2. 3D PIC SIMULATION RESULTS

The regime of laser ion acceleration from near critical density gas targets can be realized when the laser pulse propagates through a near critical density target that is much longer than the pulse itself and the pulse forms a density channel. In MVA regime, when a tightly focused laser pulse interacts with the near-critical density plasma, the ponderomotive force of the laser expels electrons and ions in the transverse direction, forming the electron and ion density channel. A portion of the electrons are accelerated in the direction of laser pulse propagation by the longitudinal electric field. The motion of these electrons generates a magnetic field, which circulates in the channel around the propagation axis. Upon exiting the channel, the magnetic field expands into vacuum and the electron current is dissipated. Some of the electrons leave the plasma channel while few of them return to sustain the magnetic field on the rear side of plasma–vacuum interface. The magnetic field displaces the electron component of plasma with regard to the ion component and a strong quasi-static electric field is generated that can both accelerate and collimate the ions. For optimum acceleration [Bulanov et al. (Reference Bulanov, Bychenkov, Chvykov, Kalinchenko, Litzenberg, Matsuoka, Thomas, Willingale, Yanovsky, Krushelnick and Maksimchuk2010a, Reference Bulanov, Litzenberg, Pirozhkov, Thomas, Willingale, Krushelnick and Maksimchukb)] in case of MVA, the laser spot size should match the size of the self-focusing channel in order to avoid filamentation. The laser pulse energy should be depleted near the target rear to create the dipole vortex at the exit side of the channel and it is not wasted to transmission.

The effectiveness of the MVA mechanism requires the efficient transfer of laser energy to the fast electrons in the plasma which are accelerated in the plasma channel along the laser propagation direction. Thus the optimal condition for efficient proton acceleration can be estimated by equating laser pulse energy in plasma waveguide to the energy of electrons. The optimum plasma length (l) can be estimated from the assumption that all laser energy is transferred to the electrons in the plasma channel. If each electron has an average energy of a _Lm _ec ², the optimum length can be written as (Bulanov et al., Reference Bulanov, Litzenberg, Pirozhkov, Thomas, Willingale, Krushelnick and Maksimchuk2010b) l = a _Lcτ(n _c/n _e) K, where K is the geometry constant (K is 0.1 in 2D case and 0.074 in 3D case) and the laser pulse amplitude (a _L), can be determined by the laser power (P) and plasma density (n _e) as a _L = (8Π (P ₀/P _c)(n _e/n _c))^1/3.

Based on the theoretical approach (as outlined previously), we demonstrate here through the 3D PIC simulations the acceleration of protons by employing the ELI-ALPS high-field (HF) laser interaction with the near critical density hydrogen plasma. The incident laser parameters are chosen comfortably within the capabilities of ELI-ALPS facility. The expected parameter regime for ELI-ALPS HF laser is as follows: Laser wavelength λ_L = 800 nm, laser energy ε_L = 34J, and pulse duration t _L = 17 fs. We consider here the interaction of a linearly polarized laser pulse with the plasma target. The laser pulse considered here is Gaussian in space and time where the beam radius is r _b = 1.042 µm obtained by focusing lens of F = 2. The tightly focused beam will initially diverge in the plasma and will expel electrons to form a plasma channel, then after the divergence of laser will stop and the most of the laser energy in plasma channel will be transferred to plasma electrons to accelerate it. The plasma target proposed here is hydrogen of density n _i = n _e = 5.22 × 10²¹ cm⁻³ (three times of the critical density n _c) and the optimum thickness of plasma target is l = 25 µm, to utilize the maximum laser energy transfer to plasma electrons [see (1)]. 3D PIC simulations were carried out using the fully relativistic electromagnetic code PIConGPU (Burau, Reference Burau2010; Bussmann et al., Reference Bussmann, Burau, Cowan, Debus, Huebl, Juckeland, Kluge, Nagel, Pausch, Schmitt, Schramm, Schuchart and Widera2013). The boundary conditions are periodic throughout the simulation. We considered here the simulation box of dimension 10 × 200 × 20 µm³ corresponding to the grid size 256 × 5120 × 512 with cell size of 40 nm. The time step is 66.7 as. The laser pulse (polarization of laser field is along x-axis) incidents on plasma along the y-axis in XZ plane. The plasma target starts at z = 0.0 µm and terminates at 25.0 µm. The 3D simulation performed in this study limits to the simulation size corresponding to the available graphical processing units (GPUs) computing facility. We considered here n _i = 3n _c to minimize the optimal plasma channel length and limit 3D simulations. In our simulations we considered intensities of the order of 10²³ W/cm² or smaller so we did not incorporate the radiation reaction force (see Gao et al., Reference Gao, Wang, Lin, Zou and Yan2012) along with the Lorentz force for proton acceleration.

A liquid hydrogen jet [Kühnel et al. (Reference Kühnel, Fernández, Tejeda, Kalinin, Montero and Grisenti2011)] might be an interesting option for MVA to obtain the near critical density plasma in laboratory with ultra-intense femtosecond pulses because, besides allowing the acceleration of protons, being a “continuously flowing” target it would allow for high repetition rate operation, which is of high relevance for applications with high repetition PW laser.

The PIC simulation results (as shown by Fig. 1), summarizes the laser pulse penetration through the near critical density hydrogen plasma and consequently the generation of magnetic field at the plasma–vacuum interface.

Fig. 1. (a) The distribution of the magnetic field of vortex structure followed by the laser magnetic field, as the laser pulse channels inside the hydrogen plasma, (b) the variation of magnetic field at plasma–vacuum interface in XZ plane, (c) scaling of magnetic field with the incident laser power, analytical result (black solid line), and simulation results (red dot); at time t = 0.2 psec.

The generation of magnetic field can be illustrated in the following manner. The relativistic electrons accelerated within the plasma channel, follow the laser pulse and exit from the rear side of the plasma channel as the pulse exist at the plasma–vacuum interface. The fast electrons leave the plasma and then return back into it under the influence of an unneutralized electric charge. The electrons form a toroidal vortex in 3D geometry (dipole vortex in 2D case) due to such motion and the electric current of vortex generates a quasi-static magnetic field. Thus in case of homogeneous plasma with sharp plasma vacuum boundary at both ends (front and rear) of plasma channel, the magnetic field produced by fast electron beam expands along the plasma slab boundary. The rapid variation in magnetic field due to the vortex motion produces a strong quasi-static electric field. The ions are accelerated to high-energies due to the quasi-static electric field at plasma–vacuum interface. The magnitude of the magnetic field can be estimated by employing the Ampere's law, B ≈ 4Πηn _eer _ch where r _ch is the plasma channel radius which is equal to the radius of focused laser beam. We further simplified the magnetic field to scale it with initial laser-plasma parameter and obtained as,

(1)$$B = 32 \times {10^{{\rm -} 8}}{\rm \eta} \sqrt {{a_{\rm L}}{n_{\rm e}}\left\lceil {{\rm c}{{\rm m}^{{\rm -} 3}}} \right\rceil} T,$$

(in terms of Tesla). The magnitude of magnetic field (using η = 0.6 taken from the simulation) is ~20.0 × 10⁴T, which is close to the simulation results as shown by Figure 1c. Thus the magnetic field (1) increases with the focused laser field and the plasma density, and consequently the strong magnetic field can be generated in laboratory by utilizing the high power laser interaction with the near-critical plasma. The evolution of the magnetic field depends dominantly on the scale on which the plasma density varies. In case of an inhomogeneous plasma with a density gradient at plasma–vacuum interface, the magnetic vortex moves down the density gradient and expands in forward direction due to the decrease in plasma density and in lateral direction due to force acting on the vortex in the $\nabla nX\Omega $ direction (Nycander and Isichenko, Reference Nycander and Isichenko1990). Due to the forward and lateral expansion of magnetic vortex structure the electron density in the current filament decreases to a value where the condition of charge quasi-neutrality fails to hold. Consequently the current filament undergoes a Coulomb expansion and leads the vanishing of magnetic field. The abrupt decrease in magnetic field induces a strong electric field which accelerates the ion beam to high-energies. PIC simulation studies (Bulanov et al., Reference Bulanov, Dylov, Esirkepov, Kamenets and Sokolov2005; Nakamura et al., Reference Nakamura, Bulanov, Esirkepov and Kando2010) of high power laser interaction with gas target have shown the relevance of plasma density gradient for ion acceleration, however in this study we focus the ion acceleration utilizing the ultra-short–ultra-intense laser pulse interacting the near critical density plasma with sharp plasma–vacuum boundary.

We show the evolution of electron and ion density in Figure 2 at 200 fs, when the laser pulse exits from the plasma channel. A strong wakefield of the order of several GV/cm is generated in the plasma, as the ultra-intense laser propagates in the near critical density plasma. The bunch of trapped electrons is then accelerated to 100 s of MeV along the channel axis. As fast electron beam ejects out of the plasma into vacuum, the most energetic electrons escapes, forming a plasma potential barrier, which prevents further acceleration of low energy electrons.

Fig. 2. Evolution of (a) electron density (n _e) and (b) ion density (n _i) at time 200 fs, after pulse exits the plasma channel. The electron and ion densities are normalized to relativistic modified critical plasma density ${n_{\rm cr}} = \sqrt {{\rm \gamma}}{n_{\rm e}} $ where γ is the relativistic factor (here γ ≈ 12) as its variation is shown by the color bar. The X, Y and Z-axis are shown in units of laser wavelength.

As these electrons are pulled back into plasma (return current), it generates a magnetic vortex field of magnitude several megagauss (as shown in Figure 3a). The expansion of the magnetic field produces a longitudinal inductive electric field (by Faraday's Law), which accelerates the protons further. The longitudinal electric field (as shown in Fig. 3b) attains the magnitude of the order of TV/m.

Fig. 3. Evolution of (a) magnetic vortex field (B_z) and (b) longitudinal electric field (E_y) at (200 fs) just after pulse exits the plasma channel. The color bar shown for B_z is expressed in units of 8.52 × 10⁴ T and for E_y in units of 25.5 TV/m (E _L~500 TV/m). The Y and Z-axis are shown in units of laser wavelength. The dark red and blue lines ahead of B_z and E_y correspond to the laser field.

The ion filament maintains over long distance because of dominant nature of focusing longitudinal field over the diverging magnetic vortex field. This electric field accelerates and collimates protons from the thin proton filament which is formed along the propagation channel. Figure 4 shows the longitudinal and transverse proton momentum along the propagation direction (Y-axis) at different time instant to demonstrate the acceleration of protons at the rear side of plasma target in vacuum.

Fig. 4. Longitudinal momentum (p _y) (a,b) and transverse momentum (p _z) (c,d) of accelerated protons along the propagation direction (Y-axis) at time instant (a,c) 0.2 psec (b,d) 0.53 psec. The Y-axis is shown in units of laser wavelength (0.8 µm) and longitudinal and transverse proton momentum is expressed in units of m _ec.

Figure 4a, c shows the acceleration of protons at time instant 0.2 psec where protons are accelerated to high energy from longitudinal electric field generated due to the expansion of magnetic field. Figure 4c, d further explain the extended acceleration of protons at time instant 0.53 psec, since as time advances the magnetic field dissipates and the acceleration of proton is due to the energy transfer from leading electrons through the rest-over laser field. The electrons and protons accelerated along the Y-axis are expelled by the laser field in transverse direction and the transverse expansion is favored because of coulomb explosion of particles at axis. From the phase space data (as shown in Fig. 4) one can obtain the proton angular distributions. For a specific energy range, the distribution can be calculated by calculating the number of particles going in a given angle found from

(2)$${\rm \theta} = {\cos ^{{\rm -} 1}}\left( {{\,p_y}/\sqrt {{\,p_x^2} + p_y^2 + p_z^2}}\, \right)$$

The energy spectrum of the accelerated proton bunch which is propagating close to the axis, shown in Figure 5 at time t = 0.2 psec (a) and 0.67 psec (b). The maximum cut-off energy, we obtain in this case is around 1.1 GeV at 0.67 psec. In the inset we show the energy spectrum of all protons, which are propagating along laser direction at different angle. The longitudinal momentum of protons is maintained from t = 0.2 ps to 0.67 psec since the responsible factor for their acceleration is the longitudinal electric field.

Fig. 5. Energy distribution of protons propagating close (−5° to + 5°) to axis at 0.2 psec (a) and 0.67 psec (b). Inset plot: Proton energy distribution while considering the all accelerated protons.

We studied further the proton density (energy) at different time instant during the acceleration process (as shown in Fig. 6), after the laser field exits from the plasma channel. At time 167 fs when the ions form thin filament, they are accelerated to energy of 0.16 GeV and at time 200 fs the ions gain energy 0.35 GeV due to the strong longitudinal electric field produced due to time varying magnetic field. The ring shaped proton distribution (as shown in Fig. 6b) can be explained as an expansion of proton filament, which is formed due to ponderomotive expulsion of electrons from the region which is having a dimension of the order of the laser spot diameter. In Figure 6c we show the angular distribution of protons, which indicates that, the high energy protons are well collimated with the small divergence angle. It is worth to note that the results shown in Figure 5 correspond to the protons propagating close to the ring and the results shown in Fig. 5a is corresponding to Fig. 6b at 0.2 psec.

Fig. 6. The proton density distribution corresponding to different maximum proton energy (at different time instant t = (a) 0.17 psec, (b) 0.2 psec corresponding to propagation direction at y = (a) 26 µm and (b) 48 µm. (c) The angular distribution of protons at 0.2 psec, where inset shows 3D view of proton energy distribution. The color bar corresponds to ion density normalized with critical plasma density. In (a) the pink background corresponds to unperturbed proton density since plot (a) corresponds to plasma–vacuum interface at the rear side of the plasma channel.

The generation of magnetic field at plasma–vacuum interface is shown in Figure 3a and subsequently the quasi-static electric field accelerating protons in Figure 3b. However the simulation results (see Fig. 3a, b) also show the electric and magnetic field of transmitted laser field, which is not completely depleted in the plasma channel. Thus the protons accelerated by the MVA mechanism at plasma–vacuum interface may also get influenced by the transmitted laser fields. As the pulse exits the plasma channel (as shown in Fig. 2a), the leading hot electrons propagate in vacuum and form a bow in front of the protons. The magnetic field upon exiting the channel pushes further protons in respect of electrons and collimated proton bunch follows the leading electrons (co-moving with the laser field). Thus leading electrons transfer their energy to proton through the charge separation field and electrons are continuously getting energy from the laser pulse, which extends the acceleration length to obtain the high proton energy. We explore further the mechanism of acceleration of proton by investigating the evolution of magnetic field (Fig. 7a) and longitudinal electric field (Fig. 7b) which is responsible for the acceleration of electrons and protons. At initial acceleration stage the ions gain energy from the longitudinal electric field which is produced by the expansion of magnetic field. The magnetic field is maintained for 106 fs after the laser pulse exits the plasma channel (as shown in Fig. 7a, since the magnetic field of vortex structure exits along the Y-axis for distance 32 µm starting from 30 to 70 laser wavelength, which corresponds to 106 fs time period). In this time period the protons gain energy 670 MeV. After this time the acceleration continues at slower rate due to the expansion of magnetic field. The post acceleration mechanism which is responsible after this time is due to longitudinal electric field between electrons, which is co-moving with laser field (as shown in Fig. 7c, d) and the accelerated protons. The proton energy density (Fig. 7c) and electron energy density (Fig. 7d) at time 0.53 psec is shown to reveal the post acceleration process along the propagation axis. Thus as time advances the magnetic field dissipates and the dominant acceleration of proton is due to the energy transfer from leading electrons.

Fig. 7. The evolution of (a) magnetic vortex field (B_z) (b) longitudinal electric field (E_y), (c) electron energy density, and (d) ion energy density; along the propagation direction (Y-axis) followed by the laser field at 530 fsec (just before where we get the maximum proton energy). The Y-axis are shown in units of laser wavelength. B_z and for E_y are expressed in the same units as in Figure 3 while the electron and ion energy densities are expressed in units of n _cm _ec ².

Following the model of Bulanov et al. (Reference Bulanov, Esarey, Schroeder, Leemans, Bulanov, Margarone, Korn and Haberer2015) the maximum proton energy from MVA can be written as ${E_{\rm P}} = {\rm \eta \gamma} _{\rm e}^2 {a_{\rm L}}{m_{\rm e}}{c^2}$ where the proton energy (E _P) scales with the laser power as P ^0.67 and ${{\rm \gamma} _{\rm e}} = \left( {\sqrt {2/1.84}} \right){(2P/K{P_{\rm c}})^{1/6}}{({n_{{\rm cr}}}/{n_{\rm e}})^{1/3}}$ is the Lorentz factor of accelerated bulk electrons (which can be obtained from the condition that electron velocity is equal to the group velocity of an electromagnetic pulse propagating in a waveguide).

In Figure 8, we show the dependence of maximum proton energy on the laser power for the optimized value of laser-target parameter. We obtained the scaling of proton energy with the laser power as E _P ~ P ^0.67, which is consistent with the model result.

Fig. 8. The dependence of maximal proton energy on the laser power. The black line curve shows the maximum proton energy from simulation results, while the blue dot corresponds to the laser power where the simulation is performed.

The proton energy scaling demonstrates that few hundred MeV–GeV protons can be obtained by employing the high power laser (from 100 TW–2 PW) interaction with the hydrogen near critical density plasma. The 3D simulation results in the considered regime show the laser-to-proton conversion efficiency ~1% (for protons with energy >100 MeV and cut-off energy ~1 GeV). The highest TNSA proton energy (~70 MeV) has been obtained by using single shot 80 J laser pulse and specially designed targets, where the conversion efficiency (for proton >4 MeV energy) was also about 1% [Gaillard et al. (Reference Gaillard, Kluge, Flippo, Bussmann, Gall, Lockard, Geissel, Offermann, Schollmeier, Sentoku and Cowan2011)]. In our case even PW range laser has repetition rate about 10 Hz, it means that our laser will have at least ten times higher efficiency in the production of fast protons.