We review experimental neutron imaging of inertial confinement fusion sources, including the neutron imaging systems that have been used in our measurements at the National Ignition Facility. These systems allow measurements with 10 µm resolution for fusion deuterium–deuterium and deuterium–tritium neutron sources with mean radius up to 400 µm, including measurements of neutrons scattered to lower energy in the remaining cold fuel. These measurements are critical for understanding the fusion burn volume and the three-dimensional effects that can reduce the neutron yields.
Since Eddington first speculated that nuclear fusion of hydrogen atoms produces the energy in the sun,1 understanding and controlling fusion has remained a compelling goal. To obtain a net energy from fusion requires sufficient heating and confinement of the fuel to burn the fuel.2,3 While gravity provides the confinement in the sun, terrestrial quantities of fusion fuel require other methods such as inertial confinement4,5 and magnetic confinement.6,7
This work focuses on imaging for inertial confinement fusion (ICF), and while thermonuclear explosives use inertial confinement,4 here, we are generally discussing inertial confinement fusion using lasers, whereby a laser heats and compresses a small capsule of fusion fuel to the required temperatures and pressures, either directly through ablation and implosion of the capsule,5,8 or indirectly by heating a hohlraum and creating x rays that ablate and implode the capsule.9,10
To understand the performance of these imploded capsules, it is useful to determine the spatial distribution of the burning fuel since this can indicate how the fuel has assembled and whether three-dimensional effects, such as non-uniform compression of the fuel or fill-tube jets,11 have affected the burn volume. For ICF, the burn region is small. A recent indirect-drive experiment at the National Ignition Facility (NIF) exceeded the Lawson criterion for ignition, producing a capsule gain of 5.8 and a yield of 1.35 MJ. The mean neutron hotspot radius for that experiment was 51–59 µm, depending on the viewing angle,12–14 and the primary fusion region in other experiments can be smaller still, with a mean radius of less than 30 µm.15,16
To obtain spatial information about these small burn regions requires imaging the emitted particles, preferably using particles that could only come from the fusion burn and that can escape the region without excessive modification by the environment. While fusion plasmas emit great quantities of x rays, the x-ray emission is sensitive to the plasma composition and temperature and is altered by the opacity of the system.
The most common fusion reactions in ICF are deuterium–deuterium (DD) fusion17–19
and deuterium–tritium (DT) fusion20,21
Thus, the fusion burn region also emits neutrons, gamma rays, and charged particles such as protons and alpha particles. Any of these particles can, in theory, be used as probes of the fusion burn region. The small gamma branching ratios and complications from reactions in any surrounding materials makes imaging the burn region with gamma rays difficult, and while charged particle imaging22–25 has been used, the charged particles can interact strongly with the plasma. The neutrons, however, are excellent probes of the fusion burn region because the neutron production is largely limited to the actual burn region, and neutrons have minimal interaction with the materials of the plasma. Moreover, the neutrons that do interact typically change energy and can be separated from non-interacting neutrons by energy thresholding or gating.26,27
Neutrons, however, are challenging to image on these spatial scales. While refractive optics exist for cold neutrons,28 the index of refraction for fusion neutrons of any ordinary material under laboratory conditions is essentially one, so, lenses do not exist for fusion neutrons. Instead, fusion neutron imagers must rely on straight-line propagation and a small aperture in a barrier to form an image, i.e., on a pinhole camera for neutrons.29,30
In an ideal pinhole camera such as that in Fig. 1, with a pinhole in an infinitely thin and perfectly absorbing barrier, the geometry sets the magnification, M, of the image, which is the distance from the image plane to the aperture, l2, divided by the distance from the source plane to the aperture, l1:
For a circular aperture of diameter, d, without diffraction, one can define two points in the source plane to be resolved when the circular images cast by those points just touch in the image plane, so, the resolution for the aperture is
In this ideal system, the image will also be nearly translation invariant with respect to source location, exhibiting only a decrease in amplitude with the cosine of the angle from the axis of the aperture centered at the aperture location. We note too that the aperture need not be round. While round apertures are often useful for pinhole imaging because the resolution is the same in any direction in the images, the aperture opening can be any shape that is small relative to the resolution required. It can be triangular, square, a ring, or even a random shape. For penumbral or coded aperture imaging, which we will discuss later, the aperture may even be large relative to the source.
The challenge, however, is that real barriers for fusion neutrons are neither infinitely thin nor perfectly absorbing. The long mean free path of fusion neutrons in any aperture material means that the apertures must be thick to produce contrast, and the penetration of the neutrons through the edges of the aperture makes the system translation variant. The shape of the image depends on the location of the source relative to the aperture. Neutron detection is also difficult since the neutrons lack charge and have low interaction cross sections. The detectors must be thick enough to guarantee sufficient interaction and energy deposition to allow detection.
Since the apertures used for ICF imaging are often of the order of the source size or are penumbral apertures that are larger than the source, image reconstruction techniques are often required to provide more robust estimates of the source shape and size. While reconstruction techniques developed for thin apertures such as Fourier analysis31,32 and Weiner filtering31,32 can sometimes be used, the penetration of the neutrons through the material of the thick apertures, which creates a translation-variant point spread function, often requires reconstruction methods that can account for the translation variance.
In this work, we will review the development of fusion neutron source imaging, discuss the apertures and detectors used, and describe the current state of ICF imaging at the National Ignition Facility. We note that we will not be discussing the field of cold or thermal neutron imaging,33 or fast neutron radiography.
DEVELOPMENT OF FUSION NEUTRON IMAGING
The first fusion neutron imagers were fielded on nuclear tests in 1957 during Operation Plumbbob.34,35 The measurement was called a Pinhole Neutron Experiment (PINEX) and a pinhole was used in an opaque neutron shield to cast an inverted image on plates of material.29,30 Neutrons with sufficient energy activated the plate material to form radioactive nuclei. An image could then be read out by detecting the decay emission either through autoradiography or by cutting up the plate and counting. As electronic cameras became more available, detection could also be achieved using scintillators to convert the neutrons to visible photons, which could then be imaged with the cameras—a method originally developed by Anger36,37 for medical gamma-ray imaging. For nuclear testing, this method was typically called an Electronic Pinhole Neutron Experiment (El PINEX) and could use multiple cameras to view a single scintillator.30
APERTURES FOR NEUTRONS
Since the aperture in the barrier is the image-forming optic for a neutron imaging system, the design, manufacture, and metrology of the aperture are crucial, and the penetration of fusion neutrons drives the design beyond the thin plate of the metal that is typical for an optical pinhole. The propagation of a fusion neutron through a barrier can generally be treated using the idea of the mean free path, λ, the average propagation distance before a strongly scattering or absorbing event. The mean free path depends on the interaction cross sections of the neutron with the atoms that make up the barrier and the density of those atoms. For a 14.1 MeV neutron, for instance, the mean free path in gold at solid density is ∼3.2 cm. In general, apertures for fusion neutrons are made of alloys of high-density metals, such as gold40,42,58,61,62 or tungsten.31,43,46,63
Now, if we consider the propagation straight through a barrier of thickness, t, and mean free path, λ, the fraction of neutrons transmitted without scattering or absorption is
Thus, for a thickness of t = λ, the transmission of the barrier is ∼37% of the incoming number of neutrons. Compared to paths through a completely open aperture, this barrier would produce an image with only ∼2.7 contrast ratio. To obtain a contrast ratio of 100, the barrier would need to be 4.6λ. Thus, a gold barrier needs to be 14.7 cm thick to produce a contrast ratio of 100. Since gold has one of the shortest mean free paths for fusion neutrons, apertures made of most other materials would be significantly longer.
Thick pinhole apertures
The first step to make the image of Fig. 1 more realistic is to incorporate a thick barrier. We generally refer to the opening in the barrier as an aperture, but these devices are frequently also called collimators or pinholes. Simply extending the aperture shape into a thick material to produce a tube, however, does not create an aperture that performs well. As shown in Fig. 2(a), the field of view of a long, straight aperture in a thick barrier is small,64 and the vignetting of the long, straight tube creates a strongly translation-variant point spread function.
To improve the field of view, while retaining a thick aperture, researchers have developed several aperture designs. See Figs. 2(b)–2(e). One option, shown in Fig. 2(b), is a single-sided, conical aperture that tapers along the axis of the aperture from one end to the other to produce a larger field of view. These apertures can be straightforward to manufacture with small openings to obtain resolution of small features.58,65,66 The main difficulty with using these apertures is that penetration at the smallest part of the opening effectively makes the aperture appear larger and displaces the apparent apex inside the body of the aperture. The point spread function (PSF) of a single-sided taper is also soft, that is, less strongly peaked at the image plane.
Several designs for apertures exist with a more well-defined apex and a more strongly peaked point spread function. These designs include biconical apertures, straight apertures with conical tapers,29,64 and apertures with hyperbolic or toroidal segments. Stepped designs, which use stacked disks with holes drilled in them to mimic these designs, have also been used.30 Variations of these designs as slit apertures have also been used for pulsed-power driven sources that were expected to be weak, and were more cylindrical than spherical.67–69 Even with these more careful designs, the neutron penetration often makes the effective size of the pinhole aperture larger than the physical opening, which must be considered when predicting the resolution of the system.
Penumbral and annular apertures
So far, we have largely discussed pinhole apertures, that is, apertures with openings smaller than the source, which produce an image of the source. The low solid angle of these apertures, however, can limit their ability to image weaker sources. To improve the solid angle and particle statistics, other forms of coded-aperture imaging70,71 have also been used, such as penumbral or annular apertures.
For a penumbral aperture,72 the opening is larger than the source, so the information about the source is encoded in the penumbral shadow of the aperture. The image of the source image is then reconstructed from the data, either through filter techniques or other forms of image reconstruction, which will be discussed below. For neutrons, this form of imaging was introduced using conical tapers,40 but biconical31,32,49 and toroidal segments have been used as well.41–43,61,62
Annular or ring apertures, which were initially developed for x-ray imaging,73–76 use a biconical plug inside a biconical penumbral aperture to block the central region of the penumbral aperture.77 These apertures eliminate the flat center of the image that occurs when using an open penumbral aperture and lower the secondary scattering in the detector, which can improve the signal-to-noise ratio.
Arrays of apertures
Even properly designed thick apertures have a field of view that can be small compared to the location uncertainty of the source in ICF systems or compared to possible misalignments of the aperture axis to the source location. Moreover, the translation variance of the point spread function can have a significant impact on the reconstruction of the source if the location of the source is not well known.78 One means of overcoming this issue is to build arrays of apertures—a method used for x-rays and gamma rays79 to produce a larger effective field of view. The NIF NIS uses this method, starting with a 23-aperture array,58 and it now uses arrays with up to 87 apertures.80 The array includes 12 penumbral apertures dedicated to gamma-ray imaging and 72 pinhole apertures and three penumbral apertures dedicated to neutron imaging.
Since the individual apertures point to different locations at the source plane, the relative intensities of the individual images can be used to determine the location of the source relative to the axes of the apertures. This localization of the source then allows for translation-variant PSFs to be used to reconstruct a single source using multiple apertures.81,82
Manufacturing and metrology of apertures
For modern ICF experiments, one challenge is to create thick apertures that can provide sufficient resolution, typically of the order 10–15 µm. For lower resolution and larger apertures, the holes may be drilled with conventional methods and then stacked to produce an aperture.30 For the small apertures at NIF, milling machines are used to scribe multiple grooves along the length of gold layers, which are then stacked to produce the apertures.80,83 Since any reconstruction can only be as good as the knowledge of the aperture, the metrology of these apertures is also crucial. The metrology can require significant effort and involve coordinate measuring machines, optical coordinate measuring machines, and microscopy of the ends of the arrays.84 Even with excellent metrology, it can still be difficult to determine the relative pointing of all the apertures at the source plane with the desired accuracy, and self-characterization of the pointing using Maximum Likelihood algorithms on multiple images from the entire aperture array may prove useful.85
As discussed earlier, the first detectors for neutron imagers on nuclear tests were plates of material such as zirconium and aluminum.29,30 The plates were activated by the neutrons and read out either through autoradiography using a film or by cutting the plate in pieces and using counting detectors to read out the activation. While these detectors have the advantage of activation thresholds that reduce low-energy neutron backgrounds, the low sensitivity and difficulty of readout made them less practical for ICF imaging.
Plates of scintillator material, 0.5–1 cm, which could be imaged optically, gated with a microchannel plate, and recorded with cameras, were used for neutron imaging on later nuclear tests.30 These were better suited to ICF, but since the lower neutron fluxes from ICF sources required even thicker, up to 50 mm, scintillators, many detector systems for ICF have relied on segmented scintillators,41 which guide the light to a single plane for optical imaging. Later variations on this idea have also used scintillating fibers86 or liquid-filled capillary detectors.60,87 While the first neutron imaging system at NIF used a BCF-99-55 fiber scintillator,58 the six gated detectors for the equatorial imagers on NIF currently use EJ-262 plate scintillators, 1–5 cm thick, and require a sophisticated lens design to collect the emission uniformly across the scintillator.88 An additional advantage of gated detection with scintillators is the ability to select the energy range and separate down-scattered neutrons from un-scattered neutrons.89 This down-scattered neutron imaging is used as a probe for the remaining cold fuel in laser ICF implosions.90
While the proton recoil range for neutrons interacting with hydrogen usually sets the ultimate resolution at the detector for plastic scintillators, manufacturing issues and crosstalk for segmented or fiber scintillators can limit the spatial resolution at the detector to ∼1 mm. The corresponding fixed pattern noise can also complicate the analysis for systems with segmented scintillators.
When energy gating is not required, image plates developed for x-ray radiography can be used to record a neutron image. They are sensitive to neutrons91 and have excellent spatial resolution when used directly since the coating layer is thin. When used with high-density polyethylene (HDPE) converters, as is routinely done at NIF to improve the sensitivity,92 the resolution is limited by the spread of the recoil protons in the HDPE.
Obtaining better resolution at the detector has also driven the use of track detectors for imaging. Gel bubble detectors, which can be read out with a pulsed laser diode, have demonstrated an average bubble size of 60 µm and 25 µm location precision.52–55 CR-39 plastic track-etch detectors, which can be read out using a microscope, have also been used with high-density polyethylene converters,68,69 for imaging pulsed-power driven fusion sources, but they have not yet been used for laser ICF.
IMAGE RECONSTRUCTION METHODS
As discussed in Sec. I, neutron source imaging for ICF requires a thick aperture with sufficient magnification to be compatible with the resolution of the spatially sensitive detector that is being used. The goal is to determine the neutron source distribution S from the recorded image I. Here, we assume that the source is optically thin so that the source projection along the line of sight is essentially an integration through the source distribution. We also assume that the image I can be described by the integral over the source plane of the source distribution S multiplied by a known or estimated point spread function (PSF) Q that accounts for all physical processes between the source and the detected image (aperture image formation, detector response function, etc.).
Here, A represents the source plane, and and are the image plane and source plane coordinates, respectively. In discretized form, the signal in pixel i, j of image I is given as
For many imaging systems such as x-ray pinhole imagers, the associated PSFs preserve the source shape, and the image is a direct representation of the source distribution. Typical ICF neutron apertures, however, do not preserve the source shape. The corresponding recorded image tends to be dominated by the system PSF and appears to be a blurred radiograph of the aperture. When solving for the source distribution in these instances, the process is typically referred to as “reconstruction.” In effect, all NIS analyses attempt to solve Eq. (5) by reconstructing S from the recorded image I.
Solving an equation of this type is typically referred to as an “inverse problem.” Many inverse problems are defined to be “ill-posed,” which means that solutions may not exist, may not be unique, or may be unstable to small changes in initial conditions. Neutron image reconstruction is an ill-posed problem. The PSF Q is not known exactly, but is estimated, so, solutions to the above equation do not necessarily exist for the estimated PSF. Additionally, aperture geometries can result in a vanishing PSF for some spatial frequencies. Source distributions with arbitrarily large amplitudes at these frequencies can be added to the solution and still obey the above equation, so infinitely many solutions exist. Image noise, particularly for images with low signal-to-noise ratios, can also lead to very unstable neutron source reconstructions.
Image noise is a particularly difficult aspect inherent to NIS. The imaged object is generally less than 100 µm in diameter. To resolve such a small object, the image must be magnified, which is accomplished with large aperture–detector distance. While the magnification helps resolve smaller features of the implosion, the detector solid angle drops off approximately with the square of this distance. Moreover, the small neutron interaction cross section of most detector materials means that only a fraction of the incoming neutron image is detected. As such, neutron images tend to suffer from poor neutron statistics, especially at lower yields, that is, Poisson noise from finite neutron statistics is substantial and can lead to unstable neutron source reconstructions.
A variety of methods have been developed to carry out neutron image reconstructions. For ICF campaigns that use penumbral apertures, Fourier analysis,41,78 in combination with Weiner filtering,31,32,45–47 has been employed in Refs. 45–47, 62, and 93. The thrust of this approach is that the PSF is assumed to be translation-invariant, and Eq. (4) can be rewritten in terms of a 2D convolution of Q and S : I = Q × S. Then, S can be determined by dividing the Fourier transform of I by the Fourier transform of Q : , where F and F−1 denote Fourier and inverse Fourier transforms, respectively. The Weiner filter provides a means to de-emphasize or reweight frequency bins based on their signal content. Adaptive Weiner filters have also been implemented, where the filter locally varies in the frequency space, based on local variance within a neighborhood of frequency bins.45–47 Fourier analysis has also been used in separate campaigns that fielded annular, as well as penumbral, apertures for neutron imaging of ICF.51,77,87,94 In this approach, auto-correlation of the image and the calculated PSF in the Fourier space is used to reconstruct the source distribution.
Fourier reconstruction is a useful approach for some inverse problems. This technique uses a point-by-point division of the Fourier transforms of the image and the PSF, but there are limitations to the technique when applied to image reconstruction. It relies on the assumption that the PSF is translation-invariant. While penumbral aperture PSFs are less sensitive to source shifts than pinhole PSFs, this assumption can be problematic when the aperture is not very well aligned with the source (>50 μm).78 It is also difficult to impose physical constraints on the solution, and it is not possible to reconstruct a single source from multiple images simultaneously. Smoothness and non-negativity of the reconstructed source distribution is generally achieved via pre-processing and smoothing the recorded image, which can artificially enlarge the apparent source size. Noise amplification, especially when signal-to-noise ratios are low, is a drawback to any approach that does not use signal smoothing or solution regularization. Thus, when the PSF vanishes, but noise results in nonzero frequency bins, that noise is amplified.
In addition to Fourier approaches, iterative routines that minimize a given cost function have also been developed for neutron source reconstruction via penumbral imaging. Genetic algorithms92–99 randomly search for a solution in a framework that borrows principles from biological evolution.95,96 Entropy maximization22 and heuristic optimization97–99 use a parameterized cost function that balances solution smoothness and recorded image-model differences (residuals).
A similar iterative approach to neutron source reconstruction for ICF was expectation maximization (EM),81,100–102 which was originally thought out in a Bayesian framework.103 This is also known as the Lucy–Richardson method.104,105 In this framework all quantities are probabilities, so, it works only on non-negative functions, which naturally imposes non-negativity to the solution. This was a specific case of the more general method of expectation maximization.
In the EM framework, the algorithm iterates an expectation step, followed by a maximization step. In the expectation step, the expectation value is computed for the functional that is being maximized using the current approximation (image being reconstructed), the measurement, the forward model, and the noise model for the measurement. Then, on the maximization step, this value is maximized on the set of model parameters.
In Expectation Maximization Maximum Likelihood (EMML), the functional we want to maximize is the likelihood and, for Poissonian noise, can be written as106
where Pij is the probability that a particle emitted in source plane pixel i will be detected in detector pixel j, yi is the observed signal in detector pixel i, λj is the source intensity in source plane pixel j, and is the estimated source intensity in source plane pixel j for EM iteration k with the iteration number shown in parentheses. Note that y is a vectorized version of the image I, P is essentially the PSF Q, and λ is a vectorized version of S in (5).
This functional is separable, and using the optimization transfer technique, the iteration becomes
This is exactly the formula that follows from the Bayesian formalism.
The routine iterates through solutions for such that . If the initial estimate of λ is positive for all values of λi, then the solution is guaranteed to be positive, preserving the physical constraint of non-negativity. All entities are non-negative, so, it imposes non-negativity of the solution.
While the EMML approach is quite robust, there are some disadvantages to the approach. Determining the appropriate stopping condition is difficult. With too few iterations, the routine has not found the optimal solution, and over-iteration effectively results in fitting to image noise. Care must be taken to ensure that the stopping criterion does not result in either of these two conditions. Another drawback to this approach is that it does not require a smooth solution, which can result in nonphysical solutions. A generalized extension of this approach, however, can ameliorate these issues.
In traditional EM, the parameter set that maximizes the likelihood function is found during the maximization step. Generalized Expectation Maximization (GEM)106–108 requires only that the updated parameter set found in the M-step result in a larger value of the likelihood function than the current estimated parameters. Additionally, GEM uses a penalty term in (6),
where N represents all neighborhoods of pixels. is termed an energy function and typically depends on relative source pixel values, and β is the regularization parameter. This energy function is referred to as the type of prior that the reconstruction assumes. For neutron reconstructions, typically Gibbs priors109–111 are used. An example of one such function is , where i and j are neighboring elements. The advantage to the generalized approach is that the solutions tend to preserve solution smoothness, the routine tends to converge faster, and over-iteration is much less of a problem.106,Figure 3 shows an example reconstruction for NIF shot N210808.14
Estimating uncertainties inherent to source reconstructions is difficult both, to quantify and visualize, or communicate in a meaningful way. Residuals between the recorded detector image and the reconstructed source projected through to the detector illustrate where (at the detector plane) discrepancies lie. Interpretation, however, of residuals is often qualitative. An alternative approach is to parameterize the source shape via Fourier or Legendre modes112 and estimate reconstruction uncertainty using the variation in the shape parameters. For NIF NIS, the reconstructions typically include a fit of the 17% contour to the sum of Legendre polynomials up to the fourth order. The individual Legendre polynomial coefficients are the fit parameters.
There are three kinds of errors that are quantified for neutron source reconstructions: systematic, statistical, and metrological. Systematic errors refer to errors in the reconstruction that are associated with incorrect knowledge of aperture pointing. This is determined by shifting the estimated aperture pointing ±10 μm in both x and ydirections in the source plane. For each shift, the source is reconstructed with an updated PSF, and the 17% contour is fit to the sum of Legendre polynomials. The variation in each Legendre polynomial coefficient is then reported as the systematic error for the given coefficient.
Statistical error refers to the uncertainty in source reconstruction due to noise fluctuations and is quantified with a set of statistical trials (see Fig. 4). First, the reconstructed source distribution is projected to the detector. Then, a sampling of the characterized noise distribution is added to the projection (simulated detector image), and a neutron source is reconstructed from this simulated detector image. This process is repeated for some numbers of noise samplings, and the variation on the fitted Legendre polynomial coefficients is reported as statistical error. It should be noted that aperture selection plays a role in statistical error. At NIF, the sources typically have a mean Legendre P0 of 30–50 µm at the 17% contour. For the existing NIS, the pinholes are generally used to reconstruct sources with yields yn > 1014. At lower yields, the statistical uncertainties inherent to pinhole reconstructions become quite substantial. Instead, penumbral images are used to reconstruct the neutron source at yields down to ∼1013. Images produced by penumbral apertures have larger signal-to-noise ratios than those produced by comparable NIF pinhole apertures. The tradeoff is in resolution. With typical penumbral apertures used for NIS at NIF, the best attainable resolution is larger than the corresponding pinhole apertures by 5–10 μm.
Metrological error captures the uncertainty in the reconstruction stemming from uncertainty in the metrology used to characterize the individual pinholes or penumbras used for the given reconstruction. The individual pinholes and penumbras were inspected using an Optical Coordinate Measuring Machine (OCMM). The OCMM data are incorporated into the NIS analysis toolkit and used to construct 3D models of the pinholes and penumbras (see Fig. 5). PSFs are calculated using these 3D models, so, small uncertainties in the OCMM data propagate through PSF calculations to the source reconstruction. To quantify these uncertainties, reconstructions are repeated with subsets of the pinholes chosen for reconstruction, i.e., if ten pinholes are used for reconstruction, the reconstruction is repeated some numbers of times with five or six pinhole subsets. The statistics of the Legendre polynomial fits to the 17% contour of the set of reconstructions are used to calculate the metrological error. If the metrology for a particular pinhole(s) is skewing the reconstruction, this analysis should capture that variation. Typically, these errors are comparable to estimated statistical errors at yields of ∼1015 neutrons. Below these yields, reconstruction errors are primarily dominated by statistical uncertainties.
Three-dimensional neutron source reconstruction
The reconstruction procedure outlined above can also be applied to 3D reconstructions of the source distribution.113 With neutron imaging systems deployed along multiple lines of sight, 2D source reconstructions can be used to constrain a 3D reconstruction. This technique is referred to as “limited-view tomography.” Typically, the detectors are sufficiently far from the source such that the paraxial approximation is valid. In 3D, Eq. (4) becomes113
where is the intensity measured by a detector pixel at position with projection direction , and S is the 3D source distribution (see Fig. 6).
As with the 2D reconstruction, the 3D reconstruction (see Fig. 7) is also an ill-posed problem, with many more unknowns than independent equations. Typically, tomography relies on dozens to thousands of detector views to reliably reconstruct source distributions in 3D. As such, any such limited-view 3D reconstruction should come with the caveat that the reconstruction only represents a probable source distribution, but may not represent the true source distribution. It should also be noted that the orientation of the detector views also impacts the reconstruction. These 3D reconstructions are not necessarily free from artifacts introduced by the orientation of the detector views.113
To solve (9), the source volume must first be parameterized in such a way that it allows one to calculate detector contributions. A voxel-by-voxel approach,113 a spherical harmonic decomposition,114 a cylindrical harmonic decomposition,115 and a Kaiser–Bessel “blob” based approach have all been implemented for 3D neutron source reconstruction. The reconstructions are then typically carried out in a GEM framework by maximizing (9) with respect to the 3D source distribution/parameterization.
Cold fuel density reconstruction
The current NIS at NIF captures primary (14.1 MeV) and downscattered (6–12) MeV neutron images. With a 3D reconstruction of the primary source S and a downscattered detector image Ids, cold fuel density can be inferred via the following relation:
where is the intensity measured by a detector pixel at position with projection direction , ρ is the density of the cold fuel, and Φ is the local scattered flux and is related to the primary source distribution S.26 The geometry can be seen in Fig. 8. This approach also utilizes a GEM-based approach to solve for ρ in the above equation. A reconstruction of the cold-fuel density for shot N210220 is shown in Fig. 9.
SUMMARY AND FUTURE DEVELOPMENTS
In this Review, we have described the development of neutron imaging for inertial confinement fusion at facilities around the world, including discussion of thick apertures, the effects of penetration of the edges of aperture, and the variety of detectors that have been used.
We have also discussed reconstruction of ICF neutron sources, including noise amplification and the need to accommodate translation-variant point spread functions and the development of expectation maximization, expectation maximization maximum likelihood, and generalized expectation maximization algorithms, which can impose physical constraints such as positivity on solutions.
Since multiple imaging lines of sight now exist at the National Ignition Facility, we described the use of expectation maximization and generalized expectation maximization for 3D reconstructions of neutron sources with a limited number of views. Inference of cold fuel density from a 3D primary neutron source distribution and a 2D downscattered neutron source reconstruction is included in that discussion.
In the future, we expect that the recent indirect-drive experiments that exceeded the Lawson criterion for ignition at the NIF12–14 will drive continued development of neutron source imaging. Improvements may come from continued development of detectors and apertures, including perhaps coded apertures with scatter and partial attenuation coded apertures.116 While researchers already use DT primary neutron images, DT downscattered neutron images and 3D reconstructions to understand the burn volume and remaining cold fuel, as ICF yields increase at NIF and other facilities, related measurements may also develop into useful diagnostics. These may include MeV gamma ray imaging to investigate the remaining ablator, upscattered neutron imaging, and spatially resolved temperature measurements.
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory, under Contract No. DE-AC52-07NA27344, and by Los Alamos National Laboratory, under Contract No. 89233218CNA000001.
Conflict of Interest
The authors have no conflicts to disclose.
D. N. Fittinghoff: Conceptualization (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). N. Birge: Formal analysis (equal); Investigation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). V. Geppert-Kleinrath: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Writing – review & editing (equal).
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