Abstract
Keywords
1 Introduction
A very recent experiment[1] at the Lawrence Livermore National Laboratory (LLNL) National Ignition Facility (NIF) resulted in fusion energy yield of about 1.3 MJ, largely in excess of the fuel energy, and about 70% of the laser pulse energy. LLNL’s scientists deem this is the threshold of fusion ignition. The above experiment was conducted using the indirect-drive (ID) approach[2]. However, the direct-drive (DD) approach[3] may have advantages compared with the ID approach[4,5].
Firstly, the efficiency of laser energy coupling with the plasma corona is significantly larger, requiring a lower laser energy for achieving fuel ignition. Furthermore, the ID approach is intrinsically non-symmetric, with laser beams overlapping at the entrance of the hohlraum and propagating over long plasmas before irradiating the internal hohlraum surface; this produces undesired plasma instabilities (e.g., crossed beam energy transfer (CBET)) and suprathermal or hot electrons (HEs), on one side, and a non-uniform X-ray irradiation of the capsule, on the other. A symmetric irradiation scheme appears therefore a necessary precondition for reducing long-scale implosion asymmetries and for achieving a higher control of laser–plasma interaction (LPI).
Among the DD schemes, shock ignition (SI), proposed by Betti et al.[6], is presently one of the most promising, and has therefore been investigated in many recent works. Here, the fuel is ignited by a strong shock driven by an intense laser spike (
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Reaching a detailed comprehension of LPI in the SI regime is important, not only because a large fraction of laser energy can be diverted out of the plasma via scattered light (SBS and SRS), therefore increasing the laser energy requirements, but also because some instabilities (SRS and TPD) result in the generation of HEs, which propagate through the compressed pellet and can affect the shock strength and preheat the fuel, thus preventing fuel ignition.
Several recent experiments aimed at the investigation of LPI and HE generation in conditions relevant for SI – that is, laser intensities in the range of
In the present paper, we describe the results obtained in an experiment aimed at investigating the LPI of a laser pulse focused at an intensity of approximately
2 Experimental setup
The experiment was carried out at the Vulcan laser in the Target Area West (TAW), at the Rutherford Appleton Laboratory. Four heating beams (
The interaction beam (
Figure 1.Top, pinhole camera images of hard X-ray emission from plasmas induced by the heating pulses, by the interaction pulse and by all the beams. Middle, sketch of the beam configuration in the vertical plane and target structure. Bottom-left, timing of the laser beams (relative intensity is arbitrary). Bottom-right, image plate signal, obtained in calibration shot on copper targets, acquired by the Bremsstrahlung cannon (BSC) looking at the rear side of the target; the intensity is proportional to photostimulated luminescence (PSL).
Multilayer targets were used during the experiment. The laser beams impinged on a 12 μm layer of polyvinylidene chloride (PVDC)
Two high-resolution X-ray spectroscopic diagnostics combining spectral and 1D spatial resolution were implemented. The Cu K
The macroscopic parameters of the plasma corona were studied via analysis of H- and He-like Cl spectra emitted from the PVDC coating. They were acquired by a second spectrometer, equipped with a spherical mica crystal with the bending radius of 150 mm, protected by 13-μm-thick Kapton foil. The instrument covered the spectral ranges from 4.16 to 4.53 Å and from 3.33 to 3.63 Å, diffracted in the fourth and fifth diffraction orders, respectively, and was set to look at the target at an angle of 17.5° ± 0.5º versus the target surface. The spectra were again recorded on the BAS-MS IP, digitized and corrected with respect to the wavelength-dependent crystal reflectivity and filter transmission. The wavelength calibration was based on the ray-traced dispersion relation and tabulated wavelengths of the dominant X-ray lines.
In addition to K
Laser–plasma instabilities were investigated by means of calorimetry and time-resolved spectroscopy of light backscattered in the cone of the focusing optics of the interaction beam. Light was collected behind the last turning mirror of the laser transport line, separated in four different channels, and sent to two calorimeters and two time-resolved spectrometers. Spectral filters were placed in front of the calorimeters to select light scattered by SRS (
3 Interaction conditions
The interaction conditions of the main laser pulse with the plasma corona were modelled by using the DUED[34] hydrodynamic code. 2D maps of density and temperature were simulated for the cases when no heating beams were used and when the main beam was delayed by
When only the interaction beam is used, the density profile at the laser peak shows a change of slope around
Figure 2.Density and temperature profiles obtained from hydrodynamic simulations carried out with the DUED code: (a) 2D map of electron density in the high-density region taken at the peak of the interaction pulse, in the case of delay time ns; (b) longitudinal profiles of electron density at different times of interaction in the case of delay time ns; (c) longitudinal profiles of electron density and temperature taken at the peak of the interaction pulse for delay times ns (blue lines), ns (red lines) and for the case where heating beams are not used (black lines); (d) transverse density profiles taken at different distances from the target surface in the same conditions as (a); (e) temporal evolution of the dip in the density profile.
When heating beams are used, the main pulse impinges on a long pre-formed plasma corona with a density scalelength of several hundreds of μm, increasing with the time delay
In contrast, the interaction conditions at densities lower than
Spatially resolved X-ray chlorine spectroscopy allowed us to calculate the plasma temperature in different regions of the plume. A typical spectral line-out, referring to the K-shell emission from H- and He-like dopant Cl atoms at 800 μm from the target surface, is shown in Figure 3. It includes well-resolved lines in spectral ranges 3.3–3.7 Å and 4.1–4.6 Å, given by the fifth and fourth crystallographic orders, respectively. The temperature is here obtained by the ratio of
Figure 3.X-ray spectrum of K-shell chlorine emission from H- and He-like atoms. The measured spectrum includes the contributions of the fourth and the fifth diffraction orders of the crystal.
Local conditions of interaction, that is, plasma temperature and density gradient, could be here also affected by the onset of filamentation, driven by the self-focusing of the speckles produced by the RPP. The relevance of filamentation can be estimated by considering a density/temperature perturbation size of the order of the speckle size
The validity of the above estimation can be corroborated by calculating the spatial growth rate
4 Experimental results
4.1 Laser–plasma instabilities
Light backscattered at
The SRS calorimeter measured no signal in the shots where only the heating beams or only the interaction beams were used. The former result suggests that no spurious signal produced by the heating beams affects the calorimetric measurements in shots with both heating and interaction beams. The latter observation can be explained by the fact that the plasma produced by the interaction beam is too steep to drive convective SRS. Considering the limits of detection of the optical line, this means that the backscatter obtained by using only the interaction beam was lower than 0.5
Figure 4.Experimental results: (a) SRS reflectivity versus laser intensity for a set of selected data with fixed values of ns and μm; (b) SRS reflectivity versus the delay time between heating and interaction beams; (c) typical SRS spectrum; (d) time profile of SRS light in shots with different delay times . In subplot (a), error bars of 20% are also reported for reference.
Detailed information about the timing of parametric instabilities and the plasma density where they are driven can be inferred by time-resolved spectroscopy of backscattered light. A clear
Figure 5.Time-resolved SRS spectra acquired in two different shots by the Hamamatsu C7700 (on the left) and C5680 (on the right) streak cameras. Time resolution is 18 ps. The laser fiducial in the left-hand image allowed us to determine the absolute timing of the interaction laser peak. The left- and right-hand panels show the SRS spectra scattered at low and high plasma densities, respectively. The dashed line shows the spectral shift of SRS scattered light with time at densities around
The central SRS wavelength does not shift with time, indicating that local conditions where SRS is driven are stationary. This suggests that local pre-formed plasma conditions, as determined by the heating beams, namely the temperature and density profiles in regions
In a few shots, an additional SRS signal is observed at wavelengths larger than 780 nm, as shown in the right-hand image of Figure 5. This signal is much weaker than that observed at shorter wavelengths, and consists at maximum of a few percent of the main SRS reflectivity. Here SRS begins approximately 200–300 ps after the laser peak and lasts for a few hundred picoseconds. This signal peaked at wavelengths moving with time from
4.2 Hot electrons
The energy and the number of HEs were here retrieved by BSC measurements. All IPs were scanned after approximately 20 minutes from exposure to reduce the uncertainties produced by the signal decay with time and corrected for the time fading, as in Ref. [42]. The signal was detected only in the first five or six IPs, despite the adoption of several filtering configurations and the addition of extra lead shielding for noise reduction. The first IP was also discarded because it was expected to be affected by plasma self-emission. On the other IPs, the signal was obtained by extracting the photostimulated luminescence (PSL) from an ROI centred on the exposed part of the IP and by subtracting the background measurement taken from an unexposed IP. However, due to the significant non-uniformity of the background signal, a line-out was taken across each IP and top hat functions, including both a constant and a linear background terms, were used to fit both the signal and background regions. The extracted data were fitted by synthetic signals produced with a combination of Geant4[43] simulations for both the detector response and Bremsstrahlung emission from HEs propagating through a cold, unexpanded multilayer target, as shown in Figure 6(a). The injected electrons had a Maxwellian distribution of the form
Figure 6.(a) Synthetic Bremsstrahlung X-ray emission spectra obtained by HE populations of different temperatures, resulting from Geant4 simulations. (b) Typical heat map of for different (,) combinations; the grey line represents the locus of minimum , while the projections on the axes show that the optimal fit consists of keV and J.
A residual sum minimization analysis between experimental and synthetic signals was carried out by varying both the electron temperature
The results obtained for both cannons revealed temperatures in the range of 7–12 keV with an uncertainty of approximately
A low value of the HE temperature is in agreement with the results obtained by Cu K
5 Discussion
Experimental results show that LPI occurs far from the critical density region, where SRS is predominantly driven at density
5.1 SRS at low densities
SRS is mainly driven in the electron density range between 0.03
In order to depict the physics of SRS in the conditions of interest, in the following sections we try to disentangle various issues affecting the plasma response, tackling progressively (i) the effect of beam smoothing with the RPP, (ii) the relevance of kinetic effects and, finally, (iii) the role of filamentation.
5.1.1 Beam smoothing and the role of laser speckles
In order to account for the SRS driven in a multispeckle focal volume, as produced by the RPP, we consider a simplified model consisting of SRS growth in independent laser speckles. The model, described in detail in the Appendix, assumes that each laser speckle contributes incoherently to the backscattered light from SRS, according to its local intensity
According to the model, the overall SRS reflectivity can be written as
As shown in Figure 7, the SRS backscattering is strongly dominated by the high-intensity tail of speckle distribution, that is, by the second term in Equation (2). By considering the simplified probability density
Figure 7.Values of total SRS reflectivity (blue) calculated by using
The practical expression for the convective SRS gain
In the shots where only the interaction beam was used, relying on the nominal laser intensity and on the density scalelength
When heating beams are used, the density scalelength
Via the latter expression we can evaluate the gain in individual laser speckles. For
It is instructive to calculate the SRS gain along the time profile of the laser pulse by using the model described above, applying Equation (4). In Figure 8, the time evolution of the SRS gain at
Figure 8.Time evolution of SRS gain calculated by using
By applying the model from Equation (2) with
5.1.2 Validation of the model and role of kinetic effects
In order to validate the model described above and to investigate the relevance of kinetic effects on the SRS growth, we performed 2D and 3D simulations with the wave-coupling code SIERA[58] (CEA and CPHT), computing the plasma response of an optically smoothed laser beam. Simulations consider a linear density ramp in the range 0.02
While the model described in the previous section yields already a satisfactory agreement with the experimental results in terms of SRS backscattered energy, numerical simulations reveal further features: (i) the spectral width of the backscattered light is found in the window roughly in between 650 and 750 nm, in agreement with the experiment; (ii) SRS exhibits an overall bursty behaviour in the spectral emission.
SIERA simulations (in agreement with the PIC simulations described in the next section) show that kinetic effects in intense speckles affect the SRS light spectrum. The excitation of trapped electrons leads, in fact, to a broader SRS spectrum and, in addition, produces a shift towards lower EPW frequencies, and thus towards shorter wavelengths in the scattered light, in the range of 650–700 nm, as shown in Figure 9. The bursty behaviour in the simulations occurs on a very short, ps time-scale, which is not resolved in the experiment because of the 7 ps time resolution limit. The bursts shown in Figure 9 are still a substructure of the SRS bursts in the experiments, and they reflect the amplification of the scattered light over the dominating density range of SRS growth, and disruptions due to transient pump depletion in intense speckles.
Figure 9.Spectra of the backscattered light from simulations with SIERA for an RPP beam at W/cm2 and for a density ramp 0.02 0.10, left/right subplots without/with taking into account kinetic effects.
The time-average of the bursty behaviour in the multiple-speckle simulations yields SRS reflectivity values of
These results also suggest that the model described in the previous section is adequate at the high laser intensities used in this experiment, where the role of kinetic effects on reflectivity is marginal. For lower average beam intensity, the situation could be different. In the model of Equation (2) we have assumed the SRS gain from Equation (4) by evaluating the linear Landau damping value. Our simulations, and those in other work[55,60], suggest that HEs emerging from high-amplitude EPWs in intense speckles massively modify the distribution function of neighbouring lower intensity speckles. In this case the local and most likely lower value of Landau damping, as well as the EPW frequency shift determined from the local electron distribution, needs to be considered. The latter may favour a more vigorous onset of SRS even in lower intensity speckles, such that our model would underestimate the SRS reflectivity for lower average beam intensity.
SIERA simulations with kinetic effects also depict the evolution of the electron energy distribution function via the EPW model used in the code[58]. The model for non-linear EPWs is here certainly incomplete with respect to more thorough approaches concerning particle trapping[27]. However, in the frame of validity of the model, it is possible to deduce the complementary electron distribution function
Figure 10.Complementary distribution function as a function (blue line) of the electron energy from a typical simulation with SIERA for an RPP beam at W/cm2 and for a density ramp 0.02 0.10. The tail of the distribution evolves versus HE distribution (black line) with 8 keV here, that is, in the range of the experimentally observed values. The position of the energy corresponding to the EPW phase velocity is indicated (dotted line at approximately 5 keV).
5.1.3 Self-focusing in intense laser speckles
In order to analyse in more detail the physics in single intense speckles, we have also carried out simulations with the 2D3V PIC code EMI2d of CPHT[61–63].
As mentioned earlier, the optically smoothed laser beams used in the experiments contain speckles with up to 8–10 times the average beam intensity
In the simulations carried out with the EMI2d code, we focused on the micro-physics around an intense speckle with
EMI2d simulations were carried out with and without mobile ions, in order to discriminate the impact of ponderomotive self-focusing. As expected, the self-focusing provokes a local density depletion in the speckle, such that both backscattering instabilities (SRS and SBS) develop preferentially in the periphery of the speckle, but not in its centre, as shown in Figure 11.
Figure 11.Normalized transverse field strength (upper subplot) and electron density perturbations (lower subplot) as a function of space for a single laser speckle with peak field strength 0.1. The self-focusing speckle shows an elongated structure on axis (180) with a field strength higher than the initial value . The laser light enters from the left, 0. The dashed lines in both subplots indicate the zones where SRS is amplified: the density perturbations (note that the range shown in is reduced with respect to the upper subplot) exhibit plasma wave oscillations in the periphery of the field hot spot, but not in the depleted zone inside. SRS-driven EPWs are in particular present at the rear on axis and in the side wings. SBS-driven ion acoustic waves are found in the front part (PIC simulations show generally too high values).
Here, however, a clear distinction is visible between the actions of SBS and SRS, which is due to their different time-scales. While SBS grows at times of the same order as the self-focusing, SRS has a much faster growth[51,63], allowing the SRS to adapt very rapidly to ponderomotively induced profile modifications. In this way, SRS (re-)establishes rapidly in the arising filamentary structure, preferentially in the rear of the self-focused speckle in which the plasma density is not (yet) depleted. SRS therefore continues to contribute vigorously to the backscatter process and leads eventually to higher backscatter rates than without self-focusing, as already reported in previous experiments[65]. The limited capacity of SRS mitigation by optically smoothed laser beams was demonstrated by Fernández et al.[66].
Let us mention that SBS is also seen in our PIC simulations with mobile ions, but starts from an unnaturally high noise level because of the limited number of particles; in contrast to SRS, however, SBS is mitigated by the ponderomotively depleted density profile[64].
5.2 SRS at higher densities
We have investigated the effects of filamentation driven at low densities and of pump depletion produced by the high SRS reflectivities
The origin and the time evolution of the weak SRS observed at densities around
Looking at the longitudinal profiles of density given by the hydro simulations (Figure 2(e)), it is evident that the densities of interest correspond to the region of the cavity drilled by the interaction beam. The density at the bottom of the cavity decreases from
The density in a cavity can be described by a parabolic profile
In our case, a scalelength value
6 Summary
In the present experiment, we characterized laser interaction with a long plasma corona with density scalelengths up to 450 μm and intensities at
7 Conclusions
Results described in this paper confirm the importance of speckle pattern and intensity distribution in determining local conditions of interaction and the growth of parametric instabilities. The long-scale plasma and the high laser intensity, driving the self-focusing of speckles, produce strong SRS amplification at low plasma densities. Even though the higher predicted plasma temperatures of full-scale SI are expected to enhance kinetic effects and to shift SRS outburst at slightly higher densities, the present results suggest that pump depletion and plasma smoothing prevent or at least strongly decrease the impact of non-collisional LPI at a quarter critical density, consequently limiting the generation of electrons, with energies able to preheat the fuel capsule being inhibited. Overall, our results suggest that a good control of both filamentation and SRS remains a key factor for the success of the SI scheme. Plasma smoothing induced by filamentation could be a route to produce a beneficial incoherence on the beam, needed to prevent the loss of laser energy in low-density plasma regions via non-collisional processes. However, a reduction of SRS driven in regions where plasma smoothing is still inactive, which could be obtained by using large bandwidth lasers, is required to avoid the large SRS reflectivity, probably reaching 40
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