INTRODUCTION
With the development of high power lasers, it has become possible to generate MeV proton beams in experiments (Kluge et al., Reference Kluge, Gaillard, Flippo, Burris-Mog, Enghardt, Gall, Geissel, Helm, Kraft, Lockard, Metzkes, Offermann, Schollmeier, Schramm, Zeil, Bussmann and Cowan2012; Gaillard et al., Reference Gaillard, Kluge, Flippo, Bussmann, Gall, Lockard, Geissel, Offermann, Schollmeier, Sentoku and Cowan2011). The accelerated proton beams have potential applications in several fields, including hadron therapy (Fritzler et al., Reference Fritzler, Malka, Grillon, Rousseau, Burgy, Lefebvre, d'Humiéres, McKenna and Ledingham2003), proton beam-driven fast ignition (Roth et al., Reference Roth, Cowan, Key, Hatchett, Brown, Fountain, Johnson, Pennington, Snavely, Wilks, Yasuike, Ruhl, Pegoraro, Bulanov, Campbell, Perry and Powell2001), injectors for conventional particle accelerators (Krushelnick et al., Reference Krushelnick, Clark, Allott, Beg, Danson, Machacek, Malka, Najmudin, Neely, Norreys, Salvati, Santala, Tatarakis, Watts, Zepf and Dangor2000), etc. However, most of these applications demand ion beams of high quality, such as narrow energy spread, high energy, low beam divergence, etc. Efforts have been made to improve energy spectra, collimation, and laser-beam coupling (Morita et al., Reference Morita, Esirkepov, Bulanov, Koga and Yamagiwa2008; Renard-Le Galloudec D'Humieres, Reference Renard-Le Galloudec and D'Humieres2010; Yu et al., Reference Yu, Zhou, Jin, Cao, Zhao, Hong, Li and Gu2012; Zhou et al., Reference Zhou, Gu, Hong, Cao, Zhao, Ding, Zhang, Cai and Mima2010).
For currently available laser intensities, the dominant mechanism for high energy ion acceleration in laser-plasma interaction is target normal sheath acceleration (TNSA) (Wilks et al., Reference Wilks, Langdon, Cowan, Roth, Singh, Hatchett, Key, Pennington, MacKinnon and Snavely2001), in which protons on the rear side of target are accelerated by electrostatic field of fast electrons penetrating the rear surface and escaping into vacuum. Therefore, a critical challenge of the TNSA mechanism is to increase the efficiency of converting laser energy into hot electrons, which further set up an electrostatic field for proton acceleration.
Sentoku et al. (Reference Sentoku, Mima, Ruhl, Toyama, Kodama and Cowan2004) found that hollow cone targets have high energy conversion efficiency from laser pulse to hot electrons via PIC simulation. After that, cone structure targets have been intensively investigated, e.g., flat-tip cone (Kluge et al., Reference Kluge, Gaillard, Flippo, Burris-Mog, Enghardt, Gall, Geissel, Helm, Kraft, Lockard, Metzkes, Offermann, Schollmeier, Schramm, Zeil, Bussmann and Cowan2012), cone-wire (Ma et al., Reference Ma, Sawada, Patel, Chen, Divol, Higginson, Kemp, Key, Larson, Le Pape, Link, MacPhee, McLean, Ping, Stephens, Wilks and Beg2012), double cones (Cai et al., Reference Cai, Mima, Zhou, Jozaki, Nagatomo, Sunahara and Mason2009), conical nanobrush (Yu et al., Reference Yu, Zhou, Jin, Cao, Zhao, Hong, Li and Gu2012), cone-funnel (Ban et al., Reference Ban, Gu, Kong, Li, Zhu and Kawata2012), etc. However, these relatively complex targets are hard to fabricate. Recently, Zhou et al. (Reference Zhou, Gu, Hong, Cao, Zhao, Ding, Zhang, Cai and Mima2010) demonstrated that higher energies and better collimation proton beams can be achieved using hollow and high-Z material cone shaped target coated with an additional sub-micron and relatively low density proton layer compared with plane target. In fact, there are mainly two effects in a cone target. The first one is the focusing of laser pulse inside the cone target (Sentoku et al., Reference Sentoku, Mima, Ruhl, Toyama, Kodama and Cowan2004), and the second one is the interaction of the laser pulse with the cone walls to generate high energy electron current (Kluge et al., Reference Kluge, Gaillard, Flippo, Burris-Mog, Enghardt, Gall, Geissel, Helm, Kraft, Lockard, Metzkes, Offermann, Schollmeier, Schramm, Zeil, Bussmann and Cowan2012; Gaillard et al., Reference Gaillard, Kluge, Flippo, Bussmann, Gall, Lockard, Geissel, Offermann, Schollmeier, Sentoku and Cowan2011). In this paper, we use the two-dimensional fully relativistic electro-magnetic particle-in-cell (PIC) code FLIPS2D to simulate the interactions between laser and hollow double-layer cone targets and study how the inside diameter (ID) of the tip size of cone target influence the proton beam characteristics (e.g., energy spectrum and beam divergence).
PIC SIMULATIONS
The simulations have been performed using PIC code FLIPS2D (Zhou et al., Reference Zhou, Gu, Hong, Cao, Zhao, Ding, Zhang, Cai and Mima2010). The code applies finite difference methods to solve the field equations, a Boris's method to solve the motion of particles (Birdsall Langdon, Reference Birdsall and Langdon1991), and charge conservation method of first-order algorithm to make the simulation faster (Umeda et al., Reference Umeda, Omura, Tominaga and Matsumoto2003). The simulation box volume x × y = 60 λ × 40 λ is divided into 2400 × 1600 cells and the time step is 0.0125 τ, where λ = 1μm is the laser wavelength and τ is the laser period. Seven particles are used in one mesh, and the total number of particles is about 2 × 107. The boundary conditions of fields are reflected on the left hand boundary, and boundary conditions of particles are absorption and then reemission. A p-polarized Gaussian laser pulse propagates along the x axis from the left boundary, and transverse focal spot diameter is 6 λ full width at half maximum. The laser beam rises up in 5 τ, with a sinusoidal profile, after which it maintains its peak intensity for 20 τ, and then falls to zero in another 5 τ. The peak intensity I 0 = 3.5 × 1019 W/cm2 , which corresponds to a normalized vector potential a 0 = 5.0.
Our target is shown in Figure 1. The width of the cone wall is 3 µm, the cone angle is 30° (the optimum cone opening angle for fast ignition (Nakamura et al., Reference Nakamura, Sakagami, Johzaki, Nagatomo, Mima and Koga2007b) and the ID size varies from 0λ to 10λ. The electron density is 10n cr in the main part of the target and is 1n cr in the proton layer behind the rear surface of the cone substrate, where n cr = ω2m e/4πe 2 is the critical density.
Schematic diagram of cone-shaped substrate and the coated proton layer.
HOT ELECTRONS PRODUCTION AND SHEATH FIELD ON THE TARGET'S REAR SURFACE
From Poisson' equation, the accelerating electric field acting on the proton is E accl = T e/e[max(L n, λD)], where T e is the hot electron temperature, L n is the local scale length of the expanding plasma, and λD is the Debye length (Wilks et al., Reference Wilks, Langdon, Cowan, Roth, Singh, Hatchett, Key, Pennington, MacKinnon and Snavely2001). The magnitude of the accelerating electric field E accl depends on the temperature of the escaping hot electrons as well as on local Debye length in the plasma, which in turn depends on the plasma electron temperature and density.
Figure 2 shows the time-integrated energy spectrum of electrons (0.5–20 MeV) behind the rear side of cone target at t = 50τ, and is fitted by a Maxwellian distribution. The hot electron temperatures are given by 4.1 MeV, 3.3 MeV, 3.0 MeV, and 0.8 MeV for cases ID = 0 λ, 3 λ, 6 λ and 10′λ, respectively. Total number of hot electrons passed through the rear side is almost the same, but the component of high energy electrons significantly increases with decreasing ID size.
Time-integrated energy spectrum of the hot electrons (0.5–20 MeV) observed at the rear side of the cone target for, respectively, a 0 λ 3 λ6 λ and 10 λ ID size of cone target at t = 50 τ.
Figure 3 shows the dependence of average hot electron energy (0.5–20 MeV) behind the rear side at the the initial locations in the cone target at t = 50 τ. We can see that the average hot electron energy at the same original location increases with decreasing the ID size. When we reduce the ID size, the interaction surface area of cone wall with laser pulse increases. Furthermore, the number of electrons extracted from the cone wall per wavelength is N e = (a 0n c λL/π) sin θ (Sentoku & Downer, Reference Sentoku and Downer2010). θ is the angle between the cone surface and the propagation direction of the incident laser light, and λL is the laser wavelength. With the decreasing of the ID size, the laser field is intensified inside the cone target (corresponding to an increase of a 0), and therefore the number of higher energy hot electrons extracted from the cone wall increases. These hot electrons from the cone wall can gain forward momentum via the v × B term of the Lorentz force or vacuum heating (Brunel, Reference Brunel1987), and then are guided along the surface toward the cone tip by the strong self-generated quasistatic magnetic field. Subsequently, these hot electrons guided to the cone tip may be further accelerated by the ponderomotive force in the laser propagation direction (Nakamura et al., Reference Nakamura, Mima, Sakagami and Johzaki2007a). Meanwhile, the low energy components of hot electrons are mainly generated from the cone tip, and are accelerated by the ponderomotive force.
The average hot electron energy (0.5–20 MeV) observed at the rear side of the cone target vs the original location in the cone target for, respectively, a 0 λ, 3 λ, 6 λ and 10 λ ID size of cone target at t = 50 τ.
When these electrons propagate through the target and enter the rear vacuum, they generate a strong sheath field according to Ampere's law, ∂E/∂t = −J. Since the protons are accelerated directly by the sheath field at the rear surface of the target in the TNSA mechanism, a comparison of the longitudinal electric field E x (x, y) is shown in Figures 4 and 5.
The longitudinal sheath field E x (x) (the cut is longitudinal on the propagation axis (x = 0)) for, respectively, a 0 λ, 3 λ, 6 λ and 10 λ ID size of cone target at t = (a) 50 τ, (b) 60 τ, (c) 70 τ, (d) 80 τ.
The longitudinal sheath field E x (y) (the transverse cut right at the vacuum-target interface (x = 39 μm)) for, respectively, a 0 λ, 3 λ, 6 λ and 10 λ ID size of cone target at t = (a) 50 τ, (b) 60 τ, (c) 70 τ, (d) 80 τ.
From Figure 4, we find that E x (x) shows an exponential decay in the longitudinal direction, because the amplitude of shield field on the rear surface is proportional to the temperature of hot electrons. As the ID size decreases, higher energy hot electrons expand faster into the vacuum, resulting in a higher amplitude and more broadly distribution of the electric field on the rear surface of the cone target. The stronger sheath field effectively accelerates the protons to higher energies.
Figure 5 shows that the transverse profile of longitudinal electricfield E x (y), which determines the divergent properties of electrons, and therefore influences the divergence of the protons beams. With decreasing the ID size, the transverse width of Ex (y) is reduced and the strength of Ex (y) is increased, which leads to a smaller divergence angle of the proton beam. From above analysis, we find that the ID size of cone target can significantly influence the electron beam trajectory, and further affect the sheath field of the target rear surface.
DEPENDENCE OF PROTON YIELD ON ID SIZE OF THE CONE TARGET
The sheath field at the rear surface is responsible for the proton acceleration. In Figure 6, we show the time-integrated energy spectrum and angular distribution of proton emission observed from the cone targets and a plane target at t = 100 τ. The emission angle ϕ = arctan(v y/v x). For a plane target, the high-Z layer is 16 λ wide and 3 λ thick, and the low density proton layer is same as that in the cone target, i.e. 1 λ wide and 0.1 λ thick.
Four cases of different ID sizes of cone targets and a plane target at t = 100 τ (a) proton energy spectra, (b) proton emitted angle ϕ.
From Figure 6 we can see that the proton peak energy increases significantly and has a slightly decreasing divergence angle with decreasing ID size. For a plane target (dashed line), the proton peak energy is E peak = 3.4 MeV, and the energy spread is E FWHM/E peak = 73.5 %. From Figure 6a, it can be seen that the peak energy ranges from 3.8 MeV to 11.8 MeV, for different ID sizes, and corresponding energy spread E FWHM/E peak ranges from 63.2% to 41.5%. For the case of an ID = 0 λ, the proton peak energy is approximately three times that using a plane target. In Figure 6b, as the ID size decreases, the emission angle ϕFWHM of the proton beam decreases from 23° to 15°, which can be attributed to the distribution of the longitudinal sheath field in the transverse direction for proton acceleration as shown in Figure 5.
CONCLUSIONS
In conclusion, we have performed FLIPS2D simulations to study how the ID size affects proton beams generated in double-layer cone target. For a fixed opening angle of cone target (30°), decreasing ID size can both enhance the laser focusing effects and increase laser interaction with the cone wall. The combined effects of a larger effective surface area and a further intensified laser field lead to higher coupling efficiency to hot electrons, which therefore result in a higher amplitude and more broadly distribution of the sheath field behind the rear surface of the cone target. As a result, a higher energy and narrower divergence angle proton beams are achieved by reducing the ID size of double-layer cone target. The result indicates that the case of ID = 0 λ cone target is favorable for accelerating higher energy and lower emission angle proton beams.
ACKNOWLEDGMENTS
We thank Prof. Dino Jaroszynski for helpful discussions. This work is supported by National Natural Science Foundation of China under Grant Nos 11174259, 11175165.
