Introduction
Ion acceleration using multi-petawatt (PW) laser system has been of interest in recent times (Tajima and Dawson, Reference Tajima and Dawson1979; Esarey et al., Reference Esarey, Schroeder and Leemans2009; Blaga et al., Reference Blaga, Xu, DiChiara, Sistrunk, Zhang, Agostini, Miller, Mauro and Lin2011; Qiao et al., Reference Qiao, Kar, Geissler, Gibbon, Zepf and Borghesi2012; Bulanov et al., Reference Bulanov, Wilkens, Esirkepov, Korn, Kraft, Kraft, Molls and Khoroshkov2014; Gonzalez-Izquierdo et al., Reference Gonzalez-Izquierdo, King, Gray, Wilson, Dance, Powell, Maclellan, McCreadie, Butler, Hawkes, Green, Murphy, Stockhausen, Carroll, Booth, Scoot, Borghesi, Neely and McKenna2016; Steinke et al., Reference Steinke, van Tilborg, Benedetti, Geddes, Schroeder, Daniels, Swanson, Gonsalves, Nakamura, Matlis, Shaw, Esarey and Leemans2016). One of the main drivers behind these studies is the use of energetic ions for biomedical purposes (Schardt et al., Reference Schardt, Elsasser and Schulz-Ertner2010; Ohno, Reference Ohno2013). It includes the potential treatment of tumors by carbon and proton radiotherapy. In laser-based ion acceleration, strong charge-separation electric fields cause the ions to be accelerated to high energies over very short distances. When a PW laser strikes a solid target, copious numbers of hot (>MeV) electrons are launched into the target. As these electrons leave the rear side of the target, a strong charge-separation electric field is set up which scales as E ~ T e/λD, where T e and λD are the hot electrons temperature and the corresponding Debye length. The field may reach ~100 TV/m and accelerates ions at the rear surface (Beg et al., Reference Beg, Bell, Dangor, Danson, Fews, Glinsky, Hammel, Lee, Norreys and Tatarakis1997; Wilks et al., Reference Wilks, Langdon, Cowan, Rooth, Singh, Hatchett, Key, Pennington, MacKinnon and Snavely2001; Daido et al., Reference Daido, Nishiuchi and Pirozhkov2012; Macchi et al., Reference Macchi, Borghesi and Passoni2013; Culfa et al., Reference Culfa, Tallents, Wagenaars, Ridgers, Dance, Rossall, Gray, McKenna, Brown, James, Hoarty, Booth, Robinson, Lancaster, Pikuz, Faenov, Kampfer, Schulze, Uschmann and Woolsey2014, Reference Culfa, Tallents, Korkmaz, Rossall, Wagenaars, Ridgers, Murphy, Booth, Carroll, Wilson, Lancaster and Woolsey2017). This process is called the target normal sheath acceleration (TNSA) (Snavely et al., Reference Snavely, Key, Hatchett, Cowan, Roth, Phillips, Stoyer, Henry, Sangster, Singh, Wilks, MacKinnon, Offenberger, Pennington, Yasuike, Langdon, Lasinski, Johnson, Perry and Campbell2000; Passoni et al., Reference Passoni, Bertagna and Zan2010; Macchi et al., Reference Macchi, Borghesi and Passoni2013).
The laser pulse has momentum which it can deliver to a target. The ion acceleration based on laser beam's pressure is called radiation pressure acceleration (RPA) (Robinson et al., Reference Robinson, Zepf, Kar, Evans and Bellei2008; Henig et al., Reference Henig, Kiefer, Markey, Gautier, Flippo, Letzring, Johnson, Shimada, Yin, Albright, Bowers, Fernández, Rykovanov, Wu, Zepf, Jung, Liechtenstein, Schreiber, Habs and Hegelich2009; Sorbo et al., Reference Sorbo, Blackman, Capdessus, Small, Slade-Lowther, Luo, Duff, Robinson, McKenna, Sheng, Pasley and Ridgers2018). The corresponding radiation pressure may go up to 2I/c. For a 10 PW laser, the radiation pressure is ~3 × 1013 atm, where I is the laser intensity and c is the speed of light. In this mechanism, the electrons are pushed inwards in an over-dense target. This motion of electrons leaves a charge separation behind and creates an electrostatic field which in turn acts on the background ions and accelerates them. Further details of ion acceleration by RPA can be found elsewhere (Robinson et al., Reference Robinson, Zepf, Kar, Evans and Bellei2008; Henig et al., Reference Henig, Kiefer, Markey, Gautier, Flippo, Letzring, Johnson, Shimada, Yin, Albright, Bowers, Fernández, Rykovanov, Wu, Zepf, Jung, Liechtenstein, Schreiber, Habs and Hegelich2009; Sorbo et al., Reference Sorbo, Blackman, Capdessus, Small, Slade-Lowther, Luo, Duff, Robinson, McKenna, Sheng, Pasley and Ridgers2018; Macchi et al., Reference Macchi, Borghesi and Passoni2013). In RPA, the ion energy scales with I, whereas in TNSA, it scales with I 1/2 which shows that RPA acceleration mechanism is favorable at higher intensities.
It is found that ion acceleration beyond the RPA and TNSA energy scaling is possible. The mechanism involved is called breakout afterburner (BOA) (Yin et al., Reference Yin, Albright, Hegelich, Bowers, Flippo, Kwan and Fernandez2007, Reference Yin, Albright, Bowers, Jung, Fernandez and Hegelic2011; Jung et al., Reference Jung, Yin, Gautier, Wu, Letzring, Dromey, Shah, Palaniyappan, Shimada, Johnson, Schreiber, Habs, Fernandez, Hegelich and Albright2013). Acceleration by BOA takes place when the target goes transparent after its density has decompressed sufficiently for it to become under-dense. In fact, in the case of PW laser pulses, the density need only decompresses to a 0 times the critical density due to the relativistic mass increase of the electrons. Ion acceleration in the transition to relativistic transparency typically takes place via several phases. Initially, the ion acceleration is due to TNSA as the laser-heated electrons traverse to the rear side. This leads to modest ion energies. This is then followed by the enhanced TNSA phase. In this phase, the laser field heats the background electrons to sufficiently high temperature which decreases the plasma frequency ωp due to the electron's relativistic mass increase. This heating also leads to target expansion and consequent reduction in the electron density, further decreasing ωp. This leads to the penetration of the laser field in the bulk of the target and initiates the BOA phase. The target electrons are then volumetrically heated, boosting the longitudinal electric field. In the BOA phase, non-linear processes via the growth of electromagnetic instabilities enhance the energy coupling into the ions (Yin et al., Reference Yin, Albright, Bowers, Jung, Fernandez and Hegelic2011). Relativistically induced transparency (RIT) is critical for BOA to occur. RIT is observed when 
$a_0 \gg \omega ^2_{\rm p}\ell /2c{\rm \omega_L}$, where ωL is the laser frequency, ℓ is the target thickness, and a 0 is the normalized laser amplitude. Previous experimental studies have shown the acceleration of carbon ions up to 60 MeV per nucleon by this mechanism (Hegelich et al., Reference Hegelich, Pomerantz, Yin, Wu, Jung, Albright, Gautier, Letzring, Palaniyappan, Shah, Allinger, Horlein, Schreiber, Habs, Blakeney, Dyer, Fuller, Gaul, Mccary, Meadows, Wang, Ditmire and Fernandez2013). Although this represents major progress, it is still less than what is needed for ion beam therapy (~400 MeV/u).
A recent experimental study with laser intensities of ~ 1021 W cm−2 showed that protons can reach up to ~100 MeV with target thicknesses of ~100 nm (Higginson et al., Reference Higginson, Gray, King, Dance, Williamson, Butler, Wilson, Capdessus, Armstrong, Green, Hawkes, Martin, Wei, Mirfayzi, Yuan, Kar, Borghesi, Clarke, Neely and McKenna2018). It was shown by particle-in-cell (PIC) simulations that protons were accelerated by both RPA and TNSA, which is called hybrid acceleration for linearly polarized (LP) laser pulses (Higginson et al., Reference Higginson, Gray, King, Dance, Williamson, Butler, Wilson, Capdessus, Armstrong, Green, Hawkes, Martin, Wei, Mirfayzi, Yuan, Kar, Borghesi, Clarke, Neely and McKenna2018). Another study with circularly polarized (CP) and LP lasers showed that a CP laser beam is more effective at generating higher energy particles (Zhang et al., Reference Zhang, Shen, Li, Jin, Wang and Wen2007). In this paper, a study of C +6 ions and electrons acceleration for ultra-thin targets illuminated by ultra-intense (5 × 1022 W cm−2) laser will be presented, considering both LP and CP laser pulses. This is in contrast to recent work investigating ion acceleration with next-generation lasers which has not focused on ion acceleration during the transition to relativistic transparency ( Duff et al., Reference Duff, Capdessus, Sorbo, Ridgers, King and McKenna2018; Sorbo et al., Reference Sorbo, Blackman, Capdessus, Small, Slade-Lowther, Luo, Duff, Robinson, McKenna, Sheng, Pasley and Ridgers2018). This study has been carried out by 2D PIC simulations. We have investigated the effect of target thickness and laser polarization on particle acceleration and angular divergence of carbon ions.
Simulation setup
We have performed 2D PIC simulations with the relativistic electromagnetic code EPOCH (Ridgers et al., Reference Ridgers, Kirk, Duclous, Blackburn, Brady, Bennette, Arber and Bell2014; Arber et al., Reference Arber, Bennett, Brady, Lawrence-Douglas, Ramsay, Sircombe, Gillies, Evans, Schmitz, Bell and Ridgers2015). As the simulated laser irradiances were well below the threshold for Quantum Electrodynamics (QED) effects (Ridgers et al., Reference Ridgers, Brady, Ducluos, Kirk, Bennett, Arber, Robinson and Bell2012), they were not included in our simulations. A thickness scan of solid carbon targets using 2D simulations was carried out. Gaussian CP and LP laser pulses with a peak irradiance of I = 5 × 1022 W cm−2 were used in the simulations. The laser pulse struck the target at normal incidence, which was along the x-axis. The beam was focused to a 3 μm focal spot at y = 0. We assumed a fully ionized planar target of carbon.
The simulation domain had a size 13 μm × 20 μm and was discretized into 6500 × 2000 cells. This corresponded to a cell size of 2 nm in the x-direction and 10 nm in the y-direction. The box along the x-axis extended between −3 and 10 μm and in the y-axis, between −10 and 10 μm. A total of 300 particles/cell for electrons and 100 particles/cell for C6+ were used in the simulations. Open boundary conditions were used for both particles and fields. The laser wavelength (λ) and pulse duration were 0.82 μm and 25 fs, respectively. The corresponding normalized vector potential 
$a_0=8.5\times 10^{-10}\lambda [{\mu} {\rm m}]\sqrt {I_0[{\rm W\,cm}^{-2}]}$ was ~156. The initial electron density in the target was 6.78 × 1029 m−3 which is 400 n c, where n c = 1.1 × 1027(λ[μm])−2 m−3 is the critical electron density. The corresponding initial ωp ~ 5 × 1016 rad/s and the skin depth ℓp ~ 6 × 10−9 m. The carbon target thickness was varied from 10 to 600 nm with a step-like density profile.
We have used short (25 fs) Gaussian laser pulse without prepulse. This was done to make sure that the target of few nm thickness remains intact when the main pulse arrives at the target. As a finite prepulse of high enough magnitude will destroy the nm thick target before the arrival of the main pulse, the study of particle acceleration will be affected (Yin et al., Reference Yin, Albright, Bowers, Jung, Fernandez and Hegelic2011).
PIC simulation results and discussion
Previous work shows that the polarization of the laser pulse affects the particle acceleration (Scullion et al., Reference Scullion, Doria, Romagnani, Sgattoni, Naughton, Symes, McKenna, Macchi, Zepf, Kar and Borghesi2017). Which is confirmed by our simulations. Figure 1 shows the carbon number density for CP and LP laser pulses at different time steps for the target thickness of 120 nm. CP accelerates the ions by RPA, whereas LP causes strong electron heating and so the TNSA mechanism dominates the acceleration of the ions. In LP, carbon acceleration happens in both directions, whereas in CP, carbon is mostly accelerated in the forward direction.
Fig. 1. Time evolution of carbon ion number density (i) for CP laser and (ii) for LP laser pulses, for a 120 nm thick target. A striking difference in acceleration mechanism between LP and CP is noticeable. CP is dominated by the RPA, whereas TNSA dominates in LP.
The energy spectra of forward accelerated electron and carbon ions for both CP and LP for different target thicknesses are shown in Figure 2. We can see that maximum kinetic energy (KE) of the accelerated carbon ions is higher in the case of a CP laser (~5 GeV) than for LP (~3 GeV). On the other hand, LP generates higher energy electrons than CP. This difference is thickness-dependent and varies by up to 300%.
Fig. 2. Energy spectra of carbon ions (left) and electrons (right) for (i) CP and (ii) LP laser beam for different thicknesses.
Figure 3 shows the mean KE of forward accelerated C6+ ions as a function of time for different target thicknesses. From Figure 3, we can see the highest mean KE for both polarizations is obtained when the target thickness is around ~100 nm. This increase in the average KE is more pronounced in case of CP.
Fig. 3. Average KE of forward accelerated carbon ions, at different time steps for the CP (a) and LP (b) laser beams. Legends show the thickness of the target in nm.
This demonstrates the possible role of BOA in ion acceleration in our simulations for both polarizations. In this mechanism, the increase in KE happens when the target becomes transparent to the laser; due to this, the laser pulse penetrates the target and further accelerates the ions via volumetric heating of electrons and generation of electrostatic fields. Transparency occurs due to the decrease in the density and it occurs mainly via three phenomena: (1) the ponderomotive force, which is the force due to laser electric field amplitude gradient, (2) the lowering of the critical density due to the increase in electron mass by fast oscillation in the laser field, and (3) the target expansion due to heating of electrons, which in turn decreases the electron density. All the above-mentioned phenomena collectively reduce ωp. The sudden increase in ion KE by three to four times is not seen for the thicker targets >200 nm and for much thinner targets <40 nm.
BOA and laser polarization
To understand the effects of both laser polarization and target thickness on particle acceleration, electron and carbon ion energy spectra were examined.
The longitudinal electrostatic field, the laser field, and the density of C6+ and electrons for the thickness of 120 nm are shown in Figure 4 at various times. All these quantities are averaged over the size of full width half maximum (FWHM) of the laser pulse. The initial target location is at x = 0 μm. Up to 50 fs, we can see that the ion acceleration is due to TNSA, RPA, and the enhanced TNSA. BOA starts between 50 and 75 fs, when the target becomes transparent to the laser. The expansion of the target due to laser ponderomotive pressure and electron heating, in addition to the relativistic mass increase of the electrons, makes the target transparent. In the transparency phase, the laser penetrates into the target and ions inside the plasma are accelerated by the laser field which leads to an increase in the electric field component in x-direction E X (see Fig. 4 at 75 fs). It can be seen that BOA has generated a strong longitudinal E X field and ions are co-traveling with the peak amplitude of the E X. We can see that during the transparency phase, the electron density is ~2 × 1029 m−3, which is ~100n c. As γ ~100, this is the threshold for relativistic transparency. We have seen BOA caused by this transition to transparency for targets of thickness ranging from 40 to 200 nm. However, for thicker targets, it is observed that BOA phase does not occur for CP laser pulses. For these thicknesses, the laser failed to make the target transparent and the acceleration is due to RPA or TNSA.
Fig. 4. Time evolution of longitudinal electrostatic electric field E X, transverse laser field E Y, C6+ ions density, and electron density for 120 nm target for both CP (top row) and LP (bottom row). Left scale is for E X and the right-hand scale represents the carbon and electron density.
From Figure 4 (50 fs), we can see that transparency occurs earlier for LP than CP. This leads to a large initial acceleration of carbon atoms in LP but in CP, it takes place at a later stage. The early transparency in LP quickly leads to the BOA phase. Whereas in the CP case, the BOA phase comes later. The required strong E x field component for the acceleration process lasts up to 100 fs. However, in LP, it lasts for a short duration. The effect of this can be seen from high KE of C6+ at 50 fs in LP, but after 50 fs, CP laser acceleration is strong, caused by the large electric field at later times.
Evidence of BOA acceleration has been seen up to the thickness of 200 nm but for larger thickness (400 and 600 nm) transparency and thus BOA do not occur for CP laser pulses (see Fig. 4). However, for the LP case, the target is still transparent so the laser can penetrate through the target and BOA conditions still valid for the thicknesses >200 nm. In CP, the hot electron generation is quenched as compared to LP for these thicknesses. Up to 200 nm, the CP laser pulse generates large cutoff energy C6+ ions; however, for the thicker targets, LP laser pulse takes over and causes a sizeable increase in the cutoff energy than the CP laser pulse.
For CP laser pulses and ultra-thin targets with the irradiances I ~ 1021 W cm−2, it is observed that the RPA (Esirkepov et al., Reference Esirkepov, Borghesi, Bulanov, Mourou and Tajima2004; Robinson et al., Reference Robinson, Zepf, Kar, Evans and Bellei2008; Tamburini et al., Reference Tamburini, Pegoraro, Piazza, Keitel and Macchi2010) is the dominant mechanism for the acceleration of protons and heavy ions. For the irradiance studied here (I = 5 × 1022 W cm−2), we see that when the laser is CP, we can still observe the RPA mechanism – except for the optimum target thicknesses where BOA is observed. LP also accelerates ions with RPA and TNSA (hybrid mechanism) mechanism for such high irradiances (Higginson et al., Reference Higginson, Gray, King, Dance, Williamson, Butler, Wilson, Capdessus, Armstrong, Green, Hawkes, Martin, Wei, Mirfayzi, Yuan, Kar, Borghesi, Clarke, Neely and McKenna2018).
The transmission of the laser pulse through the target depends on the pulse duration and intensity along with the thickness of the target. Low-intensity pulses need to have a long pulse duration, on the other hand, high-intensity laser can transmit through the target even at short pulse duration (Petrov et al., Reference Petrov, McGuffey, Thomas, Krushelnick and Beg2017). Petrov et al. (Reference Petrov, McGuffey, Thomas, Krushelnick and Beg2017) presented a simple formula from their set of simulations for RPA to RIT transition, 
$\tau _{{\rm FWHM}}\sim 210\sqrt {I[{\rm W/cm}^2]/10^{21}}$, where τ FWHM is the required pulse duration for the transparency to occur for a given intensity I. This estimation is for a 20 nm Au target but it can be scaled to align with the thicknesses of the carbon target. Au has electron density ~ 2500n c and carbon target has 400n c, it allows us to take carbon of up to 125 nm. For the former equation given above, by using our PIC simulation parameters, τ FWHM ~ 30 fs can be obtained which is similar to our simulated pulse duration. We have seen RIT occurring up to a 200 nm thick target with a pulse duration of 25 fs, consistent with the formula.
It can be seen that the maximum KE peaks for a 120 nm thick target and then decreases with increasing target thickness (see Fig. 5). In that case, relativistic transparency of the plasma is suppressed and the laser is reflected backward, which causes a drop in the energies of the particles (see Fig. 2). We have shown that laser facilities like ELI-NP (ELI, 2019) are potentially able to accelerate C6+ ions up to 5 GeV energies if the laser is CP (see Fig. 5).
Fig. 5. Maximum energy of carbon ions for LP and CP laser beam at different target thicknesses. The peak KE reaches is observed for the same thicknesses for both polarizations.
C6+ ions and electrons temperature and mean KE
We have determined the electron and ion temperature (kT) by fitting the electron and ion spectra with exponential of the form exp (−KE/kT). The C6+ ion and electron temperature are shown in Figure 6 for the CP and LP laser pulses. For thicknesses of ~100 nm, the temperature of electrons is ~40 MeV and for ions, it is ~1 GeV for a CP laser pulse. In the case of a LP laser pulse, the electrons have the same temperature as the CP laser around the ~100 nm, but the carbon ion temperature is less than the temperature achieved with CP. The temperature increases until ~100 nm thicknesses, then it decreases for thicker targets. It suggests that the laser does volumetric heating for targets of up to 200 nm thicknesses. On the other hand, for thicknesses around 400 nm and more, the target becomes impenetrable and volumetric heating does not occur, which brings the temperature down for both species. The C6+ ion energies have the same trend as the temperature (see Figs 5 and 6b).
Fig. 6. Comparison of (a) electrons and (b) carbon ions temperatures for LP and CP lasers at different target thicknesses.
To further analyze the mechanism behind the ion acceleration in CP and LP, we studied the average KE per cell for both C6+ ions and electrons. For ℓ = 120 nm target, Figures 7 and 8 show the mean KE of C6+ ions and electrons, respectively. We can see that the radiation pressure of the laser and laser electric field with such irradiances is sufficient to push the electrons and C6+ ions through the target for ultra-thin foils. The laser first starts accelerating electrons at the front surface of the target (light pressure) in both polarization cases. CP laser confines and moves together with all charged particles while LP laser first pushes electrons forward sets up a sheath field, and due to charge separation, the ions accelerate both in causing the target to expand (see Figs 7 and 8).
Fig. 7. Time evolution of mean KE of carbon ions for (i) CP, (ii) LP laser beam, for 120 nm target thickness.
Fig. 8. Mean KE of electrons, at different time steps for (i) CP, (ii) LP laser beam, for 120 nm target thickness.
For such thin foils, LP pulses create a hybrid acceleration mechanism which is the combination of RPA and TNSA. Initially, the electrons and C6+ ions are accelerated at the front surface of the target (indicating that RPA is the accelerating mechanism), then the TNSA effect on the process shows that ions are accelerated at the back of the target. On the other hand, for the CP laser, acceleration starts at the front surface as well and accelerated charged particles pushed into the target. It shows that in CP laser, acceleration is due to RPA and BOA. The same can be concluded from Figure 8, which shows mean KE per cell of the electrons.
Laser polarization and angular distribution of C6+
The angular distribution of carbon ions is important for future applications. Thus, we have also investigated the C6+ ion angular distributions for both CP and LP laser pulses and various thicknesses.
Figure 9 shows the angular distribution of the accelerated carbon ions of KE >500 MeV for different foil thicknesses, for CP and LP laser pulses. We see that laser polarization has no effect on the angular divergence of the C6+ ions. It is seen that the angular divergence we report here is in agreement with the previously reported results (Petrov et al., Reference Petrov, McGuffey, Thomas, Krushelnick and Beg2017). Regardless of polarization, thicknesses greater than 200 nm are more effective at creating ion beams with small angular distribution as compared to thinner targets. However, increasing target thickness reduces the maximum ion energy and temperature. From Figure 9, we can see that in the case of a CP laser pulse, there are no carbon ions which propagate in the backward direction; however, for an LP laser pulse, a significant number of carbon ions do propagate in the backward direction due to the enhanced electron heating and target expansion for LP.
Fig. 9. Carbon ions angular divergence for (a) CP and (b) LP laser beam.
Conclusion
In conclusion, we have investigated the effect of target thickness and laser polarization on particle acceleration and angular divergence in next-generation laser-matter interactions (I = 5 × 1022 W cm−2) by using EPOCH 2D PIC simulations. It is found that the BOA process occurs for targets with thicknesses between 80 and 200 nm, which causes a sudden increase in average ion energy. It is seen that ions can be accelerated ~5 GeV energies with a target thickness of ~120 nm by using 10 PW class lasers with CP laser pulses and up to 3 GeV with LP laser pulses. We have shown that the present-day lasers should be capable of generating C6+ ions up to 400 Mev/u, which is the required maximum energy for the carbon radiotherapy to treat tumors. We showed that laser acceleration mechanisms have a strong dependence on target thickness as well as laser polarization for 10 PW laser–solid interactions. On simulating different thicknesses, we have found that for the maximum electron and ion energy and temperature, there is an optimum target thickness (~120 nm) and no dependency on laser polarization for such high irradiances was seen. We have also found that ions are better collimated for target thicknesses of 400 nm or more. We have reported that LP laser turns the target transparent for the thicknesses >200 nm which helps to generate more energetic particles via BOA mechanism, while in CP laser pulse, BOA is not effective for carbon acceleration with those thicknesses (>200 nm). Besides, for CP laser pulses, carbon acceleration takes place in the forward direction only, whereas for LP laser pulses, acceleration in both forward and backward directions occurs.
Acknowledgments
The research was supported by (i) TUBITAK research project 116F042, 118F077 and by Karamanoglu Mehmetbey University Research Project 40-M-16. This work was in part funded by the UK EPSRC grants EP/G054950/1, EP/G056803/1, EP/G055165/1 EP/M018156/1, and EP/ M022463/1, and (ii) the Extreme Light Infrastructure Nuclear Physics (ELI-NP) Phase II, a project co-financed by the Romanian Government and the European Union through the European Regional Development Fund and the Competitiveness Operational Programme (1/07.07.2016, COP,ID 1334).
