Three-dimensional effects play a crucial role during the hot-spot formation in inertial confinement fusion (ICF) implosions. A data analysis technique for 3D hot-spot reconstruction from experimental observables has been developed to characterize the effects of low modes on 3D hot-spot formations. In nuclear measurements, the effective flow direction, governed by the maximum eigenvalue in the velocity variance of apparent ion temperatures, has been found to agree with the measured hot-spot flows for implosions dominated by mode $\u2113=1$. Asymmetries in areal-density (*ρR*) measurements were found to be characterized by a unique cosine variation along the hot-spot flow axis. In x-ray images, a 3D hot-spot x-ray emission tomography method was developed to reconstruct the 3D hot-spot plasma emissivity using a generalized spherical-harmonic Gaussian function. The gradient-descent algorithm was used to optimize the mapping between the projections from the 3D hot-spot emission model and the measured x-ray images along multiple views. This work establishes a platform to analyze 3D low-mode core asymmetries in ICF.

## I. INTRODUCTION

An inertial confinement fusion (ICF) implosion^{1} involves the compression of a deuterium–tritium (DT) spherical shell to achieve high hot-spot ion thermal temperatures and pressures. The goal is to increase the rate of alpha-particle heating and exceed the rate of total energy loss within the hot spot to achieve ignition.^{2–8} In experiments, the hot-spot formation is degraded by three-dimensional effects including target offsets, laser illumination nonuniformities, defects in capsule fabrications, laser–plasma instabilities, and so on.^{9,10} Quantifying the effects of various nonuniformities is an important step to optimize implosion performance and achieve ignition in ICF.^{11–15}

In this work, a platform to reconstruct 3D asymmetries in ion temperatures, areal densities, and hot-spot shapes is developed. By performing a diagonalization analysis for the velocity-variance matrix,^{16} variations in apparent ion temperatures are found to be governed by the effective hot-spot flow residual kinetic energy ($RKEhs$) along corresponding eigenvector directions. Through *DEC*3*D* modes $\u2113=1$ to 2 simulations,^{12} quasi-isotropic flows from mode 2 are shown to produce small ion-temperature asymmetries even when the translational hot-spot flow velocities are small. Variations in areal-density measurements along different lines of sight (LOS) are found to satisfy a semi-analytic mode-1 *ρR* model. A 3D x-ray emission tomography method is developed to reconstruct 3D hot-spot plasma emissivity profiles using a generalized spherical-harmonic Gaussian (SHG) function. Modal parameters are determined by a gradient-descent algorithm to optimize the mapping between modal projections and x-ray images measured at different LOS's. Applications of the 3D hot-spot reconstruction for OMEGA implosions dominated by low modes $\u2113=1$ to 2 are presented.

The following paper is organized into three sections. Section II describes a comprehensive 3D analysis for nuclear measurements including apparent ion temperatures, hot-spot flows, and areal densities. Section III describes the treatment for the 3D x-ray emission tomography method and its application to reconstruct 3D hot spots for OMEGA implosions dominated by low modes, followed by the conclusion in Sec. IV.

## II. THREE-DIMENSIONAL ANALYSIS OF NUCLEAR MEASUREMENTS

### A. Ion-temperature asymmetries

In ICF experiments, hot-spot apparent ion temperatures are inferred from the widths of neutron energy spectra.^{17–19,22,23} The Doppler effect boosts neutron velocities along different flow directions, producing a signature shifting of the mean neutron energies. By measuring the first moment of neutron energy distribution at four different LOS's, the hot-spot flow velocity vector is reconstructed.^{24}

The broadening of the width of a neutron velocity spectrum is caused by thermal motions and relative motions of deuterium and tritium ions in D–T nuclear fusion reactions as well as the Doppler effect that boosts the center of mass frame of D and T ions by fluid motions. The non-relativistic and relativistic treatments to infer apparent ion temperatures using moments from neutron velocity spectra are described by Refs. 17 and 22. It was found that the effect of flows on the width of neutron velocity spectra is characterized by the second moment^{17,22} of the neutron velocity distribution in terms of a velocity variance $Var[v\u2192\xb7d\u0302LOS]$. Here, $v\u2192$ is the plasma fluid velocity vector in the laboratory frame and $d\u0302LOS$ is the LOS unit vector. Both spherically symmetric flows and three-dimensional flows are sources of the velocity variance. The apparent ion temperature produced by D–T nuclear fusion reactions within a 3D distorted hot spot is given by

where $Tth$ and $MDT$ are the ion thermal temperature and the total reactant mass for D–T nuclear reactions, respectively.

Three-dimensional effects of hot-spot flows on apparent ion temperatures are analyzed by diagonalizing the velocity-variance matrix. The term $Var[v\u2192\xb7d\u0302LOS]$ is first decomposed^{16} into six different components $\sigma ij=\u27e8(vi\u2212v\xafi)(vj\u2212v\xafj)\u27e9$, where indices *i *=* *1, 2, 3 stand for the *x*, *y*, *z* component in Cartesian coordinates, respectively. $v\xafi$ is the burn-averaged hot-spot flow velocity along the *i*th direction. $d\u0302LOS=d1x\u0302+d2y\u0302+d3z\u0302$ is the LOS unit vector with components $d1=sin\u2009\theta \u2009cos\u2009\varphi ,\u2009d2=sin\u2009\theta \u2009sin\u2009\varphi $, and $d3=cos\u2009\theta $, where *θ* and $\varphi $ are the LOS's polar and azimuthal angles, respectively.

The use of bra-ket notation simplifies the diagonalization analysis. The LOS unit vector is denoted by a column-vector notation called a ket $|d\u27e9=[d1,d2,d3]T$, while the transpose (T) of the ket is denoted by a row-vector notation called a bra $\u27e8d|=|d\u27e9T$. The velocity variance along a given LOS equals the expectation value

with respect to the velocity-variance matrix

Because the velocity variance matrix commutes, $\sigma \u0302$ is Hermitian. Hence, it can be diagonalized by three orthonormal eigenvectors $|ei\u27e9$, each associated with a real non-negative eigenvalue $\lambda i\u22650$. By inserting two identity matrices $I\u0302=\u2211i=13|ei\u27e9\u27e8ei|$ into the expectation value $\u27e8d|I\u0302\xb7\sigma \u0302\xb7I\u0302|d\u27e9$ in Eq. (2), $\sigma \u0302$ is diagonalized,

The eigenvalue $\lambda i=\u27e8ei|\sigma \u0302|ei\u27e9$ is the expectation value with respect to each eigenvector. The inner product $\u27e8d|ei\u27e9$ is the projection of the LOS unit vector onto an eigenvector. The diagonalization implies a rotation of Cartesian coordinates into a new base of orthogonal coordinates spanned by the three eigenvectors such that $|e1\u27e9=x\u0302,\u2009|e2\u27e9=y\u0302$, and $|e3\u27e9=z\u0302$. In the rotated Cartesian coordinates, the inner products are $\u27e8d|e1\u27e9=sin\u2009\theta \u2009cos\u2009\varphi ,\u2009\u27e8d|e2\u27e9=sin\u2009\theta \u2009sin\u2009\varphi $, and $\u27e8d|e3\u27e9=cos\u2009\theta $, respectively. Both polar and azimuthal angles *θ* and $\varphi $ are measured in the rotated Cartesian coordinates, respectively.

Since diagonal elements $\sigma ii=\u27e8(vi\u2212v\xafi)2\u27e9$ are the square of hot-spot flow velocity fluctuations, an eigenvalue *λ _{i}* measures the nontranslational part of the hot-spot residual kinetic energy $RKEHS,inontrans=MHS\sigma ii/2$ along the

*i*th eigenvector direction, where $MHS$ is the burn-averaged hot-spot mass. The trace of $\sigma \u0302$ gives the total nontranslational hot-spot residual kinetic energies

which is a physics invariant under the special orthogonal SO(3) transformation as described above. As a result, apparent ion temperatures measured at different LOS's are linked together

to conserve the physics invariant $RKEHS,totalnontrans$.

The effective hot-spot flow along an eigenvector direction contains hydrodynamic effects of both isotropic and anisotropic flows. The minimum of all eigenvalues gives the isotropic component of the hot-spot residual kinetic energy, i.e., $RKEHS,isonontrans=Min[\lambda 1,\lambda 2,\lambda 3]MHS/2$, which broadens the widths of neutron velocity spectra isotropically in all LOS's. Turbulence^{19} and radially symmetric flows^{20,21} belong to hydrodynamic isotropic flows. The hot-spot flow of mode 2 is quasi-isotropic because the hot-spot residual kinetic energy (RKE) along and perpendicular to the axis of rotational symmetry are different. The relative kinetic energy in the two-body D–T nuclear fusion reaction contributes another isotropic term $Var[\kappa ]$ in the second moment of neutron velocity distributions, which is negligible when compared with ion thermal velocities and hot-spot flow velocities^{17,22,25} for typical ICF implosions with ion thermal temperatures ∼5 keV.

#### 1. Mode $\u2113=1Ti$ *asymmetries*

Figure 1(a) shows the impact of isotropic flows on the $Ti$ and flow relation. *DEC*3*D* mode-1 simulations in black show a vanishing $Ti$ asymmetry when anisotropic hot-spot flow velocities are small. To match the trend of experimental data in OMEGA implosions, a small amount of mode-2 perturbation, i.e., 2% velocity perturbation at the beginning of the deceleration phase, is added together with a mode-1 simulation. The resulting red curve gives a nonvanishing $Ti$ asymmetry at small velocities. Figure 1(b) shows the hot-spot flow velocity vectors plotted over the mass density profile for a *DEC*3*D* modes $\u2113=1$ to 2 simulation for the left-most data in red in Fig. 1(a). The jet, anisotropic flow, induced by mode 1 flows vertically downward, whereas the vorticity induced by mode 2 elongates the hot–spot shape horizontally. Mode 2 contains two components of hot-spot RKE, which are parallel $RKEHS,||$ and perpendicular $RKEHS,\u22a5$ to the axis of the rotational symmetry, respectively. Since hydrodynamics along the axis of the rotational symmetry is 3D and 2D along the perpendicular direction, the nonzero difference between $RKEHS,||$ and $RKEHS,\u22a5$ produces the $Ti$ asymmetry for mode 2. At the same time, hot-spot flows of mode 2 are quasi-isotropic since they do not shift the mean neutron energies to be measurable from the first moment of neutron velocity spectra.

At large flow velocities $\u226550\u2009km/s$, the $Ti$ asymmetry is dominated by mode 1 in Fig. 1(a). The diagonal form of mode-1 apparent ion temperatures in Eq. (6) can be simplified by omitting eigenvalues *λ _{x}* and

*λ*, which are perpendicular to the effective flow eigenvector direction $|eeff\u27e9$ or $|ez\u27e9$ in rotated Cartesian coordinates. At large $Ti$ asymmetries, apparent ion temperatures are described by an approximate mode-1 model

_{y}The isotropic eigenvalue $\lambda iso=Min[{\lambda i}]$ is a result of the approximation $\lambda x\u2248\lambda y\u226a\lambda z$, whereas the anisotropic eigenvalue is $\lambda anisoeff=\lambda eff\u2212\lambda iso$. The dot product $\u27e8d|eeff\u27e9=cos\u2009\theta LOS\u2212eff$ gives a cosine variation for angles $\theta LOS\u2212eff$ measured between the LOS and the effective flow axis. This phenomenon is studied by fitting apparent ion temperatures along the effective flow axis using the mode-1 ion-temperature model

The dimensionless variable $T\u0302=(TLOS\u2212Tminfit)/\delta Tfit$ is plotted against the angle $\theta LOS\u2212eff$ in Fig. 1(c). As explained in Eq. (7), the term $Tminfit=Tth+MDT\lambda iso$ fits the minimum apparent ion temperature, whereas the term $\delta Tfit=MDT\lambda anisoeff=Tmax\u2212Tmin$ fits the difference between the maximum and the minimum apparent ion temperatures. A good agreement with experimental data is observed, indicating the presence of mode 1 in OMEGA cryogenic implosion experiments.

#### 2. Mode $\u2113=2Ti$ *asymmetries*

When a hot spot is strongly perturbed by mode 2 with the condition $\lambda \u22a5\u226b\lambda ||$, where $\lambda \u22a5\u2261\lambda x\u2248\lambda y$ and $\lambda ||\u2261\lambda z$, Eq. (6) can be simplified into

where $\theta LOS\u2212eff$ is the angle between the LOS and the effective flow unit vector along the axis of rotational symmetry for mode 2. In this special case, $Ti$ asymmetries for mode 2 satisfy a sine-square variation along the axis of rotational symmetry.

### B. Areal-density asymmetries

Since it has been suggested that OMEGA implosions are dominated by mode 1 as indicated in Fig. 1, asymmetries in areal-density measurements are expected to satisfy the semi-analytic mode-1 *ρR* model,^{8}

where the cosine term is the dot product $\u27e8d|v\u27e9=cos\u2009\theta LOS\u2212flow$ between the LOS and the hot-spot flow unit vectors. This empirical *ρR* model is derived from *DEC*3*D* mode-1 simulations. As discussed in Ref. 8, $Ti$ and *ρR* asymmetries are related by constructing the arithmetic-mean (AM) and harmonic-mean (HM) areal densities. By substituting the *ρR* model from Eq. (10) into the definitions for $\u27e8\rho R\u27e9AM=\u222b0\pi (\rho R)LOSd\theta /\pi $ and $\u27e8\rho R\u27e9HM=1/(\u222b0\pi (\rho R)LOS\u22121d\theta /\pi )$, the AM and HM areal densities $\u27e8\rho R\u27e9AM=(\rho R)0$ and $\u27e8\rho R\u27e9HM=(\rho R)02\u2212[\delta (\rho R)]2$ are obtained, respectively, so that the areal-density variation in Eq. (10) is

In $DEC3D$ deceleration-phase simulations for mode-1, *ρR* and $Ti$ asymmetries are related^{8} as follows:

where $RT=Tmax/Tmin$ is the ion-temperature ratio. The two parameters are obtained from simulations leading to $\alpha =\u22120.3$ and $\beta =\u22120.47$, respectively. Therefore, the degradation of areal-density measurements induced by mode 1 along a given LOS is a unique function of the $Ti$ asymmetry and a cosine variation

To demonstrate this analysis, the 1D simulated areal density is decomposed into a product of 1D simulated variables^{14,15} to account for systematic modeling errors. As reported by Ref. 15, components *X _{i}* of the 1D kernel $K1D=a0\u220fi=1NXiai$ are best given by the convergence ratio $CR1D$, the adiabat $\alpha 1D$, the in-flight aspect ratio $IFAR1D$, and the beam-to-target-radii ratio $Rb/Rt$. Three

*ρR*measurements

^{26}are currently available in OMEGA including one magnetic-recoil-spectrometer (MRS) at the port P10 and two neutron time-of-flight (nTOF) detectors at ports P7 and H10. The nTOF infers

*ρR*from back-scattered neutrons with energies 3.3–4 MeV for H10 and 3.5–4 MeV for P7, while the MRS infers

*ρR*from forward-scattered neutrons with energies 9–11 MeV. The H10 areal densities

^{27}are fitted with the following mode-1

*ρR*model:

The 3D kernel $K3D=RT\alpha +(RT2\alpha \u2212RT2\beta )1/2\xb7\u2009cos\u2009\theta LOS\u2212flow$ is the sum of all terms on the right-hand side of Eq. (13). As shown in Fig. 2, the effect of the 3D kernel improves the standard deviation of the fit by 2% for the *ρR*-over-clean. H10 areal densities^{27} are inferred from a six-parameter fit of a neutron spectrum model that is benchmarked with *IRIS*3*D* and are found to be more sensitive to the angular dependence of mode-1 *ρR* variations when compared with P7 and MRS areal densities.

## III. THREE-DIMENSIONAL HOT-SPOT X-RAY EMISSION TOMOGRAPHY

### A. Gradient-descent optimization

In this section, we combine the nuclear measurements with the 3D hot-spot shape asymmetry measured from x-ray diagnostics. The spectral radiation intensity $I\nu $, in units of $J/s/m2/Sr/Hz$, arrived at a detector is described by the radiation transfer equation^{28–30}

where $\kappa \nu $ is the opacity, in units of $1/m$, *s* is the distance traveled by the radiant energy, and *ν* is the photon frequency. X-rays emitted by an optically thin hot spot with electron temperatures $\u2265$5 keV travel straight to the detector. The radiation intensity is attenuated slightly because of the absorption of photons within the cold dense shell. The absorption term $\u2212\kappa \nu I\nu $ in Eq. (15) is neglected. This assumption implies that each pixel $I\nu ,LOS(x,y)=\u222bPLOS\epsilon \nu ds$ of an x-ray image measured at a given LOS is the sum of all photons emitted by the plasma within the 3D hot spot along a straight-line path $PLOS$. This ray-trace process is described by a straight-line projection operator $P\u0302LOS$ symbolically as follows:

In the ray-trace step, projected images as described by Eq. (16) are spiky because the direct projections of 3D source points $\epsilon \nu (x,y,z)$ do not align themselves uniformly on a 2D detector plane along a given LOS. The conservative upwind scheme^{31} is applied to reallocate the weight of a projected point to four nearest cells on the detector plane. This treatment produces smooth projected images effectively.

The reconstruction for an arbitrary 3D hot-spot plasma emissivity $\epsilon \nu $ is treated by a complete set expansion. Both non-orthogonal and orthogonal polynomial expansions are considered. Expansion coefficients in the 3D plasma emissivity model are determined by a gradient-descent optimization algorithm.^{32} The model performance is monitored by a loss function $L$, which is the fit error between projections and normalized x-ray images,

Here, *N* is the total number of x-ray images and $P\u2192$ is a multi-valued parameter vector that contains all expansion coefficients. The average of root-mean-square (rms) deviation for each projection is chosen as the loss function to monitor the modal performance. Since all images are normalized with respect to their peak intensities, the rms percentage reported in this work refers to the residual magnitude relative to the unity for each image. The parameter vector is updated by a differential evolution equation

so that $P\u2192$ is pointed toward the steepest descent direction according to a given learning rate *ε*, which has the same units of $dP/dt$, at each time step *t*. The gradient vector $\u2207\u2192L(P\u2192)=\u2211i=1M\u2202L/\u2202Pie\u0302i$ is the sum of all linear responses for every component of the parameter vector. Here, $e\u0302i$ is the orthonormal unit vector and *M* is the total number of fitting parameters. The *i*th linear response

is calculated by perturbing the parameter vector by a differential change $dP\u2192i$, which contains zero entities except the *i*th component $dPi>0$. The optimized parameter vector for Eq. (18) is

A large initial learning rate $\epsilon =0.2$ is used to speed up the optimization. *ε* is reduced by half sequentially whenever the condition $L(t+\Delta t)>L(t)$ occurs. The decreasing learning rate implies that the optimized parameter vector approaches the neighborhood of a local minimum $Lminlocal$ of the loss function, where the curvature of $L$ varies slowly in the multi-parameter space.

### B. Three-dimensional hot-spot plasma emissivity model

The signature skewness in measured x-ray images implies mode 1 because the spatially asymmetric electron temperature profile results in an uneven emission along the flow direction. Figure 3 shows the similarity of lineouts along the direction of the maximum skewness in a measured x-ray image for the shot 82 723 and a *DEC*3*D* mode-1 synthetic x-ray image, produced by the post-processing code *SPECT*3*D*.^{34} This shot has a large $Ti$ asymmetry with $Tmax/Tmin=1.81$. As shown by the red dashed curve in Fig. 3(a), the skewed x-ray image can be fitted better by a skewed super-Gaussian (SG) model

than an elliptical super-Gaussian model, which has zero skewness parameters *S _{x}* and

*S*in Eq. (21). Here, $\Delta x=x\u2212x0,\u2009\Delta y=y\u2212y0,\u2009R=AB$ is the hot-spot radius,

_{y}*C*is a coefficient that controls the rotation of the ellipse centered at (

*x*

_{0},

*y*

_{0}), and

*η*is the SG exponent.

The distribution of kurtosis in measured x-ray images reflects the degree of compression of the hot spot by Rayleigh–Taylor (RT) spikes. Figure 4(a) shows the kurtosis at different directions for a *DEC*3*D* mode-2 synthetic x-ray image. The maximum kurtosis $Kmax$ occurs along the lineout of the pair of converging RT spikes for this oblate mode-2 configuration, whereas the minimum kurtosis $Kmin$ occurs along the lineout on the equator, where the RT bubble of mode 2 expands radially outward. Figure 4(b) shows a *DEC*3*D* mode-10 synthetic x-ray image. In the nonlinear stage of Rayleigh–Taylor instability growth, the RT spikes along the north and the south poles merge with the neighboring spikes, producing a stretched hexagon hot-spot shape. The saddle point $Rs\u224812\u2009\mu m$, at which the curvature of the lineout changes sign as indicated by two vertical dashed lines in Fig. 4(c), is shown close to the radius of the clean hot spot surrounded by converging RT spikes in Fig. 4(d).

The x-rays emitted by hot spots during the compression phase are dominated by free–free transitions^{29} of electrons at high temperatures. In this regime, electron temperatures $Te$ can be unfolded from measuring x-ray images subject to different spectral filters, i.e., taking a derivative^{35,36} of the free–free plasma emissivity $\epsilon \nu $ with respect to the photon energy $E\nu $,

Equation (22) is valid for thermalized electrons satisfying Maxwell–Boltzmann velocity distribution,^{29} which is valid for hot spots near stagnation. Figure 4(e) shows the 2D spatial profile of $Te$, a slice cutting through the center of the hot spot perturbed by mode 10 in a *DEC*3*D* simulation. A ring caused by the low-temperature cold bubbles is shown to enclose the high-temperature hot spot. The Kirkpatrick–Baez (KB) x-ray microscope^{33} in OMEGA produces four x-ray images, each separately filtered to allow imaging in different selected photon energy ranges, named by channels A, B, C, and D, respectively. The sensitivity energy range is about $2\u20138$ keV. Two synthetic x-ray images, generated from *SPECT*3*D*^{34} with KB spectral filters C and B, respectively, are shown in Figs. 4(h) and 4(i). For convenience, filters C and B are labeled in the equations as 1 and 2, respectively. The mean photon energy allowed by the filters are $E1=4.7$ keV and $E2=5.8$ keV, respectively. The hot-spot electron temperature is unfolded from Eq. (22) using the finite differencing, followed by unfolding the electron number density $ne$ from the plasma emissivity^{29}

where $IE1(x,y)$ and $IE2(x,y)$ are spectral filtered x-ray images, with transmission coefficients *T*_{1} and *T*_{2}, respectively. Unfolded $Te$ and $ne$ are shown in Figs. 4(f) and 4(g). The ring of cold bubble and spike structures is shown to be recovered from the unfolded $Te$. A regular pattern of spots is shown, reflecting the effective positions of RT spikes of mode 10 stacked up along the *x* axis. Equation (23) can be used to reconstruct the 3D hot-spot electron temperature and densities profiles once the 3D hot-spot plasma emissivity $\epsilon \nu $ is reconstructed.

To reconstruct complex hot-spot shapes, a generalized spherical-harmonic Gaussian function is derived. The 3D plasma emissivity profile $\epsilon \nu (r,\theta ,\varphi )$ is first transformed into a unique exponential function

The origin of the polar coordinate system is chosen as the peak emission position. The logarithm of the emissivity gives the information of a radial lineout profile $f(r,\theta ,\varphi )$ along given polar angles *θ* and $\varphi $, which is decomposed into a power series in terms of the contour radius $r(\theta ,\varphi )$,

Expansion coefficients *σ _{n}* are determined by the optimization algorithm as described in Sec. III A. The power series reproduces the SG model in Eq. (21) for 1D implosions because the expansion coefficient

*σ*

_{4}captures the 1D SG exponent $\eta \u223c4$ while

*σ*

_{0}captures the peak intensity

*I*

_{0}. The information of a varying contour radius $r(\theta ,\varphi )$ is decomposed into real spherical harmonics

where *R* is the radius. The rotation of a given $\u2113$ mode is obtained by the summation over all azimuthal modes $\u2212\u2113\u2264m\u2264\u2113$, in which expansion coefficients $A\u2113m$ are stated by Wigner D-matrix^{37} to specify the rotation. The varying mode amplitudes with contour levels suggest that the mode amplitude $A\u2113m$ is a function of the radius. A second power series is devised to store this information

Coefficients $A\u2113m1$ and $A\u2113m2$ are the linear and parabolic corrections to each mode amplitude $A\u2113m0$. The generalized spherical-harmonic Gaussian function that reconstructs complex hot-spot shapes is

Equation (28) captures the behavior of varying mode amplitudes with contour levels and gives 3D hot-spot shapes, which are similar to synthetic 3D hot spots as simulated by *DEC*3*D*.

The performance of the SHG model is compared with the eigenmode decomposition^{38,39} (EMD) using eigenfunctions $\Psi \u2113mk(r,\theta ,\varphi )$ from isotropic quantum harmonic oscillators

where $Lk(\alpha )(x)$ is the generalized Laguerre polynomial^{39} with $\alpha =\u2113+1/2$, and $Hn(x)$ is Hermite polynomial^{39} with a non-negative mode number *n*. The width *σ* of the Gaussian weight function defines the normalized coordinates $r\u0302=r/\sigma ,\u2009x\u0302=x/\sigma ,\u2009y\u0302=y/\sigma $, and $z\u0302=z/\sigma $ such that $r\u0302=x\u03022+y\u03022+z\u03022$. The first line in Eq. (29) is the complete set expansion in the polar form, whereas the second line is in Cartesian form. Both mode amplitudes $A\u2113mk$ and the Gaussian width *σ* are determined by the algorithm in Sec. III A.

Eigenfunctions in Eq. (29) are given by the multiplication of a 3D kernel $K3D(r,\theta ,\varphi )$ with a 1D normal Gaussian function

This leads to the capture of horizontal and vertical noises, e.g., mirror scatterings in measured x-ray images, when Cartesian eigenfunctions are used. The artifact of negative emission can be caused by the oscillating radial kernel varying from negative to positive values. In contrast, the 3D kernel of the SHG model in Eq. (26) is encoded into an exponential function, implying that the capture of background noises decays from the center to the radially outward direction. As shown in Fig. 5(a), the KB image for the shot 77 068 with a skewed hot spot is reconstructed by the SHG model, resulting in a smooth variation of contours from the peak emission center to the radially outward direction.

Eigenfunctions in Hermite–Gaussian or Laguerre–Gaussian forms are solutions^{40} to the paraxial wave equation, waveguide, and quantum harmonic oscillator problems but are not solutions to surface waves in spherical geometries that prevail in ICF hydrodynamic instabilities. The similarity of SHG modal images with x-ray images for 3D hot spots perturbed by single-mode hydrodynamic instabilities facilitates identifying the dominant mode. The single-mode $\u2113=1$ to 6 hot-spot shapes for synthetic x-ray image simulated by *DEC*3*D* are shown in Fig. 5(b). One can easily validate that the single-mode images produced by the SHG model in Eq. (28) resemble the synthetic x-ray images, simulated for single-mode deceleration-phase RT instabilities. The gradual change of mode amplitudes is shown by black contour curves, supporting the argument to introduce the second power series expansion in Eq. (27) to capture this behavior.

A benchmark test is presented in Fig. 5(c). Three synthetic x-ray images at the bottom row are measured at three orthogonal LOS's for a *DEC*3*D* mode-1 simulation. The SHG model in Eq. (28) is optimized by matching modal projections with all synthetic x-ray images. The optimization takes about 20 time steps to reduce the loss function to an acceptable level. The modal performance with rms 3.18%, 3.19%, and 5.23% for modal projections at *x*, *y*, and *z* LOS's, respectively, are shown on the upper row. The resulting mode-1 hot-spot shape reconstructed by SHG model is shown on the right-hand side in Fig. 5(c). The 2D slice bounded by a red frame, cutting through the center of the hot spot and normal to the *x* axis, shows a clear signature of skewness reflecting a mode-1 hot-spot reconstruction.

#### 1. Hot-spot reconstruction

The 3D emission model is used to reconstruct the hot spot for shots 94 017 and 96 806 on OMEGA. Four current x-ray diagnostics are used: the KB, the single line-of-sight time-resolved x-ray imager (SLOS-TRXI), the spatially resolved electron temperature (SRTE), and the time-resolved KB-framed x-ray imagers. As shown in Fig. 6, the slice cut through the hot-spot center is indicated by a red frame to reflect interior contours along to the red arrow direction. Measured x-ray images and modal projections at different LOS's are shown on the first and second rows, respectively. The optimization begins with a 3D normal Gaussian function defined by setting all expansion coefficients at zero except *σ*_{2}, as explained in Sec. III A. A Cartesian mesh with 40 pixels along each axis is used. Under the resolution of 2 *μ*m per pixel, the loss function starts flattening after 40 time steps in the steepest descent optimization. Reasonable mappings defined by the rms between measured x-ray images and projections are obtained. Shot 94 017 has an rms of 5.84% and 4.73% for KB and TRXI images, respectively. Shot 96 806 has an rms of 6.89%, 6.62%, and 3.05% for KB, TRXI, and SRTE, respectively. A skewed 3D hot spot is reconstructed for shot 94 017, which is consistent with the large measured ion-temperature ratio $Tmax/Tmin=1.78$ caused by a large mode-1 perturbation with an amplitude of 7% from the beam mispointing. An elliptical 3D hot spot is reconstructed for shot 96 806, which is consistent with the large measured ellipticity in x-ray images as a result of mode 2. Among various mode-2 sources on OMEGA, the beam-to-beam laser power imbalance contributes mode-2 perturbations with a systematic orientation over shots.

Figure 7(a) shows the effective flow reconstruction as prescribed in Sec. II A. The five measured ion temperatures are shown to be fitted well by a cosine-square curve as given by the mode-1 $Ti$ model in Eq. (8). The opposite direction of the effective flow $(\theta ,\varphi )eff=(99\xb0,\u2009292\xb0)$ is $(\theta ,\varphi )effopp=(81\xb0,\u2009112\xb0)$. The projection of the mode-1 hot-spot plasma emissivity for shot 94 017 produces rotated ellipses on KB and TRXI x-ray images. As explained in the Appendix, knowing rotation angles from the projection of a 3D prolate hot spot at two LOS's can be used to solve for the direction of the major axis. By substituting rotation angles $\xi KB=\u221241\xb0$ and $\xi TRXI=42\xb0$ in Eq. (A8), the major axis for shot 94 017 is $(\theta ,\varphi )ma=(69\xb0\xb13\xb0,\u2009107\xb0\xb14\xb0)$, which is reasonably close to the opposite direction of the effective flow. The reconstructed major axis $(\theta ,\varphi )ma$, indicated by a red arrow in Fig. 7(b), is consistent with the orientation of the skewed hot spot.

## IV. CONCLUSION

In summary, a technique for 3D hot-spot reconstruction from experimental observables has been developed. Through diagonalizing the velocity-variance matrix in apparent ion temperatures, the behavior of apparent ion temperature asymmetries was quantized into effective flows which include hydrodynamic effects of both isotropic and anisotropic flows. For effective flows in the mode-1 configuration, the resulting mode-1 $Ti$ model can be used to explain the observed $Ti$ asymmetries in OMEGA implosions. This result implies the presence of mode 1 in OMEGA direct-drive implosions. A semi-analytic mode-1 *ρR* model was derived to reconstruct *ρR* asymmetries in terms of a cosine variation along the hot-spot flow axis and a unique function of $Ti$ asymmetries. The resulting 3D kernel was shown to improve the *ρR*-over-clean prediction for measured areal densities on OMEGA. A 3D hot-spot x-ray emission tomography method was developed to reconstruct the 3D hot-spot plasma emissivity using a generalized spherical-harmonic Gaussian function. The 3D data analysis for $Ti$, flows, *ρR*, and 3D hot-spot shape asymmetries provides a satisfactory agreement with experimental signatures caused by mode-1 implosion asymmetries.

As the next step, the 3D hot-spot reconstruction technique will be extended to neutron images. The proof-of-principle tests will be conducted using synthetic neutron images generated by *DEC*3*D* hydrodynamic data, followed by applications to knock-on deuteron images in OMEGA or neutron images in NIF.

## ACKNOWLEDGMENTS

This material is based upon work 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 and DOE Grant No. DE-SC0022132. The support of DOE does not constitute an endorsement by DOE of the views expressed in this paper. 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

**Ka Ming Woo:** Conceptualization (equal), Data curation (equal), Formal analysis (equal), Investigation (equal), Methodology (equal), Resources (equal), Validation (equal), Visualization (equal), Writing – original draft (equal), and Writing – review and editing (equal). **Timothy J. B. Collins:** Supervision (equal). **Wolfgang Theobald:** Supervision (equal). **Rahul Shah:** Validation (equal). **Owen Michael Mannion:** Conceptualization (supporting). **Dhrumir Patel:** Conceptualization (supporting). **Duc Cao:** Data curation (equal). **James P. Knauer:** Supervision (equal). **Vladimir Yu. Glebov:** Supervision (equal). **Valeri N. Goncharov:** Supervision (equal). **P.B. Radha:** Supervision (equal). **Riccardo Betti:** Supervision (equal). **Hans George Rinderknecht:** Supervision (equal). **Reuben Epstein:** Supervision (equal). **Varchas Gopalaswamy:** Resources (supporting). **Frederic J. Marshall:** Resources (supporting). **Steven Ivancic:** Resources (supporting). **E. Michael Campbell:** Supervision (equal). **Cliff A. Thomas:** Supervision (equal). **Christian Stoeckl:** Supervision (equal). **Kristen Churnetski:** Data curation (equal). **Chad James Forrest:** Data curation (equal). **Zaarah Lyla Mohamed:** Data curation (equal). **B. Zirps:** Data curation (equal). **Sean Patrick Regan:** Supervision (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX: RECONSTRUCTION OF MAJOR AXIS

When the unit vector of the major axis of a prolate elliptical hot spot given by $v\u0302=sin\u2009\theta \u2009cos\u2009\varphi x\u0302+sin\u2009\theta \u2009sin\u2009\varphi y\u0302+cos\u2009\theta z\u0302$ is projected onto a 2D detector plane along a given LOS, described by a unit vector $d\u0302$, its shadow forms a projection vector $p\u2192$ (see Fig. 8). The unit vector of the projection $p\u0302$ and the LOS unit vector $d\u0302$ is orthonormal so that the decomposition of $v\u0302$ is

The horizontal and vertical axes of the detector plane are parallel to $e\u0302\varphi $ and $\u2212e\u0302\theta $, respectively, whereas the normal of the detector plane, i.e., $d\u0302$, is parallel to $e\u0302r$. As shown in Fig. 8, the polar coordinates are constrained by the cross product $e\u0302\theta \xd7e\u0302\varphi =e\u0302r$, with the center located at the target chamber center. Horizontal and vertical components of the vector $p\u2192$ are related to the major axis vector $v\u2192$ through

and

respectively. Equations (A2) and (A3) contain two unknown angles *θ* and $\varphi $ that specify the major axis unit vector $v\u0302$ uniquely. The projection $p\u2192=(v\u0302\xb7p\u0302)p\u0302$ is obtained from Eq. (A1) such that

and

where both orthonormal unit vectors $e\u0302\theta $ and $e\u0302\varphi $ are known from $d\u0302$. The projection unit vector $p\u0302$ is known from measuring the rotation angle of the rotated hot-spot shape in the measured x-ray image on the detector plane. In this approach, the reconstruction for the unit vector of the major axis $v\u0302(\theta ,\varphi )$ is valid for the skewed 3D hot spots distorted by mode 1 and the prolate 3D hot spots distorted by mode 2. By expressing $e\u0302\varphi =\u2211i=13\beta ix\u0302i$ and $p\u0302=\u2211i=13pix\u0302i$ in Cartesian coordinates, where *β _{i}* and

*p*are projection coefficients for orthonormal vectors $x\u03021=x\u0302,\u2009x\u03022=y\u0302$, and $x\u03023=z\u0302$, Eq. (A4) becomes

_{i}By dividing both sides of Eq. (A6) by a factor $cos\u2009\theta $ and introducing two new variables, $X(\theta ,\varphi )\u2261\u2009tan\u2009\theta \u2009cos\u2009\varphi $, and $Y(\theta ,\varphi )\u2261\u2009tan\u2009\theta \u2009sin\u2009\varphi $, Eq. (A6) becomes

where the projection coefficients *p _{i}*,

*β*and the dot product $D\u2261p\u0302\xb7e\u0302\varphi $ are known for a given LOS. As shown in Fig. 8, the dot product $D=cos\u2009\xi $ equals the cosine of the rotation angle

_{i}*ξ*, measured clock-wise from the horizontal axis $e\u0302\phi $, on the detector plane. When projections of the major axis at two different LOS's are measured, Eq. (A7) is described by

where the superscripts “1” and “2” refer to $LOS1$ and $LOS2$, respectively. Hence, the angles *θ* and $\varphi $ for the unit vector of the major axis can be inverted by solving $X(\theta ,\varphi )$ and $Y(\theta ,\varphi )$ in Eq. (A8).