1. INTRODUCTION
Laser-driven proton fast ignition (FI) of fusion targets (Roth et al., 2001) is a possible way toward inertial fusion energy (IFE) promising high efficiency and good control of the particle beam energy deposition in a compressed DT fuel (Deutsch, 2003, 2004). On the other hand, a critical point in the proton FI approach is the particle beam production efficiency, which is typically a few times lower than in the alternative approach using laser-driven electron beam (Kodama et al., 2001). An extremely challenging task is attaining the proton beam parameters required for ignition (Atzeni et al., 2002; Temporal et al., 2002): the beam intensity, ∼ 1020 W/cm2, the proton current density, ∼ 1013 A/cm2, the beam power, ∼1 PW (PetaWatt), the proton pulse duration, ∼10 ps, and the mean proton energy, ∼ 5 MeV. Potentially, such extreme proton beam parameters can be achieved by using the fast proton beams generated by the target normal sheath acceleration (TNSA) mechanism (Wilks et al., 2001). Achieving the required proton beam intensities with sufficiently high efficiency with this method can encounter severe difficulties. One of the reasons is that the relatively low density of accelerated protons at the source (in a close vicinity of the rear target surface), which typically is ∼ 1019 cm−3 (Snavely et al., 2000; Hatchett et al., 2000; Allen et al., 2003), or lower. As the relation between the ion beam intensity, Ii, the ion density, ni, and the ion (mean) energy, Ei, can be expressed by the following equation (Badziak et al., 2005b):
(A is the atomic mass number), for near optimum proton energy, Ei ∼ 5 MeV (Atzeni et al., 2002; Temporal et al., 2002), the required beam intensity, Ii ≥ 5 × 1019 W/cm2, can be attained at ni ≥ 1022 cm−3. To achieve such a proton density, the proton beam produced by TNSA at the rear target surface, has to be focused on a spot with diameter dif < (1/30)di, where di is the beam diameter at the rear surface, and dif ≤ 50μm (Atzeni et al., 2002; Temporal et al., 2002). Assuming the proton production energetic efficiency is ∼10% and a reasonable laser driver energy is ∼100–200 kJ, we concluded that in order to deliver 10–20 kJ of energy to the compressed fuel for ignition (Atzeni et al., 2002; Temporal et al., 2002), significantly more than 50% of the total number of produced protons should be focused in the required spot. Taking into account the angular and energetic dispersion of the produced protons it seems to be a very difficult task.
The other laser-based method of producing collimated high-density ion beams (plasma blocks), having potential to be used for FI, is skin-layer ponderomotive acceleration (S-LPA) (Badziak et al., 2004a, 2004b, 2005a, 2005b; Hora et al., 2004; Hora, 2005). In this paper, basic properties of proton beams produced by S-LPA are described and the possibility of using such beams for FI is considered.
2. PROPERTIES OF ION BEAMS PRODUCED BY S-LPA
S-LPA employs strong ponderomotive forces induced at the skin-layer interaction of a short laser pulse with a thin preplasma layer (of Lpre << df ) produced by the laser prepulse in front of a solid target (Lpre—the preplasma layer thickness, df —the laser focal spot diameter) (Badziak et al., 2004a, 2004b, 2005a, 2005b). The main short laser pulse interacts most intensely with the plasma in the skin layer near the surface of the critical electron density nec and the geometry of the interaction is almost planar (since Lpre << df ). The high plasma density gradient in the interaction region induces two opposite ponderomotive forces which break the plasma and drive two thin plasma blocks toward the vacuum and the plasma interior, respectively. As the ion density of the forward-accelerated plasma block is ni ≥ nec /z (z is the ion charge state), even at moderate mean ion velocity, the ion current density and the ion beam intensity of the block can be very high. The time duration τis of the ion flux flowing out of the interaction region (being the ion source) is approximately equal to the laser pulse duration τL and the block area Ss is close to the area of the laser focal spot Sf . Due to almost planar acceleration geometry, the angular divergence of the ion beam is small.
A comparison of parameters of a forward-accelerated proton beam at the source, attained in our S-LPA experiment with 0.5-μm polystyrene target at subrelativistic laser intensities (see Badziak et al., 2004b, 2005a, 2005b, for details), with the ones achieved in recent TNSA experiments with relativistic intensities (Snavely et al., 2000; Zepf et al., 2003; Cowan et al., 2004; Borghesi et al., 2004) is presented in Table 1. For calculation of the proton current density, js, and the proton beam intensity, Iis, at the source from experimental data we used the expressions (Badziak et al., 2004b, 2005b):
where Q is the total charge of fast protons measured in the far expansion zone, Ss is the fast proton source area, τL is the laser pulse duration, and Ei is the mean energy of fast protons. The proton density at the source, ni, was calculated from Eq. (1). As the angle cones within which the protons were recorded were essentially different in different experiments, the values of the proton beam parameters were recalculated to the same angle cone fixed as 10° (only protons with relatively small angular divergence, say ≤10°, are expected to be focused in the required small spot). From the results of the comparison presented in Table 1, the following conclusions can be reached: (a) the proton beam intensities at the source generated within a fixed angle cone by subrelativistic S-LPA are comparable to those produced (within the same angle cone) by TNSA at relativistic laser intensities and much higher laser energies, (b) the proton current densities at the source generated within a fixed angle cone are significantly higher for S-LPA than those for TNSA, and (c) the proton densities at the source produced by S-LPA are about a thousand times higher than those generated by TNSA within the same angle cone. It is worth noting that the “useful” (for focusing) part of the proton density in the TNSA beams is distinctly smaller than the “total” proton density, which, for example, for the PW experiment is estimated to be ∼ 1.5 × 1019 cm−3 (the beam divergence is ∼ 40° (Snavely et al., 2000)).
Parameters of proton beams produced in various experiments. (a) Cowan et al., 2004; (b) Zepf et al., 2003; (c) Borghesi et al., 2004; (d) Snavely et al., 2000
Although the current densities and intensities of proton beams produced by S-LPA at subrelativistic laser intensities are fairly high, to achieve MeV proton energies and beam intensities ≥ 5 × 1019 W/cm2 required for FI, S-LPA at relativistic laser intensities has to be used. The possibility of production of a high-density plasma (proton) block in such a regime was examined with the use of one-dimensional (1D) and two-dimensional (2D) two-fluid relativistic hydrodynamic codes for IL λL2 up to 1019 Wcm−2 μm2. Figure 1 and Figure 2 present the results of 2D simulations for the cases when the Gaussian laser pulse of τL = 0.25 ps, IL λL2 = 3 × 1018 Wcm−2 μm2 and df = 20 μm (Fig. 1) or df = 5 μm (Fig. 2) interacts with hydrogen preplasma of a linear density profile and of 3 μm thickness at ne = nec. For both cases, near the laser beam axis, there are produced high-density (∼5 × 1021 cm−3) proton beams of the proton energy ∼ 150 keV, the proton current density ∼3 × 1011 A/cm2, and the proton beam intensity ∼ 5 × 1016 W/cm2. However, the spatial structure of the proton beam essentially depends on the ratio df /Lpre. In the case of df = 20 μm, when the necessary condition for S-LPA, namely df >> Lpre, is fulfilled quite well, we observe a low-divergence proton beam of Gaussian-like spatial structure. On the other hand, at df = 5μm (df ∼ Lpre) a fairly complex multi-bubble structure of the proton beam is formed. Such proton beam structure is a consequence of the complex structure of the laser field in plasma, which is created, first of all, due to reflection of the light wave by the curved overdense plasma surface of the small radius of curvature. Fortunately, the proton beam spot diameter required for FI is estimated to be ∼ 30–50 μm (Temporal et al., 2002), so fulfilling the condition df >> Lpre is feasible at realistic contrast ratio of a laser pulse driving the proton beam.
Results of a numerical simulation of high-density plasma (proton) block generation by S-LPA obtained with the use of 2D two-fluid relativistic hydrodynamic code. The laser pulse of τL = 0.25 ps, λL = 1 μm, IL = 3 × 1018 W/cm2 and df = 20 μm interacts with hydrogen preplasma of 3 μm thickness at the critical density.
As in Figure 1 but at df = 5μm.
3. FAST IGNITION OF A FUSION TARGET BY A PLASMA BLOCK
The potential of a plasma (proton) block as a fast ignitor was tested for the FI by Plasma Blocks (FIPB) scheme (Badziak et al., 2005b) presented in Figure 3. In the considered case of FIPB, the laser beam (beams) driving the plasma (proton) block is focused on the planar target, placed in the vicinity of the tip of a guiding cone protecting both the laser beam and the plasma block against the influence of DT plasma, while it is compressed (Norreys et al., 2000; Kodama et al., 2001). For the above scheme, we estimated the intensity, Ih, the total energy, Eh, the pulse duration, τh, and the mean proton energy, Ei, of the proton flux heating the compressed DT fuel by using the formulae (Badziak et al., 2005b):
where: Iis and js are the proton beam intensity and the proton current density at the source, respectively; υi—the mean ion velocity, τL—the laser pulse duration, lth—the distance from the target to the high-density fuel, rh—the radius of heated fuel, g = f (lth /df ,θi)—geometrical factor, θi—the proton beam angular divergence, γ = (1 + 3SIL /Irel)1/2—the relativistic Lorentz factor, S—the dielectric swelling factor, IL—the laser intensity in vacuum and Irel ≈ 4.1 × 1018/λL2, [W/cm2, μm]—the relativistic intensity. The scaling laws Eqs. (5)–(7) were verified with 2D hydrodynamic code for IL λL2 ≤ 1019 Wcm−2 μm2 and reasonable quantitative agreement (within the factor 2) between numerical and analytical calculations was found. Then, Eqs. (2)–(7) were used for a rough estimate of the parameters of the proton flux heating a precompressed DT fuel according to the scheme presented in Figure 3. For the calculations, we assumed: lth = 100 μm, df = 40 μm, τL = 5 ps, λL = 0.53 μm, S = 1.5, θi = 5°, z/A = 1 (protons). We calculated the proton flux parameters as a function of energy of the laser pulse driving the proton beam generation. The result of the calculations is presented in Figure 4. It can be seen that parameters of the proton flux required for ignition of the precompressed (ρ ∼ 300–400 g/cm3) DT fuel, namely Eh ∼ 10–15 kJ, Ih ∼ 1020 W/cm2, Ei ∼ 5–8 MeV (Atzeni et al., 2002; Temporal et al., 2002), are attainable at laser energy ∼ 80–100 kJ. This laser energy can be even lower, if we take into account the fact that part of directed energy of laser-produced fast electrons, which is not used up for proton acceleration, can potentially be utilized for additional heating of the fuel. However, consideration of such hybrid, proton-electron FI scheme is beyond the scope of this paper.
The idea of FI by plasma block produced by S-LPA.
Parameters of proton flux heating the fuel as a function of ps laser energy.
4. CONCLUSIONS
In conclusion, it has been shown that S-LPA makes it possible to produce plasma (proton) blocks of proton densities about a thousand times higher than those of proton beams produced by the TNSA mechanism. Such a high-density plasma (proton) block driven by S-LPA at relativistic laser intensities can be considered as a fast ignitor of fusion targets. The rough estimates show that using the plasma block, the ignition threshold for precompressed DT fuel can be reached at the picosecond laser energy ≤ 100 kJ. Contrary to the scheme of Roth et al. (2001), the ignition seems to be feasible without ballistic focusing of the proton beam. In the case of increasing heating beam intensity, the focusing of the S-LPA beam using a shaped target or employing several overlapped beams generated from planar targets (Badziak et al., 2005b) can be considered.
ACKNOWLEDGMENTS
This work was supported in part by the State Committee for Scientific Research (KBN), Poland under Grant No. 1 PO3B 043-26, by the International Atomic Energy Agency in Vienna under Contract No. 11535/RO, as well as by the European Communities under the Contract of Association between EURATOM and IPPLM.
