A novel method for bringing sample elements to hydrodynamic conditions relevant to the base of the solar convection zone is investigated. The method is designed in the framework of opacity measurements and exploits the temporal and spatial stability of hydrodynamic parameters in counter-propagating Double Ablation Front (DAF) structures. The physics of symmetric DAF structures is first studied in 1D geometries to assess the influence of tracer layers in the target. These results are used to propose an experimental design compatible with the OMEGA [Boehly et al., Opt. Commun. 133(1-6), 495–506 (1997)] laser. Radiative-hydrodynamic simulations conducted using the Chic code [Breil et al., Comput. Fluids 46, 161–167 (2011).] in 2D-axisymmetric geometries suggest that a Fe sample can be brought to an electron temperature of ∼160 eV and electron number density of ∼1.35 × 1023 cm−3. These parameters are reached during a 500 ps window with temporal variations of the order of 10 eV and 1022 cm−3, respectively. This allows for potential time-integrated spectral measurements. During that time, the sample is almost at local thermal equilibrium and 2D spatial gradients in the sample are less than 5% in a 360 μm diameter cylindrical volume, including the potential effects of Hot Electrons (HE) and typical uncertainties related to target fabrication and laser performances. The effects of HEs are assessed using an inline model in Chic. The HEs are found to deposit most of their energy in the cold and dense ablator between the two fronts, leading to a small efficiency loss on the DAF parameters. The calculations also suggest that negligible amounts of unabsorbed HEs are present in the probed volume, thus not affecting the atomic properties of the sample. Potential extensions of the current design to higher sample temperatures within the OMEGA capabilities are briefly discussed.
I. INTRODUCTION
Seismic investigations have allowed to probe the solar interior with unprecedented accuracy, giving access to radial profiles of sound speed and plasma density.3–5 Comparisons of these measurements with state-of-the-art solar models have highlighted significant discrepancies at the Base of the Convection Zone (BCZ).6 In this region, located at 7 (with the solar radius), there is a thermal transport transition from a radiation dominated regime to a convection dominated one. Several potential sources for these discrepancies have been identified in the literature:8 (i) an underestimation of energy dissipated into macroscopic motions in the radiative zone, leading to an incomplete energetic balance in the Solar Standard Model, (ii) incorrect Rosseland mean opacities, which in turn affect the energy transfer from the sun center to the surface, and (iii) incorrect treatment of the radiative acceleration of elements towards the sun surface, leading to incorrect central abundances. The last two points are related to spectral opacities, which are a central component of solar models.
The accuracy of opacity tables used in solar models can be verified by dedicated experiments and further development of atomic physics codes. A few heavy elements constitute the bulk of the relative opacity at the BCZ, mainly:9 16O, 56Fe, 20Ne, and 58Ni. The detailed opacity of these solar elements in the 750–1300 eV range10 constitutes the main contribution to the Rosseland mean opacity at the BCZ. The plasma in that region is at an electron temperature , density and electron number density , at Local Thermodynamic Equilibrium (LTE).11,12 Z-pinch experiments have aimed to reproduce these solar conditions using intense X-ray sources to simultaneously heat and backlight Fe samples. These notably produced FeMg plasmas at Te = 200 eV and ne = 3.8 × 1022 cm−3, parameters not far from solar conditions.10,13 However, significant discrepancies in the measured spectral opacities have been found compared to calculations with atomic physics codes.13 While several hypotheses have been proposed to explain these differences, such as experimental issues and bound-free or ionization rate calculation problems in atomic physics codes, they have not yet been resolved.
Given the set of discrepancies observed between (i) solar simulations and helioseismic measurements and (ii) Z-pinch experiments and atomic physics calculations, there is a strong motivation to develop a novel experimental platform for single-element spectral opacity measurements in solar conditions. Experiments on various laser facilities (HELEN,14 NOVA,15 OMEGA,16,17 LULI2000,18 ORION19) or on pulsed-power devices (Z pinches at the Sandia laboratory10,13 and Imperial College20) have put in evidence the various challenges and different techniques to be developed to produce reliable benchmarks.6,9,13 As such, a convincing measurement of the elemental opacities should also allow unambiguous comparison with atomic physics codes. This notably requires to: (i) reproduce the relevant electron temperature and density, (ii) produce a uniform plasma to provide a measurement free of gradients, (iii) diagnose the plasma conditions during the experiment, (iv) ensure steady state compared to hydrodynamics and instrument resolution, (v) ensure close-to LTE conditions, and (vi) be robust versus experimental uncertainties. In this paper, we present a laser-based experimental scheme that aims at fulfilling these points and that is readily compatible with the OMEGA1 laser system specifications.
In order to achieve the solar plasma conditions at the BCZ and provide a convincing opacity measurement, we propose to make use of two counter-propagating double ablation fronts (DAF) generated in the plasma using a specific laser/target configuration. The DAF is a structure created by the concurrent propagation of a slow electronic ablation (EA) front and a fast radiative ablation front.16,21 These structures are typically obtained by irradiating a medium-Z target with high-intensity lasers (I > 1015 W/cm2).22 The DAF structure is advantageous in that the spatial gradients of hydrodynamic quantities between the two fronts are small, of the order of 10%.23 Additionally, the plasma in this region is close to LTE, with Tr ≃ Te = Ti. Using a DAF target for an opacity measurement was first proposed in Ref. 23. However, the analysis was limited to 1D geometries, did not detail the influence of a tracer presence and did not consider realistic laser configurations. The present paper extends on this previous work by revisiting the 1D analysis and extending the study to 2D-axisymmetric geometries in realistic laser/target configurations.
The technique proposed in this paper possesses distinct properties with respect to prior methods used for production of uniform samples for opacity measurements. Namely, short-pulse techniques tend to allow for comparable densities and higher plasma temperatures, but in much smaller volumes and without ensuring steady state.24,25 Heating using proton beams produces much lower temperatures albeit with higher (solid) densities.26 Similarly, shock heating experiments can approach the solid densities but at lower temperatures.17 Long pulse direct drive planar foil techniques offer higher temperature conditions,27 but tend to depart from LTE and yield lower densities. Long pulse indirect-drive techniques using high-Z holhraums tend to produce higher temperature plasmas but with ten-fold lower densities. In these experiments, particular care must be taken to shield the sample from emission lines originating from the high-Z holhraum plasma. Similarly, heating a sample using a Z-pinch driver is currently subject to similar density limitations as in the indirect-drive approach.10
The paper is structured as follows: First, the concept of the DAF structure is recalled in Sec. II. A 1D study of the DAF hydrodynamics is then presented in Sec. III. The study focuses on the effects of tracers in single-sided and double-sided irradiation configurations in Sec. III A and on DAF sensitivity to target and laser parameters in Sec. III B. From the conclusions drawn in 1D geometries, we present in Sec. IV a design for the OMEGA laser. This is achieved using 2D-axisymmetric simulations accounting for realistic efficiency loss mechanisms. The details of longitudinal and lateral spatial gradients in the target are explored. The scheme robustness versus experimental uncertainties is then presented in Sec. V, considering the effects of target defects and laser performances. Finally, we assess the DAF sensitivity to the presence of supra-thermal electrons generated by the non-linear laser plasma interaction (LPI) in Sec. VI.
II. THE DOUBLE ABLATION FRONT STRUCTURE
A. General principle of the DAF
Laser beams in the long pulse and medium intensity range (100 ps–10 ns, Iλ2 ∈ [1013–1017] W μm2/cm2 with I the laser intensity and λ its wavelength) deposit most of their energy in plasma by the process of inverse Bremsstrahlung. This energy deposition occurs in a low-density (ne ∈ [0.1 – 1]nc, with nc the critical density) and high-temperature (∼1–5 keV) corona. In these conditions, the coronal plasma emits large amounts of X-rays, mostly through the Bremsstrahlung process, and to some extent through electron-ion recombination and line emission. These X-rays propagate out of the optically thin corona in all directions and notably toward the cold and dense target. Energy transport from the corona to the target is supported by both the plasma electrons and the Bremsstrahlung photons. These transport mechanisms induce decreasing temperature and increasing density towards the target interior.
The location where electron and radiation temperatures are equal is defined as the Electronic Ablation (EA) front. This region is usually at low temperatures and densities. However, depending on the materials, the EA front can form in an intermediate regime with higher densities and temperatures. In that case, a transition layer develops around the EA front, which efficiently emits additional Bremsstrahlung photons. This so-called Electronic Radiative Ablation (ERA) front is necessary for the development of a clearly distinct DAF structure.21 This requirement limits the choice of materials that can be used to generate the DAF.
From the ERA front toward the target, the plasma is dense and cold. This is an optically thick layer with respect to low energy K-shell emission lines of the coronal plasma (typically ∼50–350 eV depending on the ablator), where the associated photon mean free path is less than the layer thickness. The Radiative Ablation (RA) front is formed where radiation transport becomes zero in the cold target. Because the RA front is faster than the ERA front,22,28 the structure formed in-between is temporally stable. In this radiation dominated layer, electron and ion temperatures are equal, and matter and radiation are near equilibrium (Tr ≃ Te = Ti). The radiation flux in this layer is accompanied by a spatial gradient in density and temperature, which is typically a 10% variation between the two fronts.23
As mentioned earlier, only certain materials will produce a DAF structure when subject to an intense laser flux. Theoretical work conducted in 1D geometries21 suggests that for a clear double front to be formed, the plasma during the interaction must be opaque to radiation from the corona and have a small enough Boltzmann number, where the Boltzmann number is the ratio of convective thermal energy flux to radiation energy flux. 1D radiative-hydrodynamic simulations8,28 show that suitable targets are medium-Z materials such as Al, CHBr, SiO2, Si, Ti, and Cu. In steady state, the simulations suggest that the density of the plateau region scales as an ideal gas21 ρ ∝ P/(T(1 + Z)), with P the plasma pressure and Z the species ionization, while the radiative temperature scales as28 Tr ∝ I0.25. As such, achieving a given set of plasma parameters in the DAF plateau (in the absence of tracers) is both a matter of material choice and laser configuration.
B. The DAF in the context of an opacity measurement
In the framework of an opacity measurement, the proposed idea is to insert a sample material of interest into a DAF target such that the sample will be located in between the two fronts at the time of measurement. This is illustrated in Fig. 1. Around the spectral region of interest (here ∼1 keV), DAF targets without samples can be made optically thin (see Sec. IV D). As such, they are compatible with typical target/backlighter configurations. Diagnosis of the plasma temperature and density close to the sample can be obtained by introducing spectroscopic tracers such as Mg, Al, or Se. Inference of the specific sample transmission can be obtained by comparing target transmissions with and without samples, or by using hydrodynamic simulations coupled to spectral-processing to remove the contributions from ablators and tracers to the total transmission. Ideally, both techniques can be combined to constrain the results, and hydrodynamic parameter measurements from the tracer can be used in conjunction with backscatter diagnostics to constrain the hydrodynamic simulations. Finally, a target layout similar to that of Ref. 10 can be used, with a sample spanning only half of the target width, thus providing simultaneous measurement of the target transmission with and without the sample.
As mentioned earlier, the DAF plateau is optically thick to emission lines from the target coronal plasma below ∼500 eV. These photons contribute to the thermalization inside the sample. Higher-energy emission lines from the plasma are consistently taken into account in the framework of hydrodynamic-radiative models. Calculations suggest that (i) the net radiation flux in the plateau is small and characterized by Te = Tr ± 5% (see Sec. IV D) and (ii) the photon distribution in the sample deviates slightly from a Planckian. Hence, the sample region is not strictly at Thermodynamic Equilibrium. However, the typical structure of the plateau region (low gradients, Tr ≃ Te = Ti, see Sec. III) ensures that in this region the plasma is close to LTE as defined in Ref. 29. This holds when considering the presence of higher-energy emission lines from the coronal plasma since these were included in hydrodynamic-radiative calculations leading to Te ≃ Tr. In addition, the typical temporal and spatial stability reached in the DAF plateau region (see Secs. III and IV D) ensures that LTE is maintained in the sense of a quasi steady state and quasi-homogeneous plasma.30 Ultimately, this hypothesis of LTE will have to be verified by comparing measurements with LTE opacities and collisional-radiative calculations.
III. PHYSICS OF DAF STRUCTURES IN 1D GEOMETRIES
A. Dynamics of DAF targets in the presence of tracers
For simplicity, the target/laser configurations considered here are comparable to that presented in the OMEGA design in Sec. IV. Simulations of the interaction are conducted with the Chic radiative hydrodynamic code. Chic is a 2D code based on a cell-centered Lagrangian discretization method.2 The physical processes considered in this section are a flux-limited electron energy transport in the Spitzer Harm approximation, multi-group thermal radiation transport with average atom opacities,31 ray-tracing with inverse Bremsstrahlung only, and realistic equation of state (SESAME32 and Ref. 33). Electron heat flux limitation is set to 4% of the free streaming. This value was chosen by comparison with benchmarked 2D calculations with non-local electron conduction presented in Sec. IV A. Note that plasma parameters reported here are significantly lower compared to the 1D simulations presented in Ref. 23. This is mostly due to a lower choice of flux limitation and to some extent to the higher mesh resolutions used throughout this study. Variations of the flux limitation parameter are not explored in this 1D study, whose aim is only to highlight the DAF physical processes in our configuration. All subsequent 2D simulations are performed with a non-local heat transport model.34
We consider a 12 μm Si bulk target irradiated by 3ω laser light at normal incidence and 1.46 × 1015 W/cm2 laser intensity. The laser pulse is a 2 ns square with 100 ps rise time. The temporal evolution of the electron temperature and mass density in the target is illustrated in Fig. 2. The associated longitudinal gradients in hydrodynamic parameters for several cases are presented in Fig. 3. The behavior of a bulk Si target, e.g., in the absence of tracer layers, is illustrated in Fig. 3(a). A shock is launched at the beginning of the interaction and exits the target after 250 ps. A reflected shock can be seen propagating back, which has typically no influence. The ERA front is readily identifiable as a steep temperature discontinuity on the laser side. In this region, the plasma temperature decreases rapidly from its large value in the corona to below the radiative temperature. The RA front is located behind the second large temperature gradient. In this region, the density reaches a maximum value and the radiative flux is close to zero. The so-called DAF plateau is located in the spatial and temporal window between these two fronts. In the plateau, longitudinal temperature gradients are smaller than in the rest of the target. Considering tracer layers that would be inserted in the center, the temperature gradients there are ∼5 eV/μm in the t = 700–1300 ps time window.
These longitudinal gradients can be reduced significantly by counter-propagating two DAF structures. This is achieved by irradiating both sides of the target simultaneously, as illustrated in Fig. 1. The resulting temperature gradients are shown in Fig. 3(b). The stability region in the target center is bracketed by the encounter of the RA and ERA fronts. Compared to the single-sided case, the target center reaches small gradient conditions sooner, in the t = 450–1100 ps window. The temperature gradients are also significantly lower, below 1 eV/μm.
In the framework of an opacity measurement, tracer layers are added to the target center. The role of such tracers is to provide a diagnostic sensitive to the local electronic temperature inside the sample as a function of time, as discussed in Sec. IV C. Calculations for a {Si6 μm/Al250 nm/Fe50 nm/Al250 nm/Si6 μm} target are shown in Fig. 3(c). Temperature profiles (not given here) show that the tracer layers heat up synchronously with the surrounding Si in the radiation-dominated layer. However, the calculations show that lateral gradients are higher in the sample than for a case without tracers, reaching ∼20 eV/μm. The Fe sample being ∼180 nm wide in the steady state density region, this corresponds to a ∼4 eV temperature variation along the sample thickness. The stability of ablator plasma parameters in the plateau region is unaffected by the presence of tracer layers.
The pressure perturbations in the target, shown in Fig. 3(d), illustrate the various hydrodynamic processes at play during the DAF interaction. At t = 0, the laser launches a shock and radiatively preheats the high-Z sample, while the ablator is still transparent [black circle in Fig. 3(d)]. This preheat leads to an increase in temperature and outward expansion of the tracer layer, thus launching a pressure perturbation in the Si ablator [blue arrow in Fig. 3(d)]. The preheat and density decrease can be seen on the time history of the Fe sample parameters in Fig. 4(a) around t = 0. The pressure perturbation from the radiative preheat of the tracer collides with the laser-generated shock and is partially transmitted toward the laser side. Upon reaching the target center [first white circle in Fig. 3(d)], the laser-generated shock re-compresses and heats the tracer layers, as illustrated in Fig. 4(a) with an upward arrow. In this double-sided configuration, the two shocks generated on either side of the target arrive simultaneously at the tracer layers. Mistiming of the shocks is discussed in Sec. V.
A fraction of the shocks are reflected on the tracer layers toward the laser side. Upon reaching the converging ERA fronts, part of the reflected shocks are transmitted to the corona (white dashed arrow) and part are reflected back toward the tracer. The second white circle highlights the location of the second tracer layer re-compression, also indicated by a downward arrow on the time-history of the tracer layers. The efficiency of the second re-compression depends on the ablator material, as illustrated in Fig. 4(a) in the case of Ti. Further shock rebounds do not have an impact on the tracer layers. The density and temperature in the sample then equilibrate toward an asymptotic value after a few oscillations. At this point, the pressure in the target is constant between the converging ERA fronts and equal to that of the laser ablation pressure. As such, the density and temperature equilibrium in the sample arises from a balance between heating and expansion due to the radiation flux and pressure from the surrounding ablator material.
The stability of the plasma parameters in the plateau region can be altered by changing the ablator material, as illustrated in Fig. 4(b) for a 3.1 μm Ti ablator. The Ti case produces a sharper transition layer, leading to a more efficient DAF structure with significantly lower temperature gradients and longer DAF stability. In this case, the lateral gradients in the sample are less than 1 eV/μm—the Fe sample being ∼200 nm wide in the steady state density region. While this would tend to favor Ti over other ablators, additional factors come into consideration, as discussed in Secs. IV and V.
B. Sample parameter sensitivity to laser/target configuration
For a given target composition and laser incidence, the hydrodynamic parameters reached in the tracer layers can be set by using a variety of free parameters. Considering a case with normal laser incidence, Si ablator material, and a 1D framework (e.g., neglecting all 2D related losses), the density and temperature reached in the sample are mainly functions of (i) the time-dependent radiation flux in the plateau region, driven by the coronal temperature Te, and (ii) the plasma pressure set at the ablation front PA, that keeps the tracer layers from expanding. Both parameters are function of the laser intensity:35 and with λL the laser vacuum wavelength and ηabs the laser energy absorption fraction. Note that these are given for planar laser-driven ablation flows in the absence of anomalous absorption processes. In addition to the laser intensity, the radiation flux seen by the tracer depends on the buried depth of the tracer layers, and as such on the ablator thickness. Radiative-hydrodynamic simulations in 1D geometries are used to illustrate how variations around a baseline configuration impact the hydrodynamic parameters reached in the Fe sample. The baseline configuration is the same as in Sec. III A: {Si6 μm/Al250 nm/Fe50 nm/Al250 nm/Si6 μm} target with normal incidence double-sided irradiation at Iref = 1.46 × 1015 W/cm2 during a 1 ns square with 100 ps rise time. The reduced pulse length is more representative of the OMEGA design at higher laser intensities presented in Sec. IV.
The incidence of varying the ablator thickness is shown in Figs. 5(a) and 5(b) for single-sided irradiation and Figs. 5(c) and 5(d) for double-sided irradiation. Both configurations show similar behavior. With a thinner ablator, the shock arrives sooner to a denser tracer layer and compresses it to higher densities. However, the steady state density in the DAF plateau only varies slightly. This illustrates that the steady state is not governed by shock compression and heating, as is also suggested in Sec. V. The ablator thickness changes the radiation flux received by the buried tracer, which affects the asymptotic temperature and density. Since the laser ablation shortens the ablator in time, this adds a temporal dependency on the steady state. For thick ablators, the steady state needs significant durations to be established, which may become an issue for short laser pulses.
The effect of laser intensity at a fixed ablator thickness is shown in Figs. 6(a) and 6(b). Increasing the laser intensity simultaneously increases the sample temperature and density, as was suggested by the dependency of PA and Te on I. The stability window starts sooner because of the increased radiation flux, which can compensate the time delay seen in cases with thicker ablators. As such, the choice of the laser pulse length imposes an upper bound on the ablator thickness and lower bound on laser intensity, as illustrated in Figs. 6(c) and 6(d).
A few general rules can be used to design a DAF target for an opacity measurement: (i) density is mainly controlled by the plasma pressure keeping the sample layers from expanding, and as such scales mainly with laser intensity, (ii) temperature is affected by both laser intensity and ablator thickness: the former changing the radiative flux generated in the corona and the latter controlling the flux reaching the buried layers, and (iii) thin targets and intense pulses both shorten the temporal stability windows, but also make the sample reach steady state more rapidly. We now present a DAF target designed for the OMEGA laser aiming to reach the conditions at the solar BCZ for the purpose of opacity measurements.
IV. REALISTIC DAF TARGET DESIGN FOR THE OMEGA LASER
In this section, the simulation framework is extended to 2D-axisymmetric geometries in order to study the influence of (i) transverse heat conduction effects, (ii) sample homogeneity in the transverse direction, (iii) sample preheat by hot electrons (HEs) generated by the non-linear Laser Plasma Interaction (LPI), (iv) scheme robustness in the presence of hydrodynamic perturbations from laser imprint or surface roughness, and (v) scheme robustness in the presence of experimental uncertainties, such as ablator layer imbalance, ablator density imbalance, and laser mistiming between both sides. The laser configuration is designed from the specifications of the OMEGA1 laser system and its chamber and constrained by the currently available spectrometers.
A. Modeling framework
Simulations in the 2D-axisymmetric framework are conducted using the radiative-hydrodynamic code Chic2 described in Sec. III A. Contrary to 1D simulations in Secs. III A and III B, we now use the Paraxial Complex Geometrical Optics (PCGO) package to describe the laser propagation.36 In addition to laser refraction and inverse Bremsstrahlung absorption, the model accounts for resonant absorption of laser light at the critical density from the laser the turning point. The PCGO package also contains an inline model for the transfer of energy from the laser field to HE populations. This is used and discussed in Sec. VI.
The DAF structure being driven by X-rays generated in the coronal plasma, it is crucial to model accurately the plasma temperature. To do so, our model was first benchmarked against experimental data from an OMEGA experiment conducted in similar conditions. In this experiment, a planar {CH5 μm/Al1 μm/CH5 μm} multi-layer target was irradiated with 1 ns square beams at 1015 W/cm2. The temperature in the Al sample was diagnosed from 1s-2p absorption spectroscopy.17 In this configuration, the Al is first heated by a shock propagating in the target, similar to the first part of the DAF framework (see Sec. III A) and then heated at small fluxes by the photons emitted in the corona. While this is not a DAF target, radiative heating contributes significantly to the measured temperature in the buried tracer. Notably, the modeled Al temperature was found to be highly sensitive to the coronal plasma temperature. The code benchmark showed that experimental results were best reproduced by describing heat conduction using a non-local thermal transport model coupled to the description of transport inhibition by self-generated magnetic fields.37 Furthermore, best results were obtained when using fully Lagrangian hydrodynamics, which is related to the artificial heat diffusion that Arbitrary Lagrangian-Euler (ALE) schemes tend to produce. Simulations presented in this section are conducted with minimal ALE relaxation, non-local thermal transport, and heat transport inhibition by self-generated magnetic fields.
B. Laser configuration
It was shown in Sec. III B that the efficiency of sample heating in the DAF plateau depends strongly on laser intensity. While the spatial gradients parallel to the target normal are suppressed by the use of a symmetric double ablation front scheme, the gradients in the target plane are mostly determined by the laser spot size. As such, the illumination scheme must produce high intensities and large laser spots.
We consider a tentative backlighter/target/spectrometer configuration aligned along the P6-P7 axis of the OMEGA chamber. The spectrometer is placed in P6, with its blast shield aperture 10 cm from the interaction target. The backlighter is positioned 5.5 mm behind the DAF target. Each side of the target is irradiated by a 1 ns square pulse with 5 and 10 beams at 42° and 58.5° of the target normal, respectively. The experimental configuration is shown in Fig. 7 and the laser pulse is shown in Fig. 8. The beams are equipped with E-IDI-300 phase-plates, focused 1 mm behind the target, and additionally smoothed by Polarization Smoothing (PS) and Smoothing by Spectral Dispersion (SSD). The amplitude modulations, included in the simulations, originate from the SSD system and lead to actual beam powers reaching an average of 0.42 TW/beam for a request of 0.5 TW/beam.
Laser intensity being a key parameter for the DAF concept, the caustic of the E-IDI-300 beams was measured in dedicated experiments in order to build a custom beam model in Visrad.38 This is discussed in the appendix. Accounting for SSD and PS and considering a peak power of 0.42 TW/beam, the illumination configuration produces a super-Gaussian on-target intensity distribution of order 2.1, 175 μm radius at 1/e, and peak intensity of 5.6 × 1015 W/cm2. Finally, the scheme allows the use of 1500 μm diameter targets, which is large enough for the interaction beams to be fully intercepted, and small enough for the backlighter beams to pass around the target without clipping (see Fig. 7). The backlighter beams irradiating the target are not equipped with phase plates, thus providing small spots at high intensities. Preliminary results for this scheme, obtained on OMEGA, have demonstrated that the backlighter outshines the target self-emission when using at least two beams (in that case, on a Sm micro-dot backlighter). This is encouraging for the demonstration of the scheme feasibility.
C. Target configuration
1. Ablator material
A variety of mid-Z ablator materials can be used to create DAF structures, with various efficiencies (see Sec. II). In the context of sample opacity measurements, material choice is mostly constrained by the absence of spectral lines or absorption features in the spectral range of interest. This applies for both the ablator and the tracer material used for characterization of the DAF hydrodynamic parameters. Several ablators have been assessed in Ref. 23 for a single-sided configuration in 1D: Ti, Cu, Si, Al2O3, SiO2, and Al. In general, lower or higher Z materials were not found to produce a well-defined DAF structure for our laser intensities. Note that doped ablators such as CHBr do produce DAF structures and may be considered in the future. Potential ablators considered here are limited to those studied in Ref. 23.
The Al2O3 and SiO2 ablators possess oxygen atoms which have spectral features in the range of interest for Fe, and as such cannot be used. Cu must also be excluded as an ablator since it produces low density and low temperature DAF stability regions. Finally, if Al is to be used as a tracer layer for hydrodynamic characterization (see Sec. IV C 2), it follows that it cannot be used as an ablator. We show in Fig. 9(a) the spectral opacities of 100 nm samples of Fe, Al, Si, Se, and Ti at Te = 200 eV and ne = 1023 cm−3 in the 200–2500 eV range (Se is shown here as it may be used as a tracer material). While the Ti-ablator produces high DAF performances in the double-sided configuration (see Sec. III A), it exhibits absorption features below 1.1 keV. On the contrary, Si shows only continuum absorption in that range and as such is a suitable ablator candidate. As shown in Sec. III B, the ablator thickness is chosen in conjunction with the on-target intensity. 2D simulations presented below suggest that an ablator thickness of 6 μm for an on-target intensity of 5.6 × 1015 W/cm2 can create a DAF structure and bring the Fe sample to 160 eV at 1.35 × 1023 cm−3 during about 500 ps.
2. Tracer material
The temperature and density of the DAF stability region are diagnosed using tracer materials placed around the sample. In the {CH5 μm/Al1 μm/CH5 μm} planar target experiments mentioned in Sec. IV A, absorption spectroscopy of the satellite lines of the Al Kα in the 1400–1700 eV range allows us to infer temperature and density in the buried Al sample (Te = 36 ± 4 eV and ρ = 3 ± 2 g/cm3).17 The same spectral range is chosen for diagnostics of the plasma parameters here. Using a separate energy range for the Fe measurement and hydrodynamic characterization can be achieved by using two different spectrometers. A streaked spectrometer can be used to verify the temporal stability of the DAF scenario, and a time-integrated (on ∼200 ps scales) spectrometer can be used for high-resolution spectroscopy of the sample.
The tracer materials were chosen with the aim of characterizing a Te ∈ [150–200] eV and ne ∈ [1–2] × 1023 cm−3 plasma in the 1400–1700 eV range. At these high densities and temperatures, the Al spectra are dominated by C-like, B-like, and Be-like features, as shown in Fig. 9(b). While Al may be a good candidate for the early interaction, for similar conditions to those in the {CH5 μm/Al1 μm/CH5 μm} experiments,17 it is not sensitive enough during the DAF plateau. Various tracer materials with higher sensitivity at higher temperatures were investigated, amongst Ge, Se, Ga, Br, As, and Zn. All these materials have temperature and density dependent L-shell absorption features in the 1400–1700 range. Spect3D40 calculations suggest that Se possesses the highest temperature and density sensitivity, while also being free of absorption features in the 900–1300 eV range [see Fig. 9(a)]. Additionally, Se is comparatively easy to manufacture in targets and thin layers are sufficient to produce the absorption features. Typical L-shell absorption in our spectral range and for various DAF plateau parameters is shown in Fig. 9(c). 2D Chic simulations post-processed in 3D with the atomic physics code Spect3D have been used to compute the optimal tracer thickness for overall target transmission in the 0.2–0.8 range. Two designs are considered, based on Al and Se tracers: {Si6 μm/Al250 nm/Fe50 nm/Al250 nm/Si6 μm} and {Si6 μm/Se100 nm/Fe50 nm/Se100 nm/Si6 μm}.
It is important to mention here that other tracers may ultimately be more suitable to the hydrodynamic characterization of the DAF structure. Notably, the experiments in Ref. 10 have reported using Mg as a tracer near BCZ conditions, albeit at lower densities and not in a DAF structure. For the purpose of this paper, we are more interested in the influence and interaction of realistic tracer layers with a DAF structure and a sample, than in the definitive choice of a tracer material.
D. Sample hydrodynamic parameters
Simulations using the target and laser configuration described above have been conducted in the 2D framework. The evolution of hydrodynamic parameters reached in the Fe sample is shown in Figs. 10(a)–10(c). The Se-tracer and Al-tracer based targets in the double-sided irradiation configuration reach similar performances during the DAF plateau of Te ≃ 160 eV and ne ≃ 1.35 × 1023 cm−3. In both cases, the temporal stability region of the DAF starts 500 ps after the start of the interaction beams and extends for 500 ps. Slightly higher sample temperatures are reached in the Al tracer case, although with higher standard deviations from gradients. The single-sided irradiation case with Al tracer reaches Te ≃ 135 eV and ne ≃ 0.87 × 1023 cm−3. The stability region of the DAF starts around 700 ps and extends for a smaller duration of 300 ps. The standard deviation due to lateral gradients also appears larger during the transient period before the DAF stability. In both double-sided irradiation cases, the plasma parameters vary temporally by at most 5% in the stability window. It is also observed that the radiative temperature does not deviate more than 5% from the electron temperature in the temporal stability region, as illustrated in Fig. 10(d). This provides an adequate framework for comparisons with stellar conditions and atomic physics codes. Additionally, it is worth mentioning that there are no significant effects of the SSD power modulations of the laser pulse on the plateau parameters, compared to the other hydrodynamic processes presented in Sec. III. This is notably due to the relation between the radiative temperature in the sample and the laser intensity, which was observed to scale as a blackbody Tr ∝ I1∕4 in Refs. 22 and 28. The total target transmission in the Se case and at the time of DAF stability is shown in Fig. 11, illustrating that the target remains transparent above photon energies of 600 eV. This was designed to allow transmission of backlighter photons for the Fe spectroscopy around 1 keV and tracer spectroscopy around 1.5 keV.
Density and temperature profiles computed in the 2D framework are more stable in time compared to the results obtained for 1D geometries, in which oscillations were present due to shock reflections between the ERA front and the tracer layers (see Secs. II and III B). In the 2D calculations, the curvature of the ablation front caused by the finite laser spot size prevents efficient reflections. The lack of oscillations may also be related to a lower numerical resolution. (The numerical cost and mesh stability requirements of the 2D configuration prohibit from using the same high resolutions as in the 1D geometries.) This is not believed to be an issue since shock reflections only change the early time history of the plasma parameters but not the asymptotic values, as shown in Sec. III B.
The DAF parameters reached in the 1D framework at normal incidence and Ipeak = 1.46 × 1015 W/cm2 are higher than in the 2D realistic configuration with almost four times as much on-target laser intensity. This constitutes a significant decrease in DAF efficiency for the realistic cases. The largest efficiency loss can be attributed to the difference in laser incidence θ. For a plane wave propagating in an exponential density gradient of the form (with L the associated scalelength) the fractional laser absorption is41 , where ν is the inverse Bremsstrahlung collision frequency at the critical density and c the speed of light. For typical plasma parameters considered here and a laser incidence at 42° and 58.5°, this corresponds to a reduction factor in the absorption fraction in the 1.5–3 range. A second source of efficiency loss arises from the finite extent of the focal spot which leads to heat flow in the target plane, thus decreasing the coronal plasma temperature and the DAF efficiency.
These simulations conducted in a practical configuration suggest that reaching the electron density in solar conditions11,12 is possible, with a significant margin on either side of that value depending on the targets. However, the temperature of is not reached in the present design. Simulations have shown that in 2D geometries and for the OMEGA configuration, increasing the laser intensity at fixed energy tends to be inefficient to reach higher plasma temperatures in the sample, since it is accompanied by smaller spot sizes and higher lateral heat losses. Instead, increasing the laser energy by adding more beams rapidly increases the conditions reached in the DAF, while maintaining its lateral homogeneity. This may be achieved by using the backlighter beams to heat the main target, adding 5 beams at 23° incidence on each side. The backlighter source may be generated using the OMEGA-EP TOP9 beam without phaseplate to generate an ns-length source. This scheme also scales favorably to larger laser systems such as National Ignition Facility (NIF) or Laser Mégajoule (LMJ), where significantly more laser energy is available so that large laser spots at high intensities can be used. Preliminary calculations conducted in the LMJ framework have shown the ability to reach the Te = 200 eV range during more than 1 ns, using only a few laser quads.
E. Probed volume and lateral gradients
In the OMEGA configuration considered here, the backlighter source is located 5.5 mm from the target, itself 10 cm from the spectrometer blast shield entrance. The backlighter source considered is a 100 μm diameter Sm micro-dot illuminated by beams without phase plates of the Gaussian profile and 1/e radius of 80 μm. In these conditions, the estimated probed volume in the DAF target is a cylinder ∼300 μm in diameter. The lateral homogeneity of the Fe plasma for the {Si6 μm/Se100 nm/Fe50 nm/Se100 nm/Si6 μm} case is shown in Fig. 12. It is found that using the defocused E-IDI-300 smoothed beam setup for the interaction, the standard deviation during the temporal stability window of the DAF is smaller than 5% in a 470 μm diameter probed volume for the density ρ and in a 520 μm diameter probed volume for the temperature Te. While the equivalent Tr map is not shown here, the relation demonstrated in Fig. 10(d) illustrates that Tr is also homogeneous at least in a 300 μm volume. Similar homogeneity values are found in the Al tracer case. The stability regions suggest that sufficiently homogeneous conditions can be achieved experimentally for opacity measurements, with small longitudinal and lateral gradients.
V. SCHEME ROBUSTNESS
The robustness of plasma parameters reached in the DAF framework is now assessed. Some of the typical sources of uncertainties relevant to this configuration are (i) manufacturing defects in layer thicknesses, (ii) effects of target surface roughness on hydrodynamics and small-scale mixing, and (iii) laser pointing and timing errors.
A. Pointing and timing errors
The RMS pointing error on OMEGA is of the order of 15–20 μm that is small compared to the on-target intensity profile of 175 μm 1/e radius. While the random pointing errors of the 15 beams on each side should not drive 3D imbalances with respect to target acceleration, it may slightly decrease the peak on-target intensity and hence the DAF efficiency.
The precision in relative timing of the laser pulses is of the order of 25 ps RMS. It was shown in Sec. II that the shocks converging in the target from each side arrive at the same time in the sample and drive it to a higher density than in a single-sided configuration. 1D simulations of laser-drive mistiming between the two sides of the DAF target have been conducted. The DAF efficiency is not seen to be affected by mistimings of 50 ps, a performance that is well within the OMEGA specifications when considering that bundles of 15 beams are overlapped. This robustness is related to the asymptotic density in the sample that is not determined by shock compression but by ablation pressure in the target.
B. Target defects: Asymmetries and surface roughness
The fabrication of DAF targets is subject to difficulties related to the Si ablators. The manufacturing process starts from a single crystal Si membrane on which tracer and sample layers are coated. Since no glue can be used, as oxygen lines would be present in the opacity measurement, the final layer of the Si ablator must also be coated. This leads to a Si structure that is different between both sides, corresponding to a ∼10% density difference. An additional 10% error on ablator thicknesses can be expected. Different ablator densities lead to shocks traveling at different velocities, hence reaching the sample at different times. The same effect is obtained for ablators of different thicknesses. As such, these sources of uncertainties are similar to the laser mistiming evoked in Sec. V A. Simulations suggest that the DAF efficiency is not affected by 10% ablator thickness differences and 10% ablator density differences, or a combination of both.
The surface roughness of the Si ablators is expected to be the highest on the coated side, with peak to peak perturbations of the order of 100 nm. 2D simulations with a sinusoidal surface modulation of 100 nm amplitude and 12.5 μm have been conducted in the double-sided irradiation configuration, where the target is stationary. The increasing density of the ablator/tracer and tracer/sample interfaces reduces the amplitude of the perturbed shock as it propagates toward the sample. These interfaces are mostly free from perturbations at the time of the DAF stability window. As such, there is no significant increase in lateral gradients. This is an advantage of using a target configuration with increasing densities from the ablator to the tracer and to the sample (that is the case for Si ablator targets but not Ti ablator targets).
VI. LPI-GENERATED HEs
The high laser intensities considered in the DAF scheme lie well within the non-linear regime for the laser-plasma interaction. While cross-beam energy transfer is not expected to be an issue due to the 3D irradiation configuration, the target is at risk from Hot Electrons (HE) generated by Two Plasmon Decay (TPD) and Stimulated Raman Scattering (SRS). The laser absorption may also be decreased by Stimulated Brillouin Scattering, which is not considered here since it would simply lead to a DAF efficiency loss. Such a contribution may typically be in the 1%–10% range and can be assessed in an actual experiment.
The PCGO-based LPI module in the Chic code allows us to estimate the fraction of energy transferred from the optical field to Maxwellian HE tails and compute their propagation in the plasma from their point of origin. Inline simulations are used to estimate the impact of LPI-HEs on DAF plasma parameters, longitudinal and lateral gradients, homogeneous volume size, and atomic physics processes for the opacity measurement.
2D simulations for the Se tracer target using this module suggest conversion efficiencies of 0.8% in forward TPD-HEs at 57 keV average temperature, 4.1% into forward SRS-HEs at 38 keV average temperature, and negligible amounts to resonant absorption HEs. In the model, SRS-HEs are propagated from 0.2 nc in a 25° cone along the laser direction, and TPD-HEs from 0.25 nc in a 90° cone along the laser direction. Given the high incidence angles of the beams, combined with the laser refraction in the corona, most of the HEs reaching the sample deposit their energy at least 50 μm away from the target axis [see Fig. 13(a)]. This non-homogeneous heating source reduces the radius of the 5% stability region for lateral gradients (defined in Sec. IV E) from 235 μm to 180 μm for the density and from 260 μm to 250 μm for the temperature. These values remain within the 150 μm radius probed volume of the backlighter/target/spectrometer configuration. The structure of the DAF and the longitudinal gradients in the plateau are unaffected by the amount, energy, and location of HEs relevant to our configuration.
In general, energy transferred from the laser to the HE population may be deposited both in front and behind the ablation front, depending on local density profiles and HE energy. Here, simulations suggest that this results in a ∼5 MBar decrease in ablation pressure, which decreases the asymptotic density reached in the sample by ∼1022 cm−3 (∼7.5%). The sample located inside the DAF structure lies between thick layers of dense and cold ablator materials. Most of the HEs being generated by SRS with moderate energies, they mainly deposit their energy in the cold Si ablator and not in places that could increase the DAF efficiency, e.g., the corona or the sample. The net effect is a decrease in plasma temperature in the sample of 2–5 eV (∼3%). The fraction of the highest-energy electrons that deposit energy in the sample is of the order of 10−5% of the incident laser power, as shown in Fig. 13(a). This small HE-induced heating does not compensate for the efficiency loss due to decreased collisional absorption and ablation pressure.
Finally, a fraction of higher-energy HEs propagate in the Fe sample layer without being absorbed, as shown in Fig. 13(b). These free HEs are in the 30–90 keV range in average temperature (Note that contrary to the source model that generates the HEs with an exponential distribution,42 the energy mentioned here is a flux-weighted average on the many mono-energetic HE beamlets propagating in the Fe sample.) and with a normalized density on the order of 10−4%ne. Such small fractions are negligible in terms of atomic physics processes for an opacity measurement.
VII. CONCLUSIONS
A platform for stellar opacity measurements in hydrodynamic conditions relevant to the BCZ in the sun has been investigated for laser conditions available on the OMEGA laser.1 It relies on counter-propagating double-ablation front structures to bring a material sample to high temperature and densities, with the aim of reaching conditions of the order of Te ∼ 200 eV and ne ∼ 1 × 1023 cm−3 at LTE and in steady state.
With that objective, the physics of double ablation front targets in the presence of tracer and sample layers has been investigated in 1D geometries using the Chic radiative hydrodynamic code. The hydrodynamic parameters reached in a sample inserted between a pair of counter-propagating ablation front are found to be mostly dependent on laser intensity and laser ablation pressure. Because shock compression is not a main factor in the asymptotic density reached in the sample, the scheme is robust to shock rebound between the fronts and shock mistiming effects. The latter can notably arise from experimental uncertainties such as imbalances in laser drive, ablator thickness, and ablator density. In the 1D framework, the symmetric DAF approach is found to produce high temperatures (∼150–200 eV) and high electron densities (∼2–4 × 1023 cm−3) depending on the target and laser configuration. The standard deviation of sample plasma parameters along the longitudinal direction is typically less than 2%–3%. Temporal stability is achieved during periods of ∼500 ps, with temporal variations of the plasma parameters in the 1%–20% range depending on the configurations. The ablator material, thickness, laser intensity, and irradiation scheme are free parameters to be tuned to adjust the efficiency of the stable DAF structure.
From the conclusions derived in the 1D study, an extension to 2D-axisymmetric geometries has been conducted. An experimental scenario suitable for the OMEGA laser system has been designed, extending on a similar configuration for shock heating experiments.17 The Chic code was first benchmarked against the latter experimental results in order to constrain numerical models for heat, laser, and radiation transport. A laser irradiation configuration has been designed based on an optimization between high laser intensity and large spot size. Accuracy in laser intensity computations was achieved by using dedicated caustic measurements for beams smoothed by E-IDI-300 phase plates in the presence of SSD and PS. The target materials were chosen to allow for unperturbed spectroscopic measurements in the 750–1300 eV range relevant to the solar opacity of the BCZ. The 2D simulations using realistic laser and target configurations suggest that a Fe sample can be brought to ∼160 ± 5% eV and ∼1.35 × 1023 ± 8% cm−3 during 500 ps, with longitudinal and lateral gradients less than 5% in a 360 μm diameter cylindrical volume. These specifications account for uncertainties in target fabrication and laser performances, as well as the effect of LPI-generated HEs. Here, the LPI-HEs reduce the sample hydrodynamic parameters by reducing the ablation pressure and laser absorption fraction. Reaching higher temperatures is expected to be possible by using OMEGA-EP as a backlighter, thus making more energy available for the main target.
The laser/target configuration has been designed with the possibility to perform time-dependent characterization of the sample temperature and density using a streaked spectrometer. Two tracer materials have been considered: absorption spectroscopy of the Kα satellites of Al or L-shell absorption spectroscopy of Se. While Al spectroscopy is expected to be more sensitive during the early part of the DAF structure, Se spectroscopy is more adapted for the high-temperature–high-density conditions (i.e., in the range of interest of the solar BCZ) of the temporal stability region. Ideally, opacity measurement of the sample of interest can be conducted simultaneously using a short time-integrated high-resolution spectrometer centered around the Fe lines at 1 keV. The concurrent usage of two different spectrometers will allow constraining hydrodynamic simulations during the whole pulse and simultaneously conduct a broadband higher resolution measurement of the sample opacity.
The experimental configuration presented here shows that the counter propagating DAF structure should be well suited as a measurement platform for solar opacities relevant to the BCZ. While a range of numerical and experimental uncertainties have been investigated, experimental verification of the hydrodynamic scenario is required. An investigation of the scheme will allow to adjust the numerical platform and refine a design to probe the opacity of various elements at various temperatures and densities. While this scheme was designed for mid-range ns-class lasers, it is expected to scale faborably to larger laser systems such as NIF or LMJ. The significant increase in available laser energy will allow for larger laser spots and higher intensities, thus extending the parameter domain and further reducing the potential gradients in hydrodynamic parameters.
ACKNOWLEDGMENTS
The authors wish to acknowledge C. Blancard and E. Le Bel for the fruitful discussions about this work. This work received funding from Project No. ANR-12-BS05-0017 from the French National Research Agency. It was also partially performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344.
APPENDIX: CAUSTIC OF BEAMS SMOOTHED BY E-IDI-300 PHASE PLATES
The OMEGA DAF target design involves beams equipped with the E-IDI-300 phase plates, which produce elliptical spots that couple efficiently to planar targets at an angle. The focal spots produced by these phase plates being small, the overall on-target intensity distribution is enlarged by shifting the best focus behind the target. The beam defocusing dictates both the laser intensity on target and the lateral target size, which must be small enough to allow adjacent beams to propagate. As such, determining the optimal defocusing distance requires a precise knowledge of the caustic of beams smoothed by the E-IDI-300 phase plates.
Dedicated measurements of the beam caustic were conducted for different positions of the final focusing optic. These results are compared to illumination calculations conducted with Visrad in Fig. 14(a) for the beam major and minor radii, and (b) for the envelope intensity. It is found that Visrad overestimates the best focus peak intensity by about 20% and the best focus super-Gaussian order by a factor of ∼2. Disagreement is also observed up to 3 mm away from the focal plane, where the baseline Visrad model now underestimates the peak laser intensity by about 30%.
These differences are thought to be related to the lack of Polarization Smoothing modeling in Visrad. In order to compute accurately the on-target illumination, we have defined a custom beam model in Visrad with major and minor radii closer to measurements [see Fig. 14(a)]. The remaining disagreements in beam radii between measurements and the custom beam approach are related to limitations in the description of the beam Rayleigh range in Visrad. However, the custom beam approach correctly describes the 1/e area of the ellipse and the intensity of the peak of the laser field [see Fig. 14(b)]. The resulting on-target intensity distribution for the overlap of the 15 beams is shown in Fig. 14(c). The custom beam model predicts a peak intensity of 5.56 × 1015 W/cm2 for a defocusing of 1 mm. This accurate computation of laser intensity is of importance for the correct design of a DAF target and the correct interpretation of measured spectral opacities.