1. INTRODUCTION
Laser-driven fast electron transport in solid targets has been studied by experiments (Baton et al., 2003; Key et al., 2004; Koch et al., 2001; Kodama et al., 2001; Roth et al., 2005; Santos et al., 2002; Stephens et al., 2004), theory (Fill, 2001; Tikhonchuk, 2002), and simulations (Bell & Kingham, 2003; Bibi et al., 2004; Davies, 2002; Gremillet et al., 2002, Honrubia et al., 2005). Experiments have shown the propagation of fast electrons with a relatively large divergence, that is, within a cone angle of 40° (Key et al., 2004; Stephens et al., 2004). However, the hybrid PIC codes developed to simulate these experiments does not reproduce this divergence when standard simulation parameters are used (Stephens et al., 2004). Therefore, the capability of these codes to simulate fast electron transport in large scale dense plasmas has not been fully exploited until now. Here, we found that the initial electron angular distribution is very important to put hybrid PIC simulations closer to experiments.
We focused our study on recent experiments of heating aluminum foil targets by fast electrons (Key et al., 2004; Stephens et al., 2004; Martinolli et al., 2004). Previous calculations (Honrubia et al., 2004) have shown temperatures at the target rear surface to be two or three times higher than measurements (Key et al., 2004). These differences can be explained by the excessive beam collimation typically found in hybrid PIC codes, which relies on the initial angular distribution used for fast electrons (Bell & Kingham, 2003; Honrubia et al., 2005). Here, we used a suitable angular distribution based on the ejection angle of ponderomotively accelerated electrons in order to reproduce the temperatures, Kα yields and effective propagation angles measured in those experiments. This distribution has also been used by Stephens et al. (2004) in the context of the Monte Carlo collisional simulations and by Alonso (2003) in the context of hybrid PIC simulations. With such a distribution, low energy electrons are injected into the simulation box with large divergence angles while high energy electrons are injected within a cone half-angle of 20°–30° as in standard hybrid calculations (Davies, 2002). This energy-dependent initial divergence of fast electrons was evidenced in experiments by Gremillet et al. (1999), and has been recently included in simulations by Evans et al. (2005), and Town et al. (2005) by adding a constant thermal spread to the forward momentum distribution of fast electrons.
This paper is organized as follows. In Section 2, we briefly present the simulation model used. In Section 3, we describe the experiment analyzed, the parameters used in simulations, and the results obtained. Finally, the conclusions and future developments are summarized in Section 4.
2. SIMULATION MODEL
Simulations have been carried out with the hybrid PIC code described in Honrubia et al. (2004, 2005) run in two-dimensional r-z cylindrical geometry mode. Fast electrons are injected in the simulation box with Gaussian distributions in radius and time, and an exponential distribution in energy. Temperatures of this last distribution are computed by means of the ponderomotive scaling formula kTf = me c2[(1 + I18 λμ2/1.37)1/2 − 1] (Wilks et al., 1992), where Tf is the fast electron temperature, I18 is the laser intensity in units of 1018 W/cm2, λμ is the wavelength in microns, and me is the electron rest mass. Fast electron angular distribution is given by the energy-dependent ejection angle of ponderomotively accelerated electrons, θf = tan−1[(2/(γ − 1))1/2] (Quesnel & Mora, 1998), where γ is the relativistic Lorentz factor, in such a manner that fast electrons are injected with an angle uniformly distributed between 0 and θf . Notice that the energy dependence of the cone half-angle θf through γ leads to an almost isotropic electron distribution at low energies, and to a highly forward peaked distribution at high energies. Notice also that the γ factor depends on the radial coordinate and time by means of the Gaussian distributions of the laser intensity assumed.
The ejection angle formula combined with the Gaussian distributions of the fast electron pulse in radius and time gives the overall initial angular distribution shown in Figure 1 for the laser pulse parameters used in this paper. Notice that this distribution is far from the Gaussian one, showing a plateau up to 20° and a FWHM (full width at half maximum) of, approximately, 40°, much higher than the typical 20°–30° divergences used in “standard” hybrid simulations. The top-hat angular distribution used by Davies (2002) is also shown for comparison.
Overall angular distribution of the fast electrons injected in the simulation box with the laser pulse parameters described in Section 3. The curve labeled with FWHM corresponds to electrons injected within the FWHM of the Gaussian distribution in radius, while the curve labeled by wings corresponds to electron injected at larger radii. The distribution used by Davies (2002) for all electrons is also shown.
Resistivities are computed as in the model of Eidmann et al. (2000) assuming separate temperatures for electron and ion species. Tables generated by MPQeos (Kemp & Meyer-ter-Vehn, 1998) are used to obtain temperatures and ionization of the background plasma from Coulomb energy deposition and Ohmic heating due to the return current.
The numerical parameters used are a time step of 1.5 fs and a cell width of 0.5 μm in both axial and radial directions. 106 particles are injected in 2 ps, with a peak rate of 1450 particles per time step at 1 ps. Fast electrons are reflected at the front and rear surfaces to simulate the refluxing or recirculation of fast electrons within the target.
3. RESULTS
The targets analyzed consist of an aluminum transport layer with thickness ranging from 10 to 320 μm, followed by a 25 μm copper layer to absorb radiation coming from the laser interaction region, and a 20 μm aluminum fluor layer, as depicted in Figure 2 (Key et al., 2004; Stephens et al., 2004). These targets were illuminated by a CPA laser beam focused onto a 10 μm diameter spot (FWHM), with a mean irradiance at FWHM of 2.5 × 1019 W/cm2 and a pulse duration of 1 ps (FWHM). The mean kinetic energy of fast electrons is 1.5 MeV (FWHM) and the laser-to-fast-electron conversion efficiency is assumed to be 33%. The diagnostics used were Kα spectroscopy of the aluminum fluor layer resolved in space and thermal self-emission to measure the temperature distribution at the target rear surface.
Targets used in the fast electron heating experiments analyzed in this paper (Key et al., 2004; Stephens et al., 2004).
In the next Sections, we first analyze the effective propagation angle and fit the size of the fast electron source to reproduce as much as possible the size of the Kα distribution at the rear surface of the target. We then check the attenuation of the Kα yield with target thickness and the temperature distribution at the target rear side. Our goal was to reproduce the measurements of these three diagnostics with the same set of simulation parameters.
3.1. Effective propagation angle
Propagation of fast electrons in solid targets depends strongly on their initial angular distribution. If the standard Gaussian distribution with a FWHM of 20° is used, previous simulations of the experiments considered in this paper have shown that the beam is resistively collimated by the self-generated azimuthal magnetic field along distances of, approximately, 150 μm (Honrubia et al., 2004). This result agrees with the analytical model developed by Bell and Kingham (2003), which predicts beam collimation for the electron beam parameters shown above. It does not agree, however, with the experimental results reported by Stephens et al. (2004), which clearly show that electrons propagate in the target with an effective cone angle of 40° rather than within a cylinder with a diameter of the order of the laser focal spot.
We used the energy-dependent angular distribution discussed in Section 2 and shown in Figure 1 to simulate fast electron propagation in the target. With this distribution, the fast electron current density spreads strongly when electrons penetrate the target, resulting in a lower current density and field generation. This in turn leads to the beam collimation over a shorter distance, 60–70 μm, than that found in standard hybrid PIC simulations (150 μm), as depicted in Figure 3. For higher penetrations, the self-generated fields have a decreasing importance, and fast electron transport is mainly dominated by the angular spread with which fast electrons come out from the collimated region, and by Coulomb collisions with the background plasma. The map of the computed Kα distribution for the 300 μm target is depicted in Figure 4, where the diameter of the Kα spot shown corresponds to an effective propagation angle of 40°, in good agreement with experiments.
Fast electron current density in the 300 μm Al target shown in Figure 2. Fast electrons are injected at the left of the simulation box.
Computed Kα image at the rear surface of the 300 μm target shown in Figure 2.
The radii of the Kα spot obtained by simulations are compared with experiments in Figure 5. Simulations with the energy-dependent angular distribution reproduce quite well the effective propagation angle seen in the experiments, but present two differences with the experimental curve, namely: (1) the computed Kα spot radii are, approximately, 30 μm below experiments for all thicknesses; and (2) the almost constant Kα spot size found in simulations for targets thinner than 50 μm due to the beam collimation. These two differences can be reduced by using a fast electron source larger than the laser focal spot. This assumption is supported by the experimental results depicted in Figure 5, which show Kα radii much larger than the laser spot for targets as thin as 10 μm. Therefore, we increased the radius of the fast electron source with the limitation that the temperature at the target rear surface has to be similar to the temperature measured in the experiments, as will be explained in Section 3.3. With the parameters used here, we found a good agreement between simulations and experiments on this temperature when the laser focal spot is multiplied by a factor of 2.5, resulting in a fast electron source radius of 12.5 μm (FWHM) instead of the 5 μm of the laser spot. Fast electrons are injected into this “extended” source with a kinetic energy and an impinging angle that depend on the local laser intensity at the 5 μm focal spot, but with a radius 2.5 times larger. This is equivalent to assume a radial drift or lateral transport of fast electrons in the plasma at the target front side from the laser spot to the fast electron spot. Notice that this radial drift is more important for low energy electrons generated at the outermost zones of the laser spot. Stephens et al. (2004) pointed out some plausible physical mechanisms that can induce this lateral transport. Indeed, this effect was studied by a number of authors more than 20 years ago (Fabro & Mora, 1982; Forslund & Brackbill, 1982). However, since a theory to explain quantitatively the radial drift observed in current high-intensity experiments has not been developed yet, the increase of the size of the fast electron source used here can be considered as a fitting of our simulations to the experimental results. Simulations show (see Fig. 5) that when this “extended” source is used, the computed Kα radii increase, approximately, the same amount that the fast electron source, reducing the difference between computed and experimental Kα radii from 30 to 20 μm for almost all thicknesses. This is in agreement with the predictions of the analytical model of Bell and Kingham (2003) about the weak dependence of the self-generated azimuthal magnetic field (∝R−3/5), the beam collimation parameter (∝R2/5) and then the effective propagation angle on the beam radius R.
Spot radius of the Kα photons emitted by the Al fluor layer as a function of the thickness of the Al transport layer. The dotted line corresponds to 5 μm fast electron spot radius at the front surface and the dashed line to 12.5 μm.
3.2. Kα yield
The computed Kα yields of the aluminum fluor layer are depicted in Figure 6. The experimental points have been plotted for comparison. The electron range obtained from simulations is 260 μm (for target thicknesses from 100 to 250 μm), in good agreement with the range of 250 μm measured in the experiments (Martinolli et al., 2003). The details of the radial distribution of the Kα photons are shown in Figure 4.
Kα yield of the Al fluor layer as a function of the thickness of the Al transport layer in the target shown in Figure 2.
As was pointed out in previous calculations (Honrubia et al., 2004), the yield of the Kα photons with target thickness depends, to a large extent, on electron refluxing. In these calculations, we used fast electron reflection at the rear surface, free boundary at the front surface and found that Kα yield of 520 keV electrons can be fitted as a function of the thickness z of the aluminum transport layer as exp(−z/z0), where z0 = 300 μm is the so-called “effective range.” Here, the mean energy is higher (1.5 MeV), but the effective range is even lower (260 μm) due to the full refluxing used in the present calculations, which reduces this “effective range” or attenuation of the Kα yield with target thickness, as discussed in Honrubia et al. (2004).
3.3. Target heating
Target heating is mainly due to the resistive energy dissipation or Ohmic heating by the plasma return current. This mechanism heats the plasma up to keV temperatures in the first tens of μm. At higher depths, self-generated fields are less important and plasma temperatures are much lower, reaching approximately 30 eV at 100 μm depth and of 3–4 eV at 300 μm, as shown in Figure 7. It is also remarkable in this figure, the separation between collimated and non-collimated zones in the transport layer, and the width of the radial temperature profile at the rear side, which is in good agreement with the Kα radial distribution shown in Figure 4.
Temperature distribution of the 300 μm target shown in Figure 2, 10 ps after the end of the laser pulse. Fast electrons are injected at the left of the simulation box.
In spite that fast electrons have an important initial divergence, which in turn leads to smaller current densities and self-generated fields, the energy transferred to the plasma by Ohmic heating is a significant fraction of the fast electron pulse energy. For instance, in the case of the target with 100 μm aluminum transport layer, 16.5% of the energy carried by fast electrons is deposited by Ohmic heating. Thus, self-generated fields have to be taken into account even in large beam divergence cases.
Since we have assumed full refluxing in the present calculations, fast electrons pass through the target until they deposit all their energy. The balance between the energy deposited by Coulomb collisions with the background plasma and the Ohmic heating due to the return current is shown as a function of the target thickness in Figure 8. Notice that less than 18% of the fast electron pulse energy is deposited as Ohmic heating. This is almost a factor of 2 lower than the energy deposited when a Gaussian distribution with an initial divergence angle of 20° (FWHM) is used (Honrubia et al., 2004).
Relative importance of the Ohmic heating in the targets sketched in Figure 2.
The temperature distribution at the target rear surface was measured for bare aluminum targets without fluor layers (Key et al., 2004; Stephens et al., 2004). We have performed simulations of these targets with the same parameters used in the former cases in order to compare our simulations with the experiments. The temperature profile at the rear surface of a 100 μm aluminum target is depicted in Figure 9. The peak temperature is 31 eV, and the FWHM of the spot 60 μm, close to the temperature of 30 eV and the spot of 70 μm measured in the experiments. This good agreement between simulations and experiments can be explained by the energy-dependent angular distribution of fast electrons used and confirms that the temperature overestimations typical of the hybrid PIC codes are due to an excessive collimation of the fast electron beam.
Temperature profile at the rear surface of a 100 μm single layer Al target 10 ps after the end of the laser pulse.
4. CONCLUSIONS
The main advantage of the hybrid PIC modeling as developed by Davies (2002) and Gremillet et al. (2002) is the large space and time scales that can be simulated with these codes with a reasonable computational cost. However, the full exploitation of this advantage has been hampered until now by the discrepancies found by several authors between predictions of these codes and the results of Kα experiments. Here, we found that using a suitable initial angular distribution of fast electrons and fitting the size of the fast electron source, hybrid PIC simulations can almost fully reproduce simultaneously for the first time the three independent diagnostics used in the experiments analyzed, namely divergence angle, Kα yield and temperature at the target rear surface. However, a theory to explain and predict the radial drift of fast electrons at the target front side as a function of the laser beam parameters is needed to determine in general the size of the fast electron source.
To have a well characterized fast electron source is important not only for simulation of Kα experiments, but also in other applications such as proton acceleration at the target rear surface, where resistive collimation of fast electrons determines to a large extent proton spectra.
ACKNOWLEDGMENTS
This work has been supported by the research grant FTN2003-06901 of the Spanish Ministry for Education and Science and the “keep-in-touch” activities of EURATOM.
