Quasi-monoenergetic carbon ions generation from a double-layer target driven by extreme laser pulses

High quality energetic carbon ions produced via laser-plasma have many applications in tumor therapy, fast ignition and warm dense matter generation. However, the beam achieved in current experiments is still limited by either a large energy spread or a low peak energy. In this paper, a hybrid scheme for the generation of quasi-monoenergetic carbon ions is proposed by an ultra-intense laser pulse irradiating a double-layer target. Multi-dimensional particle-in-cell (PIC) simulations show that the carbon ions are first accelerated via laser piston mechanism in the former carbon layer and then further accelerated by Coulomb repulsion force in the attached neon target. Since electrons are bunched synchronously in longitudinal and transverse direction by radiation reaction during the whole acceleration process, a quasi-monoenergetic carbon ion beam is eventually produced. In the following stage, the neon target provides the Coulomb field required for the continuous acceleration of the carbon ions which helps to prevent the carbon ion layer from diffusion. It is demonstrated that quasi-monoenergetic carbon ions with peak energy of 465 MeV u−1, energy spread of ∼13%, a divergence of ∼15∘, and laser-to-ion energy conversion of 20% can be achieved by using a laser pulse with intensity of 1.23 × 1023 W cm−2. An analytical model is also proposed to interpret the carbon ion acceleration, which is fairly consistent with the PIC simulations.


Introduction
Laser-driven quasi-monoenergetic ion beams have quite important applications in high energy density physics [1], medical therapy [2,3] and inertial confinement fusion [4,5] due to their unique properties of ultrashort duration and localized energy deposition. For example, it is found that quasi-monoenergetic carbon ions can lower ignition energies about 25% as compared to protons with a Maxwellian distribution [6]. Especially, a laser-driven quasi-monoenergetic ion beam can solve the problem of nonuniform heating of warm dense matter (WDM) samples with exponential energy spectrum [7]. In addition, high quality ion beam is an effective way to kill cancer cells because of the Bragg peak. Especially, tumor therapy requires ion energy of more than 400 MeV u −1 for carbon ions [8] which have higher relative biological effectiveness than protons. At present, a novel concept for radiotherapy FLASH, which can reduce the side effects on healthy cells due to very high dose rates, receives increasing attention [9,10] and FLASH radiotherapy with carbon ions has been demonstrated in vivo [11]. Thus, the generation of high quality carbon ions is in high demand.
In the past few decades, various acceleration mechanisms have been put forward for quasi-monoenergetic ion acceleration from the interaction of a laser pulse with solid targets. The main acceleration regimes include radiation pressure dominant (RPD) acceleration, relativistic induced transparent (RIT) acceleration and Coulomb repulsion (CR) acceleration, etc. In RIT [12][13][14][15], although the ion beams get accelerated as quasi-monoenergetic in the early evolution, the energy spectrum usually diverges quickly [16]. Moreover, double-layer target in CR regime is demonstrated to be an effective method to obtain monoenergetic ion beams as well [17][18][19][20]. However, due to the radial distribution of the Coulomb field by repulsion of ion cores, the ion divergence is generally large. The RPD regime also promises the generation of high-energy quasi-monoenergetic ion beams with a high conversion efficiency [21][22][23][24], while there are high requirements for the matching condition of laser pulse and target. In recent years, the hybrid acceleration of ions has attracted great attention and has been demonstrated in theory and experiments [18,[25][26][27]. For instance, a quasi-monoenergetic proton beams can be generated through combining RPD and laser wake-field acceleration [28]. Experimentally, near 100 MeV proton beam can be obtained by a hybrid acceleration mechanism with the combination of RPD and target normal sheath acceleration using a linearly polarized laser pulse [29][30][31]. However, the laser-to-ion energy conversion efficiency is typically only a few percent and the energy spread is typically Boltzmann distribution in the experiments. Moreover, the achieved maximum energy in experiments [32] is limited by ∼48 MeV u −1 for carbon-ions. These are still far away for the demand of the particular applications.
With the rapid developments of laser techniques [33,34], laser-plasma interaction has entered the extreme laser regime. Nowadays, the peak intensities of laser pulses have achieved 10 23 W cm −2 in the laboratory [35]. Meanwhile, high power laser facilities, such as ELI [36], XCELS [37] and SEL [38], are expected to get the laser power higher than hundred-PW. With such extremely-intense laser pulse, the influence of quantum electrodynamics effects must be considered, including the production of pairs, radiation reaction (RR) and so on [39][40][41][42], which can strongly influence the motion of electrons and further the ion acceleration. It is a great concern about how these influence the laser-driven ion acceleration and how quasi-monoenergetic ion beams can be obtained under extreme laser pulse. Over the past years, RR has been studied extensively in the laser-plasma community [43][44][45][46]. It has been found that RR is quite significant in the near-critical density plasma [47], and has more impact on the proton acceleration for a linearly polarized laser pulse than a circularly polarized one because the RR force has increased by four orders of magnitude due to the J × B driven longitudinal oscillations [48][49][50]. Recently, it has been reported that as a circularly polarized drive laser with intensity of 8.56 × 10 23 W cm −2 collides with another intense linearly polarized laser pulse of the same intensity, a GeV monoenergetic proton beam can be produced via RPD and quantum radiative compression method [51].
In this paper, we proposed a hybrid acceleration regime that combines laser piston (LP) and CR. In this regime, a linearly polarized laser pulse of intensity about 10 23 W cm −2 is incident on a carbon and neon double-layer target. At first, the carbon ions are accelerated to high energies through LP, and the neon target, as a buffer layer together with RR, prevents the diffusion of carbon ions and electrons in space. Subsequently, the carbon ions are further accelerated by the CR while electrons are bunched by radiation reaction force, so that a higher density carbon and electron layer can propagate over a longer distance. During the whole interaction process, carbon ions stay in the acceleration phase, and the radiation reaction effects play a positive role for the ion beam acceleration and confinement. Finally, a quasi-monoenergetic carbon ion beam can be generated with peak energy of ∼450 MeV u −1 , which may find wide applications in tumor therapy, fast ignition and WDM generation.

Theoretical model
The schematic diagram of the scheme is shown in figure 1(a). Figures 1(b)-(d) show the number density evolution of carbon and neon ions in the double-layer targets. A one-dimensional model is developed to describe the whole acceleration process, which can be divided into two stages. The first stage can be described by the LP. A linearly polarized laser-irradiated carbon foil is pushed forward by the radiation pressure, because the laser pressure is much greater than the thermal pressure when the laser intensity is significantly greater than the relativistic threshold [52], as shown by the green dots in figures 1(b) and (c). The second stage can be described by a double-layer CR model [53,54]. The CR occurs because of the excess of positive charges near the end of the laser pulse when the carbon ions pass through the neon target, inducing the space charge separation field. Meanwhile, electrons are bunched transversely and longitudinally by radiation reaction force, which contributes to lower both the ion divergence and energy spread. When the ultra-intense laser pulse expels electrons from the neon target, the remaining neon ions lead to a strong CR electric field that can accelerate the carbon ions. The presence of the second neon target prevents the relativistic transparency of the ultra-thin carbon target, and prolongs the carbon ion acceleration. Besides, the spatial diffusion of the carbon ions can be also suppressed, forming a tight ion layer moving forward with the laser pulse. During this stage, the drive laser pulse pushes the electrons out of the neon target, while the neon ions receive less impetus and remain inside the target, as shown in figures 1(b)-(d).
In order to achieve stable acceleration of the carbon ions, an appropriate thickness of the neon target is required. On the one hand, the thickness cannot be too thin, ensuring sufficient time for laser interaction with the neon target. Otherwise, the interaction enters the relativistic transparent regime, in which electrons and neon ions (orange) at t = 15T0, 27T0 and 30T0 respectively. The blue curve is the carbon density at on-axis, and the black curve is the electron density at on-axis. and ions diffuse in space rapidly. Therefore, the minimum thickness of the neon target should satisfy L ⩾ ct 1 , where t 1 is the time of relativistic transparency for the single-layer carbon target [55]: where N = n e /n c is the normalized electron density, n c = m e ω 2 0 /(4π e 2 ) is the critical density, a 0 = eE 0 /m e cω 0 is the normalized laser intensity, ω 0 is the laser frequency, τ is the laser pulse duration and l is the target thickness, C s ∼ = (Zm e c 2 a 0 /M i ) 1/2 , and M i is the ion mass. On the other hand, the thickness of target cannot be too thick, which requires L ⩽ 20λ in simulations, guaranteeing the carbon layer to pass through the neon target and further accelerated by CR force. Moreover, in the double-layer CR model, the optimal thickness of the accelerated ion layer satisfies [53]: In the LP stage, the radiation pressure depends on the laser reflectance [56]. Assuming the relative amplitudes of reflected wave to be ρ and the frequency of the reflected and incident laser pulse to be ω ′ and ω, respectively. The radiation pressure is written by P = ( . Assuming a complete laser light reflection by the foil, the foil motion equation can be written as [52] dp where p is the momentum of ions, and c is the speed of light in vacuum. Considering the initial condition In addition, the ion energy in the CR stage can be estimated by ε CR =´eE c dx, where E c [57] is the electric field from CR, where a 1 = 2π Z i en i L, and a 2 = 2π n i Z 2 i e 2 /M i . Here Z i , n i are the ion charges and density, respectively. In our scheme, the energetic carbon ions cross the second layer of the neon target, and the CR is mainly provided by neon ions behind the carbon layer. It should be noted that a portion of electrons pass through the second layer together with the laser pulse, while some remain inside the target, as shown in figure 1(d), which diminish the CR effects. Here, we let σ denote the proportion of electrons being expelled, and the ion energy obtained after the two acceleration stages is approximated by We know that under extremely-intense laser field conditions, a large number of photons can be radiated during the acceleration, so that the RR effects cannot be ignored any more. It is shown by simulations that the electrons are bunched longitudinally and transversely by the RR, so n e increases instead of decreasing, which can slow down the electron and ion diffusion processes. On one hand, the incident laser interacts with the electrons in the target and pushes the opaque carbon layer forward at stage I. On the other hand, the laser pulse passes through and keeps away from remaining electrons inside the target at stage II, so that the incident laser does not interact with the electrons in the target. Thus the RR effects mainly affect the electrons remaining in the target at stage I and the electrons moving forward outside the target with the carbon ions at stage II. With the RR effects considered, the growth trend of the maximum carbon ion energy remains the same, but the energy spectrum is compressed, resulting in a quasi-monoenergetic carbon ion beam generation.

Results and discussion
To investigate the carbon ion acceleration process and the electron dynamics, we carry out two-dimensional particle-in-cell (PIC) simulations with the open-source code EPOCH [58]. The simulation box is x × y = 50 µm × 15 µm, sampled by 10 000 × 750 grid cells involving 25 macro-particles per cell for all species. The boundary conditions are absorbing for both particles and fields. A p-polarized laser pulse is incident normally from the left side of the simulation box, and has a Gaussian profile of a 0 exp(−y 2 /w 2 0 ), where w 0 = 6.5 µm is the laser focus spot size. a 0 = 300 is the normalized laser amplitude, corresponding to the laser intensity of 1.23 × 10 23 W cm −2 . The laser central wavelength is λ 0 = 1 µm. The temporal profile of the laser complies with the 'trapezoidal' distribution (rising-plateau-falling) with a duration of 17T 0 (1T 0 − 15T 0 − 1T 0 ), where T 0 ≈ 3.33 fs represents the laser period. The target is a composite one consisting of the front layer of carbon ions and the following layer of neon ions. The carbon layer is located at x = 10λ 0 with the initial electron density of n e1 = 660n c , and the thickness of d 1 = 100 nm. The parameter setting satisfies the condition E l > 2π en e l, to ensure that the carbon ions can be accelerated effectively by the laser beam and the laser pulse could interact with the neon target. The density of gas target depends mainly on the gas jet pressure and temperature. In our scheme, the underdense neon target has a length of 10λ 0 which is greater than the minimum target thickness of L min ≈3.7λ 0 from equation (1), and density of n e2 = 10n c which can be realized by lowering the temperature or increasing the pressure of gas target in laboratories.
The red curve in figure 2(a) shows the time evolution of carbon ion energy spread ∆ε/ε peak , where ε peak is the peak energy, and ∆ε is the full width at half-maximum of the quasi-monoenergetic spectrum. At the stage I, carbon ions are first accelerated in quasi-monoenergetic manner. Besides, we notice that there exists a small peak between 10T 0 -20T 0 in figure 2(a). It originates from the development of transverse instabilities. As the relativistic laser pulse irradiates a thin foil, its strong radiation pressure pushes the foil forward at t = 15T 0 . Meanwhile the Rayleigh-Taylor-like (RT-like) instability will develop very rapidly since the laser pulse has a Gaussian transverse distribution in intensity in our scheme. At a later time of t = 18T 0 , the foil damage from the development of transverse instabilities leads to the deterioration of energy spread. Thus, ∆ε becomes larger and ∆ε/ε peak is correspondingly increased. As time goes on, ∆ε improves only slightly, but ε peak increases rapidly under the action of laser radiation pressure. Therefore, ∆ε/ε peak begins to rise quickly and thus there is a small spike in the temporal evolution of ∆ε/ε peak . Furthermore, the following acceleration by CR in the stage II leads to ion diffusion. Such a deterioration is suppressed when the effects of RR are considered. As the RR force increases in stage II, RR effects become important and the ion energy spread is determined by the competition between the CR and the RR as displayed by the curve near the dash line in figure 2(a). Finally, the carbon ions can propagate stably forward as a quasi-monoenergetic beam since the RR dominates over the CR force. We assume the electrons of neon target are completely expelled when the carbon ions just pass through the neon target at the end of the laser pulse, and take σ = 0.34 from our simulations. Figure 2(b) shows the comparison between the theorical model (equation 5) and the simulations. As expected, the maximum energy of the carbon ions agrees well with the theoretical predictions. The evolution of the energy conversion efficiency η is illustrated in figure 2(c). Apparently, it increases rapidly in stage I and gradually saturates with a maximum of over 20% in stage II. Correspondingly, the energy spread is reduced significantly from 100% to 13.7% with a final center energy of approximate 5.4 GeV as seen in figure 2(d). In addition, we calculated the divergence angle of carbon ions within the laser focus spot size, defined by arctan(p y /p x ), indicating a carbon ions beam with a divergence angle of ∼15 • .
In order to explore the ion acceleration process, we perform additional simulations with a single-layer carbon target (SLT) by using the same laser parameters. Figure 3 present the spatial distributions of the background electrons and ions after shining by the laser beam. As seen from figures 3(a) and (b), the carbon target remains opaque in the double-layer target (DLT) case, and the LP dominates at the first stage. The electron diffusion is also prevented by the buffering effect of neon target and RR. Especially, the spatial distribution of carbon ions exactly follow the high energy electrons, forming a compact layer of thin thickness. The rebuilt phase-space (p y − p x ) illustrates that the carbon ion layer displays a small spatial divergence, and is stably accelerated at the first stage (shown by the dashed circle marked by A in figure 3(d)). When the laser pulse ends at t = 27T 0 , the carbon layer just passes through the neon target. At this moment, the electron density and size of the carbon layer are 236.3n c and 0.5λ 0 , respectively, satisfying the optimal thickness condition for CR (equation (2)). Hence, the carbon ions are continuously accelerated in the second stage, as shown by the red curve in figure 3(f). As the CR dominates, some high-energy electrons move forward with carbon ions, while the remaining low-energy electrons are focused in the neon target. These electrons are bunched and trapped in the laser field by RR and radiate high energy photons as seen in figure 3(c). Meanwhile, the electric field is enhanced considerably and the carbon layer is therefore tighten. In SLT case, although the carbon ions are accelerated in quasi-monoenergetic manner resembling to the DLT case, they undergo spatial diffusion earlier and become relativistic transparent to the incident laser, as shown by the blue curve in figures 3(a), (e) and (f). Finally, the energy spectrum of carbon ions are of Boltzmann-like distribution [16]. By comparison, the energy spread of carbon ions in the DLT case is significantly reduced. Figure 4(a) shows the evolution of carbon ions (red) and neon ions (black) over time. Because of the extremely high velocity obtained in the first stage, a large number of carbon ions with high energy pass through the underdense neon target. When the laser pulse passes through the neon target, CR plays an important role due to the excess of positive charge. The carbon ions are divided into forward and backward parts, and the forward carbon ions are compressed into an over-density layer with a small longitudinal size. Meanwhile, electrons are also divided into high energy parts outside the target and low energy parts inside  Figure 4(b) shows the corresponding average electric field evolution over time, in which the red positive field results from the high density carbon layer in figure 4(a). The forward carbon ion layer can stay in the acceleration phase throughout the following process, as shown in figure 4(d). As a result, the quasi-monoenergetic characteristic of carbon ion beam can be maintained for a long time.
When the laser intensity is over 10 23 W cm −2 , the effects of RR on the electrons must be considered. Here, the motion equation of electron can be simplified in principle as dp/dt In consequence, the importance of quantum RR can be characterized by the quantum parameter 2 . It has been demonstrated that the quantum corrections cannot be ignored when χ e > 0.1 [59,60]. In order to identify the role of RR effects in the ion acceleration process in our scheme, we switch off the RR artificially in additional simulations for comparison. Especially, we trace the  electrons in both cases with RR and without RR, as shown in figures 5(a) and (b). It is shown that the electrons can remain in the laser field for a longer time with RR considered, while they diffuse in space quickly when ignoring the RR. Finally, a tighter electron bunch both in longitudinal and transverse direction Meanwhile, the RR effects outside the target become more and more significant. The trend is also clearly reflected by RR force as shown in figure 5(e). In order to explain the effects of RR, we also calculate the CR force F CR = 2π Z i e 2 n i L/(1 + 2π n i Z 2 i e 2 t 2 /M i ). Figure 5(e) shows that F L , F CR and F RR are on the same order of magnitude, which demonstrates the RR play an important role. Here, F RR outside the target increases while the F CR decreases in stage II especially after t = 32T 0 , which has been reflected in the energy spread as shown in figure 2(a). In addition, the spectrum of carbon ions is optimized remarkably by the RR effects. As seen from figure 5(g), the carbon ions are quasi-monoenergetic when switching on the RR. This has been already proved in figure 2(d).
To check the robustness of the scheme, three-dimensional (3D) PIC simulations are performed with a reduced laser focus spot size of w 0 = 3λ 0 . The simulation window is x × y × z = 50 µm × 5 µm × 5 µm, divided by 10 000 × 50 × 50 grid cells with 8 macro-particles per cell for all species. Other parameters are the same as in the 2D simulations. The results are consistent with obtained in the 2D case. Figure 6(a) shows the average electric field and carbon ion number density on the axis at t = 32T 0 . It is clearly seen that the carbon ions are divided into backward and forward ones during the interaction process. Among them, the forward parts move in the accelerating phase of the electric field of the CR. Figure 6(b) presents the energy spectrum of carbon ions. It is demonstrated that a quasi-monoenergetic carbon ion beam is obtained in the 3D simulation, with a peak energy of 2.4 GeV and an energy spread of 19%. Note that the peak energy of the carbon ions is much lower than in the 2D simulation because of the multi-dimension effects, which has been extensively investigated [21,24,40].

Effects of laser and attached target parameters
In order to study the influence of target parameters and laser intensity on the carbon ion energy, we scan the simulation parameters including the thickness d and density n of the neon target. The thickness of the second target ranges from 0 to 14 µm and density from 0 to 20n c . The simulation results in figure 7(a) show that there is a minimal target thickness L min ⩾ 3.5λ 0 required for efficient carbon ion acceleration, which agrees well with the minimum target thickness 3.7λ 0 calculated from equation (1). In addition, figure 7(b) shows the relationship between the laser-to-carbon ion energy conversion efficiency η, the laser intensity a (black curve) and the charge-mass ratio of the second target (blue curve in insert figure). With the increase of laser intensity, the conversion efficiency shows a trend of decrease after the sharp increase at first. This may be due to the fact that the acceleration process is dominated by different mechanisms when the laser intensity a 0 < π(n 0 /n c )(l/λ). In addition, the red curve in figure 7(b) shows that the maximum carbon ion energy increases as the drive laser intensity increase, providing a way to enhance the carbon ion energies further in a direct manner. Moreover, the insert figure shows that their influences on the conversion efficiency are not serious when the charge-mass ratio is 1:1. The reason is that when the second target is a proton target, the high-energy carbon layer cannot pass through the proton layer completely, but push partial protons forward, so that the CR as detailed above decreases significantly.  For the radiation pressure acceleration, it is favorable to use a circularly polarized laser pulse with sharp time rising front and transversely uniform intensity distribution to suppress the rapid development of transverse instabilities [26,61].To demonstrate the robustness and tolerance of our scheme, we performed additional PIC simulations using Gaussian and super-Gaussian laser pulse in time and kept other parameters the same as the trapezoidal laser, as shown in figure 8. We find that it still has a quasi-monoenergetic ion energy peak in super-Gaussian case, which is comparable to that with a trapezoidal time profile. For Gaussian laser pulse, the peak energy is lower due to the less laser energy. The higher energy of the ions can also be obtained by Gaussian shaped pulse of the same laser energy as the trapezoidal laser. This indicates that our scheme is therefore still applicable in a more general case. For high quality ions beam, it is very important to maintain its beam quality and charge for long distance transmission. For instance, a carbon ion beam with small divergence angle can achieve more localized energy deposition inside the cancer cells area, avoiding the damage to the surrounding healthy cells. Some optimizing methods can be used to alleviate emittance of the ion beam which can be effectively manipulated so that the ion beam can travel longer distances, such as novel target designs, laser-driven micro-lens and external electrostatic fields [1,62,63].

Conclusion
In summary, we provide an efficient hybrid acceleration scheme to produce quasi-monoenergetic carbon ions through ultra-intense laser-driven carbon and neon double-layer target. The carbon ions are significantly accelerated through the combined LP and CR. The resultant peak energy and energy spread can reach 464 MeV u −1 and 13.7%, respectively. The maximum ion energy agrees well with the theoretical model. Compared to the simple single planar target, much intense CR electric field and RR effects occur, leading to over 20% laser-to-ion energy conversion efficiency. In addition, a parameter scan over different neon target thicknesses and laser intensity demonstrates that there exists a minimal neon thickness for the hybrid scheme. This method offers possibilities to obtain more than 400 MeV u −1 quasi-monoenergetic carbon-ion beams with the future 100 PW laser facilities, which may be applied in fast ignition and cancer therapy in the future.

Data availability statement
The data cannot be made publicly available upon publication because no suitable repository exists for hosting data in this field of study. The data that support the findings of this study are available upon reasonable request from the authors.