Abstract
1 Introduction
Accelerated ion beams have a multitude of applications ranging from nuclear reactions induced by energetic heavy ions[1] to fast ignition fusion[2,3], aiding neutron production[4] and also hadrontherapy for cancer treatment[5–7]. Laser-driven ion acceleration has received much attention in recent decades, as it offers the possibility of having alternate accelerators that are smaller and more affordable as opposed to the conventional linacs, cyclotrons and synchrotron[8,9]. Experimental demonstration of ion beams by several mechanisms exhibiting different performances, such as target normal sheath acceleration (TNSA)[10], radiation-pressure acceleration (RPA)[11–13], collisionless shock acceleration (CSA)[14,15], the breakout afterburner (BOA)[16–19], etc., has already been achieved[20]. Significant efforts of innovative laser/target configurations have also been made to push the number of ion beam characteristics (energies and flux)[21], yet the highest gained energy is still less than 100 MeV/u[20,22,23]. Nevertheless, the prospects of achieving even higher ion energies with the next generation of laser sources are promising[24].
The BOA is one of the high-performance laser-driven ion-acceleration mechanisms capable of accelerating ions to relatively higher values even with state-of-the-art lasers. In this, an initially opaque, ultra-thin target (width around laser skin depth) turns transparent to the incoming laser pulse, due to lowering of the density by the expanding plasma and increase in critical density by the electron’s relativistic motion (relativistically induced transparency, RIT)[16,25]. This leads to a phase of extreme ion acceleration (BOA phase), which continues to exist until the electron density of the expanding target becomes classically underdense[26]. Buneman instability (in single ion-species targets) and ion–ion acoustic instability (in the case of multispecies targets[27]) result in an electrostatic mode structure, which is found to be instrumental in transferring the laser energy to ions via laser-induced electronic drifts[17,28]. The efficiency of this mechanism is maximized when the peak of the laser pulse arrives precisely at the onset of relativistic transparency[18,29], as opposed to the RPA-light-sail mechanism, which requires opacity in ultra-thin targets.
Experimental demonstration of fully ionized carbon ion acceleration via the BOA mechanism up to 40–50 MeV/u has been achieved using approximately 50–250 nm thick targets with the TRIDENT laser and the Texas Petawatt laser facility[18]. Also, simultaneously existing TNSA and BOA signatures in proton spectra (energy
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However, in the ultra-relativistic regime other quantum electrodynamic dynamic (QED) effects become non-negligible when the electric field of the laser in the electron’s rest frame gets closer to the critical Schwinger field (
In this paper, the impact of both RR and non-linear BW pair production (PP,
2 Simulations
We performed 2D PIC simulations using both the open-source codes EPOCH (in Sections 3 and 4) and SMILEI (in Section 3.1 and Appendix B), which include quantum RR and PP by the probabilistic Monte Carlo method[50,51]. We employ a linearly s-polarized laser pulse, impinging on the left-hand boundary with a finite spatio-temporal profile
Laser–solid pair creation by QED processes mediated in coulombic fields, such as Bethe–Heitler (BH)[54] and trident (tri) processes[55,56] is not considered in these simulations. This should be reasonable as the ratio of the electric field strength of the laser to that of the atomic nucleus at ionic Debye length is
3 Dynamics
The laser-field pushes hot electrons inside the target forward that quickly reach the non-irradiated side (rear) of the target faster than the ions. This sets up a very brief TNSA field there that kickstarts the ion acceleration from the TNSA mechanism at around 50 fs. The electrons oscillate with relativistic velocity and, thus, the effective critical density is reduced.
Moreover, as the recirculating hot electrons heat the target up, it begins to expand and the density lowers further. The target then begins to become relativistically transparent and the laser is able to penetrate through it. This marks the onset of transparency (at
In Figure 1, we plot the maximum energy gained by ions (
Figure 1.The 1D plot shows the maximum energy gained by ions (in GeV) with time in all three cases labelled. Here, a region in time is identified as , which starts at the onset of transparency and extends until the enhanced ion acceleration slows down (after which the slope of maximum ion energy begins to change to a smaller value).
Stage 1 is pre-transparency time (up to 60 fs) when the target is still intact and ion acceleration occurs with the combination of TNSA and RPA very briefly. This is determined from simulations when at least some focal part of the target becomes completely transparent to the laser and it passes through.
After around 60 fs, ion acceleration enters Stage 2, which we refer to as the BOA phase (marked as
Figure 2.These subplots show 2D spatial distributions of electrons (top row, a(i)–d(i)) and ions (bottom row, a(ii)–d(ii)) in the region (only no-QED case shown).
Figure 3.Energy-angular distribution of electrons (in a.u.) in the BOA phase without radiation reaction (a), with radiation reaction (b) and with pair production as well (c) (excluding the produced Breit–Wheeler electron density) at 80 fs.
Figure 4.The electron phase space in the no-QED case (a), the RR modelled by the corrected Landau–Lifschitz (LL) method (b) and the RR modelled by the Monte Carlo method (c) at the onset of the BOA phase.
Figure 5.Energy-angle distribution of photons (a), BW electrons (b) and BW positrons (c) at 80 fs.
Figure 6.Energy-angle distribution of carbon ions in the BOA phase without the radiation reaction (a), with the radiation reaction (b) and with pair production as well (c) at 80 fs.
Figure 7.Spectral power as a function of wave number (normalized by Debye’s length with initial temperature) and frequency (normalized by plasma frequency), , in log scale for fs and μm for all three cases (no-QED (a), RR (b) and RR+PP (c)) obtained from the simulations. The real and imaginary roots of Equation (1) (solid and dotted lines, respectively) are over-plotted to facilitate comparison.
Figure 8.Energy-angle distribution of carbon ions in the BOA phase without the radiation reaction (a), with the radiation reaction (b) and with pair production as well (c) at 130 fs. (d) The angle-averaged ion energy distribution at the same time.
Figure 9.Photons and pairs saturate after which the direct impact of QED effects can be assumed to be less significant. This justifies the dropping of the RR term in the Lorentz force from the Vlasov equation. QED effects are still captured in form of changes in plasma distribution.
Afterwards, ion acceleration enters into Stage 3, where electrons are significantly expelled and acceleration occurs due to Coulomb explosion, as also seen in Ref. [60]. In this paper, we focus on Stage 2 of ion acceleration as this is not only the stage of rapid energy gain dominating the overall accelerating mechanisms, but also the stage where QED effects reverse their energy-reduction trend from its preceding stage.
4 Early stage dynamics
4.1 Electrons
Figure 3 shows the electron’s energy-angle distribution at 80 fs (BOA phase) where the laser pulse has already penetrated the target (injecting electrons into vacuum laser acceleration by relativistic transparency[61]). Figure 3(a) shows the case where the QED effects are artificially turned off, Figure 3(b) shows the case when RR is included in the plasma dynamics and Figure 3(c) shows the case when both RR and PP are included. One can clearly see in Figure 3(a) that electrons stream diffusely at an angle and gain energy. The majority of the electrons stream in the forward direction (laser-propagation direction) and a small percentage of electrons also gain energy at the back (
4.2 Stochasticity in the RR case
In order to isolate the stochastic aspect of the RR from only the continuous frictional drag on particles, we carried out one simulation that models the RR with a corrected Landau–Lifschitz model that excludes the stochastic nature of photon emission (using SMILEI code). Figure 4 shows the electron’s momentum phase-space distribution in the no-QED case (Figure 4(a)), with the RR modelled by the corrected Landau–Lifschitz method (Figure 4(b)) and the RR modelled by the Monte Carlo method (Figure 4(c)).
Comparing Figures 4(b) and 4(c), we see that a significant reduction in the electron’s transverse momentum with the RR is common in both. Figure 4(c), which also captures stochastic effects of the RR, seems to extend the electron’s momentum in both the longitudinal and transverse directions. This is actually consistent with Ref. [63], which shows that stochasticity leads to a greater spread of the electron energy distribution. Clearly in this scenario, the collimation of electrons due to the leading term of the Landau–Lifschitz RR force (‘drift term’) dominates over the spreading out of electrons due to the stochastic (‘diffusion term’) effects, such that, compared to the no-QED case, there is an overall collimation of the beam. The subsequent ion energies due to stochastic effects are discussed in Appendix D.
4.3 Additional pair plasma
In Figure 3(c), when the RR+PP both are included, apart from a more collimated stream of electron fluid, here one can also see a higher density of electrons that also gain larger energy (around 0.6 GeV). This is due to the production of the BW pairs that occurs due to the interaction of laser photons with the emitted gamma-ray photons. One can clearly see that the created pairs have higher maximum energy than the target electrons. The angularly streaming target electrons gain more energy from the newly formed energetic pair plasma at
Figures 5(a)–5(c) show the energy-angle distribution of photons, BW electrons and BW positrons, respectively, in the RR+PP case at 80 fs. One can clearly see a large number of gamma-ray photons in the laser-propagation direction being produced in Figure 5(a) as the target turns transparent and the laser is allowed to interact with prolific electrons. In Figures 5(b) and 5(c), we see the high-energy and forward-streaming pair plasma that is responsible for the higher energy and density of electrons in Figure 3(c). Since the target is already transparent, these pairs do not accumulate at the target region and are unable to shield the incoming laser as in the cushioning scenario[64]; rather, they stream forward with the laser pulse and the ambient plasma.
4.4 Ions
Figure 6 shows the ion distribution in the same fashion as in Figure 3 and at the same time. In the no-QED case in Figure 6(a), the ions with the highest energy (around
In Figure 6(b), the highest gain in energy and the angular divergence of these high-energy ions are reduced at 80 fs when the BOA mechanism could be at play. The on-axis and off-axis ions gain nearly the same energies in this case. Further, in Figure 6(c), the angular divergence of the ions is even smaller, and the on-axis ions gain much higher energy (
It should be noted that this scenario could be similar to that of directed coulomb explosion[60] where RPA precedes the later coulomb explosion stage for the acceleration of ions. Although here a higher transparency with higher
5 Transparency stage
5.1 Identifying the RBI from simulation
The Fourier analysis of the longitudinal electric field from the simulations can shed light on the electrostatic structure of the accelerating fields in the transparency region. This has been performed for all three cases and is shown in Figure 7. Figures 7(a)–7(c) show
This BOA window is well within the resolution of the Fourier window shown in Figure 7. In this power spectrum in Figure 7, two distinct low-frequency branches can be clearly identified in all three panels (Figures 7(a)–7(c)). Clearly, one primary branch (labelled
The lower, diffuse and less-powerful branch
5.2 RBI from linear theory
The dispersion relation of the RBI[17] from the linear kinetic theory assuming relativistic cold angularly streaming plasma for the instability is given as follows:
where
It should be noted that we use the same dispersion relation for QED cases (Figures 7(b) and 7(c)) as well. This simplified treatment is still reasonable because we carefully choose the plasma parameters at the time after the production of photons and pairs has mostly saturated (see also Section 3). The major impact of the RR and PP is still well captured in the form of changes in the plasma distribution function extracted from the simulation that already includes probabilistic photon emission in plasma evolution. There are two real and two complex roots of this equation, including one high-frequency real root (starts with a positive frequency, as also in Refs. [17, 67]) and the other low-frequency real root (the negative frequency at
A good match between branch
Thus, from the spectral plots it is clear that the RR and RR+PP would enhance the RBI on account of a radiatively cooled more forward-directed electron and ion beam. Interestingly, the low-frequency real root of the same dispersion relation, which has a negative frequency for
5.3 RBI phase velocity and resonant ion velocities
Figure 8 shows the angle-energy and the
6 Conclusion
In conclusion, we investigated the effect of the RR and PP on the ion acceleration where the BOA mechanism may operate. We demonstrate how QED effects can impact the collective plasma behaviour in the early stages of laser–plasma interaction. This may lead to an enhanced phase velocity of the RBI in a later BOA stage. However, the spectra presented here could also be explained by an RPA mechanism by taking into account transverse expansion of the target[68], and a more systematic study of the ion electron and ion phase space at a smaller time scale to search for signatures of the RBI could further clarify the nature of the accelerating mechanism. Nonetheless, non-classical effects clearly modify the plasma distribution significantly in this regime and can lead to a gain of higher energy (around
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