1. Introduction

With a lot of potential applications in oncology [1], proton imaging, and inertial confinement fusion [2], laser-driven ion acceleration has drawn increasing attention these years. Several ion acceleration mechanisms have been proposed, e.g., target normal sheath acceleration (TNSA [3–5]), which is one of the best understood laser-driven ion acceleration mechanisms to date. However, the obtained maximum ion energy was limited to 67.5 MeV for protons and 10 MeV/u for heavy ions both with a quasi-thermal distribution [4, 5]. In most applications, high quality ion beams are generally desirable [6]. For example, in medical therapy for tumor, we demand a narrow ion energy spread to ensure that the ion energy is deposited over a small distance in the patient’s tissue [1]. Meanwhile, a tunable ion peak energy is very beneficial to the therapy of large size tumors. In the concept of fast ignition with laser-driven ions, it is desirable to control the ion energy spectrum as this may lead to a significant reduction of the proton energy required for ignition [2]. Despite of the relative success of the TNSA in several experiments [4,5], the beam quality of the generated protons and heavy ions is much lower than in conventional accelerators, which is insufficient for further applications in practice.

Thanks to the significant increase of the laser peak intensity (up to 10²²W/cm²) and the laser contrast (> 10⁻¹⁰), radiation pressure acceleration (RPA) or ”light-sail” regime was proposed, which holds the promise of achieving monoenergetic ion acceleration in laser thin-foil interaction [7–12]. The one dimensional (1D) flying mirror model revealed the underlying dynamics and well predicted the final ion energy [7]. Particle-in-cell (PIC) simulations showed that GeV monoenergetic collimated proton beams could be produced from a shaped foil [13, 14], a density-modulated foil [15], and a multi-layer foil [16] at a laser intensity of 10^21–22W/cm². Experimental investigations on RPA have also obtained significant progress these years. By irradiating a diamondlike carbon foil with a circularly polarized laser, 30 MeV carbon ions (C⁺⁶) were observed for the first time in experiments [17]. The recent ”light-sail” experiment has further demonstrated the quasi-monoenergetic proton generation in this regime [18–20]. However, the inevitable transverse instabilities, e.g., Rayleigh-Taylor-like (RT) instability, quickly destroyed the foil acceleration structure and deteriorated the beam quality [21, 22]. The stabilization and tunability of ion acceleration are still two critical issues in the RPA regime, which prevent the laser-driven ion beam applications [6]. Recent studies showed that, by using a C-H mixing foil we may obtain stable proton acceleration with respect to the RT-like instability [23,24]. In this case, carbon ions are immediately separated from the light protons as the laser pressure acts on the foil. Due to their smaller charge-to-mass ratio, the carbon ions move behind the protons and are widely spread in space. Finally, only a high quality proton beam is obtained, whereas stable carbon ion acceleration is still out of the reach in RPA.

In this paper, we demonstrate the simultaneous generation of monoenergetic protons and carbon ions by radiation pressure acceleration of a multi-component nanofoil. It is shown that, the light protons with the largest charge-to-mass ratio moves ahead of the carbon ions and experience a phase stable acceleration. The carbon ion layer always rides at the heavier oxygen ion front with a small electron-filling gap between the protons and carbon ions. The gap plays a key role for proton acceleration because the protons are free from the instability. The oxygen ion layer thermally expands and widely spreads in space which acts as a natural ”buffer” for carbon ion acceleration. Finally, monoenergetic collimated protons and carbon ions are obtained simultaneously. More impressive is that, the peak energy and the energy spread of both protons and carbon ions could be controlled by modulating the initial heavier ion densities in the foil.

2. 3D and 2D PIC simulations and results

We first carry out 3D simulations to demonstrate the feasibility of the scheme by using the PIC code VLPL [25]. The simulation box is x × y × z = 20λ₀ × 20λ₀ × 20λ₀, which is sampled by 5000×400×400 grid cells. Here, λ₀ = cT₀ = 1μm is the laser wavelength, c is the light speed, and T₀ = 3.3 fs is the laser period. The time step is Δt = 0.0045T₀. A circularly polarized laser pulse is incident on the foil from the left boundary at t = 0. The temporal profile of the pulse is trapeziform (linear growth-plateau-linear decrease) and the duration is 12T₀ (1T₀ + 10T₀ + 1T₀). The spatial profile is super-Gaussian with the amplitude $a=a_0\mathit{exp}\left(-r^4/{\sigma }_0^4\right)$ where a₀ = 30 is the peak laser intensity and σ₀ = 5λ₀. A planar foil is initially positioned at x = 5λ₀ with a thickness of l = 0.08λ₀ (80nm), which is composed of multi-ion-species, e.g., carbon ions (C⁺⁶), protons (H⁺), and oxygen ions (O⁺⁶). This is identical to the real chemical composition of the widely employed Formvar film in experiments [19]. Here, partially ionized oxygen ions are adopted to model the real oxygen ion state in experiments. The ion number density ratio is n_C : n_H : n_O = 1 : 1 : 3 with the corresponding electron density n_e = 125n_c, where $n_c=m_e{\omega }_o^2/4\pi e^2$ is the critical plasma density, m_e is the electron rest mass, and ω₀ = 2π/T₀ is the laser frequency. Each cell contains 27 numerical macroparticles in the plasma region. Mean-while, a₀ ∼ πn_el/n_cλ₀ is satisfied to make sure of an optimal foil acceleration in RPA [24].

Figures 1(a)–1(c) show the 3D simulation results of the laser thin-foil interaction at t = 35T₀. When radiation pressure dominates the foil acceleration at high laser intensities, electrons move ahead of ions immediately and strong charge separation fields are created. For a thin foil consisting of three ion components with different charge-to-mass ratios, C⁶⁺, H⁺, and O⁶⁺, three ion layers are thus formed as illustrated in Fig. 1(a): a compact proton layer (cyan), a thin carbon ion layer (blue), and a thermally expanding oxygen ion layer (purple). The carbon ions ride on the heavier oxygen ion’s front and a small gap forms between the carbon layer and the proton layer. This is significantly different from the two-ion-species case in Ref. [23] where the carbon ion layer is tightly adjacent to the protons and a smooth interface thus forms. Figs. 1(b)–1(c) show the ion energy spectrum and the divergency angle θ = p_⊥/p_x at t = 35T₀. We can see a pronounced energy peak for both protons (∼108MeV) and carbon ions (∼43MeV/u). The corresponding peak divergency angle is also very small, ∼ 3.6° for H⁺ and ∼ 5.4° for C⁺⁶. However, the oxygen ions exhibit a thermal-like energy spectrum with a relatively large divergency angle. In accord with the observations above, the carbon ion energy peaks at the cuttoff energy of the oxygen ions and meanwhile, is approximately equal to the adjacent proton’s energy. Such a compact acceleration structure is very stable and can be kept for a long time even after the laser pulse passes away. To our knowledge, it is the first time to simultaneously achieve such a stable monoenergetic proton and carbon ion acceleration in full 3D geometries.

 

Fig. 1 3D simulations of laser nanofoil interaction at t = 35T₀. (a) Multi-panel visualization of density distributions of protons (cyan), carbon ions (blue), and oxygen ions (purple). The circle shows the proton bunch and carbon ion bunch, respectively. (b,c) Energy spectra and divergency angles of protons (black), carbon ions (red), and oxygen ions (blue).

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For more insights into the underlying physics, we perform a series of 2D simulations. Because of circularly polarized laser pulses, 2D simulations shall reveal the same observations compared with the 3D ones, but require much less run hours. In the following 2D simulations, we keep the electron and proton densities as in the 3D case above but vary the density ratio of carbon ions to oxygen ions. Here, 4 cases are being considered: n_C : n_H : n_O=1:1:3 (cases 1), 2:1:2 (case 2), 3:1:1 (case 3), and 4:1:0 (case 4). Figs. 2(a)– 2(d) present the axial charge density (n_iq_i) distribution of ions and the corresponding axial electric field E_x at t = 15T₀. At that time, the laser-foil interacts for only 5 laser cycles and the RT-like instability is still in a linear stage [21]. Three different ion regions can be clearly recognized in the Fig. 2 together with an electron-filling gap between the protons and carbon ions. It is also found that the protons in all 4 cases are experiencing a strong linearly-decreasing electric field (E_x,max ∼ 15E₀) by which protons are accelerated in a ”phase-stable” way [9]. Finally, monoenergetic proton beams could be obtained for all 4 cases as seen in Figs. 3(a)– 3(d).

 

Fig. 2 Axial charge densitiy (n_iq_i) distributions of protons (black), carbon ions (red), and oxygen ions (blue) at t = 15T₀. (a) case 1, (b) case 2, (c) case 3, and (d) case 4. The pink shows the corresponding axis electric field E_x.

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Fig. 3 Energy spectra of protons (black), carbon ions (red), and oxygen ions (blue) within y = 5 – 15λ₀ at t = 15T₀. (a) case 1, (b) case 2, (c) case 3, and (d) case 4.

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The ion energy spectrum is very sensitive to the initial ion density ratio. With the decrease of the oxygen ion density in the foil, the proton peak energy increases from 122MeV to 162MeV and the energy spread becomes relatively smaller. On the contrary, the heavier ion (C⁺⁶, O⁺⁶) acceleration is more complicated. First, Coulomb explosion is not very important in this stage as seen from Fig. 2 and the laser pressure dominates the ion acceleration. When the oxygen ion density is lower in the foil, the carbon ions’ acceleration becomes worse. Their peak energy downshifts from 45MeV in the case 1 to 25MeV in the case 3 and then disappears in the case 4. The energy spectrum also becomes broader and approaches a thermal distribution in the case 3. By comparison, in all 4 cases the oxygen ions show a spatially expanding distribution with a thermal-like energy spectrum. We also did one more simulations by substituting C⁺⁶ with C⁺⁵ but keeping its charge density in the case 1. Stable monoenergetic proton and carbon ion acceleration were reproduced together with an expanding oxygen layer.

3. Analysis and discussions

Actually, the different ion acceleration process depends on the different evolution of the RT-like instability in the foil. We first set up a simple analytical model as shown in Fig. 4 to interpret the underlying physics. Starting from the general formulas of the RT instability in a single-ion layer, we obtain the basic equation of the perturbed pressure p seeded on the vacuum-plasma interface I as [24]

 

Fig. 4 Schematic of the laser multi-component nanofoil interaction and the evolution of the RT-like instability. The top red is protons and the bottom gray is oxygen ions with the middle blue being the carbon ions. Between the carbon ion layer and the proton layer is an electron-filling gap.

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The model also provides us with an alternative interpretation for recent observations in Ref. [20], where a C₈H₈F₂ foil was used. In their experiments, a lot of C⁺⁴, C⁺⁵, and F⁺⁵ may be produced because of the prepulse of the laser. According to our model, the observed carbon ion peak (C⁺⁶) in the energy spectrum may be attributed to the ”buffer” effect of the heavier fluorin ions (F⁺⁵) and partially ionized carbon ions, e.g., C⁴⁺ and C⁵⁺. Because the material contains only a few heavier ions, very tiny peaks of protons and carbon ions were observed.

4. Conclusion

In conclusion, we study intense laser nanofoil interaction and demonstrate the generation of monoenergetic tunable protons and carbon ions by using PIC simulations. It is shown that, the protons always move ahead of carbon ions with the latter followed by oxygen ions. A small electron-filling gap forms between the protons and carbon ions, which plays a key role for monoenergetic proton acceleration. The oxygen ion layer thermally expands and widely spreads in space, which acts as a natural ”buffer” for carbon ion acceleration. Finally, monoenergetic collimated protons and carbon ions are obtained simultaneously. By modulating the oxygen ion density in the foil, the energy spread and the peak energy of both protons and carbon ions could be controlled, which may have diverse applications in future.