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 =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.

An inertial confinement fusion (ICF) implosion1 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 DEC3D modes =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 =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.

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 moment17,22 of the neutron velocity distribution in terms of a velocity variance Var[v·d̂LOS]. Here, v is the plasma fluid velocity vector in the laboratory frame and d̂LOS 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

TLOS=Tth+MDTVar[v·d̂LOS],
(1)

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·d̂LOS] is first decomposed16 into six different components σij=(viv¯i)(vjv¯j), where indices i =1, 2, 3 stand for the x, y, z component in Cartesian coordinates, respectively. v¯i is the burn-averaged hot-spot flow velocity along the ith direction. d̂LOS=d1x̂+d2ŷ+d3ẑ is the LOS unit vector with components d1=sinθcosϕ,d2=sinθsinϕ, and d3=cosθ, where θ and ϕ 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=[d1,d2,d3]T, while the transpose (T) of the ket is denoted by a row-vector notation called a bra d|=|dT. The velocity variance along a given LOS equals the expectation value

Var[v·d̂LOS]=d|σ̂|d,
(2)

with respect to the velocity-variance matrix

σ̂=[σ11σ12σ13σ21σ22σ23σ31σ32σ33].
(3)

Because the velocity variance matrix commutes, σ̂ is Hermitian. Hence, it can be diagonalized by three orthonormal eigenvectors |ei, each associated with a real non-negative eigenvalue λi0. By inserting two identity matrices Î=i=13|eiei| into the expectation value d|ησ̂·Î|d in Eq. (2), σ̂ is diagonalized,

Var[v·d̂LOS]=i=13λid|eiei|d.
(4)

The eigenvalue λi=ei|σ̂|ei is the expectation value with respect to each eigenvector. The inner product d|ei 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=x̂,|e2=ŷ, and |e3=ẑ. In the rotated Cartesian coordinates, the inner products are d|e1=sinθcosϕ,d|e2=sinθsinϕ, and d|e3=cosθ, respectively. Both polar and azimuthal angles θ and ϕ are measured in the rotated Cartesian coordinates, respectively.

Since diagonal elements σii=(viv¯i)2 are the square of hot-spot flow velocity fluctuations, an eigenvalue λi measures the nontranslational part of the hot-spot residual kinetic energy RKEHS,inontrans=MHSσii/2 along the ith eigenvector direction, where MHS is the burn-averaged hot-spot mass. The trace of σ̂ gives the total nontranslational hot-spot residual kinetic energies

Tr[σ̂]=i=13λi=2MHSRKEHS,totalnontrans,
(5)

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

TLOS=Tth+MDTi=13λid|ei2,
(6)

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[λ1,λ2,λ3]MHS/2, which broadens the widths of neutron velocity spectra isotropically in all LOS's. Turbulence19 and radially symmetric flows20,21 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[κ] in the second moment of neutron velocity distributions, which is negligible when compared with ion thermal velocities and hot-spot flow velocities17,22,25 for typical ICF implosions with ion thermal temperatures ∼5 keV.

1. Mode =1 Tiasymmetries

Figure 1(a) shows the impact of isotropic flows on the Ti and flow relation. DEC3D 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 DEC3D modes =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, 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, 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.

FIG. 1.

(a) The difference between the maximum Tmax and the minimum Tmin ion temperatures, measured from all available LOS's, is compared with the measured hot-spot flow velocities vflow in OMEGA cryogenic implosions. (b) The hot-spot fluid velocity field v (arrows) is plotted over the 2D mass density ρ profile at stagnation for a DEC3D modes =1 to 2 simulation. (c) The difference between Texp and Tmin is plotted along the effective flow unit vector v̂floweff, which is the eigenvector direction associated with the maximum eigenvalue in the velocity-variance matrix in Eq. (3).

FIG. 1.

(a) The difference between the maximum Tmax and the minimum Tmin ion temperatures, measured from all available LOS's, is compared with the measured hot-spot flow velocities vflow in OMEGA cryogenic implosions. (b) The hot-spot fluid velocity field v (arrows) is plotted over the 2D mass density ρ profile at stagnation for a DEC3D modes =1 to 2 simulation. (c) The difference between Texp and Tmin is plotted along the effective flow unit vector v̂floweff, which is the eigenvector direction associated with the maximum eigenvalue in the velocity-variance matrix in Eq. (3).

Close modal

At large flow velocities 50km/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 λx and λy, which are perpendicular to the effective flow eigenvector direction |eeff or |ez in rotated Cartesian coordinates. At large Ti asymmetries, apparent ion temperatures are described by an approximate mode-1 model

TLOS=1=Tth+MDTλiso+MDTλanisoeffd|eeff2.
(7)

The isotropic eigenvalue λiso=Min[{λi}] is a result of the approximation λxλyλz, whereas the anisotropic eigenvalue is λanisoeff=λeffλiso. The dot product d|eeff=cosθLOSeff gives a cosine variation for angles θLOSeff 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

TLOS=Tminfit+δTfitcos2θLOSeff.
(8)

The dimensionless variable T̂=(TLOSTminfit)/δTfit is plotted against the angle θLOSeff in Fig. 1(c). As explained in Eq. (7), the term Tminfit=Tth+MDTλiso fits the minimum apparent ion temperature, whereas the term δTfit=MDTλanisoeff=TmaxTmin 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 =2 Tiasymmetries

When a hot spot is strongly perturbed by mode 2 with the condition λλ||, where λλxλy and λ||λz, Eq. (6) can be simplified into

TLOS=2=Tth+MDTλiso+MDT(λλiso)sin2θLOSeff,
(9)

where θLOSeff 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.

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 

(ρR)LOS=(ρR)0+δ(ρR)·cosθLOSflow,
(10)

where the cosine term is the dot product d|v=cosθLOSflow between the LOS and the hot-spot flow unit vectors. This empirical ρR model is derived from DEC3D 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 ρRAM=0π(ρR)LOSdθ/π and ρRHM=1/(0π(ρR)LOS1dθ/π), the AM and HM areal densities ρRAM=(ρR)0 and ρRHM=(ρR)02[δ(ρR)]2 are obtained, respectively, so that the areal-density variation in Eq. (10) is

δρR=ρRAM2ρRHM2.
(11)

In DEC3D deceleration-phase simulations for mode-1, ρR and Ti asymmetries are related8 as follows:

ρRAM=(ρR)1D·RTα,ρRHM=(ρR)1D·RTβ,
(12)

where RT=Tmax/Tmin is the ion-temperature ratio. The two parameters are obtained from simulations leading to α=0.3 and β=0.47, 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

(ρR)LOS(ρR)1D=RTα+RT2αRT2β·cosθLOSflow.
(13)

To demonstrate this analysis, the 1D simulated areal density is decomposed into a product of 1D simulated variables14,15 to account for systematic modeling errors. As reported by Ref. 15, components Xi of the 1D kernel K1D=a0i=1NXiai are best given by the convergence ratio CR1D, the adiabat α1D, the in-flight aspect ratio IFAR1D, and the beam-to-target-radii ratio Rb/Rt. Three ρR measurements26 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 densities27 are fitted with the following mode-1 ρR model:

(ρR)LOS(ρR)1D=K1D·K3D.
(14)

The 3D kernel K3D=RTα+(RT2αRT2β)1/2·cosθLOSflow 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 densities27 are inferred from a six-parameter fit of a neutron spectrum model that is benchmarked with IRIS3D and are found to be more sensitive to the angular dependence of mode-1 ρR variations when compared with P7 and MRS areal densities.

FIG. 2.

A comparison for the ρR-over-clean prediction with and without the 3D kernel in Eq. (14). The ρR measured at port H10 on OMEGA is sensitive to 3D variations induced by mode 1.

FIG. 2.

A comparison for the ρR-over-clean prediction with and without the 3D kernel in Eq. (14). The ρR measured at port H10 on OMEGA is sensitive to 3D variations induced by mode 1.

Close modal

In this section, we combine the nuclear measurements with the 3D hot-spot shape asymmetry measured from x-ray diagnostics. The spectral radiation intensity Iν, in units of J/s/m2/Sr/Hz, arrived at a detector is described by the radiation transfer equation28–30 

dIνds=ενκνIν,
(15)

where κν 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 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 κνIν in Eq. (15) is neglected. This assumption implies that each pixel Iν,LOS(x,y)=PLOSενds 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 P̂LOS symbolically as follows:

Iν,LOS=P̂LOS·εν.
(16)

In the ray-trace step, projected images as described by Eq. (16) are spiky because the direct projections of 3D source points εν(x,y,z) do not align themselves uniformly on a 2D detector plane along a given LOS. The conservative upwind scheme31 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 εν 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,

L(P)=1Ni=1N(P̂LOS·ενIν,LOSiexp)2.
(17)

Here, N is the total number of x-ray images and P 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

ddtL(P)=L(P)·dPdt,
(18)

so that P 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 L(P)=i=1ML/Piêi is the sum of all linear responses for every component of the parameter vector. Here, êi is the orthonormal unit vector and M is the total number of fitting parameters. The ith linear response

LPi=L(P+dPi)L(P)dPi
(19)

is calculated by perturbing the parameter vector by a differential change dPi, which contains zero entities except the ith component dPi>0. The optimized parameter vector for Eq. (18) is

P(t+Δt)=P(t)εL(t)|L(t)|Δt.
(20)

A large initial learning rate ε=0.2 is used to speed up the optimization. ε is reduced by half sequentially whenever the condition L(t+Δt)>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.

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 DEC3D mode-1 synthetic x-ray image, produced by the post-processing code SPECT3D.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

I(x,y)=I0·[1+erf(SxΔx/A)]·[1+erf(SyΔy/A)]×e{[(ΔxA)2+C·ΔxΔy+(ΔyB)2]η/2},
(21)

than an elliptical super-Gaussian model, which has zero skewness parameters Sx and Sy in Eq. (21). Here, Δx=xx0,Δy=yy0,R=AB is the hot-spot radius, C is a coefficient that controls the rotation of the ellipse centered at (x0, y0), and η is the SG exponent.

FIG. 3.

(a) The measured x-ray image for shot 82 723 using the KB x-ray microscope. The ion-temperature ratio is Tmax/Tmin=1.81. The dashed white line is the 17% contour. The lineout along the dashed blue line is compared with the skewed super-Gaussian fit. (b) For comparison, (b) is the synthetic x-ray image and the vertical lineout for a DEC3D mode-1 simulation.

FIG. 3.

(a) The measured x-ray image for shot 82 723 using the KB x-ray microscope. The ion-temperature ratio is Tmax/Tmin=1.81. The dashed white line is the 17% contour. The lineout along the dashed blue line is compared with the skewed super-Gaussian fit. (b) For comparison, (b) is the synthetic x-ray image and the vertical lineout for a DEC3D mode-1 simulation.

Close modal

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 DEC3D 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 DEC3D 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 Rs12μm, 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).

FIG. 4.

(a) and (b) The synthetic x-ray images for DEC3D mode =2 and =10, m =5, are compared. (c) The lineout along the vertical dashed blue line in (b) is compared with the elliptical super-Gaussian fit, given by the red dashed curve. The saddle point radius Rs is about 12 μm. (d) The corresponding mode-10 mass density profile ρ. Rtp from (c) is close to the clean hot-spot radius. (e) The 2D electron temperature profile Te for a DEC3D mode =10,m=5 simulation. The image is a slice cutting through the hot-spot center and normal to the x axis. The instantaneous neutron-averaged Te is 3.2 keV. (f) The intensity-averaged unfolded electron temperature is 2.96 keV using Eq. (23), which is close to the neutron-averaged value in (e). (g) Trace of RT spikes is shown in the unfolded electron number density profile ne. (h) and (i) Two synthetic x-ray images, generated using KB-C and KB-B spectral filters, are used to unfold Te and ne.

FIG. 4.

(a) and (b) The synthetic x-ray images for DEC3D mode =2 and =10, m =5, are compared. (c) The lineout along the vertical dashed blue line in (b) is compared with the elliptical super-Gaussian fit, given by the red dashed curve. The saddle point radius Rs is about 12 μm. (d) The corresponding mode-10 mass density profile ρ. Rtp from (c) is close to the clean hot-spot radius. (e) The 2D electron temperature profile Te for a DEC3D mode =10,m=5 simulation. The image is a slice cutting through the hot-spot center and normal to the x axis. The instantaneous neutron-averaged Te is 3.2 keV. (f) The intensity-averaged unfolded electron temperature is 2.96 keV using Eq. (23), which is close to the neutron-averaged value in (e). (g) Trace of RT spikes is shown in the unfolded electron number density profile ne. (h) and (i) Two synthetic x-ray images, generated using KB-C and KB-B spectral filters, are used to unfold Te and ne.

Close modal

The x-rays emitted by hot spots during the compression phase are dominated by free–free transitions29 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 derivative35,36 of the free–free plasma emissivity εν with respect to the photon energy Eν,

Te=εν(ενEν)1.
(22)

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 DEC3D 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 microscope33 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 28 keV. Two synthetic x-ray images, generated from SPECT3D34 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 emissivity29 

Te(x,y)IE1(x,y)/TE1×(E2E1)IE2(x,y)/TE2IE1(x,y)/TE1,
(23)

where IE1(x,y) and IE2(x,y) are spectral filtered x-ray images, with transmission coefficients T1 and T2, 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 εν is reconstructed.

To reconstruct complex hot-spot shapes, a generalized spherical-harmonic Gaussian function is derived. The 3D plasma emissivity profile εν(r,θ,ϕ) is first transformed into a unique exponential function

εν(r,θ,ϕ)=ef(r,θ,ϕ).
(24)

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,θ,ϕ) along given polar angles θ and ϕ, which is decomposed into a power series in terms of the contour radius r(θ,ϕ),

lnεν(r,θ,ϕ)=f(r,θ,ϕ)=n=0σnr(θ,ϕ)n.
(25)

Expansion coefficients σn 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 σ4 captures the 1D SG exponent η4 while σ0 captures the peak intensity I0. The information of a varying contour radius r(θ,ϕ) is decomposed into real spherical harmonics

r(θ,ϕ)=R(1+=1m=AmYm(θ,ϕ)),
(26)

where R is the radius. The rotation of a given mode is obtained by the summation over all azimuthal modes m, in which expansion coefficients Am are stated by Wigner D-matrix37 to specify the rotation. The varying mode amplitudes with contour levels suggest that the mode amplitude Am is a function of the radius. A second power series is devised to store this information

Am=k=0AmkRk.
(27)

Coefficients Am1 and Am2 are the linear and parabolic corrections to each mode amplitude Am0. The generalized spherical-harmonic Gaussian function that reconstructs complex hot-spot shapes is

lnεν(r,θ,ϕ)=n=0σnRn(1+=1m=k=0AmkRkYm(θ,ϕ))n.
(28)

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 DEC3D.

The performance of the SHG model is compared with the eigenmode decomposition38,39 (EMD) using eigenfunctions Ψmk(r,θ,ϕ) from isotropic quantum harmonic oscillators

εν==0m=k=0Amk·r̂Lk(+1/2)(r̂)Ym(θ,ϕ)·er̂2/2,==0m=0k=0AmkH(x̂)Hm(ŷ)Hk(ẑ)·er̂2/2,
(29)

where Lk(α)(x) is the generalized Laguerre polynomial39 with α=+1/2, and Hn(x) is Hermite polynomial39 with a non-negative mode number n. The width σ of the Gaussian weight function defines the normalized coordinates r̂=r/σ,x̂=x/σ,ŷ=y/σ, and ẑ=z/σ such that r̂=x̂2+ŷ2+ẑ2. 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 Amk 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,θ,ϕ) with a 1D normal Gaussian function

Ψmk(r,θ,ϕ)=K3D(r,θ,ϕ)·er̂2/2.
(30)

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.

FIG. 5.

(a) The 2D image reconstruction by the SHG model, which applies a spectrum of modes =1 to 6, is compared with the measured KB-C image for shot 77 068. The rms between the measured and reconstructed image is 2.19% relative to the normalized peak intensity. The difference between the original and the reconstructed images is provided to show the residue of the fit. (b) The synthetic x-ray images for DEC3D modes =1 to 6 are compared. (c) The benchmark test for the a DEC3D=1 hot-spot reconstruction using the SHG model. The three synthetic x-ray images measured at x, y, and z LOS's are fitted with the modal projections.

FIG. 5.

(a) The 2D image reconstruction by the SHG model, which applies a spectrum of modes =1 to 6, is compared with the measured KB-C image for shot 77 068. The rms between the measured and reconstructed image is 2.19% relative to the normalized peak intensity. The difference between the original and the reconstructed images is provided to show the residue of the fit. (b) The synthetic x-ray images for DEC3D modes =1 to 6 are compared. (c) The benchmark test for the a DEC3D=1 hot-spot reconstruction using the SHG model. The three synthetic x-ray images measured at x, y, and z LOS's are fitted with the modal projections.

Close modal

Eigenfunctions in Hermite–Gaussian or Laguerre–Gaussian forms are solutions40 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 =1 to 6 hot-spot shapes for synthetic x-ray image simulated by DEC3D 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 DEC3D 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 σ2, as explained in Sec. III A. A Cartesian mesh with 40 pixels along each axis is used. Under the resolution of 2 μm 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.

FIG. 6.

Results of 3D reconstructions for OMEGA shots 94 017 and 96 806. The 3D hot spots are reconstructed by mapping modal projections with x-ray images measured at all LOS's. The slice enclosed by a red frame is used to show interior contours.

FIG. 6.

Results of 3D reconstructions for OMEGA shots 94 017 and 96 806. The 3D hot spots are reconstructed by mapping modal projections with x-ray images measured at all LOS's. The slice enclosed by a red frame is used to show interior contours.

Close modal

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 (θ,ϕ)eff=(99°,292°) is (θ,ϕ)effopp=(81°,112°). 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 ξKB=41° and ξTRXI=42° in Eq. (A8), the major axis for shot 94 017 is (θ,ϕ)ma=(69°±3°,107°±4°), which is reasonably close to the opposite direction of the effective flow. The reconstructed major axis (θ,ϕ)ma, indicated by a red arrow in Fig. 7(b), is consistent with the orientation of the skewed hot spot.

FIG. 7.

The effective flow direction (θ,ϕ)eff=(99°,292°) is reconstructed for shot 94017 using five measured ion-temperature measurements. The reconstructed major axis direction (θ,ϕ)ma=(69°±3°,107°±4°) is consistent with the orientation of the skewed 3D hot spot.

FIG. 7.

The effective flow direction (θ,ϕ)eff=(99°,292°) is reconstructed for shot 94017 using five measured ion-temperature measurements. The reconstructed major axis direction (θ,ϕ)ma=(69°±3°,107°±4°) is consistent with the orientation of the skewed 3D hot spot.

Close modal

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 DEC3D hydrodynamic data, followed by applications to knock-on deuteron images in OMEGA or neutron images in NIF.

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.

The authors have no conflicts to disclose.

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).

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

When the unit vector of the major axis of a prolate elliptical hot spot given by v̂=sinθcosϕx̂+sinθsinϕŷ+cosθẑ is projected onto a 2D detector plane along a given LOS, described by a unit vector d̂, its shadow forms a projection vector p (see Fig. 8). The unit vector of the projection p̂ and the LOS unit vector d̂ is orthonormal so that the decomposition of v̂ is

v̂=(v̂·d̂)d̂+(v̂·p̂)p̂.
(A1)
FIG. 8.

The projection of a prolate 3D hot-spot shape onto a detector plane, located at the LOS d̂ relative to an observer viewing to the target chamber center.

FIG. 8.

The projection of a prolate 3D hot-spot shape onto a detector plane, located at the LOS d̂ relative to an observer viewing to the target chamber center.

Close modal

The horizontal and vertical axes of the detector plane are parallel to êϕ and êθ, respectively, whereas the normal of the detector plane, i.e., d̂, is parallel to êr. As shown in Fig. 8, the polar coordinates are constrained by the cross product êθ×êϕ=êr, with the center located at the target chamber center. Horizontal and vertical components of the vector p are related to the major axis vector v through

p·êϕ=v̂·êϕ
(A2)

and

p·êθ=v̂·êθ,
(A3)

respectively. Equations (A2) and (A3) contain two unknown angles θ and ϕ that specify the major axis unit vector v̂ uniquely. The projection p=(v̂·p̂)p̂ is obtained from Eq. (A1) such that

(v̂·p̂)(p̂·êϕ)=v̂·êϕ,
(A4)

and

(v̂·p̂)(p̂·êθ)=v̂·êθ,
(A5)

where both orthonormal unit vectors êθ and êϕ are known from d̂. The projection unit vector p̂ 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 v̂(θ,ϕ) is valid for the skewed 3D hot spots distorted by mode 1 and the prolate 3D hot spots distorted by mode 2. By expressing êϕ=i=13βix̂i and p̂=i=13pix̂i in Cartesian coordinates, where βi and pi are projection coefficients for orthonormal vectors x̂1=x̂,x̂2=ŷ, and x̂3=ẑ, Eq. (A4) becomes

(p1sinθcosϕ+p2sinθsinϕ+p3cosθ)(p̂·êϕ)=β1sinθcosϕ+β2sinθsinϕ+β3cosθ.
(A6)

By dividing both sides of Eq. (A6) by a factor cosθ and introducing two new variables, X(θ,ϕ)tanθcosϕ, and Y(θ,ϕ)tanθsinϕ, Eq. (A6) becomes

(p1X+p2Y+p3)D=β1X+β2Y+β3,
(A7)

where the projection coefficients pi, βi and the dot product Dp̂·êϕ are known for a given LOS. As shown in Fig. 8, the dot product D=cosξ equals the cosine of the rotation angle ξ, measured clock-wise from the horizontal axis êφ, on the detector plane. When projections of the major axis at two different LOS's are measured, Eq. (A7) is described by

[p31D1β31p32D2β32]=[β11p11D1β21p21D1β12p12D2β22p22D2][XY],
(A8)

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

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