Three-dimensional hot-spot x-ray emission tomography from cryogenic deuterium–tritium direct-drive implosions on OMEGA

Churnetski, K.; Woo, K. M.; Theobald, W.; Radha, P. B.; Betti, R.; Gopalaswamy, V.; Igumenshchev, I. V.; Ivancic, S. T.; Michalko, M.; Shah, R. C.; Stoeckl, C.; Thomas, C. A.; Regan, S. P.

doi:10.1063/5.0098977

A three-dimensional model of the hot-spot x-ray emission has been developed and applied to the study of low-mode drive asymmetries in direct-drive inertial confinement fusion implosions on OMEGA with cryogenic deuterium–tritium targets. The steady-state model assumes an optically thin plasma and the data from four x-ray diagnostics along quasi-orthogonal lines of sight are used to obtain a tomographic reconstruction of the hot spot. A quantitative analysis of the hot-spot shape is achieved by projecting the x-ray emission into the diagnostic planes and comparing this projection to the measurements. The model was validated with radiation-hydrodynamic simulations assuming a mode-2 laser illumination perturbation resulting in an elliptically shaped hot spot, which was accurately reconstructed by the model using synthetic x-ray images. This technique was applied to experimental data from implosions in polar-direct-drive illumination geometry with a deliberate laser-drive asymmetry, and the hot-spot emission was reconstructed using spherical-harmonic modes of up to ℓ = 3. A 10% stronger drive on the equator relative to that on the poles resulted in a prolate-shaped hot spot at stagnation with a large negative A_2,0 coefficient of A_2,0 = −0.47 ± 0.03, directly connecting the modal contribution of the hot-spot shape with the modal contribution in laser-drive asymmetry.

I. INTRODUCTION

In direct-drive inertial confinement fusion (ICF),¹ laser beams are focused onto a spherical target to implode a thin shell composed of an outer ablation layer (typically plastic) and an inner layer of solid cryogenic deuterium–tritium (DT) fuel. The laser-direct-drive ICF concept is studied on the 60-beam, 30 kJ, 351 nm OMEGA laser,² which has produced high-performing implosions with hot-spot pressures exceeding 50 Gbar, described in Ref. 3. The recent application of statistical modeling significantly improved the implosion performance and the neutron yield.⁴ Low-mode drive variations in the driver illumination and from target perturbations can significantly impact implosions.^5–7 The goal over the next several years is to further optimize OMEGA implosions and to demonstrate ignition-relevant implosions that when scaled to 2 MJ of laser energy would enter the burning-plasma regime.^8,9

To better understand what currently limits the performance of ICF target implosions, 3D diagnostics for x rays, neutrons, and charged particles are required to study multidimensional effects on the hot-spot formation. Three-dimensional hot-spot x-ray emission tomography is a powerful tool to diagnose low-mode asymmetries, which will help to mitigate low-mode perturbations and improve implosion performance. Previously, 3D reconstruction methods were developed using spherical-harmonic decomposition and cylindrical-harmonic expansion to reconstruct the shape of hot spots of ICF implosions.^10,11 These methods were tested on data from four time-integrated x-ray detectors on OMEGA and two orthogonal neutron detectors at the National Ignition Facility, respectively, and are discussed in Refs. 10 and 11. The work outlined in this paper is embedded in a long-term project that aims to understand the physics and multidimensional effects that currently limit the hot-spot pressure in ICF implosions on OMEGA and to help to develop strategies to increase the hot-spot pressure.

OMEGA currently has two time-gated and two time-integrated x-ray imagers that are routinely used to image the self-emission of the hot-spot region at the time of stagnation. The x-ray emission data from the four lines of sight (LOS) are combined to infer the shape of the hot-spot emission at stagnation. Future work will address the number of LOS necessary to resolve the low-mode shape of the hot spot and how the relative viewing angles of existing LOS affect the reconstruction. A method was developed to reconstruct the hot-spot emission profile using a spherical-harmonic Gaussian model¹² by defining the emissivity as a unique exponential function. This function is expanded in a power series with respect to the radial contours, which are decomposed into real spherical harmonics.

Proof-of-principle simulations with the hydrodynamic code DEC3D¹³ assuming a mode-2 perturbation were used to validate the 3D reconstruction procedure using synthetic x-ray images. This 3D model of the hot-spot emission has also been used to study several cryogenic implosions on OMEGA. Cryogenic ICF implosions with deliberate laser-drive asymmetries were used as a platform to demonstrate the technique. The experiment clearly shows the transition from a prolate-shaped to an oblate-shaped hot spot in the x-ray images at stagnation. The modal contribution in laser-drive asymmetry was directly connected to the modal contribution of the hot-spot shape for an implosion with a 10% stronger drive on the equator relative to that on the poles, which resulted in a prolate-shaped hot spot at stagnation with a large negative A_2,0 coefficient of A_2,0 = −0.47 ± 0.03.

II. HOT-SPOT X-RAY DIAGNOSTICS ON OMEGA

OMEGA currently has a combination of time-gated and time-integrated x-ray diagnostics for hot-spot imaging along four lines of sight, which enable a 3D interpretation of the hot-spot shape. These diagnostics include KBframed,¹⁴ the single line-of-sight, time-resolved x-ray imager (SLOS-TRXI),¹⁵ KB,¹⁶ and a spatially resolved electron temperature diagnostic (SR-TE). The physics behind the SR-TE diagnostic is discussed in Ref. 17. These four diagnostics are quasi-orthogonal from each other, allowing for a 3D view of the imploding ICF core. Figure 1 shows the locations of the detectors on the OMEGA target chamber and example data from each detector.

FIG. 1.

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The locations of the existing four hot-spot x-ray imaging detectors on the OMEGA target chamber are indicated by the red circles. SLOS-TRXI is located in port H4 (45°, 234°), KBframed in port H13 (105°, 342°), KB in port H12 (96°, 54°), and SR-TE in port H11 (101°, 134°). The future third time-gated x-ray imager will also be located in port H12 (96°, 54°). Example data are shown for each diagnostic and the reconstructed 3D hot-spot emission is displayed in the target chamber center. The polar and azimuthal angles for each detector are denoted in parenthesis.

KBframed¹⁴ is a Kirkpatrick–Baez x-ray microscope coupled to a framing camera. This diagnostic images the x-ray emission in the 2–8 keV energy range. The KB microscope consists of 16 mirror pairs to produce 16 images of the hot-spot x-ray emission with a spatial resolution of 6 μm. It is coupled to a multistrip, high-speed framing camera to obtain 16 time-gated images with a separation time per image of ∼15 ps and a temporal resolution of 40 ps. The core x-ray emission lasts for about 100 ps, so this diagnostic provides several time-gated images of the hot spot evolving throughout time. KBframed is located in port H13 of the OMEGA target chamber, as shown in Fig. 1.

SLOS-TRXI¹⁵ is a pinhole array camera connected to a fast-gated x-ray camera comprised of an electron pulse-dilation tube and a nanosecond-gated burst-mode hybrid complementary metal-oxide semiconductor (hCMOS) sensor. SLOS-TRXI images the x-ray self-emission of the hot spot in the 4–9 keV range and has a spatial resolution of better than 10 μm and a temporal resolution of 30 ps. With respect to the description in Ref. 15, SLOS-TRXI was significantly upgraded by improving the signal-to-noise ratio (SNR) and by increasing the number of exposures. This has been achieved by using a higher number of pinholes to provide about 30 images of the hot spot at one time. These images are averaged to increase the SNR per pixel element, defined as the mean of the images divided by the standard deviation, from ∼5 to >10. SLOS-TRXI has also been upgraded with an eight-frame hCMOS sensor to provide eight time-gated images of the hotspot during the core x-ray emission with a separation time per image of ∼25 ps. In contrast to KBframed, which records the framing camera images on film, SLOS-TRXI provides immediate readout from the electronic sensor, which allows the experimenters to make decisions for upcoming shots. SLOS-TRXI is located in port H4.

KB¹⁶ consists of four Kirkpatrick–Baez mirror pairs coupled to four time-integrating charge-injections devices, which provide immediate readout. KB has a spatial resolution of 5 μm and can image different energy ranges by changing the Al foil filtering of the four different channels. The spectral sensitivities of the four channels are as follows: Ch 1 = 1.9–6 keV (no Al foil), Ch 2 = 3.3–7 keV (25-μm-thick Al foil), Ch 3 = 4–7.3 keV (50-μm-thick Al foil), and Ch 4 = 4.3–7.7 keV (75-μm-thick Al foil). KB is located in port H12.

The SR-TE diagnostic consists of a penumbral pinhole array imager of 125-μm-diam circular apertures that are precision manufactured by General Atomics with an edge quality better than 3 μm. Differential filtering (four channels) is applied to infer the spatially resolved electron temperature distribution in the hot spot, which is not presently used in the 3D reconstruction. The channel with the softest filtration is used to detect photons in the energy range of 7–12 keV for the shots discussed in this paper. It records the time-integrated penumbral images of the hot spot with an image plate. Several images in each channel are summed together to improve the SNR. At these harder photon energies, a larger solid angle is required for adequate signal. A deconvolution¹⁸ process is applied to unfold the hot-spot images for each channel with a spatial resolution of 6 μm. SR-TE is located in port H11.

In the future, a fully time-resolved reconstruction of the hot-spot x-ray emission will be possible with the inclusion of a third time-gated x-ray imager that is currently being developed for use on OMEGA and will be comprised of an electron pulse-dilation tube and a gated burst-mode hCMOS sensor. This diagnostic will be located in port H12.

III. 3D RECONSTRUCTION

A 3D hot-spot emission model was developed to reconstruct the hot-spot emission profile of direct-drive implosions on OMEGA by combining the measured x-ray emission data from multiple lines of sight. The radiation transfer equation along a path $\hat{s}$ is considered for a steady-state plasma in which the temperature and density distributions and the radiation field are independent of time,¹⁹

\frac{d I_{ν}}{d s} = ε_{ν} - κ_{ν} I_{ν},

(1)

where I_ν is the spectral radiant energy density in units of (J/m²/s)/Hz/sr, s is the path length, ε_ν is the plasma emissivity, and κ_ν is the absorption coefficient in units of 1/m. The hot-spot plasma of cryogenic DT target implosions on OMEGA is optically thin for photon energies >2.5 keV, which is fulfilled for all of the x-ray imagers considered here, and the absorption term in Eq. (1) can be neglected.¹⁷ The projected x-ray image $I_{\hat{s}}$ along an observation direction $\hat{s}$ is given by

I_{\hat{s}} = Δ Ω \int_{- \infty}^{\infty} d t G (t) \int_{0}^{\infty} d ν \int_{\hat{s}} d s ε_{ν},

(2)

where ΔΩ is the detector solid angle, t is time, G(t) is the detector gating function, and ν is the photon frequency. We are not concerned with the absolute signal nor take the temporal evolution of the hot-spot emission into account in this simplified model. We also assume that the spectral response is the same for all the x-ray imagers. Further development of the model will include the temporal evolution of the hot-spot plasma emissivity and the actual spectral response functions of each detector. With those simplifications and dropping the frequency dependence on ε, Eq. (2) can be simplified to $I_{\hat{s}} \sim \int_{\hat{s}} ε d s$ ⁠. A method described in detail in a forthcoming publication¹² has been developed to reconstruct ε through a complete expansion set using both non-orthogonal polynomial and orthogonal polynomial expansions. The emissivity is described in spherical coordinates and transformed by a unique exponential function in the form ε(r,θ,φ) = exp[f(r,θ,φ)], where the origin of the coordinate system coincides with the peak of ε. Taking the natural logarithm of ε and expanding f(r,θ,φ) in a power series with respect to the radial contour function r(θ,φ) results in $\ln ε (r, θ, φ) = \sum_{n = 0}^{\infty} σ_{n} r {(θ, φ)}^{n}$ with radial expansion coefficients σ_n. The radial contour function is decomposed into real spherical harmonics $r (θ, ϕ) = R (1 + \sum_{ℓ = 1}^{\infty} \sum_{m = - ℓ}^{ℓ} A_{ℓ m} Y_{ℓ m})$ ⁠, where R is the radius and Y_ℓm are the real spherical harmonic functions. The mode amplitude A_ℓm is then expanded in a power series with respect to the radius, $A_{ℓ m} = \sum_{k = 0}^{\infty} A_{ℓ m k} R^{k}$ ⁠. Combining this results in a model describing the complex shape of the hot-spot emission in terms of a model of generalized spherical-harmonic Gaussian (SHG) functions,

\begin{align} \ln ε (r, θ, φ) & = \sum_{n = 0}^{\infty} σ_{n} R^{n} [1 + \sum_{ℓ = 1}^{\infty} \sum_{m = - ℓ}^{ℓ} \sum_{k = 0}^{\infty}) \\ \times {(A_{ℓ m k} R^{k} Y_{ℓ m} (θ, φ)]}^{n} . \end{align}

(3)

The expansion coefficients, σ_n and A_ℓm, are determined by a gradient-descent machine-learning algorithm that minimizes a loss function, which is the fit error between the model and the normalized measured x-ray images.

To reconstruct the emission profile, an initial guess is made for the solution of ε, which is a 1D Gaussian profile where all expansion coefficients in Eq. (3) are set to zero except for σ₂. This model is projected into the LOS of the x-ray detectors using a ray-tracing routine and these projections are compared with the experimental x-ray images from each diagnostic. The error between the model and the experimental images is calculated as the sum of the rms difference over the multiple lines of sight. The coefficients of the model are slightly perturbed, and this process is repeated for several iterations until the rms error is minimized. The rms for each line of sight is calculated for a fixed window of 80 × 80 μm², and is not weighted by the brightness. The relative lower signal limit above which the rms is being calculated is a few percent of the peak brightness.

A. Test on DEC3D simulation

A radiation-hydrodynamic simulation using the code DEC3D¹³ was run to test the 3D reconstruction algorithm. This simulation had a mode-2 laser-illumination perturbation imposed resulting in an oblate-shaped hot spot. The result of the simulation was post-processed using Spect3D²⁰ to create simulated x-ray images along the four lines of sight. The detector spectral-sensitivity functions were applied during the postprocessing to simulate the effects of each detector. Normally distributed random noise was added to the simulated x-ray images, and the hot-spot emission was reconstructed multiple times in a Monte Carlo simulation by varying the noise. The resulting 3D reconstructions were projected along the detector lines of sight and compared with the simulated x-ray images. The major and minor radii were calculated at the 1/e contour level, and Fig. 2 shows the comparison of these radii for the simulated and reconstructed x-ray images. The reconstruction results in a ∼7% systematically lower R_max for each LOS. The reason for this is currently unclear and is being investigated. The radii from the simulated x-ray data and the projections of the 3D model agree within the error bars, and the good agreement gives us confidence in our ability to use this technique on experimental data.

FIG. 2.

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Major radius (square symbols) and minor radius (triangle symbols) of the hot-spot images from DEC3D simulations (red) and those inferred from the 3D reconstruction model (blue) for the four different diagnostics.

B. Reconstruction of cryogenic DT polar-direct-drive implosions on OMEGA

A direct-drive ICF campaign on OMEGA was conducted with deliberate laser-drive asymmetries to study the effect of hot-spot shape asymmetries on implosion performance. The PDD²¹ beam illumination geometry was applied for subscale implosions of cryogenically layered DT targets (20% smaller size targets).²² This work will be discussed further in a future publication. The PDD illumination was achieved by using 40 of the 60 OMEGA beams, switching off 20 beams around the equator. The 40 beams are grouped in three beam rings in the upper and lower hemisphere according to their polar angles and the beams were repointed to achieve the best possible illumination uniformity.²¹ The partition of beam energies in rings 1 and 3 was varied while keeping the total laser energy constant. The magnitude of the laser-drive asymmetry was varied to produce hot spots that ranged from oblate to prolate in shape. Figure 3 shows data from three shots at stagnation from KBframed, which has an equatorial view of the capsule. Reconstructions were done for each shot during this campaign and compared to the experimental inputs.

FIG. 3.

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Experimental x-ray images from the KBframed diagnostic at stagnation of DT-cryogenic target implosions performed with the polar-direct-drive illumination geometry, where the 40 beams are grouped in three beam rings. By denoting the change in energy of ring 3 (irradiating closer to the equator) as ∆E₃ and the change in energy of ring 1 (irradiating closer to the pole) as ∆E₁, the change in ring energy ratio is defined as $Δ D : = (Δ E_{3} - Δ E_{1}) / E_{tot},$ ⁠, where E_tot is the total energy on target. The nominal ring energy partition for ∆D = 0 is E₁/E_tot = 0.25, E₂/E_tot = 0.25, and E₃/E_tot = 0.5. The beam-energy balance was varied from a stronger drive on the poles (shot 96578, ΔD = −4.6%), to a balanced drive (shot 96575, ∆D = 1%), to a stronger drive on the equator (shot 96581, ∆D = 10.6%), while keeping the total laser energy conserved.

The reconstruction of shot 96581 is shown in Fig. 4 and in the center of Fig. 1. Data from SLOS-TRXI and KBframed are chosen to be from the time of peak emission. Data from KB Ch 3 and SRTE channel 1 are used in the reconstruction. The differences in hot-spot size between the different diagnostics can be attributed to differences in time resolution. With a higher intensity of laser energy incident at the equator of the capsule, a prolate shape is expected in the hot-spot emission with the major axis aligned with the z axis of the target chamber. This is seen in the 3D reconstruction in Fig. 1. Table I lists the expansion coefficients A_ℓ,m of the model. Spherical harmonic modes up to ℓ = 3were used in the reconstruction. The large A_2,0 = −0.47 ± 0.03 coefficient indicates a strong mode ℓ = 2 in the reconstruction, which can be seen in the experimental x-ray images. The A_2,0 coefficient is negative, which indicates a prolate shape and is consistent with what we expect from the laser-drive asymmetry. Reconstructions of other shots from this campaign also show the expected overall shape and orientation of the hot spot based on the laser-drive asymmetry and will be further discussed in a future publication.

FIG. 4.

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Reconstruction of the hot-spot emission for shot 96581. The measured x-ray data are shown in the top row and the projections of the reconstructed 3D hot-spot emission are shown in the bottom row. The rms error for each line of sight is also listed.

TABLE I.

The expansion coefficients A_ℓ,m of the spherical-harmonic Gaussian function in Eq. (3) for spherical-harmonic modes up to ℓ = 3 for shot 96581.

	ℓ = 1	ℓ = 2	ℓ = 3
m = −3			0.02 ± 0.01
m = −2		−0.21 ± 0.01	0.08 ± 0.03
m = −1	0.02 ± 0.01	0.1 ± 0.01	−0.05 ± 0.01
m = 0	0.03 ± 0.01	−0.47 ± 0.03	0.03 ± 0.01
m = 1	0.04 ± 0.01	0.033 ± 0.003	−0.07 ± 0.01
m = 2		−0.24 ± 0.02	−0.09 ± 0.02
m = 3			0.23 ± 0.03

	ℓ = 1	ℓ = 2	ℓ = 3
m = −3			0.02 ± 0.01
m = −2		−0.21 ± 0.01	0.08 ± 0.03
m = −1	0.02 ± 0.01	0.1 ± 0.01	−0.05 ± 0.01
m = 0	0.03 ± 0.01	−0.47 ± 0.03	0.03 ± 0.01
m = 1	0.04 ± 0.01	0.033 ± 0.003	−0.07 ± 0.01
m = 2		−0.24 ± 0.02	−0.09 ± 0.02
m = 3			0.23 ± 0.03

IV. CONCLUSIONS

Reconstructions of hot spots from ICF implosions are useful for understanding what limits the performance of implosions and can be used to improve the implosion performance. Several x-ray imagers are available on OMEGA to image the self-emission of the hot spots of these implosions around the time of stagnation. A spherical-harmonic Gaussian model has been developed and was used to model the 3D hot-spot emissivity profile. This method was tested on simulated x-ray data and then applied to experimental x-ray data from implosions with known laser-drive asymmetries. In the future, the instrument response functions will be considered and the temperature and density profiles will also be reconstructed. The inclusion of a third time-gated x-ray imager, which is currently being developed for use on OMEGA, will make it possible to reconstruct the 3D hot-spot emissivity profile at several times during an implosion.

ACKNOWLEDGMENTS

This work was supported by the Department of Energy National Nuclear Security Administration under Award No. DE-NA0003856, the University of Rochester, and the New York State Energy Research and Development Authority. This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

K. Churnetski: Conceptualization (equal); Data curation (equal); Formal analysis (lead); Investigation (equal); Methodology (equal); Project administration (supporting); Software (equal); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). K. M. Woo: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). W. Theobald: Conceptualization (equal); Supervision (equal); Writing – original draft (supporting); Writing – review & editing (equal). P. B. Radha: Conceptualization (equal); Investigation (equal); Supervision (equal). R. Betti: Conceptualization (equal); Supervision (equal). V. Gopalaswamy: Data curation (equal); Resources (equal); Software (equal). I. V. Igumenshchev: Conceptualization (supporting); Investigation (supporting). S. T. Ivancic: Conceptualization (equal); Supervision (equal). M. Michalko: Data curation (equal); Resources (equal). R. C. Shah: Data curation (equal); Resources (equal); Software (equal). C. Stoeckl: Conceptualization (equal); Supervision (equal); Writing – review & editing (equal). C. A. Thomas: Conceptualization (equal); Supervision (equal). S. P. Regan: Conceptualization (equal); Project administration (equal); Supervision (equal).