LASER-DRIVEN BEAM-ELECTRONS PROPAGATING IN FOUR PLASMA TARGETS

In order to enhance the materials effect, we shall consider four plasma targets, namely Au⁺²⁵ (case I), Cu⁺²⁰ (case II), Al⁺¹⁰ (case III), and C⁺⁵ (case IV). The initial temperature of the plasma electrons and ions is 100 eV. The plasma electron number densities for the four cases are n _e [cm⁻³] = 1.5 × 10²⁴, 1.2 × 10²⁴, 6 × 10²³, 3 × 10²³, respectively. The laser-produced electrons are assumed to satisfy relativistic Maxwellian distribution with an average temperature of 1.5 MeV in the x direction (Solodov et al., Reference Solodov, Anderson, Betti, Gotcheva, Myatt, Delettrez, Skupsky, Theobald and Stoeckl2009; Zhou et al., Reference Zhou, Wu, Cai, Chen, Cao, Chew and He2010b). The electron beam with radius 15 μm is injected into the plasma along the z direction with an initial angular spread of 30° at (z, x) = (20,0) μm. The injected current rises to its maximum value in 20 fs and then remains constant. The maximum current density is 10¹³ A/cm².

To simulate the transport of fast electrons in overdense plasmas, we use hybrid simulation techniques (Welch et al., Reference Welch, Rose, Oliver and Clark2001; Honrubia et al., Reference Honrubia, Kaluza, Schreiber, Tsakiris and Meyer-ter-Vehn2005; Davies, Reference Davies2003; Evans, Reference Evans2006; Zhou et al., Reference Zhou, He, Cao, Wang and Wu2009) with PIC beam electrons and fluid background plasmas. The hybrid scheme relaxes the usual PIC restrictions on the temporal and spatial resolutions. The simulation box (z, x) is 100 μm × 100 μm. The mesh contains 500 × 500 uniform cells, with four injected beam electrons and up to four hundred plasma particles in each cell. The temporal resolution is δt = δz/2c (c is the speed of light in vacuum). The initially uniform plasma slab is located in 20 ≤ z[μm] ≤ 80. The vacuum regions are 20 μm both in front of and behind the foil.

The simulation results are summarized in Figures 1–4. Figures 2a, 2e, 2i, and 2m show the injected beam electrons propagating into the gold plasma at t = 267 fs. When the beam electrons are injected to the right at (z = 20 μm), as can be seen in the phase-space trajectories shown in Figure 2a of the injected beam electrons and their intensity distribution v _z (Fig. 2e), some of the energetic electrons are scattered back from the front surface into the vacuum region. A surface electric field (not shown) is induced to restore neutrality. As the fast electrons enter the gold plasma they set up a charge separation field E = ηj_r ≈ −ηj_f, where η is the plasma resistivity, j_f and j_r are the fast-electron and return currents, respectively. Figure 2a shows that the electric field in the gold plasma can be larger than 10⁹ V/m. In fact, Figure 2a shows that the positive space-charge field can reach 7 × 10¹⁰ V/m at z ≈ 60 μm, which decelerates the propagation of the beam electrons. As shown in Figure 2i, the beam current density in the gold plasma can be larger than 10¹³ A/cm². Thus, the beam current far exceeds the Alfvén current and beam transport is possible only if sufficient current neutralization is affected by a return current of background plasma electrons. Further, the Weibel instability leads to break up of the fast electron beam into small beamlets (Weibel, Reference Weibel1995). As shown in Figures 2i and 2m, the beamlets can also coalesce on a longer time scale. The beam electrons deposit their energy to the background plasma through the return current, which can be described by the fluid-electron energy equation (Glinsky, Reference Glinsky1995) ∂T _e/∂t ≈ 2J _r²/3σn _e, where σ is the electrical conductivity. Furthermore, non-uniform electric fields can give rise to a large magnetic field B _y > 10 MG. Figures 2m and 3a illustrate that B _y component of the magnetic field can reach several tens megagauss. An exponential reduction of the electric field and growth of the magnetic field can be seen in Figure 1, and the fields then saturate before the beam electrons reach the rear surface of the gold target and enter the vacuum. At the rear surface of the target, they generate a strong sheath electric field at z ≈ 80 μm, as shown in Figure 4a. Once again, the electric field energy can then exponentially increase, as seen in Figure 1.

Fig. 1. (Color online) The normalized electric and magnetic energies for four plasma materials (Au⁺²⁵, Cu⁺²⁰, Al⁺¹⁰, and C⁺⁵). The peak structures of the electric field energy for the four target materials correspond to t ≈ 237, 232, 224, and 220 fs, respectively.

Fig. 2. (Color online) Snapshots of the beam-electron behavior and the magnetic field at t = 267 fs. (a)–(d) The phase-space trajectories of the injected beam electrons. (e)–(h) The intensity distribution v _z of the beam electrons. (i)–(l) The beam-current density (in A/cm²). (m)–(p) The magnetic field component B _y (in gauss). The dashed line in all subplots gives the rear surface of the target. The solid line in (a)–(d) shows the E _z profile at x = 0, where its value is scaled by 1.5 × 10⁹ V/m. The first to fourth columns correspond to cases I–IV, respectively.

Fig. 3. (Color online) Enlargements of the magnetic field B _y (in Gauss) at t = 267 fs for Au⁺²⁰ and C⁺⁵ plasmas, respectively. (a) and (b) correspond to Au⁺²⁰. (c) and (d) correspond to C⁺⁵. The solid lines in the subplots give B _y at z = 30 μm (a), (c), and 60 μm (b), (d), respectively.

Fig. 4. (Color online) Snapshots of the electric field E _z (in V/m) at t = 267 fs. (a)–(d) The electric field structures of near the rear surface correspond to cases I–IV, respectively. (e) The electric field profile E _z (z) with fixed x = 0, and (f) E _z (x) with fixed z = 80 μm.

Figure 1 shows that the first strong peak appears approximately at t ≈ 237, 232, 224, and 220 fs for the cases I–IV, respectively. It is obvious that hot-electrons propagating in high-Z material targets are slower than in low-Z targets. There are two effects to influence the propagation of beam electrons inside the target. On one hand, the Spitzer resistivity η ≈ 10⁻²Z ln ΛT ^−3/2 (where ln Λ is the Coulomb logarithm) says that the electric field inside high-Z plasma targets by E = ηj is higher than in low-Z targets, as shown in Figures 2a–2d by the solid line. The large space-charged field appearing inside the high-Z target tend to prevent the forward propagation of beam electrons, as given in Figures 2e–2h. On the other hand, the electron-plasma collision behavior in high-Z cases I and II by ν_e,i = 4 × 10¹⁴Z ² (T _e/100 eV)^3/2 [s⁻¹] can also significantly scatter hot electrons when n _e > 10²⁴ cm⁻³. High collision rate of the beam electron with background plasmas for high-Z cases further slows down the forward propagation of beam electrons. Such stopping/scattering behavior of beam electrons by both electric fields and collision effects can be clearly observed in Figures 2i–2j for corresponding current densities.

The resistive filamentation instabilities associated with the return current of electron beams and/or the self-generated magnetic fields can be seen in Figures 2i–2p. Comparing Figures 2a–2d with 2m–2p, it is seen that the hot-electron beam is sufficiently well collimated in the low-Z foil target (for example, cases III and IV) by the self-generated resistive magnetic field. Inside the target, the magnetic field is mainly generated by the density and resistivity gradients, etc. These two mechanisms can be seen in the magnetic field growth rate (Robinson et al., 2008; Zhou et al., Reference Zhou, He and Chew2011) ∂B/∂t ≈ η∇ ×  j + ∇η ×  j. The observed large magnetic field at the edge of the electron beam or inside the filamentation beam can be associated with the term η∇ ×  j due to these electron density and/or velocity fluctuations. In order to compare the difference of resistive magnetic fields for different material plasmas, we in Figure 3 give a comparison of the enhanced magnetic field for the gold (case I) and carbon (case IV) plasmas. At the same initial plasma temperature, we have η_Au > η_c. The edge magnetic field in an early stage formed inside the gold target is larger than that inside the carbon foil, which is clearly observed in Figures 3a and 3c. The large edge magnetic field then focuses the beam electrons. With increasing the propagation distance, the edge magnetic field in case I becomes smaller and smaller (see Fig. 3b) and cannot bend all beam electrons. As shown in Figure 2a, parts of the injected electrons can escape or be scattered into the region |x| > 15 μm. However, the edge magnetic field for case IV is still large enough (>10 MG, as seen in Fig. 3d) to narrow the beam divergence. Figures 3a–3b further illustrate that the filamentation structure can be destroyed by the electron-plasma collision effect inside high-Z targets. For case of low-Z plasma materials, Figures 3c–3d show that regular filamentation beams can be kept for a relatively long distance.

When the beam electrons propagate through the target and enter the rear vacuum, they generate sheath electric field according to the ampere law ∂E/∂t = −J. The sheath electric fields at the rear target surface can be used to diagnose (Ridgers et al., Reference Ridgers, Sherlock, Evans, Robinson and Kingham2011) the distribution of the beam electrons using the relation n _b ≈ −∂E/ec∂t. In other words, the peak value of the longitudinal sheath electric field (E _z^peak(z)) can give a description of beam electrons passing through the target rear surface, and the distribution of the transverse electric field (E _z(x)) shall show the divergent properties of the injected-electron beam. Figure 4 gives a comparison of the electric field E _z(x, y) for our four cases. In cases I and II, the peak value E _z^peak(x, y) of the sheath electric field is lower that 6 × 10¹¹ V/m. In cases III and IV, the peak value can reach 3 × 10¹² V/m. Figures 4e and 4f gives E _z,C^peak/E _z,Au^peak > 5. On the other hand, Figures 4a–4d and 4f clearly show that the transverse electric fields E _z(x) for our four cases are completely different. Because large edge magnetic fields of the beam in case IV can almost control the beam propagation in the target with an initial radius 15 μm, thus the beam electrons can be concentrated in the rear surface to form a super-Gaussian distribution of the transverse electric field, as shown in Figure 4f. However, in cases I and II, the beam becomes divergent in the target, resulting in the appearance of widen transverse electric field structures. Since sheath field is responsible for proton acceleration from the rear surface, the present results indicate that higher accelerated-proton energy and lower proton emission should occur for the large-scaled carbon target.