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Abstract
Based on current laboratory laser parameters and the low density target that is induced by the inevitable prepulse, we propose what we believe to be a new scheme to enhance the proton energy by employing a laser pulse with two different peak intensities. Initially, the lower-intensity peak of the laser pulse P1, irradiates the low-density plasma target induced by the prepulse to form a significantly denser plasma target. Such a compressed high-density target is critical for supporting the subsequent main pulse P2 with higher peak intensity to drive proton acceleration. As an example, particle-in-cell (PIC) simulations reveal that when using a circularly polarized (CP) flat-top P1 with a peak intensity of approximately 1.71 × 10 19 W/cm2, full-width at half-maximum(FWHM) duration of 325 fs and a CP P2 with a peak intensity of 1.54 × 10 22 W/cm2, FWHM duration of 26.5 fs, and focal spot radius of 4 µm successively acting on a target with an initial density of 8nc, protons with cut-off energy of 940 MeV can be obtained from the cascaded acceleration scheme. Compared with the case without P1, the cutoff energy increased by 340 MeV. Owing to the intervention of P1, this scheme overcomes the limitation of laser contrast and is more feasible to be implemented experimentally.
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1. Introduction
Laser-driven proton acceleration [1–3] has gained significant attention in recent years because of its acceleration gradient, which is several orders of magnitudes higher than that of conventional accelerators. This makes the accelerators more compact and affordable than traditional accelerators [4]. High-energy protons are widely used in various fields, including cancer treatment [5], proton radiography [6], and inertial confinement fusion [7]. However, further improvement in laser proton acceleration technology is required in terms of the cut-off energy, conversion efficiency, and proton beam quality.
Several laser proton-acceleration mechanisms have been proposed. Target normal sheath acceleration (TNSA) [8–12] is currently the most robust acceleration mechanism in the laboratory. However, the proton energy achievable using this mechanism is limited, and the proton beams are generally of broad spectrum. Compared with TNSA, radiation pressure acceleration (RPA) [13–15] has the potential to accelerate monoenergetic ion beams with high energy conversion efficiency. However, the problem with RPA [16,17] lies in the experimental requirements of high-intensity high-contrast lasers, strict requirements for the parameters of the target, and the presence of transverse instabilities. Currently, no satisfactory implementation of pure RPA acceleration has been achieved experimentally. Records of the proton energy in recent years have been obtained through hybrid mechanisms, such as 94 MeV protons from the RPA–TNSA mechanism using picosecond laser pulses in the Vulcan system [18], and 150 MeV protons from various accelerating mechanisms stimulated via ultrashort laser irradiation on a solid plastic target [19]. Collisionless-electrostatic shock acceleration (CSA) [20–24] is a critical mechanism for laser-driven proton acceleration, and proton beams with high energy, high flux, and quasi-monoenergy can be obtained. Monoenergetic high-energy proton beams with an energy dispersion of approximately 1% can be generated through CSA driven by a linearly polarized (LP) CO2 laser train interacting with the gas [25]. Quasi-monoenergetic high-flux protons have been generated through CSA driven by a circularly polarized (CP) optical short laser interacting with a thin foil [20]. Moreover, hybrid proton acceleration [18–28] is a crucial mechanism for generating high-energy protons. Although the typical characteristics of different acceleration mechanisms have been observed experimentally, it is still challenging to further increase the proton energy. Higher-energy proton beams should be obtained under current laser conditions. A drawback is that the laser prepulse [29] expands the plasma and reduces its density before the main laser pulse arrives. Low-density plasmas cannot satisfactorily support the main pulse intensity, thereby limiting the quality of the generated proton beam. Hence, it is necessary to eliminate or weaken the effect of the prepulse in reducing the target density.
In this paper, we present an optimization scheme based on the current laser parameters in the laboratory for obtaining high-energy proton beams by employing a laser pulse with two different peak intensities. This scheme is illustrated in Fig. 1. First, the lower-intensity peak of the driving laser pulse P1, interacts with the initial low-density target that induced by the prepulse, resulting in its compression into a significantly denser plasma target. Subsequently, the main pulse P2, with much higher peak intensity, is used to irradiate the compressed target, driving the CSA and moving-TNSA accelerate protons, eventually obtaining proton beams with enhanced energy. As an example, particle-in-cell simulations reveal that when using a CP flat-top P1 with a peak intensity of approximately 1.71 × 10 19 W/cm2 and CP P2 with a peak intensity of 1.54 × 10 22 W/cm2, successively acting on a target with an initial density of 8nc, protons with cut-off energy of 940 MeV can be obtained from the cascaded acceleration scheme. Compared with the case without P1, the cutoff energy increased by 340 MeV. Furthermore, the yield of the high-energy protons is increased.
Fig. 1. Schematic of process. (a) Time profile of the driving pulse, the flat-top P1 interacting with the initial target from t1, and then the main pulse P2 interacting with the compressed target from t2. (b) P1 compresses the initial low-density plasma target to produce a high-density plasma target; (c) P2 interacts with the high-density plasma target to accelerate protons (The first process starts at t1, and the second process starts at t2).
2. Simulation and results
Particle-in-cell (PIC) simulations were conducted to reveal the proton acceleration mechanism. The simulation was conducted in two steps. The first step was P1 compressing the initial target by the induced shock. This shock can compress the plasma target to make it significantly denser several times. A one-dimensional PIC simulation was performed to shorten the simulation time. The simulation parameters are as follows. As an example, the driving laser pulse is CP and P1 is flat-top with a central wavelength of 0.8 µm, FWHM duration of 325 fs and a peak intensity of approximately 1.71 × 10 19 W/cm2, corresponding to a normalized amplitude of a1 = 2, where a is the ratio of the oscillating energy of the free electron in the laser field to the static energy of the electron. The initial target thickness was 3.2λ0; its density profile is shown in Fig. 2(a). The plasma density increased from 0 to 8nc in the x = 8 - 8.5λ0 region and remained at 8nc in the x = 8.5λ0 - 11.2λ0 range, where nc is the critical density for 0.8 µm lasers. The initial plasma temperature was 3.3 keV [30]. The simulation box was x: 30λ0, divided into 9000 cells, and each cell was filled with 100 particles. P1 interacting with the initial target facilitated the formation of a shock wave owing to light pressure and heat pressure, as observed from the proton momentum phase-space in Fig. 2(c). It then compressed the initial target at 125T0 to form a considerably denser target, as shown in Fig. 2(b) (where T0 = 2.67 fs is the laser period, and the simulation of this step starts from t1, and default to t1 = 0), with a density of approximately 20nc, which is 2.5 times that of the initial target. In this process, few protons are reflected and accelerated by the collosionless shock, and the maximum energy of the reflected protons is about 2.5 MeV which will not affect the next step.
Fig. 2. Process of P1 compressing the low-density initial target. (a) and (b) Density profiles of plasma at t = 0 and t = 125T0, respectively; (c) Proton momentum phase-space at t = 125T0, with the color bar representing the relative amount of protons (T0 = 2.67 fs).
The second step involves the process of P2 interacting with the compressed high-density target to accelerate the protons. Two-dimensional PIC simulations were performed to examine the acceleration process. P2 is a CP Gaussian pulse with a peak intensity of 1.54 × 10 22 W/cm2, corresponding to a normalized amplitude of a2 = 60, wavelength of 0.8 µm, FWHM duration of 26.5 fs, and focal spot radius of 4 µm, considering that the above laser parameters are accessible on 10 PW laser facilities. The target was the compressed target from the first step, with a size of 16 µm in the y-direction. The simulation box was x: 0 - 120λ0, y: −60λ0 - 60λ0, divided into 7200 × 6000 cells; each cell was filled with 15 particles. It should be noted that all the simulations of the main pulse P2 began at t2, and default to t2 = 0. The simulation results are presented in Fig. 3. Figure 3(a) shows the evolution of the proton density profiles at y = 0 with time. P2 interacting with the high-density target facilitated the formation of electrostatic shock waves owing to the dominant light pressure, and protons were reflected, as shown in Fig. 3(b), at t = 40T0, which is a typical property of CSA. Figure 3(c) shows the proton momentum phase-space distribution at last time, where the momentum of the protons further increases, although the phenomenon of proton reflection disappears. In other words, the protons were further accelerated to higher energies after CSA, which was confirmed by the proton energy spectrum. The red curve in Fig. 3(d) corresponds to the proton energy spectrum at t = 40T0, indicating that protons are accelerated to 190 MeV during the initial stage by the typical shock wave. Here one point should be noted that the velocity of the electrostatic shock driven by the light pressure is proportional to the laser amplitude [31], so there is not an obvious energy peak as shown in Fig. 1(d) because a Gaussian enveloped laser pulse is used where the laser amplitude a changes with the interaction time. The gray curve corresponds to the proton energy spectrum at last time, where the proton energy is further increased to 940 MeV after the CSA. This is because after undergoing the first stage of CSA, protons are further accelerated by the subsequent acceleration fields, which are induced by fast electrons. Another acceleration mechanism persistently accelerates protons after the CSA. In addition, an increase in the yield of high-energy protons is observed.
Fig. 3. Process of P2 interaction with high-density target. (a) Density profiles of protons in the y = 0 plane at t = 0, t = 36T0, t = 40T0, t = 43T0, and t = 45T0; (b) and (c) Proton momentum phase-spaces at t = 40T0 and t = 150T0, respectively; (d) Proton energy spectra at t = 40T0 and t = 150T0.
The acceleration fields of the protons were analyzed to reveal the acceleration mechanism. Figure 4(a) shows the integrated distribution of the electrostatic field Ex in the y-direction in the region where the high-energy protons are located at different times. At t = 40T0, the bipolar electric field associated with the presence of the shock wave was the acceleration field of the CSA used to reflect and accelerate protons to make the target move forward as a whole. The laser field Ey, and electric field Ex at t = 40T0 indicate that the electrostatic shock wave is driven by light pressure when the laser pulse is still acting on the target as shown in Fig. 4(b) and (f). After the CSA stage, the transmitted part of the driving pulse transmitted to the target and pulled the fast electrons forward, resulting in a moving charge separation field accompanying the moving target. That is, Ex continued to exist after CSA and moved in the + x direction, although it sharply decreased because of the fewer electrons, such as at different times. The field was also observed from the two-dimensional distribution of Ex, as shown in Fig. 4(g), (h), and (i). In this stage, protons were further accelerated by the stable field as the TNSA moved because this electric field was generated by the transmitted component of the driving laser pulse.
Fig. 4. Process of P2 interaction with high-density target. (a) Distribution of electrostatic field Ex integrated in y-direction in high-energy proton region at t = 40T0, t = 70T0, t = 90T0, and t = 120T0. (b), (c), (d), and (e) Two-dimensional distributions of laser field Ey t = 40T0, t = 70T0, t = 90T0, and t = 120T0. (f), (g), (h)and (i) Two-dimensional distributions of plasma electric field Ex at t = 40T0, t = 70T0, t = 90T0, and t = 120T0, respectively (T0 = 2.67 fs).
In comparison to the case without P1, P2 directly interacted with the initial low-density target, whereas the other parameters remained unchanged. The simulation of this step starts from t2, and default to t2 = 0, Fig. 5(a) and (b) show the proton momentum phase–space distributions at t = 40T0 and last time, respectively, for this case. Compared with Fig. 3(b), protons were not reflected at t = 40T0, indicating the absence of the CSA in this case. This occurred because the density was too low to adequately support P2, allowing the main pulse to penetrate the target rapidly. Compared with that in Fig. 3(c), the proton momentum in this case at last time was smaller because the protons were only accelerated in the TNSA stages without undergoing the earlier acceleration process of the CSA.
Fig. 5. Process of P2 directly interacting with the initial foil target. (a) and (b) Proton momentum phase-spaces at t = 40T0 and t = 150T0, respectively.
The cutoff energies of the proton beams obtained at last time with and without P1 are shown in Fig. 6. The cutoff energy was 600 MeV when only P2 was involved, and the cutoff energy was 940 MeV when both P1 and P2 were involved, which was 340 MeV higher than that of the case without P1. In addition, the simulation results show that the latter has a higher proton yield in the high-energy region, which results from the precompression of the initial target by P1 and the reflection of the subsequent CSA stage.
Fig. 6. Cutoff energy values of the proton beams obtained at t = 150T0 with and without P1. The black curve represents the proton energy spectrum at t = 150T0 when P2 directly interacts with the initial lower-density target. The red curve represents the proton energy spectrum at t = 150T0 when P1 and P2 sequentially interact with the initial target.
3. Discussion
In addition, to verify the optimization mechanism, the effects of the thickness of the compressed target and the polarization of the main pulse P2 have been examined.
3.1 Effect of target thickness
The process by which P2 interacted with compressed targets of different thicknesses was analyzed. By shortening or prolonging the compression time in the first step, the compressed targets for the second step with different thicknesses, such as d = 0.5λ0, 0.8λ0, and 1.1λ0, were obtained. The other target parameters were identical to those used in the above simulations. Two-dimensional PIC simulations showed two acceleration stages: first, through the CSA, and second, through the moving TNSA with the same acceleration mechanism. The proton energy spectra at last time are shown in Fig. 7. The cut-off energy values of the proton beams corresponding to targets of different thickness were 740, 835, and 940 MeV, respectively. This indicates that the cut-off energy increases with decreasing thickness of the compressed target. The simulation results show that the thickness of the compressed targets should be within an appropriate range for a specific laser intensity. If the target is very thick, the entire pulse is reflected by the target, and protons can only be accelerated during the CSA stage. However, if the target is very thin, the entire pulse will transmit through the target because the electric field is not sufficiently intense to balance the light pressure. For example, when the thickness is reduced to d = 0.3λ0, PIC simulations show that the CSA tends to disappear, and the cut-off energy of the high-energy protons decreases significantly. The blue curve in Fig. 7 corresponds to the proton energy spectrum obtained when P2 directly interacts with the initial low-density target. Benefiting from the high-density compression by P1 in the first step, protons are accelerated by the hybrid mechanism, that is, the CSA and moving TNSA. Thus, an increased cut-off energy was obtained.
Fig. 7. Cutoff energy values of the proton beams obtained at t = 150T0 with and without P1. The blue curve represents the proton energy spectra at t = 150T0 when P2 directly interacts with the initial low-density target. The purple, yellow, and red curves represent the proton energy spectra at t = 150T0 when P2 interacts with the high-density plasma target with thicknesses of 0.5λ0, 0.8λ0, and 1.1λ0, respectively.
3.2 Effect of P2 polarization
The CP pulse of P2 was analyzed in the above section. In this section, different polarizations of P2 are analyzed to examine the effect of the hybrid acceleration mechanism. When LP P2 is used, the oscillation term of the ponderomotive force heats the plasma more vigorously as it interacts with the target, resulting in a rapid transmission of the laser pulse through the target. Therefore, a higher density and thicker initial target are required than in the case of a CP pulse. In the first step, the extended duration of P1 was used to compress the initial target with a density of 10nc and thickness of 4.2λ0, the compressed target with a density of 25nc and thickness of 0.8λ0 was obtained. In the second step an LP Gaussian pulse P2 with a2 = 60$\sqrt 2 $ was used. Figure 8(a) shows the proton momentum phase-space at t = 45T0, as the protons undergo shock reflection acceleration. Figure 8(b) shows the proton momentum phase-space at last time when the reflection disappears, but the proton momentum phase-space is further increased, which is similar to that in the case of CP. The energy spectrum of the proton beam is shown in Fig. 8(c), which indicates a cut-off energy of 590 MeV when only P2 is used. The red curve represents the energy spectrum of the protons in which P1 and P2 successively interact with the initial target having a cut-off energy of 700 MeV, which is 110 MeV higher than that of the case without P1. For the same reason, the yield of high-energy protons increased. Therefore, the case of LP confirms that the precompression step is effective for increasing the proton energy.
Fig. 8. (a), (b) Proton momentum phase-spaces at t = 45T0 and t = 150T0, respectively, when P2 is LP. (c) The black curve represents the proton energy spectrum at t = 150T0 when P2 directly interacts with the initial target. The red curve represents the proton energy spectrum at t = 150T0 when P1 and P2 sequentially interact with the initial target.
4. Conclusion
Based on current laboratory laser conditions and considering the low density induced by the prepulse in the experiments, we propose a new scheme for obtaining high-energy proton beams by employing a laser pulse with two different peak intensities. First, the lower-intensity peak P1 of the laser pulse irradiates the low-density plasma target induced by the prepulse to form a significantly denser plasma target. Subsequently, the main pulse P2, with much higher peak intensity, is used to irradiate the compressed target, driving the CSA and moving-TNSA accelerate protons, eventually obtaining proton beams with enhanced energy. The two-dimensional PIC simulations demonstrate that the cut-off energy can be increased by 340 MeV with the precompression step when a CP pulse with a peak energy of 1.54 × 10 22 W/cm2 is used. The effects of the thickness of the compressed target and the polarization of the main pulse were examined, which further confirmed the effect of the hybrid mechanism on the proton energy increase owing to the high-density target from precompression. If the first pulse P1 compresses the initial target at a lower temperature, a higher-density compressed target can be obtained [30], which can support higher-intensity lasers and obtain higher-energy proton beams. These results show a promising and reliable approach to increase proton energy, considering that the laser intensity is already accessible on 10 PW laser facilities, such as ELI-NP and SULF [32–34].
Funding
National Natural Science Foundation of China (11935008, 12305274, 62075134).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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