3.1. Preplasma density profile

We use Multi1D to simulate laser ASE interaction with Pb target. The laser wavelength is 1 µm, and we assume the laser contrast ratio is 10⁻⁹ ASE intensity is 5 × 10¹⁴ W/cm², which is a typical value in ultrashort and ultraintense laser facility. The ASE intensity grows linearly from t = 0 to 0.1 ns and keeps constant for 1.2 ns. The density and thickness of Pb target is 11.34 g/cc and 200 µm, respectively. The initial temperature of electron and ion is 0.03 eV (300 K). In our simulation, the laser lost 50% of its energy around the critical density surface and electron heat flux limit factor 0.05 was used.

Figure 2 is the electron density profile of fully ionized Pb plasma at 1.2 ns, it shows that density profile increases rapidly from 1 n _c to 40 n _c and then increases slowly. Around critical density surface, a part of laser energy is transferred to high-energy electrons by resonance absorption and inverse bremsstrahlung absorption. A shock is formed due to the high-energy electrons ablating solid target. The plasma between the shock and the critical density surface expands along the inverse laser propagation direction due to high-energy electron heating. When laser light pressure is high enough, the plasma expansion would be restrained; hence, a plasma density jump around the critical density surface is formed (Attwood et al., Reference Attwood, Sweeney, Auerbach and Lee1978).

Fig. 2.

Electron density profile of fully ionization Pb target at 1.2 ns.

3.2. Photon and electron generation

We get the velocity and energy of gamma photons and hot electrons with 3D PIC code EPOCH. The laser peak intensity is 5 × 10²³ W/cm², corresponding to a normalized laser vector potential a ₀ = 602 and the laser electric field intensity profile is a = a ₀ sin²(πt/2τ)exp(−(r/r ₀)²), where τ = 30 fs is the duration and r ₀ = 2λ₀ (laser wavelength λ₀ = 1 μm) is the spot size. It is polarized along the Y-direction and propagates along the X–direction, and the total laser energy is about 706 J. The simulation box is X × Y × Z = 30 μm × 16 μm × 16 μm, with a cell size of dX × dY × dZ = 0.0 6 μm × 0.064 μm × 0.064 μm. The fully ionized Pb plasma with electron density 40 n _c is initially located from x = 5 to 30 µm. The initial temperature of electron and Pb ion is 1 keV. The macroparticle number of electron and ion per cell is 20 and 10, respectively. The field- and particle boundary conditions at the left hand of the X-direction are absorptive, and the particle boundary in the Y and Z-directions and the right-hand of the X-direction are thermal, which means that the particles are re-injected into the box with a thermal velocity at the boundaries.

In this simulation, the motion of gamma photons is suspended right after generated. We get information of gamma photons that propagate along the X-positive direction, from the simulation box at simulation time 160 fs. The hot electrons come from two parts in this simulation. The first part is the electrons stay at the simulation box at 160 fs; the second part is the electrons pass through the diagnostic surfaces from t = 0 to 160 fs. The diagnostic surface is located at the left, right, up, down, front boundary of the simulation box along the X-positive direction, respectively. The electron will be recorded when it passes through the diagnostic surface from the box inner region to the boundaries.

Gamma photons and hot electrons with their energy beyond the positron generation threshold 1.022 MeV and propagate forward are selected. Information of these gamma photons and hot electrons generated from t = 0 to t = 160 fs are collected to simulate positron generation (in Fig. 3). We selected 160 fs because it is the time that laser energy is almost completely exhausted. To estimate the influence of QED effect, we carried out simulations both with and without considering QED effects. In the case of including QED effect, the spectra of gamma photon and hot electron show that the particles with energy up to several hundred MeV are generated (in Fig. 3a). The energy conversion efficiency from laser to gamma photons is 42% [This conversion efficiency is consistent with the result reported by Ji et al. (Reference Ji, Pukhov, Nerush, Kostyukov, Shen and Akli2014b ) and Brady et al. (Reference Brady, Ridgers, Arber and Bell2014)] and is 20% for hot electrons. The angular distribution shows that hot electrons are approximately isotropically emitted, while gamma photons are emitted along the forward direction with a half-divergence angle of 0.7 rad (40°) (Fig. 3b). In comparison, without QED effect (the black line in Fig. 3), the number of the high-energy electrons is overestimated, which would lead to overestimate of positron number and cutoff energy. Thus, it is necessary to consider QED effect in PIC simulations of the interaction of 10 PW laser with plasmas. In addition, the energy conversion efficiency from laser to hot electrons is 63%, which is close to the energy conversion efficiency from laser to gamma photons and hot electrons in the QED effects included case. The angular distribution of hot electrons in this case is also approximately isotropic (in Fig. 3b).

Fig. 3.

Spectra and angular distribution of photons and electrons. (a) Spectra; (b) Angular distribution.

In the simulation with QED effect, the electrons acquire energy from laser field by the J × B acceleration mechanism (Wilks et al., Reference Wilks, Kruer, Tabak and Langdon1992) because of the momentum profile of electrons in the X-direction includes periodic structure with λ/2 as shown in Figure 4d. The synchrotron emission of gamma photons will be generated when the high-energy electrons are oscillating in the laser field (in Fig. 4c) (Brady et al., Reference Brady, Ridgers, Arber and Bell2014), and this emission of photons exert a radiation reaction force on the oscillating electrons. The oscillating electrons will be trapped in the laser field when the radiation reaction force exceeds laser ponderomotive force (Ji et al., Reference Ji, Pukhov, Kostyukov, Shen and Akli2014a ), and continuously transfer laser energy to gamma photons. The radiation reaction effect significantly reduces the electrons oscillation amplitude; this can be found by comparing the electron densities in Figures 4a and 4b; therefore reduces the number of high-energy electrons and photons shown as in Figure 3a.

Fig. 4.

Electron densities in the X–Y plane at t = 100 fs with QED effect (a) and w/o QED effect (b), respectively. (c) Gamma photon density profile at 100 fs. (d) Phase space distribution of electrons in frame (a). The unit for density is the critical density n _c.

Compared with the interaction of laser with lower intensity 10¹⁸ − 10²¹ W/cm² with plasmas, there are two advantages for positron production in the case of higher laser intensity around 10²³ W/cm². (i) Higher photon and electron energy: this improves the positrons yields because the positron generation cross-section increases with the energy of photon and electron. (ii) Higher laser energy conversion efficiency: when lower intensity laser interacts with the preplasma, hot electrons (gamma photons are rarefied and ignored) are generated by collisionless absorption mechanisms, such as resonance absorption (Kruer, Reference Kruer1988), relativistic J × B heating (Wilks et al., Reference Wilks, Kruer, Tabak and Langdon1992) and direct laser acceleration (Pukhov et al., Reference Pukhov, Sheng and Meyer-ter-Vehn1999). In these mechanisms, a significant part of laser energy is reflected near the critical density surface. This leads to an energy conversion efficiency to forward hot electrons of about 30% in general (Gibbon, Reference Gibbon2005). In the case of higher intensity laser, the laser energy is consumed by the plasma rapidly via accelerating electrons and emitting photons and almost have no laser reflection. The laser energy conversion efficiency to forward electrons and photons can reach 60%, much higher than the case of lower laser intensity obviously.

3.3. Positron characters from Pb converter

The positron production is simulated with Monte Carlo code Fluka (Battistoni et al., Reference Battistoni, Muraro, Sala, Cerutti, Ferrari, Roesler, Fasso and Ranft2007), gamma photons and hot electrons impinging a columned Pb target with radius 1 cm, whose thickness ranges from 0.05 to 1.5 cm. Positrons are recorded by the detector on the rear surface of the Pb target (in Fig. 1). Figure 5 shows the characteristics of positrons generated by interacting ultraintense laser with intensity of 5 × 10²³ W/cm² with different Pb targets. These characteristics will be used when we estimate the positron yield of relativistic collisionless shock formation in Section 3.5.

Fig. 5.

Comparison of positrons generated directly by photons and electrons in the Pb target. (a) Positrons yield versus Pb target thickness. (b) Positron energy spectrum. (c) Positron angular distribution.

Positron yield: We obtain the maximum positrons yield of 1.38 × 10¹³/kJ at the optimal Pb target thickness of 6 mm (in Fig. 5a). This positron yield is one to two orders higher than that in the works using lasers with intensity below 10²¹ W/cm² (Myatt et al., Reference Myatt, Delettrez, Maximov, Meyerhofer, Short, Stoeckl and Storm2009; Chen et al., Reference Chen, Fuiza, Link, Hazi, Hill, Hoarty, James, Kerr, Meyerhofer, Myatt, Sentoku and Williams2015a ). It is important to note that we do not consider the effect of hot electron refluxing that is considered by Chen et al. (Reference Chen, Fuiza, Link, Hazi, Hill, Hoarty, James, Kerr, Meyerhofer, Myatt, Sentoku and Williams2015a ), and can further improve positron yield.

Optimal thickness: The optimal target thickness is determined by the radiation length (L _rad) of the material, which can be seen as the average distance needed to perform one step of the cascade (either photon creation by bremsstrahlung or generation of an electron–positron pair). The radiation length is approximately given by $L_{{\rm rad}} = 1/[4{\rm \alpha} (Z{\rm \alpha} )^2 n{\rm \lambda} _{\rm C}^2 L_0 ]$ , where n is the number of atoms per unit volume, α ≈ 1/137 is the fine structure constant, λ_C = 3.9 × 10⁻¹¹ cm is the Compton wavelength and L ₀ = log(183Z ^−1/3) − f(Zα), with $f(x) = \sum\nolimits_{a = 1}^\infty x^2 /a(a^2 + x^2 )$ . Put corresponding parameters in, we can obtain the radiation lengths of Pb is 5.6 mm (Sarri et al., Reference Sarri, Poder, Cole, Schumaker, Di Piazza, Reville, Dzelzainis, Doria, Gizzi, Grittani, Kar, Keitel, Krushelnick, Kuschel, Mangles, Najmudin, Shukla, Silva, Symes, Thomas, Vargas, Vieira and Zepf2015b ), which is consistent to the optimal target thickness.

Positron Temperature and Divergence: The positrons temperature is 78 MeV obtained from positron spectra as shown in Fig. 5b. Since the positrons are emitted along the forward direction with an large half angle of 0.7 rad (40°), the pair plasma density declines rapidly as the jet propagate away from the target, which is unpropitious for the formation of relativistic collisionless shock. Certainly, there are methods to collimate electron–positron jets, such as intense magnetic field along with target normal direction suggested by Chen et al. (Reference Chen, Link, Sentoku, Audebert, Fuiza, Hazi, Heeter, Hill, Hobbs, Kemp, Kemp, Kerr, Meyerhofer, Myatt, Nagel, Park, Tommasini and Williams2015b ), or the concave rear surface target suggested by Wilks et al. (Reference Wilks, Langdon, Cowan, Roth, Singh, Hatchett, Key, Pennington, MacKinnon and Snavely2001) to collimate proton beams, which can also collimate positron beams.

3.4. Positron yield and laser intensity

In a real ultraintense laser plasma experiment, the intensity of 10 PW laser facility might be actually below 5 × 10²³ W/cm². Thus, it is necessary to study the dependence of positron yield on laser intensity from 5 × 10²²  to 5 × 10²³ W/cm². This dependence is given in Figure 6a. It can be found in Figure 6a that the positron yield per kJ through gamma photons increases with the laser intensity and that through electrons decreases with laser intensity. However, the total positron yield per kJ only has a weak dependence on laser intensity in this range.

Fig. 6.

Dependence of positron yield and preplasma density on laser intensity.

There are two main reasons for the weak dependence: (i) the correspondingly relativistic plasma critical density γn _c $({\rm \gamma} = \sqrt {1 + a_0^2 /2} )$ is ranging from 134 n _c to 426 n _c when the laser intensity changes from 5 × 10²²  to 5 × 10²³ W/cm², which is far beyond the preplasma density, which correspondingly changes from 10 n _c to 40 n _c calculated by Multi1D code and shown in Figure 6b. Thus, the laser pulse can penetrate the preplasma with almost no reflection due to relativistic transparency effect and generate hot electrons, gamma photons, and energetic ions. The PIC numerical simulations show that the forward hot electrons and gamma photons with energy beyond 1.022 MeV carry about 60% of laser energy, and this number is approximately independent of the laser intensity. (ii) The variety of positron production cross-section is slow when the energy of hot electrons and gamma photons reaches several tens MeV. These two factors lead to total positron yield per kJ have little difference with various laser intensity.

3.5. Relativistic collisionless shock formation

To generate relativistic collisionless shocks using two positron–electron counter-stream plasmas, the duration (τ) and radius (R) of pair plasmas being larger than temporal and spatial scale of the Weibel instability must be guaranteed (Chen et al., Reference Chen, Fuiza, Link, Hazi, Hill, Hoarty, James, Kerr, Meyerhofer, Myatt, Sentoku and Williams2015a ). Namely, τ > 1/Γ and R > c/ω_p, where $\Gamma = \sqrt 2 (v_0 /c){\rm \omega} _{\rm p} $ is the growth rate of Weibel instability for two counter-stream symmetric pair plasmas, where v ₀, γ ₀, and n ₀ are the velocity, Lorentz factor, and density of pair plasmas, respectively.

Firstly, we estimate the positron yield to see whether relativistic collisionless shock formation is possible in the interaction of a laser pulse of 5 × 10²² W/cm² with 6 mm thickness Pb target by the simulation case. The hot electrons, which produce positrons penetrate the Pb target with velocities ranging from 0.943c to ~c (the corresponding electrons energy exceed 1.022 MeV). The duration of hot electron beam will reach ~1 ps at the position of radiation length (6 mm) in Pb target. The majority of positrons are generated at the position of radiation length; thus, the duration of positron beam is ~1 ps, in addition, v ₀ ~ c and γ ₀ ~100. Using these parameters and the temporal and spatial conditions of relativistic collisionless shock formation, one can find that the density and radius of positron beams must satisfy n ₀ > 1.5 × 10¹⁶/cc and R > 0.4 mm. The positron yield required for relativistic collisionless shock formation is N = πR ² cτn ₀ > 2.2 × 10¹². In our simulations, the positron yield 10¹³/kJ can be achieved with 10 PW laser. Then a 10 PW laser pulse, which usually contains energy up to several hundred or even a thousand Joule can produce more than 2.2 × 10¹² positrons. This indicates that the positrons yield satisfies the formation condition of relativistic collisionless shock.

Secondly, we estimate the positron density to see if relativistic collisionless shock formation is possible. The positron beam radius R is ~4 mm when beam transport radiation length (6 mm) with half-angle of 0.7 rad. We assume that the positron number is 10¹³, the positron density ~7 × 10¹⁴/cc near the target rear surface is then obtained. This positron beam density is too lower to generate relativistic collisionless shock (1.5 × 10¹⁶/cc). However, we could use some methods, such as additional intense magnetic field and special structure target, to reduce the positron beam divergence to ~0.1 rad, and then to satisfy the formation condition of relativistic collisionless shock.