angles related to the target rear side). As expected, the highest energies and numbers of hot electrons are recorded close to the axis. The observed broad energy–angle distributions of the hot electrons are common features of laser–solid interactions.42,43

Figure 5 shows the electron spectra measured at an angle of 13.5° for different laser shots. All the distributions can be approximated by Maxwellian functions. The corresponding hot-electron temperatures (reported in Table I) greatly exceed the predictions from known scaling laws given by Wilks et al.44 (ponderomotive scaling) and Beg et al.42 According to these scalings, one would obtain under our conditions maximum temperatures of about 1.5 and 0.9 MeV, respectively, instead of the measured 8.3 MeV.

From these spectra, the total number of energetic electrons emitted can be inferred using angular distribution predicted by simulations.45,46 The values obtained are also reported in Table I. These values are underestimated considering that the lowest part of the spectrum (under 2.5 MeV) is not recorded by the SESAME spectrometers.

An undesired effect occurring in high-intensity laser–matter experiments is the generation of very strong electromagnetic pulses (EMPs) in the microwave frequency range: the very high electric fields associated with such EMPs may cause malfunctioning or even serious damage to the diagnostic tools used in the experiment. EMP generation has been studied experimentally and theoretically.47–51 It originates from the return current induced by the ejection of high-energy laser-accelerated electrons from the irradiated target. This ejection creates a deficit of negative charges inside the target, which induces an electric current through the target holder. This return current has been identified as the main source of EMPs in the microwave (100 MHz to 2 GHz) range.52 The role of escaping hot electrons has been confirmed in an experiment where the ejected electrons were collected by a cage placed around the target.53 Such a small-scale Faraday cage provides a strong reduction in EMP emission.

Because of mode mixing by reflections within the chamber, the spatial distribution of EMP energy is randomized and becomes uniform inside the chamber. Therefore, knowing the chamber volume and the magnetic field at a fixed location, one can estimate the total microwave energy emitted by the target and the target holder.

The EMP microwave spectra originate from the sum of two contributions. The first, below 2 GHz, is due to the discharge current from the target through the target holder, which acts as an antenna, as explained in Ref. 52. The second, above 2 GHz, is the low-frequency part of the THz emission from the target. This emission is due to the electrons ejected from the target. Its duration is related to the laser pulse duration. That leads to a THz emission in the range from 0 to 1000 GHz. The range between 3 and 6 GHz corresponds to the low-frequency limit,54 where the spectral energy readsdE=Q22πε0c1βln1+β1−β−2dν,(1)where β, Q, and ε₀ are respectively the ratio of the speed of the electron before deceleration to the speed of light c, the total ejected charge, and the permittivity of the vacuum.

According to Eq. (1), the low-frequency part of the THz radiated spectral energy does not depend on frequency. Therefore, knowing the electron speed before deceleration, one can retrieve the total charge. For relativistic electrons, the dependence on their initial speed in Eq. (1) is logarithmic, and therefore an order of magnitude can be obtained by using the mean (thermal) hot-electron velocity associated with Maxwellian distributions fitting the measured electron distribution. Table II reports the calculated ejected charges for the three shots under consideration.

A comparison of these values with the hot-electron numbers estimated from the electron spectra in Table I confirms that indeed a significant number of electrons with energies less than 2.5 MeV were ejected from the target. These relatively low-energy electrons were not measured with the SESAME diagnostics, but seem to make a dominant contribution to the EMP.

The SEPAGE Thomson parabola was calibrated on an accelerator with protons and carbon ions36 and was simulated using an accurate measurement of the magnetic field inside the spectrometer using a 3D Hall probe. These two calibrations, in very good agreement, allow accurate recovery of the absolute spectra of ions on each shot. Figure 6 shows the traces obtained on the imaging plate (type MS) in shot #176 [450 J/610 fs laser pulse on a 50 μm thick plastic target (CH) coated with a thin (1 μm) aluminum layer on its front side]. The lower detection limit for the high-energy channel is 7.5 MeV. Very energetic protons up to 51 MeV were recorded on this shot. Carbon and oxygen ions (C⁵⁺, C⁴⁺, and O⁶⁺) are clearly visible, whereas the trace of the highest ionized carbon, C⁶⁺, is very dim.

It is worth noting the fluctuations in the heavy–ion profile (Fig. 7). Different artifacts induced by the spectrometer have been considered as potential causes of these fluctuations, such as supply-voltage fluctuations or EMP shielding default. None, however, can explain the observed results, since they would result in transverse deformations of the parabolas. Instead, it is known that the presence of ion species with different masses and charges in the laser-driven target can lead to the formation of several ion fronts that move at different velocities. The charge separation around each front can provide an additional accelerating field and modulate the ion spectrum.55 As will be shown below, the simulation results tend to confirm this explanation.

Figure 8 plots the absolute proton distribution obtained, combining low- and high-energy channels, for all three shots as recorded in the solid angle of the entrance pinhole (6.1 × 10⁻⁸ sr for #176 and 177, and 9.7 × 10⁻⁹ sr for #178). Small bumps are observed in the proton distribution, at around 30 MeV, for shots #176 and 177. Similar features are observed in the particle-in-cell (PIC)-simulated spectra (see below) when one considers only the protons accelerated within a small opening angle along the target normal. In the experiment, only such directed protons can reach the Thomson parabola. The spectral bumps may thus be ascribed to both (i) the angular selection of collected protons and (ii) the correlation between proton energy and angle induced by the acceleration process.

Assuming an emission cone of 20° half-angle,56 we can estimate, by extrapolating the number of protons collected through the pinhole, the energy conversion efficiency of the laser into protons and the total number of protons accelerated at the target rear side. This ranges from 1.3% for the lowest-energy shot (#178) to 3.3% for shot #176, with total numbers of protons above 7 MeV of 6.1 × 10¹², 4.2 × 10¹², and 9.5 × 10¹¹ for shots #176, 177, and 178 respectively. The number of energetic protons turns out to be consistent with the number of ejected electrons inferred from EMP measurements (see Table II).

In addition to measuring the ion spectra, a stack of RCF films was mounted in a cassette on the tip of the SEPAGE diagnostic. A hole at the center of each RCF allowed the protons to enter the SEPAGE Thomson parabola channels. Here, we used HDV2 and EBT3 (higher-sensitivity) RCF films, protected by 50 μm tantalum foil, to record the entire proton emission cone. This allowed measurement of the proton beam divergence as a function of the incident proton energy. Figure 9 displays the RCF images recorded in shot #176. One can see a reduction of the divergence with increasing energy, in agreement with the trend expected for proton beams produces by the target normal sheath acceleration (TNSA) mechanism.57–59 Additionally, on the first HDV2 films (measuring low-energy protons), an annular pattern is observed, corresponding to an angle of emission between 10° and 15° from the target normal (see Fig. 9).

The images obtained on the RCF stack allow to retrieve the energy spectra of protons emitted in the direction close to the central hole (5°–8° to the target normal). Figure 8 shows the proton spectra corresponding to the raw results in Fig. 9. Despite large uncertainties due to the spatial inhomogeneity of the signal within the RCF stack and the different angles of collection (0° vs 5°–8°), the RCF measurements closely agree with the Thomson parabola and therefore validate the experimental proton spectra.

The proton energy spectra obtained in our experiment can be compared with those from previous studies conducted in high-energy, high-intensity laser facilities worldwide. Flippo et al.24 measured a conversion efficiency into energetic protons of around 2% and a maximum proton energy of around 40 MeV using a 1 kJ laser pulse focused onto 15 μm thick targets at the Omega EP laser facility. The laser intensity was around 4 × 10¹⁹ W/cm², the laser pulse duration 10 ps, and the FWHM spot size 40 μm.

Yogo et al.25 reported up to 33 MeV proton energies using the LFEX laser, which delivered 1.5–6 ps pulses at an intensity of 10¹⁸ W/cm². The laser energy was 1 kJ, the FWHM spot size was 60 μm, and the pulse was focused onto 5 μm thick Al targets. The maximum conversion efficiency into energetic protons was ∼5%. Mariscal et al.26 measured protons with energies up to 18 MeV at laser intensities around 10¹⁸ W/cm² using the ARC laser system. The FWHM spot size was about 100 μm, and the laser performance was either 1 kJ/1.6 ps or 2.6 kJ/9.6 ps. The conversion efficiency into energetic protons reached a maximum of 2% in the second configuration. More recently, Margarone et al.27 reported maximum proton energies up to 30 MeV obtained on the LFEX laser, with 25 μm thick CH targets being irradiated by laser pulses of 2 × 10¹⁹ W/cm² intensity, 2.6 ps duration, 1.4 kJ energy, and 50 μm FWHM spot size. The maximum conversion efficiency into energetic protons was estimated as 7%. These results are summarized in Table III.

We see that despite the smaller energy delivered on target (reduced by at least a factor of 2 compared with other experiments) and laser intensities not exceeding those achieved in previous experiments, we have obtained on PETAL a significantly higher proton cutoff. It is also remarkable that we have used relatively thick targets, while it is well known that using thinner targets increases proton energies (if their use is compatible with the laser pedestal in a given laser system).60,61

These results are explained by detailed numerical simulations in Sec. V.

Simulating the experimental results obtained with PETAL is a complex task that requires multiple steps. Here, we just outline the main results obtained in each step, and a detailed description of the simulation methods and the full simulation results will be published in a separate paper.

To understand the high proton energies obtained in the experiment despite the moderate laser intensity, we have modeled shot #176, which produced the maximum energy of 51 MeV. In this shot, a 450 J/610 fs laser pulse was used, leading to a measured peak intensity of 7.9 × 10¹⁸ W/cm².

The laser pulse interacts with a 50 μm thick plastic target with a 1 μm aluminum layer on the target front side. This layer was designed to reduce both preplasma expansion (by means of the higher mass of the aluminum atoms) and shine-through, which could prevent CH ionization at low intensity. The laser beam incidence on the target is 13.5° from the target normal. Owing to the use of reduced geometries in the simulation (2D Cartesian or 2D axisymmetric), this small angle is neglected and a normal incidence is assumed, which is a reasonable approximation for this experiment.

Owing to the high intensity level of the long (nanosecond-scale) PETAL prepulse, a long preplasma can be generated before the arrival of the high-intensity peak on target. This preplasma significantly modifies the interaction process and thus ion acceleration. Its role has been evaluated using a two-stage approach: first, 2D axisymmetric radiation hydrodynamics simulations were conducted with the code TROLL62 to estimate the preplasma generated. Then, using the plasma conditions obtained from these simulations, particle-in-cell (PIC) simulations were performed with the code CALDER63 (2D Cartesian geometry) or CALDER-CIRC64 (2D quasi-axisymmetric geometry) to model the interaction of the intense PETAL beam with the target and the ion acceleration.

To estimate preplasma formation with hydrodynamic simulations, two factors have been considered:(i)the experimental peak intensity measured for this shot (7.9 × 10¹⁸ W/cm²) assuming a Gaussian beam with a waist of 50 μm (in the 2D axisymmetric geometry, we cannot reproduce the nonsymmetric properties of the laser spot shown in Fig. 1);(ii)the temporal contrast, long-term (5 ns) and short-term (around 250 ps), as shown in Fig. 10.

Simulations were carried out with the code TROLL, which is a radiation hydrodynamics code, based on a Lagrangian approach with the possibility of using an arbitrary Lagrangian Eulerian (ALE) method with unstructured meshes. Radiative transport and nonlocal thermodynamic equilibrium (NLTE) atomic physics are included. Laser propagation is computed by means of a ray-tracing method, taking account of refraction and inverse Bremsstrahlung absorption. Electron thermal conduction was modeled either with a flux limiter or with a nonlocal multigroup diffusion model according to Schurtz et al.65

The hydrodynamic simulations were initialized with a homogenous cold medium at solid density, consisting of a 50 μm CH target with a 1 μm Al front layer. To provide a correct description of the laser absorption at early times, the cell size was set to 1.25 nm within a 1 μm thick layer at the target front (directly seen by the laser). Deeper into the target, the cell size increased in a geometric progression.

The long-term (5 ns before the main peak) and short-term (250 ps before the main peak) prepulses were entirely simulated up to the foot of the main PETAL peak (at t = 5300 ps in Fig. 10), which hydrodynamic simulations are unable to handle. The simulations did not include two measured short prepulses with 10⁻⁴ contrast, because they are not energetic enough to have an appreciable impact on the plasma expansion.

Simulations were performed with two different laser energies, 450 and 180 J, for a 50 μm CH target coated with a 1 μm Al layer on its front side. On-axis lineouts of the electron density profile obtained in both cases just before the main pulse peak of PETAL are shown in Fig. 11. As can be seen, after the long 5 ns prepulse, a long preplasma is formed, which does not show any significant dependence on the pulse energy. This preplasma only contains Al atoms: the Al layer has been ablated, but the rear side of the CH target has not been modified. However, a shock has compressed the CH layer, reducing its thickness to 45 μm and increasing its local density. This shock has not yet reached the target rear side when the main PETAL pulse arrives. We therefore conclude that the target rear side was not perturbed by the prepulse. Conversely, this result could explain the lower proton cutoff energy observed experimentally with a 10 μm CH target, the rear side of which is distorted by the shock according to our simulation.

As a second step, the density profile computed with TROLL was introduced as input conditions for particle-in-call (PIC simulations) modeling the interaction of a high-intensity PETAL pulse with the target. However, the large dimension of the interaction region (several hundreds of micrometers), the long interaction time (several picoseconds), and the high plasma density all imposed stringent conditions that cannot be met in 3D with a PIC code. Consequently, several simplifications were applied.

As will be shown in the following, the hot electrons are produced by the laser in the preplasma up-ramp, where the laser energy is absorbed and reflected. These hot electrons then propagate in the high-density part of the target and produce a TNSA field at the back of the target. As the laser field does not reach the dense part of the target, the description of this high-density region can be simplified.

We assume that the propagation of hot electrons in a solid target is free, owing to (i) the return current that screens electric and magnetic fields and (ii) the low atomic number of the ions (CH), which makes collisional contributions negligible. Under these two assumptions, the target thickness and peak electron density are not sensitive parameters, and they can be decreased to relax the numerical constraints. We have estimated that reducing the target thickness to 10 μm would not have an impact on the results. Moreover, the use of a maximum electron density of 10n_c is sufficient to reduce the computational load but still ensure a correct description of the interaction physics. Here n_c is the electron critical density associated with the PETAL laser. The preplasma profile presented in Fig. 11 is modeled in the PIC simulation by the fit n_e(x) = 10n_c exp[−1.1(x − 150)^0.35], where n_e is the electron density as a function of the longitudinal (along the laser propagation axis) position x in micrometers. Here we have assumed that the Al preplasma and CH target are fully pre-ionized by the foot of the PETAL high-intensity peak, and so the electron density can be deduced from the TROLL simulation by using the simulated atomic density multiplied by the atomic number. Field and collisional ionization are thus not modeled in the simulation. A comparison between the electron density profile deduced from TROLL and the fit used in the PIC simulation is given in Fig. 11(b). In the PIC simulation, the preplasma is initialized over 150 μm, followed by the 10 μm long solid target, leaving 158 μm of vacuum at the target rear side to model the ion acceleration process.

As a full 3D simulation is out of reach, we present 2D simulations performed in the Cartesian and the cylindrical axisymmetric geometries. 2D Cartesian simulations of ion acceleration in the TNSA regime provide a qualitative understanding of the physical processes involved. However, it is known that they can significantly overestimate the accelerating field produced by hot electrons and hence the cutoff proton energy. Therefore, to obtain a realistic description of the TNSA acceleration, we also ran PIC simulation in a quasicylindrical (also called quasi-3D) geometry with the code CALDER-CIRC.36 As an axisymmetric laser intensity distribution should be considered in this geometry, the focal spot was first modeled as the coherent sum of two Gaussian beams (with the same phase) characterized respectively by a normalized peak vector potential of a0,1=1.3 and a 130 μm spot size (FWHM) and by a0,2=1.1 and a 20 μm spot size. Here the vector potential is normalized to mec/e. This leads to a total a0=2.4, corresponding to an intensity of 8 × 10¹⁸ W/cm² and a total energy of 318 J.

The high proton energies obtained in the experiment, considering the moderate laser intensity reached by PETAL, are explained by efficient electron heating by the laser fields in the long density ramp induced by the prepulse. A set of complex physical processes take place during the long (ps) and wide (100 μm) laser pulse interaction. To study the laser beam evolution and the processes leading to efficient electron heating, we performed 2D Cartesian simulations using a 318 μm long and 326 μm wide box. The longitudinal and transverse mesh cell sizes were dx=dy=0.1c/ω0 (20 000 × 20 480 cells), and the time step was dt=0.059ω0−1, with ω0 being the laser angular frequency. To initialize the electron, proton, C⁶⁺, and Al¹³⁺ species, 32, 16, 4, and 4 macro-particles per cell, respectively, were used. A third-order interpolation (four-point stencil) was used for current deposition. Some smoothing techniques (two-path binomial filter + compensator) were used on the currents to reduce the noise.

The PIC simulations show that density modulations with size close to the laser wavelength are produced in the preplasma. These modulations are due to a fraction of the laser beam, which is reflected or backscattered owing to Raman scattering in the plasma. This counter-propagating field interferes with the incident laser field and induces the observed modulation (Fig. 12). In addition, the laser beam is split into several filaments in the density ramp. When the peak laser intensity reaches the high-density part of the target, focusing of the laser field inside the filaments induces strong amplification of the local laser intensity up to a value of 5 × 10¹⁹ W/cm².

In addition, the filaments are associated with the creation of density channels and the formation of strong quasistatic azimuthal magnetic and radial electric fields. Magnetic fields of several kT and up to more than 10 kT are generated in the filaments, together with electric fields in the TV/m range.

Both the backward reflected/scattered laser field and the filaments boost electron acceleration and heating. Figure 13 presents the electron phase space (distribution of electrons in energy E and coordinate x) at four subsequent time moments. At early time (1 ps), we can observe that around the laser beam center (x ∼ 25 μm, the position corresponding to the laser beam center at 1 ps), some electrons have already acquired energies close to 20 MeV. Even more-energetic electrons are observed in the denser plasma at x = 80–100 μm. These high energies cannot be explained by ponderomotive acceleration in the laser beam. The quiver electron energy in the laser field is only 1.5 MeV (Wilks’ scaling with a₀ = 0.24). However, the presence of counter-propagating laser fields at this intensity level induces stochastic electron heating,66–74 which may increase the temperature by an order of magnitude. Later, at 1.3 and 1.5 ps, when the laser intensity inside the filaments is highest, electrons reach energies close to 60 MeV. Despite the laser focusing in the filaments, the enhanced laser field alone (a₀ ∼ 6, which implies a ponderomotive electron acceleration up to 12a02mec2∼9MeV) cannot explain the high energy observed. The presence of at least two oppositely propagating electromagnetic wave is a necessary condition for efficient stochastic heating. Moreover, the presence of intense laser fields in the plasma channels along with quasistatic electric and magnetic fields induces direct electron acceleration.75–77 Other heating mechanisms can also occur, such as electrons energization during the interaction and crossing of different filaments. After crossing the target rear side (located at x = 160 μm), those energetic electrons propagate away from the target and build the TNSA field, which decelerates them while accelerating the protons. At 2 ps, most of the energetic electrons are located behind the target in the region defined by x > 240 μm, and their energies decrease below 35 MeV (Fig. 14).

The bunch of electrons defined by x > 240 μm at 2 ps is far enough from the target for them to be considered as ejected electrons. We can measure their temperature and compare it with experimental data from SESAME diagnostics. The temperature of electrons in the bunch is close to 4.5 MeV. This value is much higher than expected from scaling and is comparable to the values shown in Table I. This confirms that the stochastic heating and direct acceleration in the filaments may explain the measured temperature of the energetic electrons.

The experimental electron temperature has only been measured at 13.5° and 58.5° from the laser propagation axis (see Fig. 3). In the simulation, we selected the electrons propagating with angles 13.5° ± 1°and 58.5° ± 1°. The corresponding spectra are shown in Fig. 14. Temperatures of 7 MeV at 13.5° and 2 MeV at 58.5° are in fair agreement with the experimental values of 8.3 and 3 MeV, respectively. The differences can be explained by the simplifying assumptions in the simulations (laser transverse distribution and incidence, target properties, reduced box size, etc.) and by experimental uncertainties (laser parameters, preplasma density profile, etc.). The 2D geometry may also reduce the increase in laser intensity in the filaments in comparison with what could be expected in 3D; thus, lower intensities may produce smaller temperatures.

An energy of 50 MeV is obtained at the end of the 2D simulation (Fig. 15), but the protons are still being accelerated. Indeed, the 2D geometry is known to overestimate ion acceleration in the TNSA regime, and we would expect a final energy higher than 50 MeV if a bigger simulation box could be used (but simulations with longer box and time imply a significant increase in numerical cost).

However, the 2D simulation can already qualitatively explain the observed features of the ion spectrum. The proton and carbon-ion density maps recorded at the end of the simulation are shown in Fig. 16, together with the TNSA fields, both longitudinal and radial.

The acceleration of protons and carbon ions, which have different charge-to-mass ratios, leads to a complex structure of the TNSA field. The first layer of accelerated protons is already detached from the accelerated carbon ions. The density of the proton layer increases when going backward from the proton front up to the region located near 260 μm < x < 280 μm on the axis and near x = 220 μm at y = −150 μm. Then, the proton density is strongly reduced and there is a ∼10 μm gap before the position at which the carbon-ion density starts to rise. The electron density is much smoother, and so the structured ion density generates a nonmonotonic accelerating field. The longitudinal field E_x is maximal at the proton front, as expected, but the high proton density at the back of the first proton layer leads to a reduction in the electric field behind it. Then, the sheath at the carbon front again creates an accelerating field (E_x > 0) for ions. In addition, at this late time, a stronger acceleration of ions on the axis induces a curvature of the ion fronts. As a consequence, the normals to the ion fronts (front and back of the proton first layer, carbon-ion front, etc.) are no longer aligned in the longitudinal direction. The accelerating fields, mainly normal to the sheath, exhibit a significant transverse component, as shown in Fig. 16(d). This component tends to defocus ions that are not on-axis, and the presence of the gap induces a more complex behavior for the protons in this region.

Figure 17 reveals a correlation between the positions of the protons, their energy, and the angle of emission. The fastest protons (>40 MeV) form a bunch located on the axis, at the front of the proton layer. They mainly propagate in the longitudinal direction, but the transverse electric field that appears at the edges of the proton fronts tends to defocus the off-axis protons of this bunch. As a consequence, the angular distribution of these protons is rather flat, with a typical divergence of the order of 6°. The protons with energy around 15 MeV are mostly located off-axis, at the back of the first proton layer at y = ±50 μm and close to the front of the proton layer at y = ±150 μm. Owing to the small number of these protons on the axis, most of them are deviated by the transverse fields, and their angular distribution presents two peaks at angles of ±6°. This annular pattern distribution is clearly observed on the first RCFs of the SEPAGE diagnostic (see Fig. 9), with slightly higher angles.

The proton spectrum is thus very different if we consider all the protons or only the protons accelerated longitudinally, within a small opening angle, as shown in Fig. 15. The spectrum is globally decreasing if all protons are considered within the angle ±10° to the target normal. The high-energy tail of protons emitted with angle <1° is similar to the total spectrum, since the angular distribution is nearly flat. However, a limited number of protons that propagate longitudinally with energies between 10 and 20 MeV leads to a depression of the spectrum at these energies and a bump near 20–25 MeV. In the 2D simulation, this bump is pronounced, but it may be reduced in the 3D geometry. In addition, modeling of the target with only two ion species, H⁺ and C⁶⁺, and a maximum density of only 10n_c, may affect the simulated acceleration process.

Nevertheless, the bump around 30 MeV observed in the experiment (Fig. 8) could be justified by this qualitative description based on (i) the proton layer formation, (ii) the narrow opening angle of the Thomson parabola, (iii) the formation of a gap between the proton layer and other ions, and (iv) the transverse deformation of the ion fronts, which induces a structured TNSA field acting on protons differently depending on their position and energy.

We observe in Fig. 16 that the accelerating field E_x is modulated inside the carbon-ion layer (e.g., near x = 220 μm on the axis). This modulation is associated with the interplay between the proton and carbon-ion motion, which leads to the creation of several ion fronts, as explained in Ref. 55. As a consequence, the carbon spectrum is modulated. This effect is not seen when all the carbon ions are taken into account, but it becomes visible if ions only within a small opening angle are selected. This effect may explain the experimental spectral modulation of heavy ions seen in Fig. 6.

To better model 3D effects in the ion acceleration process, we performed a simulation in the 2D axisymmetric geometry using the code CALDER-CIRC.64 In this code, the macroparticles describing the plasma can move in a 3D cylindrical volume, but the fields are described by a truncated discrete sum of Fourier modes along the angle θ. Only the first two modes were used in this simulation. The simulation box corresponded to a 318 μm long and 326 μm wide cylinder (with radius 163 μm). This code is slower than a standard 2D simulation. Therefore, to reduce the simulation cost, the resolution was twice as coarse as in the previous simulation. The longitudinal and transverse mesh cell sizes were then dx=dy=0.2c/ω0 (10000×5120 cells), and the time step was dt=0.118ω0−1. To initialize the electron, proton, C⁶⁺, and Al¹³⁺ species, 128, 8, 8, and 8 macroparticles per cell, respectively, were used. In addition, to obtain sufficient statistics to model the accelerated layer of protons and carbon ions, 128 macroparticles per cell were used to describe the last micrometer of the target at the rear side. Some smoothing techniques were are used on the currents and fields to reduce the noise that appeared close to the axis.

The qualitative description of the interaction and acceleration processes obtained from the 2D simulation matches shows a good match with the results obtained in the quasi-3D simulation. First, laser filamentation also occurs. However, the electron heating process is less efficient in this quasicylindrical geometry with coarser resolution. As a consequence, the electron temperature is 4.5 MeV, which is significantly smaller than the temperatures measured experimentally or obtained in the 2D planar simulation. This lower temperature induces a weaker TNSA accelerating field, which explains why the proton cutoff energy at the end of the simulation is only 35 MeV. The protons are still being accelerated at that time, but, unlike in the Cartesian 2D simulation, the energy gain is very slow and we may not expect to reach much higher energy with a longer simulation. The maximum carbon-ion energy at the end of the simulation is 10.8 MeV. One of the interesting aspects of the quasi-3D simulation is that, unlike in 2D, it is possible to compare the absolute numbers of accelerated ions directly with the experimental data. Apart from the maximum energy, good agreement is found in the absolute numbers of protons, as shown in Fig. 18: about 10¹³ protons MeV⁻¹ sr⁻¹ are obtained at the lowest energies (a few MeV), a few 10¹¹ protons MeV⁻¹ sr⁻¹ are found at medium energies (25 MeV in the experimental data and 17 MeV in the simulation), and close to 10⁹ protons MeV⁻¹ sr⁻¹ are found near the energy cutoff.

The total number of protons accelerated over 7 MeV is 2.2 × 10¹², which is also in reasonably good agreement with the value estimated from the experimental data.

We have presented an experimental campaign performed at the PETAL laser facility studying proton acceleration with plastic targets of different thickness. For both electrons and protons, we were able to not only measure the spectra, but also to consistently retrieve the absolute numbers of accelerated particles, which confirms the good performance of the developed diagnostic tools.

Despite a moderate laser intensity on target (up to 7.9 × 10¹⁸ W/cm²), very energetic protons with cutoff energy of 51 MeV and hot electrons with temperatures far exceeding the values predicted by ponderomotive scaling were measured. These results can be explained by the specific features of the PETAL laser, which is characterized by a long and quite intense laser prepulse (creating a long-scale-length plasma in front of the solid target), a relatively long duration (610 fs), and a large focal spot, all of which favor long and efficient TNSA acceleration of ions. To fully understand these results, detailed simulations were performed.

First, the radiative hydrodynamics code TROLL was used in a 2D axisymmetric geometry to simulate the preplasma generated by the PETAL prepulse. Then, 2D Cartesian and axisymmetric PIC simulations were performed with the codes CALDER and CALDER-CIRC to simulate the propagation of the laser beam in the plasma and the processes of electron heating and ion acceleration. The results of these simulations show the importance of the long (100 μm scale) and low-density preplasma, which is responsible for (i) backward scattering of laser radiation, which is coupled with the forward-propagating laser beam and induces stochastic electron heating, and (ii) filamentation of the laser beam in high-density plasma leading to local amplification of the laser intensity and creation of density channels that efficiently boost electron heating. Owing to the long (ps) laser pulse duration and the large laser waist, hot electrons sustain a large TNSA field at the target rear side for a long time before electron cooling and transverse dispersion stop the acceleration. In addition, the presence of different ion species induces restructuring of the TNSA fields, with detachment of the first proton layer from the accelerated carbon ions and modulation of the ion densities and fields inside the accelerated carbon-ion layer. The curvature of the ion fronts due to the transverse variation of the acceleration strength is responsible for the divergence of the ion beams and their energy modulation, owing to the correlation between ion position and energy.

Acknowledgment. The PETAL laser was designed and constructed by the CEA with funding from the Conseil Regional d’Aquitaine, the French Ministry of Research, and the European Union.

The SESAME and SEPAGE diagnostics were realized by the CEA within the PETAL+ project coordinated by the University of Bordeaux and funded by the French Agence Nationale de la Recherche under Grant No. ANR-10-EQPX-42-01.

The Twist and EMP diagnostics were realized within the PetaPhys project funded by the LabEx LAPHIA of the University of Bordeaux under Grant No. ANR-10-IDEX-03-02.

The LMJ-PETAL experiments presented in this article were supported by Association Lasers et Plasmas and by the CEA.

We also acknowledge GENCI for granting us access to the supercomputer IRENE under Grant Nos. A0070506129 and A0060507594.