Laser pulses with intensities over 10¹⁸ W cm⁻² can produce bright X- or γ-ray radiation when striking a solid high-Z material. The laser will accelerate electrons1–4 to relativistic energies. In a high-Z material, most of these high-energy electrons will undergo bremsstrahlung.5,6 The majority of these electrons will create a large number of X- or γ-rays. These sources have applications in various fields of physics. Bright directional X-ray beams are interesting for radiography7,8 of dense plasmas and nondestructive testing. Intense γ beams open up new fields of study such as laboratory astrophysics.9 Collision of two intense γ beams will also generate electron/positron pairs through the Breit–Wheeler process.10 Up to now, linear Breit–Wheeler pairs have never been seen in the laboratory, but several schemes have been proposed to produce them at laser facilities.11–16 These schemes involve the production of a large number of γ photons by bremsstrahlung or nonlinear inverse Compton scattering. High-energy photons can also interact with the electric field of the nuclei inside the target and create electron/positron pairs through the Bethe–Heitler process. This process is considered as a means of generating pair plasmas.17–19 Energetic electrons can also produce electron/positron pairs through the trident process,19,20 but with a much lower yield. A scheme based on the production of an electron/positron plasma has also been investigated21 as a possible mechanism involved in the generation of astrophysical γ-ray bursts.22,23

In this work, we present the results of the first experimental campaign carried out at the Petawatt Aquitaine Laser (PETAL) laser facility24 to produce X- or γ-ray beams. The PETAL laser was shot at a thick solid tungsten target. A bremsstrahlung cannon25 diagnostic was placed at the rear side of the target to measure the high-energy photons produced. To model this experiment, we have used a simulation chain including hydrodynamics, particle-in-cell (PIC) and Monte Carlo simulations. We can simulate the preplasma generated at the front of the target by the PETAL laser prepulse, as well as the acceleration of electrons inside the plasma and the generation of MeV-range photons through bremsstrahlung emission. We are also interested in the production of electron/positron pairs by the Bethe–Heitler and Breit–Wheeler processes within the PETAL facility, although they were not measured during the experiment.

The remainder of this paper is organized as follows. Section II briefly describes the laser facility. Section III describes the detectors and the experimental results that were obtained during the campaign. Section IV deals with the simulation chain and the results that we obtained. Section V is dedicated to estimation of the production of electron/positron pairs on PETAL. Finally, we present our conclusions and provide some perspectives in Sec. VI.

The PETawatt Aquitaine Laser (PETAL) is a high-energy petawatt beam implemented within the Laser Mégajoule (LMJ) facility. Operating at a wavelength of 1053 nm, the interaction of PETAL with different types of targets provides bright and short particle sources, synchronized with the high-energy beams of LMJ, offering the opportunity to study matter under extreme conditions. For instance, proton radiography using PETAL has already found successful applications in research; see Refs. 26 and 27 for more details. We are also developing X-ray radiography capabilities to open up PETAL to new possibilities such as the detection of shock waves propagating through matter, as initially proposed in Ref. 28.

The PETAL laser system is based on the chirped pulse amplification (CPA)29 and optical parametric amplification (OPA)30 techniques for the front end, allowing energies of 100 mJ to be reached with a large spectral bandwidth (8 nm). The amplifying section with Nd:glass amplifier slabs then increases the energy up to a few kJ. Finally, the compressor is based on a subaperture compression scheme: the beam is split into four subapertures that are independently compressed, synchronized, and coherently combined using a segmented mirror with interferometric displacements. This scheme is able to deliver short pulses (∼500 fs) with high energy (∼kJ). In 2015, it was demonstrated that PETAL is able to provide 1.15 PW at the end of the compressor, corresponding to a 850 J, 700 fs pulse.31 Damage-prevention requirements regarding post-compressor transport mirrors under vacuum, however, constrain the on-target beam energy to be no more than 400 J (i.e., 0.6 PW) for a daily-basis use. An on-target energy boost to 650 J up to the kJ level will start in 2024.

The PETAL focusing system consists of an off-axis parabolic mirror with a 90° deviation angle and a focal length of 7.8 m followed by a pointing mirror. The on-target central focal spot is usually around 30–50 μm FWHM (1.5 times the diffraction limit) and contains an estimated 10%–20% of the total energy, corresponding to a peak intensity of up to 10¹⁹ W cm⁻². The remaining energy is distributed over a 400–600 μm-FWHM zone with a pattern of multiple rebounds likely induced by beam phase distortion from the four-subaperture compression scheme. For more details, typical PETAL focal spot profiles can be found in Refs. 32 and 33. The limited level of encircled energy is mainly due to instabilities (air, vacuum) during beam propagation to the target and the amount of correction (spatial phase) of the amplifiers’ thermal phase. The beam segmentation only introduces an increase in diffraction bounces to a value of 10% along the segmentation direction.

The PETAL pedestal long-term temporal contrast is about 10⁻⁷ in intensity over a 5 ns time-window prior to the main pulse. The short-term temporal contrast (i.e., 250 ps before the main pulse) is in the 10⁻⁵ range with one or two prepulses with 10⁻⁴ contrast. For more details of the typical PETAL prepulse contrast, see Ref. 32. Such a prepulse laser intensity generates at the front surface of the target a large preplasma that will influence the interaction of the main peak intensity pulse with the target.

In this work, we focus on two PETAL shots performed on tungsten targets of 2 mm thickness and 2 mm diameter [see Fig. 1(a)]. The laser parameters are given in Table I.

Figure 1 shows schematics of the experimental setup as well as the principles of operation of each diagnostic. The particles produced by the interaction of the PETAL laser with the targets are measured in absolute with a set of spectrometers:•The CRACC-X (Cassette de RAdiographie Centre Chambre-rayonnement X) diagnostic [Fig. 1(c)], which is mounted on an equatorial diagnostic insertion system (DIS) located at 13.5° from the PETAL axis [see Fig. 1(a)], is used to bring a shielded cassette containing a bremsstrahlung spectrometer 30 cm away from the target. The spectrometer is fabricated from a stack of various filters and imaging plates (IPs), in a way similar to that described in Ref. 34.•The SESAME (Spectre ÉlectronS Angulaire Moyenne Énergie) diagnostics [Fig. 1(d)] are two identical electron spectrometers located at the LMJ chamber wall, in the equatorial plane.³⁵ SESAME 1 is positioned on the PETAL laser axis (0°) and SESAME 2 is located at 45° from this axis [see Fig. 1(a)]. Each spectrometer uses a single permanent NdFeB magnetic dipole (0.5 T) to deflect electrons, covering the 3–150 MeV energy range. SESAME is also able to measure positively charged particles (protons, ions, and positrons) on the opposite side of the magnet. The energy range for protons is 0.7–14 MeV. Moreover, the zero-deviation axis is also equipped with a compact bremsstrahlung spectrometer downstream of the dipole.•The SPECTIX (Spectromètre Petal à Cristal en TransmIssion pour le rayonnnement X) diagnostic [Fig. 1(b)], which is also mounted on a DIS located at 108° from the PETAL axis, is a hard X-ray spectrometer based on a Cauchois geometry using two transmission Bragg crystals: quartz (10-10) and LiF(200). It covers a wide spectral range (7–150 keV) with a high enough resolution to clearly distinguish Kα1 from Kα2 emissions.³⁶

The harsh electromagnetic environment generated during a PETAL shot requires the use of IPs, which are insensitive to electromagnetic pulses, as detectors inside the diagnostics described above. Extensive work has been done to improve our knowledge of the detectors. The IPs and our scanner system have thus been absolutely calibrated on different accelerators and laser facilities.37–40

The two SESAME spectrometers enable the detection of electrons emitted from the rear side of the target at two different angles. The experimental spectra are provided in Fig. 2. An exponential fit (∼Ae^−/) of each distribution is performed above the detection threshold, estimated as 10⁸ electrons MeV⁻¹ sr⁻¹. The electron temperatures for shot SR174 are 6.69 MeV at 0° and 2.46 MeV at 45°, whereas for shot SR182 these temperatures are slightly lower, namely, 6.33 and 1.99 MeV, respectively, because the on-target intensity is lower. However, in both cases, high-energy electrons (up to ≈35 MeV) and high temperatures (a few MeV) are measured. Note that the SESAME-measured electrons are transmitted through the thick tungsten target. This means that only high-energy electrons that could escape the target are detected here. Most of the accelerated electrons cannot leave the target, and their interactions with matter should produce secondary particles such as positrons, X rays, and bremsstrahlung photons. The measurements of these secondary particle sources are further explored in Secs. III C–III E.

As already mentioned, owing to the symmetry properties of the magnetic field, SESAME is also able to detect positively charged particles. The low-energy detection threshold of 0.7 MeV corresponds to the theoretical minimal energy a proton needs to make it across the protective layer of an IP and deposit energy into the sensitive layer.38 The raw signal is therefore attributed to only target normal sheath acceleration (TNSA) protons. The experimental spectra are displayed in Fig. 3 and correspond to the raw signal present in the top panel of Fig. 1(d). The proton energy cutoff is ≈1 MeV, which is rather low in comparison with the spectra shown in Ref. 32. Obviously, the tungsten target is relatively thick and the TNSA mechanism is not efficient. The positron signal, or lack thereof, indicates that the positron beam density is lower than 10⁸ positrons MeV⁻¹ sr⁻¹ on the axis of the SESAME detector. Protons with the lowest kinetic energies deposit less energy in the sensitive layer of the IP, which results in a less intense signal when compared with protons with higher energies (closer to the Bragg peak of the energy deposition curve).

The SPECTIX spectrometer faced the irradiated surface (front surface) of the tungsten target at an angle of 72° to the PETAL laser axis. Figure 4 presents an example of a temporally and spatially integrated X-ray spectrum for shot SR182, measured using a quartz (10-10) crystal in the [8, 10] keV energy range and a LiF(200) crystal in the [50, 70] keV energy range. The L- and K-shell emission lines are unambiguously detected.36 Absolute integrals of X-ray lines measured in shots SR174 and SR182 are given in Table II. Note that the L-shell lines were not measured in shot SR174 because of uncontrolled background noise (coming from bremsstrahlung emissions). The L_α lines of shot SR182 are merged owing to the spectrometer resolution. The FWHM of the peaks is larger than the energy separation between Lα1 and Lα2. For an electron temperature of a few keV in the hot plasma region, there is no reason to have Lα2 and no Lα1.

K-shell emission spectroscopy is able to give information on the target heating and ionization induced by the propagation of hot electrons generated by the laser–matter interaction.41 For instance, we recently used this diagnostic for hot-electron population characterization in the context of shock ignition at LMJ-PETAL.28

The filters of the CRACC-X bremsstrahlung cannon are made of various materials ordered by increasing atomic number (see Table III). This setup is similar to that developed at the OMEGA EP (extended performance) facility.34

The bremsstrahlung cannon is encapsulated inside lead shielding with high-density polyethylene (HDPE) shielding of 25 mm thickness placed in the line of sight to prevent diffusion of X-ray radiation and to reduce the contribution of electrons and other charged particles interacting directly with the IP. Owing to the compactness of the CRACC-X cassette, no magnet is used to deflect charged particles. Figure 5 presents the energy deposit for each IP (as inferred using a calibration performed with Geant4 simulations of the diagnostic).

The SESAME spectrometers are also equipped with one compact bremsstrahlung cannon each at the zero-deviation axis downstream of the dipole. These cannons are made of stacks of five filters and IPs. However, the reduced number of IPs increases the relative fitting uncertainty for temperature fits to ≈15% in this case.

We would like to stress here that bremsstrahlung spectrometer analysis is strongly dependent on the spectral shape hypothesis, and the reader should be aware that several different distributions could give similar reproductions of the experimental data. Moreover, high-energy electrons are also expected to partly contribute to the IP signals, leading to additional uncertainties. Thus, a better way to exploit the CRACC-X diagnostic is to compare experimental data with a simulated response using, for instance, simulations as inputs. This approach will be discussed in Sec. IV.

In this work, simulations are used to better understand the emission and measurement of high-energy photons. As these γ photons are directly emitted by energetic electrons (≳MeV), the modeling of the hot-electron distribution is also discussed.

To simulate the photon emission of the tungsten target, we need to use a simulation chain, i.e., several simulation codes,42–45 because it is not possible to model all the physical phenomena involved with a single code. The laser prepulse generates a preplasma at the front side of the target. To simulate the generation and expansion of this plasma, we use the radiative hydrodynamics code TROLL.43 From the preplasma characteristics thereby obtained, we inject the electron density into the PIC code CALDER.42 This code allows simulation of the electromagnetic interaction between particles and the main peak of the laser pulse. The electron distribution in the 6D (3D3V) phase space obtained from the PIC simulation is then injected as input to a Monte Carlo code based on Geant4.45 In this simulation, it is possible to model individual collisions and radiation emissions inside a thick solid target and finally the γ-photon beam characteristics. Figure 6 gives a diagram of the simulation chain setup.

As inputs for our Geant4 simulations, we use the already existing hydrodynamics and PIC simulations carried out in Ref. 32. In that work, the target was made of 1 μm of aluminum and 50 μm of plastic, and the laser energy was 450 J, with a laser pulse duration of 610 fs. Simulation of a tungsten target is prohibitively expensive owing to the target thickness and density (n_e ∼ 4000n_c). That study showed that electrons were accelerated inside the aluminum’s preplasma by filamentation and stochastic heating.46 The main pulse of the target does not interact with the plastic target, but only with the aluminum preplasma. The collision, ionization, and radiation modules are not activated.

To be able to use the characteristics of the energetic electrons from these simulations (Al + CH target) in our case (W target), we need to ensure the following:(A)There is no great difference between the hydrodynamics of a tungsten and aluminum preplasma.(B)We can neglect the deceleration of the high-energy electrons by the TNSA mechanism over the duration of the PETAL pulse interaction with the target. Indeed, with the 50 μm-thick target, the electrons leaving the target at the rear side will be decelerated in the TNSA field and eventually recirculate, whereas in the 2 mm-thick tungsten target, such deceleration does not occur, and the accelerated electrons can propagate in the 2 mm target while only being subject to collisions.

To ensure that (A) holds, we perform 2D axisymmetric hydrodynamic simulations with the code TROLL. Figure 7 shows the results of 2D TROLL simulations and compares the electron density profiles obtained with targets composed respectively of a layer of aluminum in front of a layer of plastic (orange) or tungsten (blue). For the initialization of the PIC simulations, an exponential fit of the preplasma electron density profile is done over 150 μm length. This fit starts at a density of 10n_c (where nc=ω02meε0/e2 is the critical density, with ω₀ the laser angular frequency, m_e the electron mass, e the elementary charge, and ɛ₀ the vacuum permittivity) up to ∼0.01n_c. The density profile corresponds to the density inside the red vertical lines given in Fig. 7. There is a separation of 10 μm between the solid layers of aluminum and tungsten, because a shock induced by the prepulse in the aluminum + plastic target is able to push its surface by 10 μm (the target front face was initially at x = 0 cm in Fig. 7), while the heavier tungsten target is pushed less. The green curve corresponding to the aluminum profile is shifted to superimpose at n_c the density profiles of aluminum and tungsten. In the preplasma area characterized by densities lower than n_c, where most of the laser–plasma interaction, absorption, and thus production of hot electrons occur, the two preplasmas are reasonably alike. There is an area from n_c to 10n_c where the two preplasmas are different, but only over a very short distance. Moreover, the normalized laser field amplitude a₀ ≡ eE₀/m_eω₀c (where E₀ is the laser field amplitude) is close to unity (a₀ = 2.4), which implies that relativistic transparency does not allow the laser pulse to propagate much farther than the critical density. Below n_c, the preplasma densities are quite close, and so the preplasma density profiles of tungsten and aluminum can be considered quite similar in this study, which is in agreement with what was found in Ref. 47 for preplasmas of Cu, Al, Si, and CH. Other lineouts at radial positions from 10 to 100 μm indicate that over a length of 25 μm, the preplasma exhibits no significant deviation from that in Fig. 7. After a radial distance of 65 μm, the two are very much alike.

To ensure that (B) holds, we retrieve the electron phase space of the 2D CALDER simulations [the list of all the energetic macroparticles (MPs) present in the simulation box, with, for each MP, the data on its position and momentum, which allows us to take into account all the position/angle/energy correlations] at 1.5 ps after the beginning of the laser–target interaction. We have chosen this time to limit the deceleration of electrons by TNSA. The majority of the accelerated electrons are still inside the simulation box (the electrons have not yet reached the edges of the box) and are close to the target. This choice limits the deceleration of electrons in the TNSA process, since this deceleration would be much less important in the case of a thick tungsten target. The electron phase space (i.e., the list of the MPs) is injected as input to the Monte Carlo simulation code. The electrons are injected inside the tungsten target at the same position they had inside the PIC simulation at 1.5 ps. In 2D PIC simulations, we cannot immediately retrieve a real number of particles, because macroparticles in a PIC simulation are associated with a weight that takes into account the real number of particles each macroparticle is supposed to represent. The weight, in 2D, is a linear density in units of number by length. To obtain a number of particles, we need to multiply the weight by an arbitrary length. We choose to multiply by the laser spot size, because the majority of the hot electrons are accelerated by the laser field.

Figure 8 presents the electron energy spectrum of the initial electron phase space. The electron spectrum from the PIC simulations is plotted in red. It has been confirmed in Ref. 32 that the simulated hot-electron distribution emitted in the two SESAME spectrometer directions is in good agreement with the experimental data, which provides support for our approach of reusing these simulation results for γ-ray modeling. However, owing to memory constraints, the PIC electron phase-space distribution (list of MPs) has only been saved for electrons with energies >4 MeV, since only these electrons can be measured in the SESAME spectrometers. The PIC electron energy spectrum thus spans only the range between 4 and 60 MeV. As shown in Fig. 8, the PIC spectrum has a Maxwellian shape, and we therefore assume that the electron energy spectra can be fitted with two Maxwellian distributions5,48 having different temperatures. To complete the low-energy electron distribution, we generate a low-energy phase-space distribution with a Maxwellian distribution, writingdNdE=KETe−E/T,(1)where N is the number of electrons, T the temperature of the population, E the energy, and K a normalization constant. To define this distribution, we need to assume different values for the laser energy absorption. The interaction of a high-intensity laser pulse with a solid target can lead to various values49–52 of the laser energy absorption coefficient, depending on the laser intensity and the preplasma scale length. We choose to use various values of the total energy transferred in the energetic electrons:•216 J, which implies an absorption coefficient of 50% of the laser energy (T_e = 0.63 MeV);•253 J, which implies an absorption of 59% (T_e = 0.6 MeV);•300 J, which implies an absorption of 70% (T_e = 0.58 MeV);•371 J, which implies an absorption of 86% (T_e = 0.55 MeV).

To define the total phase-space distribution, it is necessary to make additional hypotheses. We assume that the electrons with an energy below 4 MeV have the same angular (θ, ϕ) and spatial distributions as the PIC electron distributions with energies included between 4 and 5 MeV. The analysis of the angular spectrum obtained from the PIC simulation demonstrates that the angular (θ) divergence of electrons, for slices of 1 MeV from 1 to 6 MeV of electron energy, exhibits a similar distribution for each slice. The reconstructed electron spectrum is plotted in Fig. 8. The spectrum in red comes from the PIC simulation. This spectrum goes from 4 to 60 MeV. It is composed of a part that goes from 4 to 7.5 MeV and has a temperature of 2 MeV and a part with a temperature of 4 MeV after an energy of 7.5 MeV. The spectrum in purple comes from the electron phase space that we define to fill the low-energy part of the electron energy distribution [defined by Eq. (1)]. This case (see Fig. 8) is an example for an initial energy of 253 J. This spectrum has an energy distribution approximated by the expression in Eq. (1) with K = 2.3 × 10¹⁵ MeV^−1/2 and T = 0.6 MeV. The blue vertical line corresponds to the limit of the generated electron phase space, since we use the PIC simulation results for energies over 4 MeV. The merger of these two electron phases spaces will be used as input in the Monte Carlo simulation code (Geant4). In the rest of this paper, if the total electron energy is not indicated, the results presented are for the case with a total electron energy of 253 J, corresponding to an absorption of 59%. We chose this energy because experiments realized with a 150 fs laser pulse (the Jupiter Laser), with a laser intensity ≈10²⁰ W cm⁻² show laser energy absorption in electrons of 60%.49 At a time of 1.5 ps, according to the 2D PIC simulation, 60% of the energy is absorbed by hot electrons. Moreover, simulations of a beam that is shorter (32 fs) but with similar intensity ≈10¹⁸ W cm⁻² find an absorption of the order of 64%.53

We inject the previously obtained electron phase space as input in a 3D Monte Carlo code named Gp3m2. This code is based on the Geant4 toolkit45 and was developed to model the interaction of charged particles inside a series of several circular layers of different solid materials. The physics taken into account in this code describes electromagnetic processes, and it uses the G4EmStandardPhysics_option4 physics list of Geant4. This physics module list includes radiation, collisions, scattering, ionization of charged particles, and electron/positron pair production and annihilation.

This Monte Carlo code will create an electron distribution with the characteristics (position, momentum) of the phase space. The PIC simulation box is 318 μm long and 326 μm wide. Since we create the particles in the Monte Carlo simulation with the same position as those of the PIC simulation, some electrons will be generated inside the tungsten target. After their initialization, the electron distribution will propagate inside the solid 2 mm layer of tungsten.

Figure 9 gives the spectrum of photons at the end of the simulation. The red curve corresponds to photons at the rear side of the tungsten target (x = 2 mm). This spectrum is decomposed in two parts: a high-energy Maxwellian distribution that goes from 2.5 to 50 MeV and a low-energy part composed of the atomic transition lines of cold tungsten like K_α and K_β. There are 1.46 × 10¹³ photons with energies exceeding 0.3 MeV and 1.01 × 10¹² with energies exceeding 2.5 MeV.

Experiments have recently been performed at the National Ignition Facility (NIF) with the NIF-ARC (Advanced Radiographic Capability) laser54 to determine the characteristics of the high-energy photons produced. They shot on different types of Au targets to produce bright MeV photons and obtained a very high dose. NIF-ARC is a high-energy petawatt laser with an intensity comparable to that of PETAL. The photon dose produced by PETAL was compared with that of NIF-ARC. To evaluate the photon dose deposition, we use the method of Ref. 55. The dose can be calculated asD=∫fγ(Eγ)EγΔΩd2μenρ(Eγ)dEγ,where f_γ is the photon distribution, ΔΩ the solid angle, d the distance of the absorption material, and μ_en/ρ the material X-ray mass energy absorption coefficient. Looking at the dose absorption in air at 1 m (d = 1 m), we can approximate the X-ray mass energy absorption coefficient, for photon energies higher than 0.5 MeV, as55μenρEγair=0.011+1.5Eγ.Calculation gives, for the total rear side photons (ΔΩ = 2π sr), a dose of 0.26 rad. Close to the laser axis, with an emission angle of 1°, a dose of 0.37 rad is estimated. To compare with other installations, an efficiency (rad/J_laser) is calculated. In our case, the efficiencies are respectively of 6.1 × 10⁻⁴ and 8.6 × 10⁻⁴ (rad/J). For these experiments, the NIF-ARC laser had a laser energy between 2602 and 1191 J and a pulse duration of 10 ps. They obtained efficiencies between 2.3 × 10⁻⁴ and 6.6 × 10⁻³, respectively. We obtained similar doses and slightly lower efficiencies than those obtained at NIF-ARC with a flat target of 0.5 mm of Au.

The inset in the right upper corner of Fig. 9 is a zoom on the low-energy part of the photon spectrum. This low-energy part is composed of different emission lines (see Table IV). The annihilation line corresponds to photons at 511 keV produced by annihilation of electrons and positrons. The other lines are emitted during atomic transitions inside the inner shells of tungsten atoms.

Table V corresponds to the number of photons that cross the front side of the tungsten target. These are obtained by integration of the front-side photon spectrum. We can compare the experimental K-shell emissions measured by the SPECTIX detector (Table II) with the data obtained from the simulation (Table V). There is good agreement between the two sets of data for the three emission lines.

To compare our photon spectrum with the CRACC-X experimental data, we simulate the CRACC-X detector using the Geant4 code. We inject into the CRACC-X detector the blue curve of Fig. 9. This spectrum is made of photons that are propagated to the detector position located at 0.5 m from the rear side of the tungsten target and inside the detector acceptance angle. This spectrum has the same shape as the total rear-side photon spectrum, but its amplitude is attenuated.

At the back surface of the tungsten target, energetic electrons will also be present. These electrons can be created inside the target by ionization, pair production, and Compton scattering. They can also come from electrons with high enough energies to be able to cross the full target. Electrons with more than 5.5 MeV will on average propagate more than 2.06 mm in tungsten. This calculation was done with the ESTAR software of the NIST56 using the continuous slowing down approximation (CSDA) range and a 19.3 g cm⁻³ density for tungsten.

Figure 10 shows the electron spectrum at the rear side of the tungsten target. The blue curve corresponds to all the rear-side electrons. In total, there are 6.34 × 10¹¹ electrons that propagate out of the target from the rear side. All the electrons that propagate out of the target do not have the same angular distribution, and only some of these electrons will have the possibility to go through the CRACC-X detector. These electrons correspond to the green spectrum in Fig. 10. As for photons, these electrons only account for a small part of the total electron spectrum. There are 1.9 × 10⁸ electrons that interact with CRACC-X.

Figure 11 presents a comparison between the measured electron energy spectra from SESAME 1 (blue dots) and SESAME 2 (red dots) and the results of the simulation at 13.5° (blue curve) and 58.5° (red curve). An exponential fit of both distributions is performed to obtain electron temperatures, which are 6.39 MeV for the blue curve (SESAME 1) and 4.09 MeV for the red curve (SESAME 2). The simulated results for SESAME 1 are in pretty good agreement with the experimental data, but we overestimate the number of electrons that are off axis.

The Geant4 simulations do not take into account the refluxing of electrons. To estimate the effect of refluxing on electron transport, we use a model proposed in Ref. 57 in which the sheath field is taken to be E≈αnhεh̄/ε0, where α is a scaling factor to account for the effect that the mean field strength felt by hot electrons in vacuum is less than that at the front side of the target, n_h is the hot-electron density, and εh̄ is the mean electron kinetic energy. We assume that the maximum length on which a hot electron can reflux is the Debye length λD∝εh̄/nh. Under the electrostatic hypothesis, an electron that can escape from this electrostatic field has an energy higher than ɛ_max = eEλ_D. Moreover we also choose a value for α = 0.5 that is the same as that in Ref. 57. This rough calculation gives an escaped-electron energy of 3.5 MeV.

In Ref. 58, the effect of electron refluxing on the K_α emission of copper was investigated using the PHELIX laser, a petawatt laser with an intensity of I_L = 2 × 10¹⁹ W cm⁻², with shots on a double foam layer of Cu and CH with a varying CH thickness of 0–300 μm. The data measurements and the refluxing prediction model were found to be in close agreement with the case without refluxing for 300 μm of plastic. It should be noted that the target in the present study is of a greater thickness and density than the plastic foam used in that experiment.

A relativistic electron traveling at a velocity close to the speed of light takes ∼6.7 ps to travel 2 mm. This implies that the laser, which has a shorter pulse, cannot reaccelerate an electron that had refluxed at the rear side of the target. Moreover, as already mentioned, to propagate through 2 mm of tungsten, an energy of more than 5 MeV is required for an electron. We can therefore neglect the effect of refluxing on photon generation. Refluxing affects the total charge ejected from the target, but it is of concern mostly for the low-energy electrons.

To compare the experimental and simulation results, we reproduce the measurements by sending our simulation data on the energetic electrons and photons emerging from the converter to a Geant4 simulation of the CRACC-X detector. The detector is modeled with its HDPE shielding, lead, IPs, and filters (see Table III). The Monte Carlo simulation calculates the energy deposited by particles in each IP.

According to results from the NIST ESTAR software,56 electrons with energies greater than 5 MeV can propagate through the HDPE shielding. This implies that the energy deposited in the IPs and filters comes not only from photons but also from electrons that can pass through the shielding. These electrons can create photons through bremsstrahlung inside the filters or interact directly with the IPs. In Ref. 59, the IP response was calibrated to different types of particles such as electrons and photons, and it was found that the contribution of the electrons and photons to the luminescence of the IP are the same for both types of particle. That implies that one can sum up the contributions of the two populations. To take into account these two contributions, the photon and the electron spectra are sent through the CRACC-X detector. These two populations correspond to the blue curve in Fig. 9 and the green curve in Fig. 10.

Figure 12(a) presents a comparison between the measured energy deposited in each IP of the CRACC-X (blue dots) and that calculated using the simulation chain with the contributions of simulated electron and photon distributions (orange, green, and red markers). The green triangles correspond to the energy deposited by the photons that came from the simulation chain. This population corresponds to the blue spectrum in Fig. 9. The photon contribution alone is not sufficient to explain the experimental results for the IPs up to the ninth IP. After the ninth IP, for which the filter is made of gold, the simulated photon energy deposition alone is in agreement with the measurements. The contribution of the electrons corresponds to the red markers in Fig. 12(a). For the first IP, the electron energy deposition is close to the photon contribution. In the first IP, the total energy deposition from electrons is 8 × 10⁹ MeV sr⁻¹ and that from photons is 9 × 10⁹ MeV sr⁻¹. The signal in each IP decreases as the electrons cross over the different IPs. The orange markers in Fig. 12(a) correspond to the combined contribution of the two populations. The sum of the energies deposited by electrons and photons from the simulation chain is therefore in good agreement with the experimental data.

Figure 12(b) presents a comparison between different levels of initial total electron energy corresponding to different laser absorbed energies (see the list in Sec. IV B) and the measurements (blue dots). Four cases are compared, and the extremes are the green and purple triangles, which correspond respectively to 50% and 86% of absorbed energy. For the first IP, the green triangles are outside the error bars. This implies that in the experiment, the laser absorption is close to 50%. The two cases in red and in purple overestimate the laser absorption by the electrons. The case of ∼60% of absorption (orange diamonds) is in good agreement with the experimental results.

We investigate numerically the possibility of producing and detecting positrons on the PETAL laser facility. In this study, the positrons are assumed to be produced by the Bethe–Heitler (BH) and linear Breit–Wheeler (LBW) processes, although it should be noted that the LBW process has never been detected in the laboratory. We investigated the configuration proposed in Ref. 12 to produce LBW pairs with PETAL. In the case of laser interaction with a high-Z solid material, the number of BH pairs produced can be important.60 It is very difficult to distinguish the signal from the BH and LBW pairs. In this section, we present the results obtained using the previous simulation chain for the production of BH positrons and an estimate of the number of LBW positrons created when two γ beams are collided. This latter case requires the existence of two targets and a division of the PETAL beam in two. The PETAL laser could offer this possibility in the future.

The BH process61 occurs when a photon of high energy interacts with the electric field of a nucleus and creates an electron/positron pair. The total cross section of the BH process, which takes into account the emission of the photon by bremsstrahlung, is proportional to the quantity Y given byY=Z4d(d−x)ρ2A2≈Z2dρA2,(2)where Z is the atomic number, d is the thickness of the material linked to the emission of a photon by bremsstrahlung, d − x is the remaining thickness that the photon covers before decaying into an electron/positron pair, ρ is the density, and A is the mass number.20 The creation of electron/positron pairs by laser–matter interaction has been studied with powerful lasers like TITAN, OMEGA and NIF-ARC62 in the USA. The study presented in this article is comparable because the PETAL laser characteristics are close to those of the NIF-ARC laser.

Figure 13 presents the positron spectrum obtained with Geant4 at the rear side of 2 mm thickness of tungsten. In total, 1.06 × 10¹⁰ positrons are produced, corresponding to an energy of 6 mJ. This spectrum shows the production of relativistic positrons with energies up to 30 MeV. The positron spectrum, like the electron spectrum, has a Maxwellian distribution with a temperature of 3.6 MeV. In the PETAL experiment described previously in Sec. III, positrons were not detected with the SESAME diagnostic, but we can give an estimation of the positron number that should interact with the SESAME 1 and SESAME 2 detectors. SESAME 1 received 4.9 × 10³ positrons and SESAME 2 received 5.1 × 10³ positrons.

Simulations are performed for various target thicknesses d = 25, 50, 75, 100, 125, 150, 175, 200, 400, 600, 800, 1000, 1200, 1400, 1600, 1800, and 2000 μm. They show that the emission occurs from the majority of the tungsten thickness. We will therefore consider that d ≫ x in the expression (2) for Y. Figure 14, which is similar to Fig. 7 of Ref. 20, presents the number of produced positrons normalized by the initial total electron energy as a function of Γ = Z²dρ/A for two cases, with and without electron/positron annihilation that creates two photons of 511 keV. In this figure, the two curves can be well fitted by a power law in Γ, i.e., Γ, with n = 0.63 ± 0.08 with pair annihilation and n = 0.71 ± 0.06 without. The exponent n that we find is in agreement with that given in Ref. 20 for an electron temperature of 2 MeV. This exponent n drops from its theoretical value of 2 to 0.71 because other collisional processes take place at these energies (∼MeV) and decrease the reserve of electrons and photons available to produce pairs.

In the Geant4 simulations of BH positron production, the electromagnetic and electrostatic fields generated inside and around the solid target are not taken into account. Electron recirculation, i.e., reflection of the electrons by the sheath field at the target surfaces, is not included in these simulations. The electrons that recirculate have the possibility to produce photons each time they pass into the material. Then, if there is much refluxing of electrons, they can produce more photons and more pairs. In Ref. 17, a simulation tool is presented based on the model described in Ref. 19 that takes into account electron reflux. Using this tool, it was found that for the OMEGA EP laser characteristics, in the case of a 1 mm-thick gold target, a 100-fold higher positron yield was obtained with electron reflux than without. That calculation was performed under the hypothesis of 100% electron reflux, which maximizes the number of positrons produced. In our case, the target has twice the thickness (2 mm). With such a thickness, the effect of reflux should be less important. With the PETAL laser peak intensity and the scaling used in Ref. 17, which depends on the intensity, a ratio of 1.25 between the numbers of positrons that are created respectively with and without reflux is obtained. Multiplying the number according to our simulation by this factor gives a total of 1.33 × 10¹⁰ positrons. This is fairly close (by less than a factor 3) to the scaling given in Ref. 60, which predicts around 3.8 × 10¹⁰ positrons for an intensity of 8 × 10¹⁸ W cm⁻² and a laser energy of 400 J. After a thickness of 1400 μm, the number of positrons begins to decline. This decline is likely linked to the fact that the target becomes too thick for the energy of the positrons produced, and the collisional effects such as disintegration become the dominant processes.

The LBW process corresponds to the collision of two γ photons,10 which annihilate to create an electron/positron pair (γ₁ + γ₂ → e⁻ + e⁺). To produce a pair, the photons must overcome the center-of-mass energy threshold εγ1εγ2≥2me2c4/(1−cosΦγ1γ2), where Φγ1γ2 is the angle of collision between the two photons. From this condition, a head-on collision (Φγ1γ2=180°) is the configuration minimizing the energy threshold. Collision of two photons going in the same direction (Φγ1γ2∼0°) does not create pairs. The total cross section of the LBW process is defined by12σγγ=π2re2(1−β2)(3−β4)ln1+β1−β−2β(2−β2),where r_e is the classical electron radius and β=1−1/s, with s=εγ1εγ2(1−cosΦγ1γ2)/2me2c4. This process is the inverse of electron/positron annihilation.

The nonlinear part of this process corresponds to the annihilation of one γ photon in an intense electromagnetic field (γ + nω → e⁻ + e⁺). This intense electromagnetic field can be a laser field. There have been many theoretical studies of the nonlinear Breit–Wheeler process,63–65 as well as an experiment at SLAC where positrons were measured.66 The next generation of petawatt lasers like Apollon67 should allow new types of experiments to be performed to study this process without the need for a linear electron accelerator. The high-intensity lasers will be used to accelerate relativistic electrons and produce high-energy photons.68

Concerning the LBW, theoretical work has been done to propose experimental laser schemes to produce and detect LBW pairs.11–13,69 The LBW process is difficult to observe in the laboratory because it requires the collision of two bright γ beams. The beams need to be intense to maximize the pair production probability, but if the beam size is too small, it is possible for the two beams to miss each other.

To produce LBW pairs using the PETAL laser, the main PETAL beam would need to be split into separate beams to create two different γ beams through bremsstrahlung emission. If an angle is set between the two lasers, the two γ beams can be collided. This scheme was proposed in Ref. 12. To calculate the number of LBW pairs produced, we use the code TriLEns.70 This is a tree code using a bounding volume hierarchy to calculate collisions between photons that produce electron/positron pairs by the LBW process. It samples the momentum phase space of photons to define volumes of particles moving in the same direction. A test of overlapping of volume is performed to find photons that could collide with each other. If there is an overlap, collisions between photons are treated. Collisions are realized in the center-of-mass frame of the colliding particles. Figure 15 illustrates the principle. The angle between the two beams is set to θ = 135° (−67.5° and +67.5°), which allows the creation of positrons from −67.5° up to 67.5°. To generate pairs of two identical photons with this angle, a photon energy of at least 553 keV is needed. This implies that on the 0° axis (the vertical axis in Fig. 15), LBW positrons will be produced. With this scheme, the signal can be polluted by BH positrons generated inside the target. These positrons will propagate mainly in the direction of the two incident γ-photon beams. If a detector is put close to the vertical axis, it will see the majority of positrons from the LBW process, allowing spatial separation of the different populations of positrons.12 In the present investigation, a distance between the target and the photon collision zone L of 0.05 cm is chosen for the collision of the two γ beams (see Fig. 15). However, because of processes such as photo-ionization and BH pair production, if the tungsten target is too thick, the γ-beam intensity at the rear side of the target will be reduced. Conversely, if the target is too thin, bremsstrahlung emission will be hampered. In a similar way to what was done for BH pair production in Sec. V A, simulations are performed for different thicknesses of the tungsten target.

Figure 16 gives the number of LBW positrons produced in the collision of the two γ beams created by PETAL. The γ beams are the same as that from the previous simulations in Sec. IV C, and two similar γ beams are collided. A maximum of 348 LBW positrons are created. This number of positrons is maximized for a target thickness of 0.8 mm. If the target if too thin (<0.4 mm), the electrons do not have time to create a sufficient number of energetic photons. Once a target thickness of 0.8 mm has been exceeded, the number of LBW positrons produced saturates to a value of about 4 × 10² and then decreases. This decrease is linked to the fact that for thicker targets, the photon divergence is higher. In the case of two conical γ beams, the number of positrons can be estimated by Np≈Nγ2σγγ/2πR(1−cosθ), where R is the collision distance and θ is the divergence (FWHM) of the γ beams.12 The greater the divergence of the beams, the fewer positrons are produced.

Figures 17(a) and 17(b) present energy and angular spectra, respectively, for a 0.8 mm-thick tungsten target. Pairs are mainly produced with energies close to 100–200 keV, with a cutoff energy around 4.5 MeV. Relativistic pairs are produced, but in a limited number, with as 43 pairs of energy greater than 1 MeV being produced. The angular distribution of the pairs shows that they are mainly produced close to the γ and laser beam axes, which are at 67.5° and −67.5°, respectively. Near the angle 0°, 1.5° pairs per degree are created instead of 2.5 on the γ-beam axis. If we compare the number of LBW and BH positrons produced, many more BH than LBW positrons are produced. The ratio of the number of BH and LBW pairs is close to 10⁸. Moreover, the BH divergence is larger than the LBW one, but they peak at the same angular values of ±67.5°. This shows that it will be difficult to measure LBW positrons with this type of experiment, because they will be polluted by the BH positrons produced inside the target and by positrons produced by photon interactions with the detector. Moreover, Fig. 17 shows that fewer LBW positrons are produced at 0°. Even if we placed a detector in this direction, the positron signal that propagates toward it would be small.

This paper has reported the first experiments on X-ray/γ-photon production at the PETAL laser facility. These experiments were carried out as part of the commissioning of the PETAL laser.

Two experiments were performed on 2 mm-thick tungsten targets with similar laser intensities (7.5 × 10¹⁸ and 5.5 × 10¹⁸ W cm⁻²). Using the SESAME spectrometers, exponential distributions of high-energy electrons were measured with high temperatures close to 6 MeV at 0° and 2 MeV at 45°. In both cases, high-energy electrons (up to ≈3 MeV) were recorded. Exponential distributions of protons were also obtained. These were produced by TNSA at the rear side of the target. However, this phenomenon was not really efficient, because of the target thickness. The cutoff energy of these proton distributions was ≈1 MeV.

X-ray/γ photons were quantified with two diagnostics: a spectrometer using quartz and LiF crystals and a bremsstrahlung cannon. K-shell lines were recorded for both shots with more than 10¹¹ photons.sr⁻¹. However, L-shell lines were only recorded for shot 182. These lines can be used to obtain information on ionization and to infer the hot-electron temperature. Bremsstrahlung cannon analysis is strongly dependent on the spectral shape hypothesis regarding the initial photon or electron distribution. Comparisons with simulations benefits the analysis and makes it possible eliminate the hypothesis on the distribution shape.

To analyze these data, we used a simulation chain composed of hydrodynamic, PIC, and Monte Carlo simulations. This stacking of simulations allowed us to model the experiment.

Monte Carlo simulations indicate that high-energy electrons escape the solid tungsten target. These electrons can interact with the CRACC-X diagnostic. Indeed, from the simulations ≈2×108 electrons interact with this diagnostic. These electrons cover a spectrum from 1 to 20 MeV. Calculations with NIST software predict that electrons with more than 5.5 MeV of energy can propagate inside the detector. The simulations predict an electron temperature of 6.39 MeV for SESAME 1 and 4.09 MeV for SESAME 2. The diagnostics show that our simulation overestimates off-axis electrons in comparison with experimental data, but results close to the laser axis are in agreement.

The simulated photon K-shell lines are in agreement with the experimental data from the SPECTIX diagnostics. The Geant4 simulations used to generate these photons assume a cold solid tungsten target. This implies that the emission occurs mostly for cold tungsten nuclei inside the solid target. We managed to produce a high-energy photon population with a total energy of 2.5 J and photon energies that reach 30 MeV with a large emission angle of the order of 88°. The calculation of the photon dose gives results comparable to those of experiments at the NIF-ARC facility.

Injection of the simulated electron and photon distributions into a Geant4 simulation of the CRACC-X detector demonstrates that the relativistic electrons that escape the target will interact with the detector and perturb the photon measurements. The contribution of the two populations are in excellent agreement with the experimental data. In future experiments, the contributions of these electrons should be taken into account to provide a better reconstruction of the photon distribution.

Simulations of the BH and LBW pair production processes were performed. These positrons come from the annihilation of photons with the electromagnetic field of the nucleus or in collisions between two photons. The estimate of the number of positrons created by the BH process is promising, because ∼10¹⁰ positrons are predicted on PETAL. Future experiments on PETAL to measure BH positrons could be designed with adapted detectors and by maximizing the pair production through increasing the electron temperature or the thickness of the target. The possibility of producing pairs through the LBW process on the basis of a previously published scheme has been investigated. This scheme requires two different high-intensity laser pulses. In this configuration, ∼10² LBW pairs are estimated. This process has never been observed experimentally, and it will be difficult to design experiments to measure pairs produced by the LBW process on PETAL, because the number of LBW positrons will be polluted by BH positrons.

The PETAL laser is capable of producing high-energy directional photons through interactions with a thick target. In a previous article on the PETAL laser, it was demonstrated that it can also produce high energy protons.32 These capabilities of the PETAL laser are crucial to enable radiography of LMJ fusion experiments.

Acknowledgment. The authors acknowledge support by GENCI France through awarding us access to HPC resources at TGCC/CCRT (Grant Nos. A0110512943 and A0130512943). The PETAL laser was designed and constructed by 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 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 CRACC-X diagnostic was realized within the PetaPhys project funded by the LabEx LAPHIA of the University of Bordeaux under Grant No. ANR-10-IDEX-03-02. We acknowledge financial support from the IdEx University of Bordeaux/Grand Research Program “GPR LIGHT.”