INTRODUCTION

Relativistically, intense laser pulses interacting with over-dense plasmas or solid targets has stimulated considerable interest in relativistic plasma physics (Cai et al., Reference Cai, Mima, Zhou, Jozaki, Nagatomo, Sunahara and Mason2009; Liu et al., Reference Liu, Gu, Wang, Zheng, Ge, Wen, Jiao, He, Wen, Huang, Wang, Zhou, Zhang, Wang, Zhou, Wang, Huang and Ni2005; Teng et al., 2010; Williams et al., Reference Williams, Favre and O'Connor2009; Zhou et al., Reference Zhou, He and Yu2008) for its practical applications, such as the generation of high-energy particle beam (Chen et al., Reference Chen, Pukhov, Yu and Sheng2009; Roth et al., Reference Roth, Cowan, Key, Hatchett, Brown, Fountain, Johnson, Pennington, Snavely, Wilks, Yasuike, Ruhl, Pegoraro, Bulanov, Campbell, Perry and Powell2001), radiography of dense plasma (Li et al., Reference Li, Séguin, Frenje, Rygg, Petrasso, Town, Amendt, Hatchett, Landen, Mackinnon, Patel, Smalyuk, Sangster and Knauer2006, Reference Li, Séguin, Frenje, Petrass, Amendt, Town, Landen, Rygg, Betti, Knauer, Meyerhofer, Soures, Back, Kilkenny and Nikroo2009; Marshall et al., Reference Marshall, Mckenty, Delettrez, Epstein, Knauer, Smalyuk, Frenje, Li, Petrasso, Séguin and Mancini2009), cancer treatment (Malka et al., Reference Malka, Fritzler, Lefebvre, D'Humieres, Ferrand, Grillon, Albaret, Meyroneinc, Chambaret, Antonetti and Hulin2004), and fast ignition (FI) in inertial confinement fusion (Tabak et al., Reference Tabak, Hammer, Glinsky, Kruer, Wilks, Woodworth, Campbell, Perry and Mason1994). The FI has the possibility of higher gain, lower driver energy, and less implosion symmetry requirements than the conventional inertial fusion. In the FI, hot electron beam is considered to be the most suitable energy source for ignition. In order to improve the coupling efficiency of the incident laser to the fuel and reduce the ignition energy requirement, the hot electrons should be collimated to propagate through the over-dense plasma and deposit their energy to a hot spot of deuterium-tritium fuel. However, numerous simulations and experiments have shown that the hot electron beams have large divergence while transporting in plasma (Green et al., Reference Green, Ovchinnikov, Evans, Akli, Azechi, Beg, Bellei, Freeman, Habara, Heathcote, Key, King, Lancaster, Lopes, Ma, Mackinnon, Markey, Mcphee, Najmudin, Nilson, Onofrei, Stephens, Takeda, Tanaka, Theobald, Tanimoto, Waugh, Van Woerkom, Woolsey, Zepf, Davies and Norreys2008; Kodama et al., Reference Kodama, Norreys, Mima, Dangor, Evans, Fujita, Kitagawa, Krushelnick, Miyakoshi, Miyanaga, Norimatsu, Rose, Shozaki, Shigemori, Sunahara, Tampo, Tanaka, Toyama, Yamanaka and Zepf2001; Stephens et al., Reference Stephens, Snavely, Aglitskiy, Amiranoff, Andersen, Batani, Baton, Cowan, Freeman, Hall, Hatchett, Hill, Key, King, Koch, Koenig, Mackinnon, Lancaster, Martinolli, Norreys, Perelli-Cippo, Rabec Le Gloahec, Rousseaux, Santos and Scianitti2004).

In the past years, several methods for collimating electrons have been introduced. These methods include the cone-wire (King et al., Reference King, Akli, Freeman, Green, Hatchett, Hey, Jamangi, Key, Koch, Lancaster, Ma, Mackinnon, Macphee, Norreys, Patel, Phillips, Stephens, Theobald, Town, Van Woerkom, Zhang and Beg2009), the two-layer targets (Wu et al., Reference Wu, Liu, Zhou and Zhu2009), the funnel-guided target (Zhou et al., Reference Zhou, Wang, Ruan, Wu, Chew, Yu and He2010), and the conical target (Galloudec et al., Reference Galloudec, D'humieres, Cho, Osterholz, Sentoku and Dit Mire2009), etc. In the cone-wire scheme, wires attached to cones have been used to collimate and guide the energetic electron beams by strong self-generated magnetic field around the wires. In the two-layer targets approach, two intense laser pulses interact with plasma consisting of two parts with different densities. The resistive magnetic field induced by the electron current generated by the first pulse along the density interface collimates the hot electrons generated by the second pulse. In the funnel-guided targets approach, a gold funnel is attached to a gold slab, and the funnel is embedded in Deuterium-Ttritium plasma. The magnetic field generated by fast electrons entering the gold funnel makes electron beam transport along the funnel. In the approach of conical target, a channel is attached to the cone target, and the electron beam can travel into the channel for a long distance. All the approaches are based on the idea that the self-generated resistive magnetic field can constrain the divergence of hot electron beams. However, few researchers have attempted to collimate the hot electrons by external magnetic field from magnetic-flux compression.

Simulations and experiments show that magnetic-flux in high-energy-density plasma can be compressed by laser-driven implosion (Gotchev et al., Reference Gotchev, Chang, Knauer, Meyerhofer, Polomarov, Frenje, Li, Manuel, Petrasso, Rygg, Séguin and Betti2009; Knauer et al., Reference Knauer, Gotchev, Chang, Meyerhofer, Polomarov, Betti, Frenje, Li, Manuel, Petrasso, Rygg and SÉguin2010; Radha et al., Reference Radha, Goncharov, Collins, Delettrez, Elbaz, Glebov, Keck, Keller, Knauer, Marozas, Marshall, Mckenty, Meyerhofer, Regan, Sangster, Shvarts, Skupsky, Srebro, Town and Stoeckl2005). Magnetic-flux compression is a viable method to generate tens of MG magnetic fields. In Gotchev et al. (Reference Gotchev, Chang, Knauer, Meyerhofer, Polomarov, Frenje, Li, Manuel, Petrasso, Rygg, Séguin and Betti2009), a compressed magnetic field >36 MG can be obtained with seed field of 56-kG. In Knauer et al. (Reference Knauer, Gotchev, Chang, Meyerhofer, Polomarov, Betti, Frenje, Li, Manuel, Petrasso, Rygg and SÉguin2010), 30–40 MG magnetic fields of the imploded core were measured. In Radha et al. (Reference Radha, Goncharov, Collins, Delettrez, Elbaz, Glebov, Keck, Keller, Knauer, Marozas, Marshall, Mckenty, Meyerhofer, Regan, Sangster, Shvarts, Skupsky, Srebro, Town and Stoeckl2005), a peak compressed field about 100 MG at the center of the hot spot was obtained by two-dimensional hydrodynamics simulation. In this paper, a method of hot electron beam collimated by external magnetic field from magnetic-flux compression is proposed. The external magnetic field can be achieved after the stage of implosion by seeding tens of kG magnetic field in a Deuterium-Tritium fuel before implosion. When the external magnetic field is produced, the hot electron tracks will be affected by the magnetic field. The hot electrons space distribution and tracks were investigated by two-dimensional particle-in-cell simulations. The simulation results show that the hot electron beams were well-collimated.

In fast ignition of inertial confinement fusion, hot electrons are divergent as they are generated and transporting in over-dense plasma. Collimating the hot electrons has become an important topic. It is known to all that the electron track will be affected when it moves in magnetic field, especially when the electron crosses the magnetic field fluxes. If the magnetic field is large enough, the electron will move in the form of helix along the fluxes. In other words, the electron will be collimated by the B-field, but the problems are how strong the field is and how to acquire it. In the scheme of cone target fast ignition, hot electrons are generated by the interaction between laser and cone target. Figure 1 show the target sketch maps of (Fig. 1a) before implosion and (Fig. 1b) after implosion. The initial and pre-compressed target radiuses are R ₀ and R _n, respectively. A magnetic field is assumed to be seeded in the target along the direction of laser transmission before the stage of implosion. We consider that the magnetic flux is conserved without diffusion, and the magnetic field is uniformly distributed in the target. In these cases, the magnetic field of pre-compressed target can be expressed as

(1)$$B_n=\displaystyle{{B_0 R_0^2 } \over {R_n^2 }}.$$

Here B ₀ and B _n are initial and imploded magnetic fields, respectively. In fast ignition, the relation between R ₀ and R _n can be written as

(2)$$R_0 \geq 10R_n .$$

So the relation between B ₀ and B _n is

(3)$$B_n \geq 100B_0 .$$

Which means the seed magnetic field can be enhanced more 100 times after implosion. In Radha et al. (Reference Radha, Goncharov, Collins, Delettrez, Elbaz, Glebov, Keck, Keller, Knauer, Marozas, Marshall, Mckenty, Meyerhofer, Regan, Sangster, Shvarts, Skupsky, Srebro, Town and Stoeckl2005), 30–40MG magnetic fields were observed. We assume that we have acquired such magnetic field, and add it to the plasma. When hot electrons are generated by the interaction of laser with guiding cone and transport in the plasma, the tracks will be affected by the magnetic field. If the magnetic field is strong enough, the electron will move in the form of helix along the fluxes. We simulate this collimation process of hot electron beams by external field using two-dimensional particle-in-cell code FLIPS2d, which has been successfully used in investigating the laser-plasma interactions (Zhou et al., Reference Zhou, Mima, Nakamura and Nagtomo2008, Reference Zhou, Gu, Hong, Cao, Zhao, Ding, Zhang, Cai and Mima2010).

Fig. 1.

(Color online) Magnetic field fluxes at the case of (a) before implosion, (b) after implosion.

The grid size of Δx = Δy = 0.1λ₀ with simulation box scales of X × Y = 30λ₀ × 20λ₀ is used, and the time step is 0.05τ₀. Here λ₀ = 1.0 μm is the wavelength of incident laser pulse. And τ₀ is the period of laser. The number of electrons and ions used in one cell are both 400. A p-polarized laser with dimensionless amplitude of a ₀ = 5.0 irradiates the target along X axis from the left boundary. The focal spot size of the p-polarized Gaussian laser is 4λ₀. The temporal profile of the laser can be described as rising up to peak intensity in 5τ, maintaining at the peak intensity in 10τ, and then falling to zero in another 5τ. A plane target with 14λ₀ wide and 10λ₀ high was used in these simulations. The plane target consists of two species: electrons and heavy ions. The density of the plasma is 50n _c, where n _c = m _e ω₀²/4π e ² is the critical density. Second-order interpolation method is used to evaluate fields and currents. The initial temperatures of electrons and ions are both 1.0 keV. The external magnetic field from magnetic-flux compression is assumed to 30 MG.

In order to investigate the effect of external magnetic field on the hot electrons, the space distributions of hot electrons have been studied. Figure 2 describes the distribution of hot electrons (0.5–5MeV) in the target under the conditions of (Fig. 2a) without external magnetic field and (Fig. 2b) with 30 MG external magnetic field. In the case of without external magnetic field, the hot electrons are divergent, which is disadvantageous in fast ignition. From Figure 2b, one can see that hot electron beam is collimated and guided in a small region along the laser axis by the external magnetic field. Figure 2c shows the hot electrons space distribution in y direction. From the figure, we can find only a small part of hot electrons distribute in the laser spot diameter under the condition of without external magnetic field, while more than 90% hot electrons are collimated in the laser spot diameter in the case of with external magnetic field.

Fig. 2.

(Color online) The distribution (0.5–5MeV) of hot electrons in the target at the time of 35τ, (a) without external magnetic field, (b) with external magnetic field, (c) hot electrons distribution in y direction.

Now we investigate the hot electron tracks under the two conditions. Figure 3 shows the hot electron (0.5–5MeV) tracks in the targets. From the figure, we can find that the tracks of hot electron are helixes under the condition of (Fig. 3b) with external magnetic field, and the hot electrons are confined in a column region of diameter no larger than the laser spot. The tracks of hot electron under the condition of with external magnetic field prove the analysis of single electron track in magnetic field, and indicate that the method of adding external magnetic field for collimating the hot electrons is successful in producing an intended result.

Fig. 3.

(Color online) Hot electron tracks in the targets, (a) without external magnetic field, (b) with external magnetic field.

In summary, we propose a method of collimating hot electron beams by external magnetic field which can be acquired by compressing seeding magnetic field in laser-driven imploding plasma. We simulated this collimation process of hot electron beams by using two-dimensional FLIPS code, and the simulations results showed that the hot electrons can be well-collimated by high (30 MG in this case) external magnetic field, which can be obtained in the laboratory. As a result, the collimation of hot electron beams in fast ignition can be realized by using external field from magnetic-flux compression.