K_α x-ray imaging of laser-irradiated, limited-mass zirconium foils

In the fast-ignition (FI) approach to inertial confinement fusion (ICF), an imploded DT fuel capsule is ignited by energy deposition from energetic particles injected into the capsule just prior to the time of peak compression.¹ In the cone-in-shell approach to FI, the shell is collapsed adjacent to the tip of a hollow cone, where a kilojoule-class, few-picoseconds-long petawatt laser pulse focused on its tip accelerates energetic (fast) electrons into the compressed core.² The cone-in-shell concept reduces the distance over which the electrons must propagate to reach and ignite the compressed part of the fuel while maintaining a plasma-free path for the short-pulse ignitor laser. Figure 1 illustrates this concept. Experiments that determine the fast–electron penetration efficiency into FI fuel capsules are essential for understanding the electron transport and optimizing the capsule design. Surrogate integrated experiments³ were performed on the OMEGA Laser System⁴ to assess the fast-electron coupling from a measurement of the neutron yield enhancement from compressed deuterated plastic shells. Those experiments only allowed us to infer a global coupling efficiency and did not show where in the compressed plastic the fast electrons deposited their energy. Imaging of K_α x-ray emission generated by fast electrons while propagating through a buried fluorescent layer is a powerful technique to determine local energy deposition.⁵ 

In integrated FI experiments, this technique has been further developed by imaging the K_α x rays emitted from fluorescent materials that are doped into the fuel.^6,7 Electrons accelerated from the cone tip propagate into the compressed capsule, where they stimulate the emission of K_α fluorescence. The fluorescing material is chosen to have an emission energy sufficient for the x rays to propagate out of the capsule with minimal attenuation. A spherically bent crystal, oriented at the Bragg angle, collects and focuses the K_α photons to an x-ray detector [charge-coupled device (CCD) or imaging plate]. The resulting image shows the spatial distribution of the K_α photons from which the spatial distribution of the fast electrons can be inferred.

The use of K_α radiation in ICF studies is well established. Most applications employ the K_α of Cu at around 8 keV. The fast-electron spreading angle in flat-foil targets, irradiated with a high-intensity laser, is routinely inferred by imaging the K_α emitted from Cu layers buried at different depths inside the foil.⁵ Another application involves backlighting the ICF fuel assembly with Cu K_α to generate radiographic information about the fuel density or its evolution.⁸ The use of Cu as a fluorescing dopant to diagnose the penetration efficiency of fast electrons in hot dense ICF fuel capsules is, however, limited by K_α line shifting and broadening.^6,9 The shifts result from changes in atomic electron energy levels caused by changes in the nuclear screening that accompanies high-temperature-induced ionization. As the lines shift in energy, they move outside the narrow acceptance bandwidth of the diagnostic imaging crystals; consequently, the detection efficiency drops. Figure 2 shows the calculated density and temperature profiles of an imploded 40 μm-thick, 870 μm-diam plastic capsule that was compressed using 20 kJ from the 60-beam OMEGA laser at the University of Rochester's Laboratory for Laser Energetics (LLE). The profiles were calculated using the one-dimensional (1-D) hydrodynamics code LILAC.¹⁰ The temperature of the plastic reaches several hundred electron volts (eV) at which point the material is partially ionized.

Previous characterizations of the background temperature and resulting spectral shift of K_α emission in thin-foil targets irradiated with high-intensity lasers and heated by ohmic return currents using 1 to 7 J pulses and 50 or 100 μm-diam, 12.5 μm-thick polyvinylidene chloride targets indicated a peak temperature of 30 eV, resulting in an average ionization of 3.5 and a spectral shift in Cl K_α (2.6 keV) of 4.5 eV (Ref. 11). The peak temperature in these experiments may have been limited by a rapid adiabatic expansion of the target material. Nilson et al.¹² irradiated Cu foils over a range of conditions in which the laser energy per target volume varied from around 10² to 5 × 10⁶ J/mm³. Using the ratio of the K_α to K_β, it was determined that the highest background temperature exceeded 200 eV. Experiments using around 250 J of laser energy in a 10 ps pulse on 5 μm-thick, 100 μm-sq Cu targets constrained from expansion by 1 μm Al tamp layers were inferred to have acquired a charge state of 14 at a temperature of 220 eV (Ref. 13). In this case, the peak of the Cu K_α shifted by ∼10 eV with respect to the cold-target case. In all cases presented above, larger-volume targets irradiated under the same conditions resulted in lower background temperatures, smaller spectral shifts, and less broadening.

Figure 3(a) illustrates how the lines of the Cu K_α doublet are predicted to shift as a function of background temperature and ionization state. The line shifts were calculated using the multiconfiguration Dirac–Fock method described in Ref. 14. The green bar indicates the typical acceptance bandwidth of a crystal used to reflect the K_α x rays. As the temperature increases, the $Kα1$ line initially undergoes a shift to a lower energy before shifting to a higher energy and eventually moving out of the acceptance bandwidth of the crystal. The shift in the $Kα2$ line follows a similar pattern. As the lines shift, there is a change in the signal that reflects from the crystal. The signal first decreases as the $Kα1$ line shifts to a lower energy, reaching a minimum at an average ionization state of 7, when the line peak moves to the edge of the green band, corresponding to a temperature of ∼35 eV. The crystal again reflects strongly when the ionization state reaches 9, corresponding to a temperature of ∼50 eV. At higher temperatures, the $Kα1$ radiation is tuned out of the acceptance bandwidth of the crystal, and the reflectivity falls to zero. When the ionization state reaches 14, corresponding to a temperature of ∼160 eV, the $Kα2$ radiation is tuned into the bandwidth of the crystal and the reflectivity briefly recovers. A similar behavior was reported in Ref. 15.

Another issue associated with imaging Cu K_α is the high optical opacity of the compressed fuel. When the fuel is compressed to 150 g/cm³, the fraction of 8 keV x rays that are able to escape is estimated to fall below 10%. The K_α emission must also be brighter than the thermal emission from the hot coronal plasma that surrounds the imploded capsule if it is to be distinguished; the spectral tail of the coronal emission can reach 8 keV. In addition, the coronal x rays above 8 keV can optically pump the Cu K_α, creating an additional level of uncertainty to the electron-induced component of the K_α emission, although this is expected to be a small effect.

The use of more-energetic x rays, such as Zr K_α at ∼15.7 keV, should mitigate the issues described above. Promising results with a novel Zr K_α imager were recently reported.¹⁶ Figure 3(b) shows how the Zr K_α lines are predicted to shift as a function of temperature and ionization state. This calculation was performed using the web-based kinetic K-shell spectroscopy code FLYCHK.¹⁷ First, note the enhanced acceptance bandwidth for imaging Zr K_α, which stems from the d spacing of the crystal lattice for the cut that is appropriate for imaging Zr K_α lines. Initially, the peak of the $Kα2$ line stays within the acceptance bandwidth of the crystal, gradually shifting to a higher energy as the ionization increases. The $Kα2$ line then tunes out of the crystal bandwidth at an ionization state of around 18, corresponding to a temperature of 200 eV and resulting in a decrease in the reflected signal. Reabsorption of the Zr K_α radiation in the foil is less of an issue since the opacity of the target material to Zr K_α is substantially smaller than for Cu K_α. For comparison, the 1/e attenuation length of Zr K_α in solid-density Ag is 28.9 μm, while for Cu K_α, it is only 4.6 μm. The thermal emission from the coronal plasma will be extremely low at 15.7 keV; consequently, the relative brightness of the electron-induced emission is enhanced over the background while the potential for optical pumping is strongly diminished. Using web FLYCHK, Fig. 4 compares the variation in the normalized reflection efficiency with the temperature of a Cu K_α imager, similar to what was used in the experiments described by Akli et al.,¹⁶ to the efficiency of a Zr K_α imager for a solid-density plasma. The precise behavior of the line shift shown in Fig. 3(a) is not repeated for Cu K_α in Fig. 4, but the general behavior is similar; this discrepancy arises from the different methods employed to perform the two calculations. Note that the absolute reflection efficiency for the Zr K_α imager is at least an order of magnitude lower than that of the Cu K_α imager so that use of the latter is favored in the cold limit.

Measurements to determine the temperature dependence of the Zr K_α emission have been carried out using a spherically curved quartz crystal on the Texas Petawatt Laser at the Texas Center for High Energy Density Science at the University of Texas at Austin.¹⁸ For the experiment reported here, the laser focused 100 J of energy in an 8 ps pulse onto a 10 μm-diam spot that contained 50% of the energy, corresponding to a mean intensity of around 4 × 10¹⁹ W/cm². To lower the on-target intensity, the laser was set up to have a longer pulse duration than its nominal 150 fs value; this lowers the fast-electron temperature and softens the associated bremsstrahlung emission, which can add significant x-ray energy to the background. The laser angle of incidence was 25°. The setup is shown schematically in Fig. 5(a).

The targets, provided by General Atomics, consisted of various-sized laser-machined Zr foils. The sizes were 3000× 3000 × 21 μm³, 1000 × 1000 × 21 μm³, 500 × 500× 11 μm³, 300 × 300 × 6 μm³, and 150 × 150 × 6 μm³. To minimize energy coupling into the supporting structure, the targets were supported only at the corners, as shown in Fig. 5(b). The laser-machined slots that form the target were all 75 μm wide. A 1 μm-thick Al layer, included in the above thicknesses, was coated onto the laser-irradiated surface of the target to prevent the ablation of Zr by the laser prepulse and to suppress the production of lines from highly ionized Zr.

A 7.5 cm-long, 1.0 mm-thick, highly ordered pyrolytic graphite (HOPG) flat crystal spectrally dispersed the x-ray emission onto a Fujifilm TR imaging plate detector filtered by a 700 μm-thick Be vacuum window. Because of the target chamber's port geometry, the center of vacuum window was ∼10.5 cm below the plane of the target corresponding to an angle, with respect to the target, of ∼7°. Consequently, the HOPG was oriented with its face at an angle of 7° to the horizontal in a position that was below the incoming laser pulse. The HOPG crystal viewed the target from the laser-irradiated side and in second order for Zr K_α. The direct line of sight to the target was blocked with a 50 mm-thick piece of lead. The imaging plate was oriented vertically and positioned just outside the vacuum window. To block ambient light, the imaging plate was wrapped in a 25 μm-thick single layer of Al foil.

A curved quartz crystal (Inrad Optics), with Miller indices (234), viewed the target from the rear side. The crystal, with a focal length of 250 mm and a diameter of 30 mm, subtended a solid angle of 7 mSr with respect to the target. The crystal focused Zr $Kα2$ x rays to a second Fujifilm imaging plate with a magnification of 6.8. The acceptance bandwidth was ∼65 eV. The $Kα2$ emission was filtered using a 700 μm-thick Be vacuum window and a 16 μm-thick Zr foil. This imaging plate was also wrapped in Al foil. The direct line of sight to the target was blocked with 15 mm of lead. The Bragg angle, rocking curve, and reflection efficiency of the curved quartz crystal x-ray imager were measured at a photon energy of 15.6909 keV, corresponding to the $Kα2$ line of Zr, using the X15A beamline from the National Synchrotron Light Source at Brookhaven National Laboratory.¹⁸ A Bragg angle of 86.8° (corresponding to an incidence angle of 3.2°), a peak reflectivity of (3.6 ± 0.7) × 10⁻⁴, and a rocking-curve full width at half maximum of 0.09° are reported in Ref. 18.

Figure 6 shows lineouts through the Zr K_α doublet from the HOPG spectrum for four target sizes: 1000 × 1000 × 21 μm³ (dashed-dotted blue), 500 × 500 × 11 μm³ (dashed green), 300 × 300 × 6 μm³ (dotted red), and 150 × 150 × 6 μm³ (solid aqua). The lineouts, which are shown spaced apart for clarity, have been corrected for variations in the laser energy and the background noise has been subtracted. Because of an issue with the shielding, the background signals produced during the shots with the largest targets were significantly higher than those produced during shots with other targets. The $Kα1,Kα2$ doublet was resolved for the 1000 × 1000 × 21 μm³ target, while for the other sizes the $Kα2$ line appears as an increasingly less prominent, low-energy shoulder on the $Kα1$ line. As expected, the ratio of $Kα1$ to $Kα2$ is 2:1. The K_β emission was weak for all targets and is not shown. Because of the uncertainty in the positioning of the imaging plate between shots, neither the absolute position of the K_α lines nor the temperature-induced shifting in their position could be determined from the HOPG spectrum. The energy spacing between the K_α and K_β peaks can be measured, however, and no changes in this energy spacing were observed. Figure 6 indicates that the lines broaden with decreasing target volume. We presume that the broadening is accompanied by a temperature-induced shift toward higher energies as a result of an ionization-driven change in the nuclear screening of the inner shell's electrons as reported in Ref. 13. Note that Doppler-induced broadening was expected to be relatively small with estimates indicating a shift of less than 2 eV for the Zr K_α line from a 200 eV emitter.

Figure 7 shows the variation in the integrated value of the $Kα1$ signal from the HOPG spectrograph as a function of the laser energy divided by target volume. The results indicate that the K_α source's intensity decreases as the target gets smaller.

Figure 8 shows images of the $Kα2$ emission through the target's rear surface for various target sizes. The images have been corrected for magnification and plotted using the same color scale. The white lines in Fig. 8(a) are lineouts through the highest intensity pixel. The $Kα2$ emission is not observed from the whole target area but is instead localized to a small volume most likely centered on the electron beam's first pass through the target where the electron density is highest. The edges and corners of the target cannot be seen in the images. The signal decreases with decreasing target volume. The peak signal decreases by a factor of ∼3 in going from the largest target to the smallest.

Figure 9(a) shows the integrated image signal as a function of a target volume, corrected for variations in the laser energy. Figure 9(b) shows the integrated signals for each image corrected for the variation in the strength of the K_α source that was measured simultaneously using the HOPG spectrometer. The results suggest that the decreasing image brightness is connected to the weakening source with no strong variation due to rising temperature through decreasing volume.

The K_α source signal decreased as the target volume was reduced. The foils were all thinner than 21 μm, which is small compared to the typical fast-electron range. Using the ponderomotive scaling¹⁹ and a laser intensity of I = 4 × 10¹⁹ W/cm², we estimate the fast-electron temperature to be T_hot = 2.3 MeV. For a typical fast electron, this corresponds to a continuously slowing down approximation range of ∼3.7 mm in zirconium or ∼90 round-trip excursions though the thickest targets. Therefore, it is expected that the K_α radiation is generated uniformly throughout the entire thickness. The thin foil also minimizes the K_α reabsorption, eliminating concerns that arise from target material opacity; the single e-folding length of Zr K_α in solid zirconium is ∼75 μm. Thermal expansion of the foil can be estimated by the sound speed $cs=(ZkBT/M)1/2$⁠, where Z is the effective unscreened nuclear charge, k_B is Boltzmann's constant, T is the temperature, and M is the Zr ion mass. Assuming that T = 200 eV and Z = 20, c_s ∼ 0.07 μm/ps. Although the highest-energy electrons circulate in the target for several picoseconds before their energy falls below the threshold for K_α generation, the bulk of the K_α emission will be exhausted within 2 ps after the end of the 8 ps laser pulse so that the total expansion during this time should not exceed 0.7 μm.

The decrease in the strength of the K_α source with decreasing target volume was likely caused by pre-plasma effects. Given the laser energy and target Z, we do not believe that L-shell depletion¹² significantly accounts for any reduced K_α emission, even in the smallest targets. The laser prepulse was not known at the time of the experiment. Subsequent measurements performed after the completion of the experiment indicate that several pencil-beam prepulses arrive a few tens of nanoseconds before the main pulse with a fractional energy in the range of 10⁻⁶ to 10⁻⁵ with millimeter-scale spot sizes. Scanning third-order cross-correlation measurements indicated an intensity contrast of ∼10⁻⁶ up to 5 ps before the peak of the pulse, allowing for the possibility of pre-plasma formation. Extreme ultraviolet images of thin-foil targets irradiated with a higher peak intensity, but the same prepulse, indicated significant ablation.²⁰ One then expects ablation during each laser shot and furthermore that the target size might influence the pre-plasma geometry: For example, the thinnest targets may experience ablation on both the irradiated and rear surfaces. The varying conditions and degrees of expansion may support varying K_α suppression via adiabatic cooling of the fast-electron population.¹¹ 

The target temperature was estimated from numerical calculations using the web-based version of the code FLYCHK. FLYCHK is a zero-dimensional spectral synthesis code designed for modeling the spectral properties of radiation emitted from hot, highly transient laboratory plasmas. The code does not assume local thermal equilibrium (LTE) as a simplifying assumption. Instead the relative abundance of each ion species is determined from the solution of coupled differential equations that include ionizing effects from a population of fast electrons and non-LTE radiation fields.²¹ 

Target temperatures were estimated by convolving the synthetic spectra generated by FLYCHK with the HOPG instrument response function, which was taken to be a Gaussian profile with σ = 29 eV width, and comparing the result to the measured spectra. The 29 eV profile width produced the best fit to the measured spectra and was determined from the 1000 × 1000 × 21 μm³ sized targets where the K_α doublet was experimentally resolved. The experiment was modeled with a minimum set of assumptions about the plasma: We assumed a bulk temperature that ranged from 10 to 500 eV in steps of 10 eV. The results were relatively insensitive to the precise value of the fast-electron temperature T_hot and the fast-electron fraction f_hot, which were set to T_hot = 1 MeV and f_hot = 0.1%. Because the laser prepulse and, consequently, the density of the Zr during the K_α emission were uncertain, we performed calculations for a range of ten densities from 0.01 g/cm³ to solid density. Since the duration of the K_α emission was short, we did not model the temporal evolution of the plasma conditions. The peak of the synthetic K_α feature was normalized to the peak of the experimentally measured spectra.

Figure 10 presents FLYCHK results for the 500 × 500 × 11 μm³ target. Figure 10(a) shows the reduced χ² $(χred2)$ statistic, which was a measure of the goodness of fit for a given synthetic spectrum over a photon-energy range that extends from the peak of the $Kα1$ feature to 300 eV to its low-energy side. This range includes the $Kα2$ feature. This statistic was computed for synthetic spectra over the full range of temperatures and densities. A good fit is signified by $χred2=1$⁠, while a poor fit is signified by $χred2≥10$⁠, which have been compounded in the figure. The black crosses (+'s) appearing at (120 eV, 0.75 g/cm³) and (250 eV, 1.54 g/cm³) highlight two different density–temperature conditions that agree well with the measured spectrum (i.e., low $χred2$⁠). Figures 10(c) and 10(d) show the K_α component of the experimentally measured spectra for the 500 × 500 × 11 μm³ target (dotted blue line), the FLYCHK prediction at the best-fit temperatures (dashed green line), and these same best-fit synthetic spectra convolved with the HOPG instrument response (solid magenta line). The x axis is displayed in units of eV relative to the peak of the $Kα1$ feature. Because of the similarity of the synthetic spectra in Figs. 10(c) and 10(d), the shape of the spectrum alone is insufficient to distinguish between these two best-fit conditions. The imager efficiency data [Fig. 9(b)] suggest, however, that none of the targets was heated to extremely high temperatures. Using this, an estimate on the upper limit of temperature can be obtained by analyzing Fig. 10(b), which shows the synthetic imager efficiency over the range of temperatures and densities. Here, the efficiency represents the integrated synthetic signal that remains within the acceptance bandwidth of the imager. The crosses (+'s) again represent the low-$χred2$ solutions identified in Fig. 10(a), and in both plots, a solid black line indicates where the efficiency falls to 60%. One can conclude from Fig. 10(b) that the temperature and density of the 500 × 500 × 11 μm³ target could not have been as high as 250 eV at 1.5 g/cm³, despite good agreement with the measured spectrum. Instead, the 120 eV temperature at the 0.75 g/cm³ synthetic spectrum is regarded as the best fit because it provides a good match to the measured spectrum and is not in conflict with the experimental results. We note that the FLYCHK results are weakly dependent on density above ∼0.5 g/cm³, and take the value of 0.75 g/cm³, simply to be the best fit for the density of the emitting material.

Best-fit temperatures and densities were computed in an analogous way for the other targets. The results indicate only a marginal increase in temperature, from 90 to 130 eV, in going from the largest to the smallest target. In all cases, the best-fit density was determined to be 0.75 g/cm³. Experiments to determine the temperature of laser-irradiated copper targets indicated a temperature ranging from ∼10 to 200 eV over a similar range of volumes.¹² We believe that this range better represents the conditions encountered during the experiment and that inaccuracies in the atomic physics of the FLYCHK model limit its applicability in the current case. We note also that the best-fit analysis described above was performed with respect to a limited range of experimental data. The high-energy side of the measured $Kα1$ feature, which exhibits a 10 to 20 eV shift compared to the synthetic spectra, was not included in the analysis. Such a shift would by itself require an unrealistically high temperature but we note a source velocity of ∼300 km/s toward the spectrometer, which measured the spectrum from the laser-irradiated side, would produce such a shift. The FLYCHK model does not account for this motion. Our inability to extend the fitting analysis to this portion of the spectrum may account for the apparent insensitivity of the temperature with target volume.

Measurements to determine the efficiency of a quartz crystal to image laser-induced K_α emission from zirconium targets that ranged in size from 3000 × 3000 × 21 μm³ to 150 × 150 × 6 μm³ have been performed. The laser provided 100 J, 8 ps-long pulses with a mean intensity of 4 × 10¹⁹ W/cm². The crystal, with Miller indices (234) and a focal length of 25 cm, imaged the Zr $Kα2$ emission through the target's rear surface. The K_α source was simultaneously measured using a HOPG crystal spectrometer that viewed the target from the front laser-irradiated surface. The ratio of the integrated image signal to the HOPG-determined source strength remained approximately constant, within the experimental uncertainties, throughout the range of target volumes used, implying that the emission remained inside the collection bandwidth of the imaging crystal. Numerical calculations that compared measured and synthetic spectra did not provide a reasonable estimate for the variation in temperature with target volume. Comparison to previous experiments with copper that used a similar range of target volumes irradiated under similar laser conditions suggest, however, a temperature, for the smallest targets, of around 200 eV. Based on this assumption, the Zr-based K_α imager is robust against temperature-induced line shifting within the temperature range of interest.

We thank Javier Jaquez of General Atomics in San Diego and the staff of the Texas Center for High Energy Density Science at the University of Texas at Austin for their support in this work. This material is based upon work supported by the Department of Energy National Nuclear Security Administration (NNSA) under Award Number DE-NA0001944, the University of Rochester, and the New York State Energy Research and Development Authority. The NNSA also supported this work via the Texas Center for High Energy Density Science through award number DE-FC52-08NA28512. Support also came from the U.S. Department of Energy Office of Fusion Energy Sciences (OFES), Fusion Science Center Grant No. DE-FC02-04ER54789; and the OFES ACE Fast Ignition Grant No. DE-FG02-05ER54839. The support of DOE does not constitute an endorsement by DOE of the views expressed in this paper.