Enhanced MVA of polarized proton beams via PW laser-driven plasma bubble Open Access

In recent years, high-powered laser facilities have become capable of delivering ultrashort pulses with an intense laser power surpassing 10 PW^1,2 and a pulse duration of less than 20 fs.^3,4 The development of PW lasers is geared toward fundamental applications, including fusion ignition systems and high-energy particle physics.^5,6 Benefiting from these intense laser facilities, highly energetic beams of electrons, ions, and photons can be produced through laser–plasma interactions.⁷ Ion acceleration driven by ultra-intense lasers in plasmas has been intensively investigated through theoretical models and experimental studies over the past decades.^8,9 Several efficient acceleration mechanisms in laser–solid interactions have been experimentally demonstrated, such as target normal sheath acceleration (TNSA) and radiation pressure acceleration.^10–13 An experimental record of generating 150 MeV proton beams from a solid foil has been reported recently.^14,15 Furthermore, ion acceleration at high repetition rates naturally occurs in hydrodynamic flows with near-critical density,^16–18 where ions can be accelerated through various mechanisms such as bubble acceleration (BA),^19,20 shock acceleration (SA),^21,22 and magnetic vortex acceleration (MVA).^23–25 

Energetic polarized particle beams play a pivotal role in high-energy physics and nuclear physics. The collision of polarized particle beams provides a unique opportunity to study the spin structure of hadrons.^26–28 Polarized particle beams can be produced in traditional accelerators, but these are expensive and huge. For instance, the proton accelerators of the RHIC and JLEIC consist of an ion linac injector with an energy of ∼200 MeV and energy boosters with hundreds of GeV.^29–31 With the rapid development of PW laser facilities, compact and economical laser plasma accelerators have come to the rescue. Polarized particles can be generated by radiation polarization of a laser-accelerated beam^32–34 or by laser acceleration in a pre-polarized target.^35,36 Notably, laser-driven proton acceleration only works with the latter method because of the significantly larger mass-to-charge ratio of protons compared with electrons.^37,38 Recently, it has become possible for targets with polarized protons to be provided by dense polarized atomic hydrogen gas jets.^39,40 To date, studies of laser-driven polarized proton acceleration using these dense gas targets have primarily focused on feasibility verification, with demonstrations of 100 MeV polarized proton beams generated with PW lasers via the MVA scheme^38,41 or boosted SA.⁴² The BA scheme has been demonstrated to be successful in producing GeV polarized proton beams with a 200 PW laser.⁴³ The production of such beams, however, becomes challenging with currently available laser power, owing to the existence of a self-injection threshold.^44,45 Current intense laser facilities provide PW pulses, while 100 PW lasers are still under construction.^1,2 Considering the requirements for polarized colliders,^26–28 it is worthwhile to pursue the goal of increasing the acceleration efficiency of polarized protons using a PW laser.

To enhance laser-driven ion acceleration in a gas jet, a variety of useful experimental techniques have been applied, including tight focusing of PW pulses near the diffraction limit^46,47 and high-density gas jet production with a hydrodynamic shock wave excited by a low-energy laser pulse.^48,49 Propagation of such intense pulses in dense hydrodynamic jets may excite bubble-like ultra-nonlinear plasma waves, where the intense oscillation of background ions becomes a part of the plasma wave. The condition of self-injection of the background ions in the plasma wave is relatively challenging for PW lasers. Nevertheless, the background ions in the plasma wave acquire significant energy, which could be expected to boost the further injection and acceleration in MVA at the rear side of the target.

In this paper, we demonstrate the generation of monoenergetic polarized proton beams with energies in the hundreds of MeV range from a PW laser in a pre-polarized hydrogen chloride gas jet within an enhanced MVA. When the intense laser pulse penetrates the target, atoms with pre-polarized nuclei are ionized and a nonlinear plasma wave is excited by a ponderomotive force. The motions of background electrons and ions induce strong electromagnetic fields in the laser plasma wave. These fields consist of the charge separation field and a vortex magnetic field, where the proton energy is enhanced and its spin precesses. Before the laser pulse leaves the target, protons are accelerated by the laser-driven plasma wave, in radial and forward directions, respectively. The dependence of proton precession on laser plasma parameters is analyzed. When the laser pulse leaves the target, the laser-driven magnetic vortex introduces a stationary electric field at the rear boundary. Protons from the density down ramp still oscillate longitudinally in the plasma wave, which may be trapped and further accelerated in the magnetic-vortex-enhanced stationary acceleration field. Our particle-in-cell (PIC) simulations indicate that monoenergetic polarized proton beams with energies of several hundred MeV can be generated using PW lasers.

When an intense short laser pulse propagates through an underdense plasma, a laser-driven bubble is formed, with electrons being completely blown out by the ponderomotive force. Figure 2(a) shows an electron bubble with positive charges surrounding the drive laser within the plasma. The density of the positive charges is ρ ≈ n_e0e. In the ultra-nonlinear scheme driven by a PW laser, the electron bubble is adjoined by a relativistic plasma channel with the same radius. In this adjoined bubble–channel structure, the electric field is strong enough to accelerate background ions.^19,57 The simulations are initialized as the intensity peak of the laser pulse reaches the position z = 0 at time t = 0. Figure 2(a) shows the charge density distribution at time t = 250τ, when the laser front arrives at a position z_f ≈ 245λ. Electrons along the laser propagation axis are expelled forward to z_f and radially within the range z_f − cT_L ≤ z ≤ z_f, thereby forming the sheath of the front half of the bubble. Under the space charge effect, background electrons are drawn back toward the propagation axis, and they enclose the bubble at z_t ≈ 220λ. Furthermore, Eq. (1) gives the plasma channel radius R_ch ≈ 4.76λ by taking the geometrical factor of 2D simulation as K ≈ 1/10,⁵⁶ which coincides with the density boundary of the plasma channel behind the bubble in the region z < z_t, as shown in Fig. 2(a). As well as the sheath of the bubble and the plasma channel, a filament composed of dense electrons and ions appears on the propagation axis at x ≈ 0, owing to the motion of ions.^58,59 This filament exhibits distinct characteristics: inside the bubble, it mainly consists of trapped electrons, owing to the difference in charge–mass ratio between electrons and ions, revealed as a negative charge density; whereas behind the bubble, the positive charges are dominated by background ions that are pulled along the propagation axis by the electron filament.

A density plot of energetic protons with energy $E>9.4$ MeV in z–x space is shown in Fig. 2(d). There are two groups of energetic protons, on and off the propagation axis, respectively. Density plots of these energetic protons at t = 250τ in z–$E$ space and in z–p_z space are shown in Figs. 2(e) and 2(f), respectively. The on-axis group with r < 2λ in Fig. 2(d) occupies the same region of the bubble z_t < z < z_f. These protons are pushed forward when the front half of the bubble passes by. Corresponding dense protons can be found within the bubble region of Figs. 2(e) and 2(f), with $E≲20$ MeV and p_z ≳ 0.15m_pc. The momentum given by Eq. (6) is about 0.16m_pc, which is qualitatively consistent with Figs. 2(c) and 2(f). Here, it should be noted that v_z,max ≪ v_f, with v_z,max = p_z,max/(γm_pc). This indicates that the on-axis group has not been trapped by the bubble, which is different from the self-injection case of BA.^43–45 The on-axis protons are accelerated in the bubble, acquiring forward velocity in the front half of the bubble, and decelerated in the rear half. In addition, the spin of the protons precesses in the plasma channel according to Eqs. (2) and (4), owing to the presence of the magnetic field. The magnetic vortex at t = 250τ is shown in Fig. 3(a). The distributions of energetic protons with respect to the longitudinal spin component s_z and transverse spin component s_x are shown in Figs. 3(b) and 3(c), respectively. The distribution of protons with respect to s_x is symmetric, since the proton spin precesses in the cylindrically symmetric magnetic vortex. Owing to the presence of the singularity of the magnetic vortex at $x 2 + y 2 =0$⁠, the spin precession of on-axis BA protons is restricted. Specifically, the spin distribution of the protons near the bubble front z → 240λ in Figs. 3(b) and 3(c), where s_z → 1 and s_x → 0, i.e., $s ⃗ → s ⃗ i$⁠.

Energy enhancement of protons via MVA is the second stage of acceleration for the on-axis protons near the rear boundary of the target. They are accelerated first when they are swept by the acceleration field of the bubble, as described by Eqs. (5) and (6). The momentum distribution of the energetic protons along the propagation axis before they leave the gas target is shown at t = 250τ in Fig. 2(f). When the bubble passes the rear boundary at z₂, the pre-accelerated protons in the plasma wave are then trapped by the stationary field of MVA and accelerated to much higher energies. When the accelerated protons pass z₂, the longitudinal momentum component reaches 0.28m_pc at t = 270τ and 0.52m_pc at t = 280τ, as depicted in Figs. 5(a) and 5(c), respectively. Figures 5(b) and 5(d) present the momentum distributions of energetic protons with respect to the spread angle θ_x = tan⁻¹(p_x/p_z), from which it can be seen that only protons with small spread are accelerated forward. The accelerated beam comprises protons within a narrow spread angle, specifically, θ_x ≲ 10°.

Figure 6(a) reveals that protons are accelerated to 550 MeV by t = 320τ, with the maximum energy appearing at z = 284λ. These protons are found along the propagation axis with |x| ≲ λ in Fig. 6(b). The electron bunch in Fig. 6(b) around the energetic proton beam corresponds to the electron filament in Figs. 2(a) and 4(a). It also expands transversely into the region x ≲ 10λ, where the magnetic vortex is evanescent. According Eq. (9), the on-axis acceleration field E_z is weakened when the negative radial gradient of the magnetic vortex moves radially from the propagation axis, as shown by the solid green curves in Figs. 6(c) and 6(d).

When the energetic protons in the plasma wave are trapped by the magnetic vortex enhanced acceleration field, a monoenergetic proton beam can be produced. Energy spectra of protons from t = 260τ to 400τ are plotted by curves of different colors in Fig. 7(a). When the laser pulse propagates in the gas jet during t ≤ 270τ, the cutoff energy is around 61 MeV, corresponding to the radially accelerated proton energy due to the plasma channel $E ch$⁠. The cutoff energy is almost doubled to 116 MeV at t = 280τ. Although the lower boundary of the MVA field in Figs. 4(b) and 6(c) stays at z₂, its amplitude decreases significantly when the negative radial gradient departs radially from the propagation axis, as illustrated in Figs. 6(c) and 6(d). As a consequence, only a portion of the protons in the plasma wave with higher longitudinal velocity component p_z, shown in Figs. 2(e) and 5(a), are trapped. As a result, energy peaks are found in the spectra at t ≥ 280τ in Fig. 7(a). With the continuous acceleration of the trapped beam, the peak energy reaches 689 MeV at t = 400τ. The production of the energy peak stems from the pre-acceleration of protons within the bubble, driven by a PW laser with an optimized plasma density. On the one hand, the effect of BA may diminish in plasmas with higher density, resulting in an energy spectrum of MVA protons that lacks a distinct energy peak.^23–25 On the other hand, self-focusing of the laser beam is weakened in plasmas with lower density, and the reduction of a₀ in Eqs. (5) and (6) decreases the acceleration of the protons.

The time evolution of beam polarization is shown by the solid blue curve in Fig. 7(d), where the solid red curve represents the evolution of the cutoff energy of the accelerated proton beam. This reveals that the protons acquire energy a₀mc²/2 before t₁ = 270τ, when the laser pulse propagates in the plasma. As illustrated in Figs. 2(f) and 3(b), the proton beam consists of protons with relatively high longitudinal momentum. Therefore, the beam retains a relatively high polarization when the laser propagates inside the target. It can be seen from Fig. 7(d) that a strong energy increase and strong depolarization appear in the time interval t₁ ≤ t ≤ t₂, when the laser passes though the rear boundary of the target and energetic protons experience MVA. The MVA almost comes to an end when the energy reaches 495 MeV at t₂ = 310τ. Subsequently, the acceleration field around the propagation axis becomes weaker, and the magnetic field almost vanishes, as seen in Figs. 6(c) and 6(d). The protons are accelerated slowly with the residual electron bunch until t₃ = 500τ, and the proton beam polarization is sustained after t₂. A density plot of protons in $E$–s_z space at t = 500τ is depicted in Fig. 8(a). The solid red curve and the dashed black curve represent the energy spectra of all protons and the beam protons, as defined by Eq. (10), respectively. The energy spectrum reveals an energy peak at 749 MeV, and the beam polarization is 66%.

To compare the spin dynamics of different protons in the accelerated beams, we trace two typical protons with the same longitudinal momentum p_z = 1.41m_pc but different longitudinal spin components s_z at the momentum peak in Fig. 7(b). These are named as proton A and B, and their locations are indicated in all the relevant figures by a green diamond and magenta square, respectively. The corresponding evolutions of their s_z are plotted by green and magenta solid curves in Fig. 7(d), from which it can be seen that the longitudinal spin component of proton A remains above 94.5% for the whole process, while that of proton B falls below 50% when t > t₂. The different spin dynamics of protons A and B can be understood from their trajectories, which are plotted by green and magenta lines in Figs. 4(a) and 6(b). It is found that both of these originate from the down ramp of the gas jet, with z₂ ≲ z < z₂ + d. More importantly, proton A originates from the propagation axis of the drive pulse and moves along it. By contrast, proton B originates from an off-axis location and moves forward with a diverging angle. When protons A and B are accelerated in almost the same electric fields, A remains along the propagation axis with zero magnetic field, while B experiences a stronger magnetic field beside the propagation axis, as shown by their locations in Figs. 4(a)–4(c) and 6(b)–6(d). The positions of protons A and B in $E$–s_z space at t = 500τ are indicated in Fig. 8(a). One may conclude that protons with initially smaller deviation from the propagation axis contribute to the portion of protons with higher polarization in the accelerated beam.

Longitudinal momentum spectra of accelerated proton beams at t = 500τ from gas jets with density ramp lengths d = 20λ and 50λ are shown in Fig. 8(b), where peaks can be seen at p_z = 1.32m_pc and 0.91m_pc, respectively. If the beam protons are defined by the same lower limit of proton momentum p_z0 = 95%p_z,peak in Fig. 8(b), one obtains beam polarizations of 67.8% and 38%, respectively. Figure 8(c) shows the corresponding evolution of the polarization of the accelerated beams and the cutoff energy for d = 20λ and 50λ. Similar characteristics are found as for the case of d = 5λ [Fig. 7(c)]: the proton beams are intensely accelerated and depolarized when the laser pulse propagates through the density down-ramp, with the corresponding time duration for beam acceleration and depolarization t₂ − t₁ depending linearly on d. The cutoff energies in Fig. 8(c) are 668 and 380 MeV for d = 20λ and 50λ, respectively. It is found that the proton beam obtained with d = 20λ has similar beam energy and polarization to the d = 5λ case shown in Fig. 7. Therefore, a density scale length of the gas jet of the order of tens of micrometers is required within an experimentally realizable gas target with sharp gradient.^65–71 

Following on from the above two-dimensional PIC simulations, to provide further understanding of the acceleration process and to verify the practical efficiency of the acceleration scheme, three-dimensional PIC simulations are conducted using the extended EPOCH code. The size of the simulation box is 180λ × 24λ × 24λ, with cells of 1800 × 240 × 240 and six pseudoparticles per cell for protons, electrons, and chlorine nuclei. The power of the driving laser power is increased to 7.7 PW, with a laser amplitude of a₀ = 200 and a pulse duration of 8τ. The plasma density and the length of the uniform plateau are set as n_e = 0.72n_c and z₂ − z₁ = 250λ, respectively. The remaining parameters are identical to those used in the two-dimensional simulations for Fig. 7.

Figure 9(a) depicts the energy spectrum of the protons from the three-dimensional simulations at t = 550τ, together with the corresponding density plot in $E$–s_z space. An energy peak is observed at 257 MeV. It is notable that the protons at the energy peak have a relatively large s_z. The beam energy is significantly lower than that predicted by the two-dimensional simulations, owing to the magnetic vortex-induced acceleration field being reduced in the three-dimensional scenario, which could be attributable to the effect of dimension on laser-induced electron heating.⁷² Nevertheless, protons are still accelerated to ∼20 MeV by the laser-driven bubble according to Eqs. (5) and (6). Their energy is significantly boosted via MVA at the rear boundary of the target. Additionally, these polarized proton beams with energy ≳200 MeV can be considered as an alternative injector for polarized colliders.^29–31 

Figure 9(b) presents a density plot in s_z–θ space of protons within the full width at half maximum (FWHM) of the energy peak, where $θ= tan − 1 p x 2 + p y 2 / p z$ is the spread angle. The corresponding longitudinal spin spectrum is shown by the solid red curve and indicates a beam polarization of 39.5%. The majority of protons are within a spread angle θ ≲ 15°. The density plot reveals that protons with smaller θ have higher s_z. If protons with θ < 10° are selected, then the beam polarization increases to 73.3%. The corresponding longitudinal spin spectrum of protons with θ < 10° is shown by the dashed red curve. The beam polarization increases further to 88.8% if only protons with θ < 5° are considered. Therefore, more stringent requirements for beam polarization can be met by using a smaller collection angle. The beam charge at the FWHM of the energy peak is 349 pC. This is reduced to 95.4 and 23.1 pC for collection angles of 10° and 5°, respectively.

Laser-driven ion acceleration in a pre-polarized hydrogen halide gaseous target is expected to generate energetic spin-polarized protons economically. When a PW laser passes through a hydrogen halide gas jet, protons are accelerated by the laser-driven relativistic channel and wakefield inside the target, as well as by the stationary electric field at the rear boundary of the target. In particular, longitudinal motion of protons in the plasma wave triggers the injection of protons into the magnetic-vortex-enhanced acceleration field. In this study, the acceleration and spin precession of protons have been analyzed on the basis of the TBMT equation and analytical models of the laser-driven blowout bubble–channel structure. It has been found that the magnetic vortex affects the precession of protons in this structure and at the rear boundary of the target. By analyzing the proton dynamics and the field structures, the dependence of the final beam energy and polarization on laser and plasma parameters has been examined. It has been found that the energy gain is determined by the pre-acceleration of the protons in the plasma wave and the post-acceleration out of the target. Compared with our initial demonstration of energetic polarized proton beam generation,³⁸ the proton pre-acceleration within the plasma wave has been enhanced using our analytical model,^44,45 resulting in a significant increase in the final energy gain. Results from multidimensional PIC simulations indicate that monoenergetic proton beams with energies in the several hundred MeV range and a polarization of tens of percent can be achieved using PW-scale lasers and a pre-polarized gaseous target. This method can serve as an alternative to a polarized ion injection linac for use in a polarized collider.^28–31 Our study has thus provided a novel approach to high-energy laser-driven polarized proton acceleration, offering a practical proposal for polarized particle colliders.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12075081 and 12404395) and the Innovation Group Project of the Natural Science Foundation of Hubei Province of China (Grant No. 2024AFA038). Bin Liu acknowledges the support of Guangdong High Level Innovation Research Institute Project, Grant No. 2021B0909050006.

The authors have no conflicts to disclose.

Zhikun Zou: Data curation (equal); Software (equal). Gan Guo: Data curation (equal); Validation (equal). Meng Wen: Funding acquisition (equal); Supervision (equal); Writing – original draft (equal). Bin Liu: Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Writing – review & editing (equal). Xue Yan: Formal analysis (equal); Resources (equal). Yangjié Liu: Conceptualization (equal); Writing – review & editing (equal). Luling Jin: Conceptualization (equal); Supervision (equal); Visualization (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.