The significance of laser-driven polarized beam acceleration has been increasingly recognized in recent years. We propose an efficient method for generating polarized proton beams from a pre-polarized hydrogen halide gas jet, utilizing magnetic vortex acceleration enhanced by a laser-driven plasma bubble. When a petawatt laser pulse passes through a pre-polarized gas jet, a bubble-like ultra-nonlinear plasma wave is formed. As a portion of the particles constituting this wave, background protons are swept by the acceleration field of the bubble and oscillate significantly along the laser propagation axis. Some of the pre-accelerated protons in the plasma wave are trapped by the acceleration field at the rear side of the target. This acceleration field is intensified by the transverse expansion of the laser-driven magnetic vortex, resulting in energetic polarized proton beams. The spin of energetic protons is determined by their precession within the electromagnetic field, which is described using the Thomas–Bargmann–Michel–Telegdi equation in analytical models and particle-in-cell simulations. Multidimensional simulations reveal that monoenergetic proton beams with an energy of hundreds of MeV, a beam charge of hundreds of pC, and a beam polarization of tens of percent can be produced at laser powers of several petawatts. Such laser-driven polarized proton beams have promise for application in polarized beam colliders, where they can be utilized to investigate particle interactions and to explore the properties of matter under extreme conditions.

In recent years, high-powered laser facilities have become capable of delivering ultrashort pulses with an intense laser power surpassing 10 PW1,2 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 beam32–34 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 scheme38,41 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 limit46,47 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.

The driving laser is a tightly focused Gaussian pulse, with a focal radius w0 = 3λ, a laser amplitude a0 = 120, a pulse duration TL = 10τ, and linear polarization along the y axis, where τ = λ/c denotes the laser cycle and λ = 800 nm is the laser wavelength. The corresponding laser power P L = ( π 2 / 4 ) P 0 a 0 2 w 0 2 / λ 2 attains 2.78 PW, where P 0 = 4 π ε 0 m e 2 c 5 / e 2 8.7 GW is the natural relativistic power unit, with ɛ0 the vacuum permittivity, and me and −e the electron mass and charge. A schematic illustrating the proton acceleration and spin depolarization in the laser-driven plasma field is presented in Fig. 1. A PW laser pulse irradiates an HCl gas target containing pre-polarized hydrogen atoms with relatively high density. The atoms are ionized rapidly, with the spin of their nuclei conserved. Such an ultra-intense PW pulse can penetrate into overdense plasma with density nea0nc/2,50 where n c = ε 0 m e ω L 2 / e 2 is the plasma critical density and ωL is the laser frequency. A plasma channel is formed when the laser power exceeds 2 P 0 .51,52 The channel radius depends on the laser power and the plasma density according to Refs. 24 and 53 
(1)
Inside the plasma channel, the magnetic field produced by the laser-driven hot electrons and returned current depends linearly on the channel radius.54 It is therefore estimated as
(2)
This magnetic vortex field reaches 105 T, which results in significant proton spin precession.
FIG. 1.

Schematic showing the drive laser irradiating a gas jet of polarized HCl. A plasma channel is formed behind the laser bubble. Some background protons are pushed longitudinally by Ez, and protons with v ch are accelerated radially by the transverse field in the plasma channel. Meanwhile, the spin of protons precesses in the magnetic vortex according to Eq. (3). Transverse expansion of the magnetic vortex enhances the acceleration field at the rear boundary of the target. The forward-moving protons are trapped and accelerated to several hundred MeV.

FIG. 1.

Schematic showing the drive laser irradiating a gas jet of polarized HCl. A plasma channel is formed behind the laser bubble. Some background protons are pushed longitudinally by Ez, and protons with v ch are accelerated radially by the transverse field in the plasma channel. Meanwhile, the spin of protons precesses in the magnetic vortex according to Eq. (3). Transverse expansion of the magnetic vortex enhances the acceleration field at the rear boundary of the target. The forward-moving protons are trapped and accelerated to several hundred MeV.

Close modal
Two-dimensional PIC simulations are performed using the EPOCH code, which has been extended to include spin effects.55 We use a moving window of size 180λ × 80λ represented by a grid of 3600 × 1600, with four pseudoparticles per cell for protons, one pseudoparticle per cell for chlorine nuclei, and four pseudoparticles per cell for electrons. Additionally, a time step according to the Courant–Friedrichs–Lewy condition with a Courant number of 0.95 is utilized in the simulations. The laser pulse is designed to focus onto the left boundary of the gas target at z1 = 50λ. The target consists of a uniform plateau of ne = 0.54nc in the region z1zz2, and edges n e exp [ ( z z j ) 2 / d 2 ] on the up-ramp side z < z1 and the down-ramp side z > z2, where j = 1, 2, d = 5λ, and z2 = 250λ. The corresponding densities of protons and chloride ions in the plasma are denoted by np = nCl = ne/18. The target thickness is the optimized laser depletion length ∼a0TLKnc/ne.56 The plasma target consists of electrons and nuclei derived by dissociation from pre-polarized HCl molecules,40 with all protons initially polarized along the axis of laser propagation s i = e z . The proton spin precession in the PIC code is described by the Thomas–Bargmann–Michel–Telegdi (TBMT) equation
(3)
where G ≈ 1.79 represents the anomalous magnetic moment of the proton, γ is the relativistic Lorentz factor, and mp is the proton mass. In the nonrelativistic limit vc, the precession frequency is determined by the first term of Eq. (3), which depends linearly on the magnetic field. In the magnetic vortex of the laser-driven plasma channel, we have
(4)

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 ρne0e. 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 zf ≈ 245λ. Electrons along the laser propagation axis are expelled forward to zf and radially within the range zfcTLzzf, 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 zt ≈ 220λ. Furthermore, Eq. (1) gives the plasma channel radius Rch ≈ 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 < zt, 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.

FIG. 2.

(a) Snapshot in zx space of the plasma charge density ρ (comprising the charge densities of electrons, protons, and chlorine nuclei) at t = 250τ. (b) Snapshot in zx space of the longitudinal electric field Ez at t = 250τ. The solid green curve depicts the longitudinal electric field along the laser propagation axis at x = y = 0. (c) Density plot in zpz space of on-axis protons (with |x| < 2λ). (d)–(f) Density plots in zx space, z E space, and zpz space, respectively, of energetic protons (with energy larger than E min = 9.4 MeV) at t = 250τ.

FIG. 2.

(a) Snapshot in zx space of the plasma charge density ρ (comprising the charge densities of electrons, protons, and chlorine nuclei) at t = 250τ. (b) Snapshot in zx space of the longitudinal electric field Ez at t = 250τ. The solid green curve depicts the longitudinal electric field along the laser propagation axis at x = y = 0. (c) Density plot in zpz space of on-axis protons (with |x| < 2λ). (d)–(f) Density plots in zx space, z E space, and zpz space, respectively, of energetic protons (with energy larger than E min = 9.4 MeV) at t = 250τ.

Close modal
The charge density distribution results in cylindrically symmetric charge separation fields Ez and Ech, corresponding to the longitudinal electric field of the bubble and the radial electric field of the plasma channel, respectively. Figure 2(b) shows that the longitudinal component of the electric field in the plasma bubble has a large amplitude at the front of the bubble z = zf. The peak field strength is estimated as60,61
(5)
with v f = c / 1 + π 2 n e / a 0 being the velocity of the dense electron layer at the front of the bubble and E0 = mωc/e the unit of electric field. In addition to the charge separation and electric field, another characteristic of an ultra-nonlinear plasma wave is longitudinal motion of background protons.44,45 The momenta of protons on the propagation axis with r < 2λ are shown in Fig. 2(c), where the maximum momentum is found in the middle of the bubble. Taking account of Eq. (5), it is found that the protons oscillate in the plasma wave with maximum momentum62 
(6)

A density plot of energetic protons with energy E > 9.4 MeV in zx 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 zpz 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 zt < z < zf. 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 pz ≳ 0.15mpc. The momentum given by Eq. (6) is about 0.16mpc, which is qualitatively consistent with Figs. 2(c) and 2(f). Here, it should be noted that vz,maxvf, with vz,max = pz,max/(γmpc). 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 sz and transverse spin component sx are shown in Figs. 3(b) and 3(c), respectively. The distribution of protons with respect to sx 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 sz → 1 and sx → 0, i.e., s s i .

FIG. 3.

(a) Snapshot in zx space of the azimuthal magnetic field at t = 250λ/c. Here, the y component of the magnetic field By represents the azimuthal magnetic field Bφ in the plane y = 0. (b) and (c) Density plots in zsz space and zsx space, respectively, of energetic protons (with energy E > 9.4 MeV) at t = 250λ/c.

FIG. 3.

(a) Snapshot in zx space of the azimuthal magnetic field at t = 250λ/c. Here, the y component of the magnetic field By represents the azimuthal magnetic field Bφ in the plane y = 0. (b) and (c) Density plots in zsz space and zsx space, respectively, of energetic protons (with energy E > 9.4 MeV) at t = 250λ/c.

Close modal
Notably, the radial acceleration of background protons in the bubble channel also seems to occur inside the gas jet. The off-axis group of protons with xRch in Fig. 2(d) is mainly accelerated by the field Ech and moves radially in the region z ≤ (zf + zt)/2. The corresponding energy shown in Fig. 2(e) is consistent with the proton energy accelerated by the strong electrostatic sheath field of the channel E ch = a 0 m c 2 .57,63 In other words, the SA protons are accelerated to a velocity v ch 2 E ch / m p during a time duration tch ≈ 2Rch/vch. This time duration is determined by the laser intensity and plasma density as
(7)
Moreover, the proton spin precesses in the magnetic vortex field given by Eq. (2). This field is roughly 8E0/c with the parameters that we have assumed and is comparable to the field shown in Fig. 3(a). Taking the precession frequency to be given by Eq. (4), the precession angle is then obtained as
(8)
Since we are assuming a the PW laser and a near-critical dense gas jet, the precession angle of the SA protons Θ becomes roughly 2 rad. The corresponding longitudinal spin component sz = cos Θ approaches −0.43, in agreement with the minimum of sz in Fig. 3(b).
When the drive pulse passes through the rear surface of the gas jet, a stationary acceleration field with magnitude given by Eq. (5) appears in the region behind the surface z > z2. The charge density, acceleration field, and azimuthal magnetic field at t = 270τ are shown in Figs. 4(a)4(c), respectively. It is found that the bubble shell expands transversely in the region z > z2, which is accompanied by transverse expansion of Ez and Bφ. To show this expansion along the transverse direction, we set the horizontal axis of the plots in Fig. 4 along the x direction. It can be seen from Fig. 4(a) that half of the bubble leaves the flattop density of the gas jet at t = 270τ. The bubble radius expands to 10λ at z > z2, in comparison with the radius Rch ≈ 5λ inside the gas jet at z < z2. The acceleration field Ez in Fig. 4(b) has a static lower boundary at z2, while its upper boundary moves with the laser-driven electron layer. It should be noted that the acceleration field is maintained at EzE0, as a result of electric field enhancement via the gradient of the magnetic vortex:
(9)
The increment in the electric field at z = 270λ is plotted by the solid green curve in Fig. 4(d). It is consistent qualitatively with the radial gradient of the magnetic vortex field shown by the dashed red curve. Proton acceleration in this acceleration field enhanced by the stationary magnetic vortex is thus called MVA.23 The axis of the magnetic vortex deviates slightly from the initial laser propagation axis, owing to hydrodynamic instability of the plasma.64 
FIG. 4.

(a)–(c) Snapshots in xz space of the plasma charge density ρ, the longitudinal electric field Ez, and the azimuthal magnetic field By respectively, at t = 270τ. The light green and magenta lines in (a) show the trajectories of two tracer protons, with diamond and square symbols indicating their positions along the respective trajectory. The solid green diamond and solid magenta square in (a)–(c) indicate the positions of each proton at t = 270τ. (d) Comparison of the electric field increment in one laser cycle, ΔEz = Ex|t=270.5τEx|t=269.5τ (solid green curve), with the radial gradient of the azimuthal magnetic field, rBφ|y=0 = xBy (dashed red curve), at z = 270λ.

FIG. 4.

(a)–(c) Snapshots in xz space of the plasma charge density ρ, the longitudinal electric field Ez, and the azimuthal magnetic field By respectively, at t = 270τ. The light green and magenta lines in (a) show the trajectories of two tracer protons, with diamond and square symbols indicating their positions along the respective trajectory. The solid green diamond and solid magenta square in (a)–(c) indicate the positions of each proton at t = 270τ. (d) Comparison of the electric field increment in one laser cycle, ΔEz = Ex|t=270.5τEx|t=269.5τ (solid green curve), with the radial gradient of the azimuthal magnetic field, rBφ|y=0 = xBy (dashed red curve), at z = 270λ.

Close modal

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 0.28mpc at t = 270τ and 0.52mpc 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 θx = tan−1(px/pz), 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, θx ≲ 10°.

FIG. 5.

(a) and (b) Density plots in zpz space and θxpz space, respectively, of energetic protons at t = 270τ. (c) and (d) Corresponding density plots at t = 280τ.

FIG. 5.

(a) and (b) Density plots in zpz space and θxpz space, respectively, of energetic protons at t = 270τ. (c) and (d) Corresponding density plots at t = 280τ.

Close modal

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 Ez 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).

FIG. 6.

(a) Snapshot of the phase-space distribution of energetic protons at t = 320τ. (b)–(d) Snapshots in xz space of the plasma charge density ρ, the longitudinal electric field Ez, and the azimuthal magnetic field By respectively, at t = 320τ. The dashed black lines indicate the position z = 284λ at which the maximum energy of protons appears. The corresponding longitudinal electric field and azimuthal magnetic field at z = 284λ are shown by solid dark green curves in (c) and (d), respectively. As in Fig. 4, the light green and magenta lines and the diamond and square symbols in (b) show the trajectories of two tracer protons, with the solid diamond and square symbols in (b)–(d) indicating their positions at t = 320τ.

FIG. 6.

(a) Snapshot of the phase-space distribution of energetic protons at t = 320τ. (b)–(d) Snapshots in xz space of the plasma charge density ρ, the longitudinal electric field Ez, and the azimuthal magnetic field By respectively, at t = 320τ. The dashed black lines indicate the position z = 284λ at which the maximum energy of protons appears. The corresponding longitudinal electric field and azimuthal magnetic field at z = 284λ are shown by solid dark green curves in (c) and (d), respectively. As in Fig. 4, the light green and magenta lines and the diamond and square symbols in (b) show the trajectories of two tracer protons, with the solid diamond and square symbols in (b)–(d) indicating their positions at t = 320τ.

Close modal

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 pz, 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.

FIG. 7.

(a) Energy spectra of protons at different times during 260τt ≤ 400τ. (b) Density plot in szpz space of protons at t = 400τ, together with the longitudinal momentum spectra of all protons (solid red curve) and beam protons (dashed black curve). (c) Longitudinal momentum spectra at t = 260τ (black curve) and 280τ (blue curve). The shaded areas indicate the protons corresponding to those in the final accelerated beam. (d) Longitudinal spin spectrum of beam protons, dN/dsz, as a function of time. The solid blue curve describes the dependence of the beam polarization P = | s | on time. The cutoff energy of the accelerated protons is plotted by the solid red curve. The dashed black curve in (a) corresponds to the accelerated beam protons shown by the black dashed curve in (b). The magenta square and the green diamond in (b) indicated the locations of the two tracer particles, as in Fig. 4. The magenta and green curves in (d) show the variations of their respective values of sz with time.

FIG. 7.

(a) Energy spectra of protons at different times during 260τt ≤ 400τ. (b) Density plot in szpz space of protons at t = 400τ, together with the longitudinal momentum spectra of all protons (solid red curve) and beam protons (dashed black curve). (c) Longitudinal momentum spectra at t = 260τ (black curve) and 280τ (blue curve). The shaded areas indicate the protons corresponding to those in the final accelerated beam. (d) Longitudinal spin spectrum of beam protons, dN/dsz, as a function of time. The solid blue curve describes the dependence of the beam polarization P = | s | on time. The cutoff energy of the accelerated protons is plotted by the solid red curve. The dashed black curve in (a) corresponds to the accelerated beam protons shown by the black dashed curve in (b). The magenta square and the green diamond in (b) indicated the locations of the two tracer particles, as in Fig. 4. The magenta and green curves in (d) show the variations of their respective values of sz with time.

Close modal
The spin polarization of the accelerated proton beam is given by P = | s | , where indicates the average over all protons in the beam. In our setup, this polarization is roughly determined by s z when the spin precession of beam protons occurs in the cylindrically symmetric magnetic vortex Bφ. The beam polarization is calculated on the basis of identifying beam protons, i.e., the protons in the accelerated beam. Since there is no energy peak in the spectra at t < 280τ in Fig. 7(a), beam protons cannot be identified with the energy spectrum when the laser pulse propagates in the gas jet. Alternatively, we identify the proton beam with protons in the peak of the longitudinal momentum spectrum, with pz > pz0. This is reasonable because, as shown in Fig. 5, both before and after the injection of MVA, the protons in the accelerated beam have relatively high pz. A density plot of all energetic protons in pzsz space at t = 400τ is shown in Fig. 7(b), where the longitudinal momentum spectra of all protons and the beam protons are shown by solid red and dashed black curves, respectively. The lower limit of the longitudinal momentum of the proton beam is given by pz0 = 95%pz,peak, where pz,peak = 1.42mpc corresponds to the peak value of the momentum spectrum at t = 400τ. The number of beam protons can be calculated from the momentum spectrum as
(10)
Since the number of protons in the accelerated beam remains stable, one can consider protons with relatively high momentum, with the same number N0, as representative of the beam protons throughout the entire acceleration process. In other words, at times t ≠ 400τ, the lower limit of momentum pz0 is determined by Eq. (10) with N0 = N0|t=400τ, rather than pz,peak. The beam protons at t = 260τ and 280τ are depicted by the shaded areas of the momentum spectra in Fig. 7(c). This reveals that only a small fraction of protons at the momentum peak at t = 260τ, while still within the target, will be trapped into the final accelerated beam. The corresponding energy spectrum of the beam protons is plotted by a dashed black curve in Figs. 7(a) and 8(a). With the above definition of beam protons, protons of the same number N0 with pzpz0 are taken into account to calculate the beam polarization for the whole process of acceleration. The beam polarization s z is thus obtained, with P = 92%, 55%, and 57% at t = 200τ, 300τ, and 400τ, respectively.
FIG. 8.

(a) Density plot in E sz space of protons at t = 500τ, together with the energy spectra of all energetic protons (solid red curve) and the accelerated beam protons (dashed black curve). The magenta square and the green diamond indicate the locations of the two tracer particles, as in Fig. 4. (b) Longitudinal momentum spectra of accelerated protons at t = 500τ for density ramp lengths d = 20λ (solid curve) and 50λ (dashed curve). (c) Evolution of maximum energy (red curves) and beam polarization (blue curves) over time for d = 20λ (solid curves) and 50λ (dashed curves).

FIG. 8.

(a) Density plot in E sz space of protons at t = 500τ, together with the energy spectra of all energetic protons (solid red curve) and the accelerated beam protons (dashed black curve). The magenta square and the green diamond indicate the locations of the two tracer particles, as in Fig. 4. (b) Longitudinal momentum spectra of accelerated protons at t = 500τ for density ramp lengths d = 20λ (solid curve) and 50λ (dashed curve). (c) Evolution of maximum energy (red curves) and beam polarization (blue curves) over time for d = 20λ (solid curves) and 50λ (dashed curves).

Close modal

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 t1tt2, 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 sz 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 pz = 1.41mpc but different longitudinal spin components sz 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 sz 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 z2z < 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 sz 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 pz = 1.32mpc and 0.91mpc, respectively. If the beam protons are defined by the same lower limit of proton momentum pz0 = 95%pz,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 t2t1 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 ne = 0.72nc and z2z1 = 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 sz space. An energy peak is observed at 257 MeV. It is notable that the protons at the energy peak have a relatively large sz. 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 

FIG. 9.

(a) Density plot in E sz space of protons from the three-dimensional simulation at t = 500τ, together with the energy spectrum of all energetic protons (solid red curve). (b) Density plot in szθ space of accelerated protons within the FWHM of the energy peak in (a). The solid red curve shows the corresponding longitudinal spin spectrum and the dashed red curve the spectrum of protons within the FWHM of the energy peak for θ < 10°.

FIG. 9.

(a) Density plot in E sz space of protons from the three-dimensional simulation at t = 500τ, together with the energy spectrum of all energetic protons (solid red curve). (b) Density plot in szθ space of accelerated protons within the FWHM of the energy peak in (a). The solid red curve shows the corresponding longitudinal spin spectrum and the dashed red curve the spectrum of protons within the FWHM of the energy peak for θ < 10°.

Close modal

Figure 9(b) presents a density plot in szθ 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 sz. 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.

1.
C. N.
Danson
,
C.
Haefner
,
J.
Bromage
,
T.
Butcher
,
J.-C. F.
Chanteloup
et al, “
Petawatt and exawatt class lasers worldwide
,”
High Power Laser Sci Eng.
7
,
e54
(
2019
).
2.
Z.
Li
,
Y.
Leng
, and
R.
Li
, “
Further development of the short-pulse petawatt laser: Trends, technologies, and bottlenecks
,”
Laser Photonics Rev.
17
,
2100705
(
2022
).
3.
W.
Li
,
Z.
Gan
,
L.
Yu
,
C.
Wang
,
Y.
Liu
et al, “
339 J high-energy Ti:sapphire chirped-pulse amplifier for 10 PW laser facility
,”
Opt. Lett.
43
,
5681
5684
(
2018
).
4.
F.
Lureau
,
G.
Matras
,
O.
Chalus
,
C.
Derycke
,
T.
Morbieu
et al, “
High-energy hybrid femtosecond laser system demonstrating 2 × 10 PW capability
,”
High Power Laser Sci. Eng.
8
,
e43
(
2020
).
5.
G. A.
Mourou
,
T.
Tajima
, and
S. V.
Bulanov
, “
Optics in the relativistic regime
,”
Rev. Mod. Phys.
78
,
309
371
(
2006
).
6.
G.
Mourou
, “
Nobel lecture: Extreme light physics and application
,”
Rev. Mod. Phys.
91
,
030501
(
2019
).
7.
E.
Esarey
,
C. B.
Schroeder
, and
W. P.
Leemans
, “
Physics of laser-driven plasma-based electron accelerators
,”
Rev. Mod. Phys.
81
,
1229
1285
(
2009
).
8.
A.
Macchi
,
M.
Borghesi
, and
M.
Passoni
, “
Ion acceleration by superintense laser–plasma interaction
,”
Rev. Mod. Phys.
85
,
751
793
(
2013
).
9.
W.-J.
Ma
,
Z.-P.
Liu
,
P.-J.
Wang
,
J.-R.
Zhao
, and
X.-Q.
Yan
, “
Experimental progress of laser-driven high-energy proton acceleration and new acceleration schemes
,”
Acta Phys. Sin.
70
,
084102
(
2021
).
10.
F.
Wagner
,
O.
Deppert
,
C.
Brabetz
,
P.
Fiala
,
A.
Kleinschmidt
et al, “
Maximum proton energy above 85 MeV from the relativistic interaction of laser pulses with micrometer thick CH2 targets
,”
Phys. Rev. Lett.
116
,
205002
(
2016
).
11.
W. J.
Ma
,
I. J.
Kim
,
J. Q.
Yu
,
I. W.
Choi
,
P. K.
Singh
et al, “
Laser acceleration of highly energetic carbon ions using a double-layer target composed of slightly underdense plasma and ultrathin foil
,”
Phys. Rev. Lett.
122
,
014803
(
2019
).
12.
I. J.
Kim
,
K. H.
Pae
,
I. W.
Choi
,
C.-L.
Lee
,
H. T.
Kim
et al, “
Radiation pressure acceleration of protons to 93 MeV with circularly polarized petawatt laser pulses
,”
Phys. Plasmas
23
,
070701
(
2016
).
13.
W. P.
Wang
,
C.
Jiang
,
H.
Dong
,
X. M.
Lu
,
J. F.
Li
et al, “
Hollow plasma acceleration driven by a relativistic reflected hollow laser
,”
Phys. Rev. Lett.
125
,
034801
(
2020
).
14.
T.
Ziegler
,
I.
Göthel
,
S.
Assenbaum
,
C.
Bernert
,
F.-E.
Brack
et al, “
Laser-driven high-energy proton beams from cascaded acceleration regimes
,”
Nat. Phys.
20
,
1211
(
2024
).
15.
J.
Bin
, “
Petawatt pulse pushes protons
,”
Nat. Phys.
20
,
1055
(
2024
).
16.
F.
Sylla
,
M.
Veltcheva
,
S.
Kahaly
,
A.
Flacco
, and
V.
Malka
, “
Development and characterization of very dense submillimetric gas jets for laser–plasma interaction
,”
Rev. Sci. Instrum.
83
,
033507
(
2012
).
17.
I.
Prencipe
,
J.
Fuchs
,
S.
Pascarelli
,
D. W.
Schumacher
,
R. B.
Stephens
et al, “
Targets for high repetition rate laser facilities: Needs, challenges and perspectives
,”
High Power Laser Sci. Eng.
5
,
e17
(
2017
).
18.
V.
Ospina-Bohórquez
,
C.
Salgado-López
,
M.
Ehret
,
S.
Malko
,
M.
Salvadori
et al, “
Laser-driven ion and electron acceleration from near-critical density gas targets: Towards high-repetition rate operation in the 1 PW, sub-100 fs laser interaction regime
,”
Phys. Rev. Res.
6
,
023268
(
2024
).
19.
B.
Shen
,
Y.
Li
,
M. Y.
Yu
, and
J.
Cary
, “
Bubble regime for ion acceleration in a laser-driven plasma
,”
Phys. Rev. E
76
,
055402
(
2007
).
20.
X.
Zhang
,
B.
Shen
,
L.
Zhang
,
J.
Xu
,
X.
Wang
et al, “
Proton acceleration in underdense plasma by ultraintense Laguerre–Gaussian laser pulse
,”
New J. Phys.
16
,
123051
(
2014
).
21.
C. A. J.
Palmer
,
N. P.
Dover
,
I.
Pogorelsky
,
M.
Babzien
,
G. I.
Dudnikova
et al, “
Monoenergetic proton beams accelerated by a radiation pressure driven shock
,”
Phys. Rev. Lett.
106
,
014801
(
2011
).
22.
D.
Haberberger
,
S.
Tochitsky
,
F.
Fiuza
,
C.
Gong
,
R. A.
Fonseca
et al, “
Collisionless shocks in laser-produced plasma generate monoenergetic high-energy proton beams
,”
Nat. Phys.
8
,
95
99
(
2012
).
23.
T.
Nakamura
,
S. V.
Bulanov
,
T. Z.
Esirkepov
, and
M.
Kando
, “
High-energy ions from near-critical density plasmas via magnetic vortex acceleration
,”
Phys. Rev. Lett.
105
,
135002
(
2010
).
24.
J.
Park
,
S. S.
Bulanov
,
J.
Bin
,
Q.
Ji
,
S.
Steinke
et al, “
Ion acceleration in laser generated megatesla magnetic vortex
,”
Phys. Plasmas
26
,
103108
(
2019
).
25.
I.
Tazes
,
S.
Passalidis
,
E.
Kaselouris
,
D.
Mancelli
,
C.
Karvounis
et al, “
Efficient magnetic vortex acceleration by femtosecond laser interaction with long living optically shaped gas targets in the near critical density plasma regime
,”
Sci. Rep.
14
,
4945
(
2024
).
26.
T.
Roser
,
L.
Ahrens
,
J.
Alessi
,
M.
Bai
,
J.
Beebe-Wang
et al, “
First polarized proton collisions at RHIC
,”
AIP Conf. Proc.
667
,
1
8
(
2003
).
27.
M.
Burkardt
,
C. A.
Miller
, and
W.-D.
Nowak
, “
Spin-polarized high-energy scattering of charged leptons on nucleons
,”
Rep. Prog. Phys.
73
,
016201
(
2009
).
28.
Y.
Zhang
, “
JLEIC: A high luminosity polarized electron-ion collider at Jefferson lab
,” in
10th International Particle Accelerator Conference (IPAC2019),
Melbourne, Australia
, May 2019 (JACoW, 2019), paper TUPRB112, pp.
1916.10.18429
/JACoW-NAPAC2019-TUPLO02.
29.
I.
Alekseev
,
C.
Allgower
,
M.
Bai
,
Y.
Batygin
,
L.
Bozano
et al, “
Polarized proton collider at RHIC
,”
Nucl. Instrum. Methods Phys. Res., Sect. A
499
,
392
414
(
2003
).
30.
J.
Martinez Marin
,
B.
Mustapha
, and
L.
Spentzouris
, “
Spin dynamics in the JLEIC ion injector linac
,” in
3rd North American Particle Accelerator Conference (NAPAC2019),
Lansing, MI
, Sep. 2019 (JACoW, 2019), paper TUPLO02, pp.
533.10.18429
/JACoW-NAPAC2019-TUPLO02.
31.
B.
Mustapha
,
Z.
Conway
,
M.
Kelly
,
A.
Plastun
, and
P.
Ostroumov
, “
Design of the multi-ion injector linac for the JLAB EIC (JLEIC)
,”
J. Phys. Conf.
1401
,
012008
(
2020
).
32.
Y.-F.
Li
,
R.
Shaisultanov
,
K. Z.
Hatsagortsyan
,
F.
Wan
,
C. H.
Keitel
et al, “
Ultrarelativistic electron-beam polarization in single-shot interaction with an ultraintense laser pulse
,”
Phys. Rev. Lett.
122
,
154801
(
2019
).
33.
A. S.
Samsonov
,
E. N.
Nerush
, and
I. Y.
Kostyukov
, “
High-order corrections to the radiation-free dynamics of an electron in the strongly radiation-dominated regime
,”
Matter Radiat. Extremes
8
,
014402
(
2022
).
34.
S.
Tang
,
Y.
Xin
,
M.
Wen
,
M. A.
Bake
, and
B.
Xie
, “
Fully polarized Compton scattering in plane waves and its polarization transfer
,”
Matter Radiat. Extremes
9
,
037204
(
2024
).
35.
M.
Wen
,
M.
Tamburini
, and
C. H.
Keitel
, “
Polarized laser-wakefield-accelerated kiloampere electron beams
,”
Phys. Rev. Lett.
122
,
214801
(
2019
).
36.
Z.
Gong
,
M. J.
Quin
,
S.
Bohlen
,
C. H.
Keitel
,
K.
Põder
et al, “
Spin-polarized electron beam generation in the colliding-pulse injection scheme
,”
Matter Radiat. Extremes
8
,
064005
(
2023
).
37.
A.
Hützen
,
J.
Thomas
,
J.
Böker
,
R.
Engels
,
R.
Gebel
et al, “
Polarized proton beams from laser-induced plasmas
,”
High Power Laser Sci. Eng.
7
,
e16
(
2019
).
38.
L.
Jin
,
M.
Wen
,
X.
Zhang
,
A.
Hützen
,
J.
Thomas
et al, “
Spin-polarized proton beam generation from gas-jet targets by intense laser pulses
,”
Phys. Rev. E
102
,
011201
(
2020
).
39.
D.
Sofikitis
,
C. S.
Kannis
,
G. K.
Boulogiannis
, and
T. P.
Rakitzis
, “
Ultrahigh-density spin-polarized H and D observed via magnetization quantum beats
,”
Phys. Rev. Lett.
121
,
083001
(
2018
).
40.
A. K.
Spiliotis
,
M.
Xygkis
,
M. E.
Koutrakis
,
K.
Tazes
,
G. K.
Boulogiannis
et al, “
Ultrahigh-density spin-polarized hydrogen isotopes from the photodissociation of hydrogen halides: New applications for laser-ion acceleration, magnetometry, and polarized nuclear fusion
,”
Light: Sci. Appl.
10
,
35
(
2021
).
41.
L.
Reichwein
,
A.
Pukhov
, and
M.
Büscher
, “
Acceleration of spin-polarized proton beams via two parallel laser pulses
,”
Phys. Rev. Accel. Beams
25
,
081001
(
2022
).
42.
X.
Yan
,
Y.
Wu
,
X.
Geng
,
H.
Zhang
,
B.
Shen
et al, “
Enhanced polarized proton acceleration driven by femtosecond laser pulses irradiating a micro-structured solid–gas target
,”
Plasma Phys. Controlled Fusion
65
,
035005
(
2023
).
43.
X. F.
Li
,
P.
Gibbon
,
A.
Hützen
,
M.
Büscher
,
S. M.
Weng
et al, “
Polarized proton acceleration in ultraintense laser interaction with near-critical-density plasmas
,”
Phys. Rev. E
104
,
015216
(
2021
).
44.
B.
Liu
,
J.
Meyer-ter-Vehn
,
K.-U.
Bamberg
,
W. J.
Ma
,
J.
Liu
et al, “
Ion wave breaking acceleration
,”
Phys. Rev. Accel. Beams
19
,
073401
(
2016
).
45.
B.
Liu
,
J.
Meyer-ter-Vehn
, and
H.
Ruhl
, “
Self-trapping and acceleration of ions in laser-driven relativistically transparent plasma
,”
Phys. Plasmas
25
,
103117
(
2018
).
46.
J. W.
Yoon
,
Y. G.
Kim
,
I. W.
Choi
,
J. H.
Sung
,
H. W.
Lee
et al, “
Realization of laser intensity over 1023 W/cm2
,”
Optica
8
,
630
635
(
2021
).
47.
P.
Wang
,
Z.
Gong
,
S. G.
Lee
,
Y.
Shou
,
Y.
Geng
et al, “
Super-heavy ions acceleration driven by ultrashort laser pulses at ultrahigh intensity
,”
Phys. Rev. X
11
,
021049
(
2021
).
48.
J.-R.
Marquès
,
L.
Lancia
,
P.
Loiseau
,
P.
Forestier-Colleoni
,
M.
Tarisien
et al, “
Collisionless shock acceleration of protons in a plasma slab produced in a gas jet by the collision of two laser-driven hydrodynamic shockwaves
,”
Matter Radiat. Extremes
9
,
024001
(
2023
).
49.
O.
Seemann
,
Y.
Wan
,
S.
Tata
,
E.
Kroupp
, and
V.
Malka
, “
Laser proton acceleration from a near-critical imploding gas target
,”
Phys. Rev. Lett.
133
,
025001
(
2024
).
50.
B.
Liu
,
B.
Lei
,
Y.
Gao
,
M.
Wen
, and
K.
Zhu
, “
Plasma opacity induced by laser-driven movement of background ions
,”
Plasma Phys. Controlled Fusion
66
,
115004
(
2024
).
51.
A. B.
Borisov
,
A. V.
Borovskiy
,
V. V.
Korobkin
,
A. M.
Prokhorov
,
O. B.
Shiryaev
et al, “
Observation of relativistic and charge-displacement self-channeling of intense subpicosecond ultraviolet (248 nm) radiation in plasmas
,”
Phys. Rev. Lett.
68
,
2309
2312
(
1992
).
52.
X. F.
Shen
,
A.
Pukhov
,
O. N.
Rosmej
, and
N. E.
Andreev
, “
Cross-filament stochastic acceleration of electrons in kilojoule picosecond laser interactions with near-critical-density plasmas
,”
Phys. Rev. Appl.
18
,
064091
(
2022
).
53.
Y.-J.
Gu
and
S. V.
Bulanov
, “
Magnetic field annihilation and charged particle acceleration in ultra-relativistic laser plasmas
,”
High Power Laser Sci. Eng.
9
,
e2
(
2021
).
54.
A.
Pukhov
and
J.
Meyer-ter-Vehn
, “
Relativistic magnetic self-channeling of light in near-critical plasma: Three-dimensional particle-in-cell simulation
,”
Phys. Rev. Lett.
76
,
3975
3978
(
1996
).
55.
T. D.
Arber
,
K.
Bennett
,
C. S.
Brady
,
A.
Lawrence-Douglas
,
M. G.
Ramsay
et al, “
Contemporary particle-in-cell approach to laser-plasma modelling
,”
Plasma Phys. Controlled Fusion
57
,
113001
(
2015
).
56.
S. S.
Bulanov
,
V. Y.
Bychenkov
,
V.
Chvykov
,
G.
Kalinchenko
,
D. W.
Litzenberg
et al, “
Generation of GeV protons from 1 PW laser interaction with near critical density targets
,”
Phys. Plasmas
17
,
043105
(
2010
).
57.
P. K.
Singh
,
V. B.
Pathak
,
J. H.
Shin
,
I. W.
Choi
,
K.
Nakajima
et al, “
Electrostatic shock acceleration of ions in near-critical-density plasma driven by a femtosecond petawatt laser
,”
Sci. Rep.
10
,
18452
(
2020
).
58.
K. I.
Popov
,
W.
Rozmus
,
V. Y.
Bychenkov
,
N.
Naseri
,
C. E.
Capjack
et al, “
Ion response to relativistic electron bunches in the blowout regime of laser-plasma accelerators
,”
Phys. Rev. Lett.
105
,
195002
(
2010
).
59.
L. L.
Ji
,
A.
Pukhov
,
I. Y.
Kostyukov
,
B. F.
Shen
, and
K.
Akli
, “
Radiation-reaction trapping of electrons in extreme laser fields
,”
Phys. Rev. Lett.
112
,
145003
(
2014
).
60.
B.
Liu
,
J.
Meyer-ter-Vehn
,
H.
Ruhl
, and
M.
Zepf
, “
Front edge velocity of ultra-intense circularly polarized laser pulses in a relativistically transparent plasma
,”
Plasma Phys. Controlled Fusion
62
,
085014
(
2020
).
61.
I.
Göthel
,
C.
Bernert
,
M.
Bussmann
,
M.
Garten
,
T.
Miethlinger
et al, “
Optimized laser ion acceleration at the relativistic critical density surface
,”
Plasma Phys. Controlled Fusion
64
,
044010
(
2022
).
62.
B.
Liu
,
M.
Shi
,
M.
Zepf
,
B.
Lei
, and
D.
Seipt
, “
Accelerating ions by crossing two ultraintense lasers in a near-critical relativistically transparent plasma
,”
Phys. Rev. Lett.
129
,
274801
(
2022
).
63.
A.
Pukhov
,
Z.-M.
Sheng
, and
J.
Meyer-ter-Vehn
, “
Particle acceleration in relativistic laser channels
,”
Phys. Plasmas
6
,
2847
2854
(
1999
).
64.
T. W.
Huang
,
C. T.
Zhou
,
H.
Zhang
,
S. Z.
Wu
,
B.
Qiao
et al, “
Relativistic laser hosing instability suppression and electron acceleration in a preformed plasma channel
,”
Phys. Rev. E
95
,
043207
(
2017
).
65.
S.
Semushin
and
V.
Malka
, “
High density gas jet nozzle design for laser target production
,”
Rev. Sci. Instrum.
72
,
2961
2965
(
2001
).
66.
K.
Schmid
and
L.
Veisz
, “
Supersonic gas jets for laser-plasma experiments
,”
Rev. Sci. Instrum.
83
,
053304
(
2012
).
67.
S.
Lorenz
,
G.
Grittani
,
E.
Chacon-Golcher
,
C. M.
Lazzarini
,
J.
Limpouch
et al, “
Characterization of supersonic and subsonic gas targets for laser wakefield electron acceleration experiments
,”
Matter Radiat. Extremes
4
,
015401
(
2019
).
68.
I.
Engin
,
Z. M.
Chitgar
,
O.
Deppert
,
L. D.
Lucchio
,
R.
Engels
et al, “
Laser-induced acceleration of helium ions from unpolarized gas jets
,”
Plasma Phys. Controlled Fusion
61
,
115012
(
2019
).
69.
O.
Zhou
,
H.-E.
Tsai
,
T. M.
Ostermayr
,
L.
Fan-Chiang
,
J.
van Tilborg
et al, “
Effect of nozzle curvature on supersonic gas jets used in laser–plasma acceleration
,”
Phys. Plasmas
28
,
093107
(
2021
).
70.
Z.-Z.
Lei
,
Y.-J.
Gu
,
Z.
Jin
,
S.
Sato
,
A.
Zhidkov
et al, “
Supersonic gas jet stabilization in laser–plasma acceleration
,”
High Power Laser Sci. Eng.
11
,
e91
(
2023
).
71.
Z.
Lécz
,
R.
Polanek
,
A.
Andreev
,
A.
Sharma
,
D.
Papp
et al, “
Hybrid acceleration of compact ion bunches by few-cycle laser pulses in gas jets of two atomic species
,”
Phys. Rev. Res.
5
,
023169
(
2023
).
72.
D. J.
Stark
,
L.
Yin
,
B. J.
Albright
, and
F.
Guo
, “
Effects of dimensionality on kinetic simulations of laser-ion acceleration in the transparency regime
,”
Phys. Plasmas
24
,
053103
(
2017
).