1. INTRODUCTION
Since the first proposal of fast ignition (FI) (Tabak et al., Reference Tabak, Hammer, Glinsky, Kruer, Wilks, Woodworth, Campbell, Perry and Mason1994) for inertial fusion energy, this concept is under intense investigation experimentally as well as in theory (Badziak, Reference Badziak, Glowacz, Hora, Jablonski and Wolowski2006; Bret & Deutsch, Reference Bret and Deutsch2006; Chen & Wilks, Reference Chen and Wilks2005; Danson et al., Reference Danson, Brummitt, Clarke, Collier, Fell, Frackiewicz, Hawkes, Hernandez-Gomez, Holligan, Hutchinson, Kidd, Lester, Musgrave, Neely, Neville, Norreys, Pepler, Reason, Shaikh, Winstone, Wyatt and Wyborn2005; Deutsch et al., Reference Deutsch, Bret and Fromy2005; Gus'kov, Reference Gus'kov2005; Hoffmann et al., Reference Hoffmann, Blazevic, Ni, Rosmej, Roth, Tahir, Tauschwitz, Udrea, Varentsov, Weyrich and Maron2005; Hora, Reference Hora2007; Leon et al., Reference Leon, Eliezer, Piera and Marinez-Val2005; Nakamura et al., Reference Nakamura, Sakagami, Johzaki, Nagatomo and Mima2006; Sakagami et al., Reference Sakagami, Johzaki, Nagatomo and Mima2006). In our experiments with cone-guided targets (Kodama et al., Reference Kodama, Shiraga, Shigemori, Toyama, Fujioka, Azechi, Fujita, Habara, Hall, Izawa, Jitsuno, Kitagawa, Krushelnick, Lancaster, Mima, Nagai, Nakai, Nishimura, Norimatsu, Norreys, Sakabe, Tanaka, Youssef, Zepf and Yamanaka2002), the imploded core plasma was heated up to ~ 800 eV with a high coupling efficiency. As the next step, fast ignition realization experiment (FIREX) project (Mima et al., Reference Mima, Azechi, Fujita, Izawa, Jitsuno, Johzaki, Kitagawa, Kodama, Miyanaga, Nagai, Nagatomo, Nakatsuka, Nishimura, Norimatsu, Sakabe, Takabe, Tanaka, Yoshida, Yamanaka, Norreys, Zepf, Krushelnic, Habara and Hall2002) has been started at the Institute of Laser Engineering, Osaka University. The mechanism of the high coupling in a petawatt laser heating is, however, not clarified yet. In most of the previous studies of the heating process, laser-plasma interaction (LPI), relativistic electron transport, and core heating processes have been individually studied, e.g., relativistic LPI and the fast electron propagation by particle code (Wilks et al., Reference Wilks, Kruer, Tabak and Langdon1992; Pukov & Meyer-Ter-Vehn, Reference Pukhov and Meyer-Ter-Vehn1997; Lasinski et al., Reference Lasinski, Langdon, Hatchett, Key and Tabak1999; Sentoku et al., Reference Sentoku, Mima, Kojima and Ruhl2000, Reference Sentoku, Mima, Kaw and Nishikawa2003a, Reference Sentoku, Mima, Ruhl, Toyama, Kodama and Cowan2004; Ren et al., Reference Ren, Tzoufras, Tsung, Mori, Morini, Fonseca, Silva, Adam and Heron2004) and hybrid code (Gremillet et al., Reference Gremillet, Bonnaud and Amiranoff2002; Campbell et al., Reference Campbell, Degroot, Mehlhorn, Welch and Oliver2003; Taguchi et al., Reference Taguchi, Antonsen and Mima2004; Matsumoto et al., Reference Matsumoto, Taguchi and Mima2006), and core heating process by hybrid code (Campbell et al., Reference Campbell, Kodama, Mehlhorn, Tanaka and Welch2005; Mason, Reference Mason2006), and relativistic Fokker-Planck (RFP) code (Johzaki et al., Reference Johzaki, Mima, Nakao, Yokota and Sumita2003; Yokota et al., Reference Yokota, Nakao, Johzaki and Mima2006). To simulate the overall physics and identify the crucial physics in the fast heating, an integrated simulation is indispensable. Though Sentoku et al. (Reference Sentoku, Kemp and Cowan2006) carried out the core heating simulations including LPI using a collisional PIC code, the implosion dynamics was not included, i.e., the imploded core profile was arbitrary. We developed a multidimensional integrated code system “Flast Ignition Integrated Interconnecting Code” (FI3 code) (Sakagami & Mima, Reference Sakagami and Mima2001, Reference Sakagami and Mima2004; Sakagami et al., Reference Sakagami, Johzaki, Nagatomo and Mima2006; Nakamura et al., Reference Nakamura, Sakagami, Johzaki, Nagatomo and Mima2006), which includes all important physics from the implosion to the core heating. In the FI3 code, the overall implosion dynamics is simulated by an ALE-CIP radiation-hydro code “PINOCO” (Nagatomo et al., Reference Nagatomo, Ohnishi, Mima, Sawada, Nishihara and Takabe2001). A collective PIC code “FISCOF” (Sakagami & Mima, Reference Sakagami and Mima2001) simulates the relativistic LPI to evaluate the time-dependent energy distribution of relativistic electron beam. The core heating is simulated with a RFP-hydro code “FIBMET” (Yokota et al., Reference Yokota, Nakao, Johzaki and Mima2006; Johzaki et al., Reference Johzaki, Nagatomo, Mima, Sakagami and Nakao2004).
In the previous works (Sakagami et al., Reference Sakagami, Johzaki, Nagatomo and Mima2006; Johzaki et al., Reference Johzaki, Nagatomo, Sakagami, Sentoku, Nakamura, Mima, Nakao and Yokota2006), on the basis of integrated simulations with FI3 code for fast heating experiments with cone-guided CD shell targets (Kodama et al., Reference Kodama, Shiraga, Shigemori, Toyama, Fujioka, Azechi, Fujita, Habara, Hall, Izawa, Jitsuno, Kitagawa, Krushelnick, Lancaster, Mima, Nagai, Nakai, Nishimura, Norimatsu, Norreys, Sakabe, Tanaka, Youssef, Zepf and Yamanaka2002), we found the density gap effect on fast electron transport. If the density of plasma located between cone tip and dense core is as low as 10n c (n c is a laser critical density), the strong two stream instability is induced there due to fast electron flow, which generates the strong static field. Due to this field, some of fast electrons having relatively low energy are reflected and trapped inside the cone tip, and bulk electrons are accelerated toward the cone tip. These confined moderate-energy fast electrons are gradually released from the cone tip with intensity of ~1018 W/cm2 even after finishing laser irradiation and contribute to the core heating. Thus, if such density gap exists behind the cone tip, the profiles of fast electrons change during propagation into the dense core, i.e., the energy spectrum are moderated and the electron beam duration becomes longer. As the results, core heating efficiency becomes high compared with the no density gap case. Even if including the density gap effect, however, the resultant ion temperature reaches only 0.45 keV, which is still lower than the value obtained in the experiments (0.8 keV) (Johzaki et al., Reference Johzaki, Nagatomo, Sakagami, Sentoku, Nakamura, Mima, Nakao and Yokota2006).
In the heating phase of FI, before main pulse irradiation, low density plasma is formed on the cone inner surface due to the pre-pulse. In the previous simulations (Johzaki et al., Reference Johzaki, Nagatomo, Sakagami, Sentoku, Nakamura, Mima, Nakao and Yokota2006), we assumed a scale length of L f = 5 µm for the pre-plasma. However, the scale length strongly depends on the pre-pulse profiles of the heating laser (intensity, duration, and focal spot), and it affects the fast electron energy distribution, and the energy coupling efficiency from heating laser to fast electron. In the present paper, in addition to the density gap effect, hence, we evaluate the pre-plasma scale length dependence of core heating efficiency using FI3 code.
At the relativistic LPI, in addition to fast electrons, fast ions are generated. We also evaluate the possibility of the accelerated ions for contributing to the core heating.
2. SIMULATION CONDITION
The details of FI3 code consisting of three codes (an ALE Radiation-hydro code “PINOCO,” a collective PIC code “FISCOF” and a RFP-hydro code “FIBMET”) are described (Yokota et al., Reference Yokota, Nakao, Johzaki and Mima2006; Sakagami et al., Reference Sakagami and Mima2001, Reference Sakagami and Mima2004, Reference Sakagami, Johzaki, Nagatomo and Mima2006; Nagatomo et al., Reference Nagatomo, Ohnishi, Mima, Sawada, Nishihara and Takabe2001; Johzaki et al., Reference Johzaki, Nagatomo, Mima, Sakagami and Nakao2004). The procedure of the present core heating simulations for a cone-guided target is as follows. First, we carried out implosion simulations for an Au cone attached CH shell target using PINOCO to obtain the compressed core profile. For heating phase, time-dependent profiles of fast electron and ion generated by the heating laser irradiation are evaluated with PIC simulations (FISCOF). The following energy transports into the imploded dense core are simulated by FIBMET where the time-dependent profiles of fast particles evaluated by FISCOF simulations are used as the external sources in fast electron and ion transports, and the imploded core profiles obtained by PINOCO simulations are used as the initial condition of bulk plasma. In the core heating, the multi-dimensional natures, e.g., geometrical effects of laser-cone interactions, magnetic field effects, spatial beam divergence, are of course important. However, full scale (time and space) multidimensional simulations are very expensive. Thus, for the present parametric study on scale length dependence, we used one-dimensional (1D) PIC, and RFP-hydro codes.
3. PRE-PLASMA SCALE LENGTH DEPENDENCE
3.1. Fast Electron Profiles
For evaluation of fast electron profiles, we carried out 1D PIC simulations for laser-cone tip interactions. In the PIC simulations, electros and ions are mobile and the collision process is not included. The density profile assumed in PIC simulations is shown in Figure 1. The cone tip is modeled by 100n c and 10 µm thickness plasma. The pre-plasma having exponential density profile (n e∝ exp(x/L f), L f is the scale length and is changed from 0 µm to 5 µm) is attached to the front surface (laser irradiation side). For introducing the density gap on the rear surface of the tip, we located low density imploded plasma, of which density is set to n e = 10n c at the contact surface and is exponentially raised up to 100n c at 24 µm away from the rear surface. Following the imploded plasma, dense core plasma is located with the density of 100n c. In the PIC simulation, we assumed Au with real mass and Z = 50 (this value is evaluated by PINOCO simulations) as the bulk ion in the whole region. Since the bulk electrons are rapidly heated up to tens of keV by the laser-plasma interactions, the fast electron profiles are not sensitive to the initial electron temperature. Thus we assumed relatively high initial electron temperature (10 keV) to enlarge the mesh size, hence to reduce the total number of spatial mesh. Contrary to this, practically-low temperature was assumed for bulk ion (300 eV) to prevent un-physical thermal expansion. The heating laser is the Gaussian pulse with the wavelength of λL = 1.06 µm, the pulse duration (full width of half maximum) of τFWHM = 750 fs and the peak intensity of I L = 1020 W/cm2. The irradiated laser energy is 0.79 J/µm2, which corresponds to 560 J when the laser spot size is 30 µm. The simulation time is 6 ps. The forward-directed fast electron profiles are observed at two different points; one inside the cone tip and the other at the 20n c point in the imploded plasma.
Fig. 1. Electron density profile of the cone tip and the imploded plasma assumed in PIC simulation.
3.1.1. Effects of Density Gap and Density Profile Steepening
In Figure 2, we show the fast electrons (E fe > 100 keV) profiles observed in the case of L f = 1.0 µm. Figure 2a is the temporal profile of beam intensity and Figures 2b and 2c are the energy spectra during laser irradiation (t = 1 ps ~1.5 ps) and after finishing laser irradiation (t = 2.5 ps ~ 3.0 ps), respectively. The density gap effect (Sakagami et al., Reference Sakagami, Johzaki, Nagatomo and Mima2006; Johzaki et al., Reference Johzaki, Nagatomo, Sakagami, Sentoku, Nakamura, Mima, Nakao and Yokota2006) is observed in Figure 2. The higher beam intensity of forward directed electrons in the cone tip than that at 20n c point indicates the confinement of relatively low energy fast electrons inside the cone tip during the laser irradiation. These confinement components are clearly found at the region of E fe < 1 MeV in Figure 2b. The longer fast electron beam duration than the heating laser duration means the release of the confined electrons from the cone tip after finishing laser irradiation and the energy of these electrons is lower than 1 MeV as is shown in Figure 2c.
Fig. 2. Forward-directed fast electron (E fe > 100 keV) profiles observed at the two different points (inside the cone tip and at 20n c point in the imploded plasma) in the case of L f = 1 µm. (a) time history of electron beam intensity, (b) energy spectra observed during laser irradiation (t = 1.0 ps ~ 1.5 ps), and (c) the spectra after finishing laser irradiation (t = 2.5 ps ~ 3.0 ps).
In a previous work (Johzaki et al., Reference Johzaki, Nagatomo, Sakagami, Sentoku, Nakamura, Mima, Nakao and Yokota2006); we assumed the homogeneous density profiles (100n c, 10n c, and 2n c) for the plasma located at the rear of cone tip. Contrary to this, in the present study, we assumed the exponential profile which seems to be more realistic. In this case, the bulk ions tend to flow into the lowest density point from both sides (from the cone tip and the denser imploded plasma region) because of the pressure imbalance. In addition, the static field generated at the rear surface of the cone tip due to the fast electron current pulls ions from the cone tip into the low density plasma. The density gap is hence filled due to these ion flows, which reduces the density gap effect. As a result, the confined electrons are rapidly released from the cone tip. It can be found, this phenomenon in the profile of fast electron intensity observed at 20n c point (Fig. 2a). There exist two peaks; the first one corresponds to the peak of laser intensity and the second one corresponds to the release of confined electrons resulting from filling up the density gap. After the second peak, thus, the fast electron beam intensity is rapidly decreased.
When the heating laser intensity is as high as 1020 W/cm2, the pre-plasma is pushed into the dense region by the laser ponderomotive force, and then the density steepening takes place at LPI region. Figure 3 shows the spatial profiles of electron density n e (blue lines) and phase plots of electrons (light blue dots) observed at t = 1 ps for the three different scale length cases (L f = 0.25 µm, 1.0 µm, and 5.0 µm).
Fig. 3. (Color online) Spatial profiles of electron density n e (blue lines) and phase plots (light blue dots) observed at t = 1 ps for the three different cases (L f = 0.25 µm, 1.0 µm, and 5.0 µm).
In short scale length cases (L f ≤1.0 µm), the pre-plasma is almost swept away and the heating laser is directly interacts with the dense cone tip at this moment. Under this situation, the returned electrons are pushed back into the cone tip by the laser oscillating ponderomotive force at the LPI region. It is hence observed the periodically-launched forward-directed electron bunches in electron phase plots (Figs. 3a and 3b). Since the energy of these electrons is not high, part of these electrons are reflected again by the static field at the density gap and goes back to the LPI region. In this way, the cone tip is filled up with relatively low energy fast electrons. These confined electrons gain the energy from the heating laser. The electrons of which energy becomes higher than the potential gap at the rear surface are released from the cone tip. The other electrons are released after laser irradiation.
3.1.2. Scale Length Dependence
The difference in fast electron profiles due to the pre-plasma scale length is clearly observed in Figure 3. In a very short scale length case (L f = 0.25 µm), the underdense LPI region is narrow, so that the energy coupling efficiency from laser to fast electrons is small. Thus, the static field generated due to the fast electron current at rear side of cone tip is weak, and then the density gap effect is weak. At the LPI region, the density steepening rapidly occurs and then the electron bunches are observed. However, the accelerated electron energy and forward-directed electron beam intensity are low because of low coupling efficiency.
With increasing the scale length, the underdense LPI region and a time required for density steepening become long. Thus, more electrons are accelerated before density steepening occurs, which enhances the confinement due to the density gap effect. As the results, in the case of L f = 1 µm (Fig. 3b), the energies of both forward-directed bunched electrons and returned electrons become high.
In the further long scale length case, a time required for density steepening is much longer. As is shown in Figure 3c (L f = 5 µm), the pre-plasma has not been completely swept away at t = 1.0 ps; the long scale pre-plasma is still remained at the LPI region. Thus, at this moment, the energy of fast electron accelerated at the LPI region is very high compared with the shorter scale length cases. These very high energy fast electrons cannot be trapped by the static field at the rear surface of the cone tip. In addition, at the LPI region, the reflection of returned electrons becomes weak; the backward acceleration is observed in the underdense region. Thus, the trapping effects due to the density gap and the density steepening become weak.
Figure 4 shows the time-integrated forward-directed fast electron spectra observed at 20n c point behind cone tip for the different scale length cases. With increasing L f from 0 µm to 1 µm, the coupling efficiency from laser to fast electrons and the generated fast electron energy are increased. In further long scale case such as L f = 5 µm, the energy of fast electron becomes so high. The number of electrons is large in high energy region (E fe > 4 MeV) compared with L f = 1 µm case. However, the stopping range of such high energy electron is much longer than a compressed core size, and then does not heat the core efficiently. Contrary to this, the number of relatively low energy electrons, which are mainly contribute to the core heating, is small since the effects of density gap and the density steepening become weak as was stated above. These results mean the existence of the optimal scale length for efficient core heating.
Fig. 4. Time-integrated forward-directed fast electron spectra observed at 20n c point in the imploded plasma for the cases of L f = 0 µm, 0.25 µm, 1 µm, and 5 µm.
3.2. Core Heating
Using the time-dependent profiles of fast electron after passing through the low-density gap region, we carried out the core heating simulations using the 1D RFP-Hydro code. As for the initial condition of bulk plasma, we use the imploded core profiles obtained by r-z cylindrical 2D implosion simulation for a cone-guided CD shell target with PINOCO. The shell has 1.06 g/cm3 solid density, 8 µm thickness, and 250 µm inner radius. The Au cone with an opening angle of 30 degree is attached to the shell. The shell is uniformly irradiated by 2 kJ Gaussian-pulse-shaped laser (the wavelength λL= 0.53 µm). The imploded core profiles at the central axis (r = 0) are shown in Figure 5. The maximum core density is 85 g/cm3 and the area density of the core is 0.07 g/cm2. The fast electrons are injected at the low density region behind the cone tip.
Fig. 5. Imploded core profiles at the central axis (r = 0) obtained by r-z cylindrical 2D implosion simulation for Au cone guided CD target. The hatched region is Au and the other region is CD. The fast electron injected point is indicated by a thick arrow.
In Figure 6, spatial profiles of fast-electron energy deposition rates at t = 2.5 ps are plotted for the L f = 1 µm case. In the low-density region around the fast electron injection point, the Joule heating is comparable to the collisional heating due to the Coulomb interactions with bulk electrons. In the dense core region, the fast electron current can be easily cancelled by bulk electron flow with small drift velocity (v d ≪ c) because of much larger density of bulk electron than that of fast electron. Thus, the field effect is negligible and the collisional heating is dominant. As for the Coulomb interactions in dense plasmas, not only short-range binary collisions in the impact parameter range of b min < b < λD but also long-range interactions in the impact parameter range of b>λD, where the collective shielding effect is included, contribute to the energy deposition of fast electrons (Deutsch et al., Reference Deutsch, Furukawa, Mima, Murakami and Nishihara1996 & Reference Deutsch, Furukawa, Mima, Murakami and Nishihara2000; Johzaki, Reference Johzaki, Mima, Nakao, Yokota and Sumita2003; Yokota et al., Reference Yokota, Nakao, Johzaki and Mima2006). Here, b min is the minimum cutoff impact parameter, being on the order of the distance of the closest approach, and λD is the Debye length. The temporal evolution of ion and electron temperatures averaged over ρ > 10 g/cm3 region, <T k > =∫ρ>10g/cm3T k (x)R DD (x)dx/∫ρ>10g/cm3R DD(x)dx where k denotes ion or electrons, and R DD(x) is the DD reaction rate at position x, are shown in Figure 7 for the case of L f = 1 µm. Due to the collisional heating, the bulk electron is heated first, and then the bulk ion is heated via the temperature relaxation. Thus, the electron temperature reaches maximum (<T e> max = 0.84 keV) at first (at t = 3.6 ps), and then <T i> max = 0.72 keV is obtained by 3 ps delay.
Fig. 6. Energy deposition profile of fast electron at t = 2.5 ps for the case of L f = 1 µm.
Fig. 7. Temporal evolution of electron and ion temperatures averaged over dense core region (ρ > 10 g/cm3) in the case of L f = 1 µm.
The results of core heating simulations by varying L f are summarized in Figure 8. Figure 8a shows the scale length dependence of energy carried by the fast electrons (total and E fe > 2 MeV, E fe < 2 MeV components) evaluated by PIC simulations and the energy deposited by fast electron inside the fuel plasma evaluated by the following RFP-hydro simulations. The right axis shows the corresponding energy coupling efficiencies. In Figure 4b, the maximum value of <T i> is plotted as a function of L f. With increasing L f up to 1.5 µm, the energy coupling from heating laser to fast electron becomes large, so that the deposited energy and the resultant core temperature increase. The scale length becomes long furthermore, the total beam energy gradually decreases. However, the higher energy component (E fe > 2 MeV) increases, and then the low energy component (E fe < 2 MeV) that is effective in core heating, decreases faster than the total beam energy. As a result, the deposited energy in the fuel plasma and the resultant core temperature also decrease. These results indicate that the core heating efficiency depends not on the total beam energy but on the beam energy of low energy component (E fe <2 MeV). In the present simulations, the optimum scale length for core heating is L f = 1.5 µm. In this case, the energy coupling from the heating laser to the core is 14.9% and ion in the core is heated up to 0.86 keV (0.48 keV rising).
Fig. 8. Pre-plasma scale dependence of (a) total fast electron energy (• with a broken line), and its E fe > 2 MeV (Δ with a broken line) and E fe < 2 MeV (○ with a broken line) components and deposited energy of fast electron inside the fuel, and (b) maximum value of core ion temperature <T i>max.
3.3. Double Scale Length Case
In the above discussion, we assumed single scale-length pre-plasmas, i.e., the density profile of pre-plasma was assumed as n e ∝ exp(x/L f) from the underdense region to the 100n c region. The results show that the long scale length for underdense region is in favor of high coupling efficiency at the early stage of heating laser irradiation. The underdense plasma is rapidly swept away, and then the heating laser directly interacts with overdense plasma. In this phase, the short scale length plasma is desirable since the generated fast electron energy is relatively low and the returned electrons are completely pushed back into plasma by the laser oscillating ponderomotive force. Thus, we introduce a double scale length for pre-plasma structure, i.e., we assumed long scale length for underdense region and short scale length for over dense region. In practical, the double scale length structure is observed in the results of radiation-hydro simulations for low intensity laser (~1011 W/cm2) and plasma interactions, i.e., the underdense region has a relatively long scale length (~10 µm or more) and the overdense region has a short scale length (~1 µm). For estimating the core heating properties in such cases, we carried out the simulations by assuming double scale length pre-plasmas. In order to compare with the single scale length cases, the scale length was assumed as L fu = 5 µm for the underdense region and L fo = 1 µm (or 1.5 µm) for the overdense region.
Figure 9 shows spatial profiles of electron density n e and phase plot observed at t = 1 ps for the double scale length case (L fu = 5 µm and L fo = 1 µm). Compared with the L f = 1 µm single scale length case, the double scale length plasma has the long underdense plasma, so that in the early stage of laser irradiation, more fast electrons are generated. The static field generated at the density gap, thus, becomes slightly stronger, which makes the density gap effect remarkable a little. But, the underdense plasma is rapidly swept away by the laser ponderomotive force. As for the density steepening, hence, the scale length of overdense region determines the characteristics, so that the density profile at t = 1 ps is almost the same as that in the L f = 1 µm single scale length case.
Fig. 9. (Color online) Spatial profiles of electron density n e (blue lines in log scale) and phase plots (light blue dots in linear scale) observed at t = 1 ps for the double scale length case (L fu = 5 µm and L fo = 1 µm).
The time-integrated fast electron spectrum for the double scale length case (L fo = 1 µm) is plotted in Figure 10, together with those for the single scale length cases (L f = 1 µm and 5 µm cases). The spectrum for the double scale length case is almost the same as that for L f = 1 µm single scale case except for the low energy region (E fe < 2 MeV). The number of electrons in the low energy region is slightly large in the double scale case because of the increase in the gap effect. Using the fast electron profiles obtained in the double scale cases, the core heating simulations were carried out. The obtained results are plotted in Figure 11, together with the results of single scale cases. It is found that in the double scale cases, the core heating efficiencies and the resultant core temperatures become high compared with those in the single scale cases because of increase in the electron number in the low energy region (E fe < 2 MeV).
Fig. 10. Time-integrated fast electron spectra for the double scale length case (L fu = 5 µm and L fo = 1 µm) and for the single scale length cases (L f = 1 µm and 5 µm cases).
Fig. 11. Pre-plasma scale dependence of (a) total fast electron energy (black) and its E < 2 MeV component (blue) and deposited energy of fast electron inside the fuel, and (b) maximum value of core ion temperature <T i>max. The circles are the single scale length cases and the triangles are the double scale length cases. The results for the double scale length cases are plotted by triangles with lines.
Even if the single scale length is assumed, we obtained the heated core temperature comparable to the value obtained at the experiments when 1 µm ≤ L f ≤ 2 µm. In the case of the double scale length, which seems to be more practical, the core heating becomes more efficiently.
4. CONTRIBUTION OF FAST IONS
In the intense LPI, the bulk ions are also accelerated. At the LPI region, fast ions are generated by sweeping potential (Sentoku et al., Reference Sentoku, Cowan, Kemp and Ruhl2003b). At the rear side of the cone tip, the electric field built up due to the fast electron current acts on the ions to accelerate toward the dense core. Those accelerated ions have the possibility for contributing to the core heating.
In the present PIC simulations, all the ion is assumed as Au with Z = 50 and irradiated laser intensity is 1020 W/cm2. The maximum energy of ion accelerated at the LPI region is ~ 600 MeV, which well agrees with the scaling for the acceleration by sweeping potential (Sentoku et al., Reference Sentoku, Cowan, Kemp and Ruhl2003b). On the other hand, the ions are at the rear surface up to ~100 MeV. The energy coupling efficiency from laser to fast ions is 0.6%–2.3% for the single scale length case (L f = 0 µm–5 µm). However, the stopping range of such a high-Z and heavy ion is too short to heat the dense core. The range of 500-MeV (100-MeV) Au ion (Z = 50) evaluated with continuous slowing down model is 0.027 g/cm2 (0.0083 g/cm2) in a CD plasma with T = 1 keV, ρ = 50 g/cm3. The accelerated ions hence deposit their kinetic energies and momenta in the cone tip and the low density imploded plasma before reaching the dense core. As the results of core heating simulations including the fast ion transport, where the fast ion beams evaluated by PIC simulations are injected at the same point as the electron beam injection point, all of the fast ions stopped in the low density imploded plasma and do not contribute to the core heating.
5. CONCLUDING REMARKS
On the basis of the FI3 integrated simulations, we evaluated the dependence of core heating properties of cone-guided targets on the pre-plasma scale length. From the simulations where we assumed the single scale length for the density profile of pre-plasma and located the density gap at the rear surface of cone tip, we found that the core heating efficiency is very sensitive to the pre-plasma scale length and there exists the optimum scale length.
At the LPI region, the density steepening occurs due to the strong laser ponderomotive force. When the scale length is too short (≪1 µm), the underdense plasma is rapidly swept away and the steepening occurs quickly. Thus, the energy coupling from laser to fast electron and the resultant core heating efficiency are low. On the other hands, when the scale length is too long (≫1 µm), it takes a long time to steepen the density at the LPI region, which means the wide underdense plasma remains in long time. The generated fast electrons energy is hence so high that the confinement effects due to the density gap at the rear surface of cone tip becomes weak. Then the number of relatively low energy fast electrons (E fe < 2 MeV), which mainly contributes to the core heating, is decreased, and the core heating efficiency becomes low. It is found that the optimum scale length of pre-plasma is L f = 1.5 µm when the heating pulse intensity is 1020 W/cm2 and the duration is 1 ps. In this case, we obtained the energy coupling from heating laser to core of 14.9% and heated core temperature of 0.86 keV.
To increase the energy coupling efficiency, we introduced the double scale length for pre-plasma profile; a long scale length (5 µm) in the underdense region and a short scale length (~1 µm) in the overdense region. It was fond that the density steepening is characterized by the scale length of overdense region since the underdense plasma is rapidly swept out. Thus, the spectrum of fast electrons generated in the double scale length case is almost the same as that in the single short scale length case except for the low energy region (E fe < 2 MeV). The number of electrons in the low energy region is slightly large in the double scale case because of the increase in the gap effect. As the results, the core heating efficiencies and the resultant core temperatures become high compared with those in the single scale length case, e.g., the energy coupling from heating laser to core of 16.7% and heated core temperature of 0.94 keV are obtained when the scale length of underdense and overdense region are assumed as 5 µm and 1.5 µm, respectively. The obtained ion temperature is almost the same as that measured at the experiment (Kodama et al., Reference Kodama, Shiraga, Shigemori, Toyama, Fujioka, Azechi, Fujita, Habara, Hall, Izawa, Jitsuno, Kitagawa, Krushelnick, Lancaster, Mima, Nagai, Nakai, Nishimura, Norimatsu, Norreys, Sakabe, Tanaka, Youssef, Zepf and Yamanaka2002).
The above results indicate that the pre-pulse profile (both the spatial and temporal profiles) of the heating laser, which determines the pre-plasma structure, is very important when the heating pulse duration is ~1 ps. Even in the longer pulse case such as the heating laser at FIREX-I (10 ps), the pre-plasma scale length affects the fast electron profiles through the density steepening effect at the LPI region and the density gap effect at the rear surface of cone tip. Thus, for further study, in addition to the multi-dimensional features, the estimation of pre-plasma profiles using realistic pre-pulse profile is required.
We also evaluated the fast ion contribution to the core heating. The fast ions (Au, Z = 50) are also generated by both static potentials at the laser irradiation surface (~600 MeV) and at the rear surface density gap (~100 MeV). However, these ions do not contribute to the core heating because of the short range.
Acknowledgments
This work was supported by MEXT, the Grant-in-Aid for Creative Scientific Research (15GS0214) and partially the Grant-in-Aid for Encouragement of Young Scientists (B) (17760666). We are grateful for the support of the computer room of ILE and the cyber media center at Osaka University.
