Invited article: X-ray phase contrast imaging in inertial confinement fusion and high energy density research

While absorption-based x-ray imaging has been in use for over 100 years since discovery of x-rays,¹ where relative differences in the absorption of transmitted x-rays provide the image contrast, it was only appreciated in the past 25 years that relative differences in the phase of transmitted x-ray waves could also produce image contrast, either by overlap and interference of the waves after propagating a sufficient distance to a detector,² or by use of a special analyzer prior to detection.³ In either case, a coherent source of x-rays is required, with spatial coherence being the main requirement,⁴ either by using third generation x-ray sources such as synchrotrons,⁵ a microfocus x-ray laboratory source,⁴ or fourth generation coherent light sources such as x-ray free electron lasers.⁶ The use of phase differences in the object to produce the majority of image contrast has been collectively called x-ray phase contrast imaging (XPCI) or sometimes referred to as refraction-enhanced imaging (REI). It has been successfully used in various fields ranging from biology, medicine, physics, materials science, and others. More recently, XPCI has found use in inertial confinement fusion (ICF) and high energy density (HED) research, first to characterize ICF capsules and HED targets,^7–9 and later used in dynamic experiments¹⁰ requiring ultrafast x-ray pulses and high temporal resolution (<100-ps), high spatial resolution (∼micron-scale or less), and sufficient coherence and brightness of the x-ray source. In this Review article, XPCI image formation theory is presented, its various uses in ICF and HED research are discussed, and the unique requirements for ultrafast XPCI imaging are given, as well as current challenges and issues in its use. Image formation theory and x-ray coherence requirements are given in Sec. II, and the characterization of ICF and HED targets using XPCI is discussed in Sec. III. Various examples of its use in dynamic ICF and HED experiments are presented in Sec. IV by using laser-plasma generated x-ray sources, synchrotrons, and x-ray free electron lasers (XFEL). Current challenges, issues, and opportunities are identified in Sec. V.

Phase sensitive x-ray imaging techniques rely on the overlap and interference of wave fields as they propagate due to spatial gradients in the phase of the object. The complex index of refraction for x-rays is n = 1 − δ + iβ, where the real part 1 − δ accounts for refraction effects by the object, and the imaginary part β accounts for absorption by the object. The complex phase shift due to transmission through the object, with respect to vacuum propagation, is given by $Φ=2π/λ∫n−1dz$⁠, which is a projection of the complex index of refraction of the object along the propagation direction z. Since both δ, β are proportional to the object density, density gradients in the object produce phase gradients in the transmitted wave, which are visualized with XPCI, and often provide higher image contrast than just absorption alone. In this section, the theory of image formation for x-ray phase contrast imaging (XPCI) is presented, both from a geometric optics perspective (refraction only) in Sec. II A, and using full scalar diffraction in Sec. II B.

A simple geometric optics approach is instructive to understand XPCI image formation. Consider a collimated, uniform beam of x-rays incident on a thin, pure phase object (no absorption), as shown in Fig. 1. After traversing the object, the rays deviate by some small angle due to refraction by the object. When a ray deviates by an angle θ at the object plane z = 0, then its transverse coordinates at $x,y$ are shifted to the transverse coordinates $x′,y′$ at an image plane located at distance z, and the shifted positions can be written simply as

and since ray deviation is related to the gradient of the phase front, then

which maps x to x′, and y to y′. The area element $∂x∂y$ is mapped to $∂x′∂y′$ at image plane z. Assuming uniform intensity at the object plane $x,y,0$ and ignoring cross-terms, the intensity at the image plane z is proportional to the ratio of the area elements $∂x′∂x∂y′∂y−1$⁠, which are written as

which is the Jacobian of the coordinate transformation. The intensity at plane z is then given by

where $∇⊥2φ$ is the Laplacian of the phase for the transverse $x,y$ coordinates only, and Eq. (6) is valid in the geometric-optics approximation only when $λz2π∇⊥2φ≠1$⁠, otherwise a caustic is formed at the singularity. In the limit of small argument $λz2π∇⊥2φ≪1$⁠, then we can write

which was derived by Pogany et al. in Ref. 4 using a first-order simplification of diffraction theory.⁴ From Eq. (7), it can be easily seen that the intensity is unperturbed at z = 0 for a pure phase object, and the intensity due to phase curvature (non-zero second derivative) increases with increasing z. To first order x-ray, phase contrast imaging can be viewed as an object consisting of small “lenslets” that slightly focus or defocus the x-rays by refraction after the rays propagate a finite distance from the object. While this geometric optics simplification offers insight, it is overly simplistic and is often necessary to consider the full diffraction theory when applying the formalism to experimental data, as discussed next in Sec. II B. Equation (6) can also be easily modified to include absorption in the object.¹¹ Accurate material interface locations and lateral dimensions can be obtained in the geometric-optics regime for very small propagation distances z, given by the maximum slope of an intensity profile near the interface region, i.e., at the boundary between the dark and white bands as in Fig. 4(a). For larger propagation distances, the phase gradient causes a substantial lateral shift due to refraction, as is apparent from Eqs. (2) to (3) and becomes even more complicated when diffraction effects are considered. In these cases, phase retrieval is required to obtain accurate lateral sizes and interface locations.

Using scalar diffraction theory, consider now a monochromatic plane wave E₀e^−ikz, where the wavenumber k = 2π/λ, propagating along the z-axis through a “thin object” such that small angle refraction within the object can be ignored. The phase of the transmitted electric field just past the object, defined as the plane z = 0, is simply given by $φx,y=k∫δx,y,zdz$⁠. Similarly, the absorption coefficient is given by $μx,y=k∫βx,y,zdz$ so that the transmitted complex electric field at the plane z = 0 can be written as $Ex0,y0expiφx0,y0−μx0,y0$⁠, which then propagates through free space for z > 0 according to the Helmholtz wave equation, whose solutions in the paraxial limit (E varying slowly with z) yield the scalar diffraction integral,⁴ 

where

As a physical picture, the resulting electric field at some plane z > 0 is simply a convolution of the transmitted field at the object plane $Ex0,y0$ with an optical transfer function $hx,y$⁠, which is the paraxial approximation of a spherical wave in the limit z ≫ x, y. Since scalar diffraction integral is a convolution, Fourier analysis can be used to solve Eq. (8). Let $Hu,v$ represent the 2D Fourier transform of $hx,y$⁠, so that

where $u,v$ are the spatial transform coordinates of $x,y$⁠. Following Ref. 4 to simplify for illustrative purposes, consider only the 1D case $Hu=expiπλzu2$⁠, and assume χ = πλzu². Then, expand as $Hiχ=cosχ+isinχ$⁠, where the real part (cosine term) can be viewed as responsible for absorption contrast, i.e., when z = 0, then χ = 0, and absorption contrast is maximized. Alternatively, the imaginary part (sine term) can be viewed as responsible for phase contrast and reaches its first maximum when χ = π/2 or when λzu² = 1/2. The function $Hu,v$ can be viewed as a contrast transfer function that optimizes contrast by choosing the appropriate propagation distance z given the x-ray wavelength λ and some limiting feature size a. If the highest spatial frequency u is set by the smallest feature size a that can be resolved, $u=2/a$⁠, then for the case of plane wave illumination, the optimum propagation distance between the object and detector is z_opt = a²/λ. As an example, for x-ray wavelength λ = 0.5 Å (24.8 keV) and feature size a = 5 µm, the optimum propagation distance between the object and detector is z_opt ∼ 0.5 m. The contrast transfer function terms are illustrated in Fig. 2, as well as the geometric-optics limit when the phase contrast transfer function can be approximated as sin χ ≈ χ.

Equation (8) can be generalized for the case of a point source,⁴ with source to object distance R₁, and object to detector distance R₂, such that the geometric magnification is $M=R1+R2/R1$⁠, and the effective propagation distance becomes z_eff = R₂/M. The coordinates at the detector plane are $x′,y′=Mx,My$⁠. Plane wave propagation is recovered when R₁ ≫ R₂ such that M = 1. In the limit of large magnification R₂ ≫ R₁ and finite source size s, the optimum distance between the source and object is R₁ ≈ s²/2λ given that the magnification is large enough such that the resolution at the detector plane Δx satisfies the Nyquist sampling criterion Δx/M ≤ s/2. This more generalized point source imaging geometry is shown in Fig. 3.

For propagation-based XPCI, various imaging regimes have been noted, which are best characterized by the dimensionless Fresnel number $NF=a2/λzeff$⁠, where a is a characteristic feature size in the object, and λ and z_eff are the x-ray wavelength and effective propagation distance. For N_F ≫ 1, the propagation distance is sufficiently small and feature sizes are sufficiently large that one is in the near-field or geometric-optics limit, where refraction and absorption can accurately describe the image, as in Eqs. (6) and (7). When N_F ∼ 1, one is in the holographic or Fresnel regime, and the image retains many features seen in the geometric-optics limit but substantial diffraction occurs and will be observed given sufficient detector resolution and spatial coherence. Here one must resort to scalar diffraction theory to accurately model image formation using Eqs. (8)–(10). In the limit N_F ≪ 1, for very large propagation distances, one is in the far-field or Fraunhofer regime. By inspection of Eq. (9) in the limit z → ∞, and expanding out the terms in the integral of Eq. (8), one can easily show that $Ex,y,z$ in the far-field is simply the Fourier transform of the wave at the object exit $Ex0,y0,0$⁠, a well-known result.

A forward-model using the scalar diffraction theory in Eqs. (8)–(10) with a synthetic object consisting of a 120-µm diameter C₈H₈ sphere, with up to 200 spherical voids of 2-µm diameter, is shown in Figs. 4(a)–4(c) assuming 20-keV x-rays at 1-µm resolution for the various N_F regimes.

These represent a continuum of propagation-based XPCI imaging regimes where object reconstruction, via phase retrieval, is possible. The far-field regime is often called coherent x-ray diffractive imaging (CXDI). The main focus of this review is propagation-based XPCI in the near-field (N_F ≫ 1) and Fresnel (N_F ∼ 1) regimes, but dynamic experiments using CXDI at XFELs will be briefly discussed. A related branch of phase contrast imaging, requiring specialized and accurately aligned optics, such as analyzer crystals or Talbot gratings, will only be briefly reviewed in the context of ICF and HED but is beyond the scope of this review.

As shown in Ref. 4, a major requirement for XPCI is to have sufficient source spatial coherence, i.e., a small enough source size at a sufficiently large distance, while spectral coherence is not as restrictive, a spectral bandwidth Δλ/λ ∼ 1 still yields appreciable phase contrast.⁴ Synchrotron radiation from undulators and x-ray free electron lasers (XFEL) easily satisfy both of these requirements. For benchtop laboratory x-ray microfocus sources^7,9 or laser-plasma generated x-ray sources^10,12,13 with a finite source size s, one needs the source a sufficient distance R₁ away from the object that the “bright band” due to refraction near an object or feature edge is of order or larger than the source angular divergence s/R₁, and sufficient detector resolution Δx to resolve the magnified source at the detector plane, sR₂/R₁ ≥ 2Δx. Consider the simple case of refraction of x-rays near the edge of a cylinder of radius r with real part of refractive index n = 1 − δ, and a detector situated at a distance R₂, where the transverse projected cylinder thickness $Tx=2r2−x2$ for x ≤ r, and $Tx>r=0$⁠. The phase is $φx=δkTx$⁠, and its derivative is $∂φ/∂x=−2δkxr2−x2−1/2=φ′x$⁠. The deflection angle of rays is given by $φ′x/k$⁠. While $φ′x$ contains a singularity at the edge when x = r, the angular deflection can be evaluated very near the object edge, e.g., x ≈ 0.98r, which is conveniently written as $x=5r/26$ so that the deflection angle near the edge due to refraction is $φ′5r/26/k=10δ$⁠. The refraction from the edge must be larger than the source angular divergence in order to see the bright band from refraction. This yields requirements that incorporate both x-ray source spatial coherence and detector resolution: $10δ≥s/2R1$⁠, and sR₂/R₁ ≥ 2Δx. References 11 and 14 use a different approach to arrive at similar requirements for spatial coherence and detector resolution.

Noise due to photon counting statistics can affect the ability to observe the contrast in a detected image, whether absorption or phase contrast, and one typically wants the contrast to be a few times larger than the percent noise level, defined to be the inverse of the signal-to-noise ratio 1/SNR. With image contrast defined as $CI=Imax−Imin/Imax+Imin$⁠, and with I_max and I_min being the local maximum and minimum intensities for an object or given feature, one wants the contrast to be at least $CI⋅SNR≥3$ as a rough guide. While this is easily obtained by sufficiently long x-ray exposure times in static experiments, dynamic experiments require a sufficiently bright x-ray source with a short enough x-ray pulse to avoid motion blur. As an example for an x-ray point source, consider a phase jump of error function form, $φx=φ0erfx/L+1/2$⁠, where φ₀ is the magnitude of the phase jump φ₀ = δkL, over the scale length L. Its second derivative is $φ″x=−2δkxexp−x2/L2/πL2$⁠. Using Eq. (7) for a simple estimate, and noting that $φ″x$⁠, in this case, has maxima/minima at x ≈ ±0.7L gives $φ″0.7L≈δk/2L$⁠, and intensity bright and dark bands ΔI/I₀ ≈ zδ/2L, or image contrast C = zδ/2L. As an illustrative example, consider a weak phase jump for δ = 2 × 10⁻⁶ over scale-length L = 1.5-µm, and 15 keV x-rays (λ = 0.83 Å). The optimum source to object distance z_eff ∼ R₁ ∼ 15-cm for a source size s = 5-µm, and the phase jump is φ₀ ∼ 0.23 radians (0.036 waves). The phase contrast would be C = 0.1 with these parameters, requiring SNR ≥30, or ≥900 detected photons per spatial resolution element Δx. Adequate detector resolution for a source size s = 5-µm, and detector pixel size Δx = 25-µm requires R₂ = 135-cm $M=10$⁠, and assuming a detective quantum-efficiency DQE = 0.03, one needs a photon radiant flux (photons/sR/pulse) of

or ≥10¹⁴ photons/sR/pulse for a single-pulse exposure from a nearly point source in this example. For a short-pulse laser-plasma generated x-ray source, the conversion efficiency¹² of laser energy to 15 keV x-rays is ∼10⁻³–10⁻⁴, and this example would require ∼3–30 kJ of laser energy to produce this flux of x-rays in a geometry optimized for phase contrast, assuming all laser energy is converted to x-rays with the above conversion efficiency, emitted within the small source size (5-µm). This can be quite challenging for laser-plasma generated x-ray sources, and often the source-to-object distance R₁ is chosen to be less than optimum, and various choices are made to produce higher SNR, such as a higher QE detector using an image plate, lower x-ray energy for higher x-ray conversion efficiency and larger phase effects, or the experiment only detects higher contrast image features for a given SNR.

Synchrotron and XFEL sources, with their low beam divergence ∼ μrad, produce extremely bright x-rays ∼ 10¹⁹ photons/sR/pulse for a synchrotron (∼10⁹ photons per pulse with ∼10-µrad divergence) to 10²⁴ photons/sR/pulse for a XFEL (∼10¹² photons per pulse with 1-µrad divergence), and easily meet the spatial coherence and photon flux requirements for XPCI imaging of low contrast phase features. The next sections review the use of laboratory microfocus sources to characterize ICF and HED targets using static XPCI, as well as the use of XPCI in dynamic experiments at laser, synchrotron, and XFEL facilities.

Characterizing the cryogenic deuterium-tritium (D-T) fuel layer of an ICF capsule introduced the challenge of measuring a frozen hydrogen (Z = 1) ice layer inside of a visibly opaque low-Z capsule of beryllium (Z = 4) or high-density carbon (Z = 6), where both the fuel layer and capsule are mostly transparent to x-rays. XPCI was proposed theoretically in Ref. 7 as a solution, defining the optimum x-ray source requirements using commercially available laboratory microfocus sources, as well as detector resolution and optimum distances between the source, object, and detector to maximum phase contrast. A synthetic phantom ICF capsule was forward modeled using Fresnel diffraction theory to demonstrate the expected contrast of various features, in addition to first proposing the use of XPCI in dynamic ICF and HED experiments by forward modeling an idealized synthetic shock front in comparison to absorption contrast alone. The use of XPCI in dynamic experiments will be reviewed in Sec. IV.

Initial proof-of-principle experiments demonstrating that XPCI was able to observe the solid D-T layer in a Be capsule were performed,⁸ and improved source size and detector resolution were implemented⁹ to meet the requirements for measuring the D-T fuel layer roughness for planned experiments at the National Ignition Facility (NIF). Figure 5(a) shows a zoomed in XPCI image of a 2.154-mm diameter, 105-µm thick Be shell doped with 0.9% atomic Cu, with a 187-µm thick solid D-T fuel layer from Ref. 9. Details such as capsule defects, layers from vapor deposition fabrication of the capsule, as well as the D-T fuel layer are clearly observed with 5-µm resolution in the XPCI image, and such detailed images enabled the ICF program to substantially improve the capsule quality compared with this initial prototype. A radial profile through the XPCI image is shown in Fig. 5(b) from Ref. 9, with a forward model of an idealized cryogenic Be/Cu capsule using Fresnel diffraction theory and the geometric parameters from the experiment, and is in excellent agreement with the data. The forward modeling can be used to guide XPCI experimental design for future experiments and help infer object parameters. Due to the success of the initial research in Refs. 7–9, XPCI is now used by researchers at laser facilities worldwide to characterize ICF capsules and HED targets.^15–25 

In this section, XPCI in dynamic ICF and HED experiments using laser-plasma generated x-ray sources, synchrotrons, and XFEL is reviewed.

It was soon recognized after the publication of Ref. 7 that phase contrast enhancement, often dubbed refraction enhancement, had been observed in dynamic ICF and HED experiments using pinhole-apertured laser-plasma x-ray sources that were intended to improve spatial resolution and photon statistics compared with pinhole imaging, with the unintentional benefit of edge-enhancement due to XPCI.^26–29 However, concerns about implementing the large source-to-object distances required to optimize XPCI at ICF facilities, where the facility had been designed to align lasers and targets only a few cm from the target chamber center, prevented earlier optimization for XPCI. X-ray sources generated with ns-long laser pulses have conversion efficiency¹³ from laser to x-rays of ∼few %, but concerns about the closure of the ∼10-µm scale pinhole during the ∼1-ns laser pulse limited the amount and duration of x-rays that could be produced.^28,30

To overcome these challenges, an initial proof-of-principle experiment¹⁰ was carried out at the Trident Laser facility using a laser-plasma generated x-ray source to radiograph a dynamic experiment with XPCI using 17.5 keV K-α x-rays produced by a 12.5 μm diameter Mo wire, illuminated with an intense (∼10¹⁹ W/cm²) short pulse laser of 2 ps duration. XPCI radiographs at a delay of 4.3 and 6.3 ns after shock initiation in a strongly shocked (∼6 Mbar) thick CH target, driven by a separate 2-ns laser, clearly show the curved shock front, consistent with the 600-µm diameter shock-drive laser, as well as unshocked edges of the target. While the XPCI geometry was not optimum, due to target chamber constraints, and the images suffered from poor signal-to-noise (optimization discussed in Sec. II C), this first demonstration of ultrafast XPCI in a dynamic experiment was an important development for use in ICF experiments.

Championed by the theoretical work of Koch et al.¹⁴ at LLNL, who examined optimization and use of laser-generated x-ray sources for XPCI in ICF experiments at large laser facilities, experiments using the more energetic Omega laser facility were performed by Ping et al. looking at the interface of two near-solid density low-Z materials isochorically heated by x-rays to temperatures of ∼3–10 eV, and using XPCI to determine the time-dependent density gradient at the interface, which is related to thermal transport.³¹ Such experiments are ongoing and currently being planned for the more energetic NIF laser, with the development of ∼1-µm resolution, sufficient to observe diffraction, and the small length scale associated with thermal transport in warm dense plasmas.^32,33

The thin-wire laser-plasma x-ray source for XPCI by Workman et al.¹⁰ was revisited by Antonelli, Barbato and colleagues^34,35 at the PHELIX laser to optimize source brightness, source size, and geometry using a 5-µm tungsten wire irradiated by an intense 0.5 ps laser, and imaged strong shocks driven by a second ns-laser using XPCI with 10-µm spatial resolution, limited by the detector resolution. Figure 6(a) shows an XPCI image from the experiments in Refs. 34 and 35, and a density map generated via phase retrieval and tomographic reconstruction is shown in Fig. 6(b), as well as a simulated XPCI image from a radiation hydrodynamic calculation.

An early conceptual use of dynamic XPCI was proposed to perform 1-D streaked imaging of an imploding ICF capsule to measure the shock and refraction waves in the shell and D-T fuel layer.³⁶ This long range position of utilizing XPCI in an imploding capsule required substantial development to implement on NIF,^37,38 ultimately requiring ∼100-kJ of laser energy to produce sufficient 9-keV x-rays from a 5-µm slit source with magnification M ∼ 87 at the detector. The recent results³⁹ of a forward XPCI simulated streaked image from a radiation hydrodynamics calculation and experimental XPCI streak image are shown in Fig. 7. Various shock and rarefaction waves between the carbon ablator and cryogenic fuel layer are clearly made visible by XPCI during the acceleration phase of the implosion. Similar dynamic XPCI experiments are being planned at other laser facilities.⁴⁰

Undulator radiation from synchrotrons is capable of producing ∼10⁹ x-ray photons in a mm² area in a ∼100 ps single pulse and has sufficiently low beam divergence (∼10 μrad) to have high spatial coherence required for XPCI, and spectral bandwidth ΔE/E ∼ 5%–10%, with flexible tuning of x-ray energy between 5 and 25 keV. The electron bunch structure (∼150-ns bunch separation), ∼100 ps pulse duration, and ∼1-mm field of view with ∼few μm resolution are ideally suited for multi-frame dynamic material experiments and low-pressure HED imaging with <1 Mbar pressure. Synchrotron facilities worldwide have dedicated experimental areas with dynamic drivers, such as gas guns, explosives, pulsed power, and laser-driven shocks, e.g., the Dynamic Compression Sector at APS is one of several examples.⁴⁰ 

While the ∼100 ps pulse duration has applicability in the very lowest pressure conditions of ICF and HED, the long-duration bunch structure in a synchrotron, e.g., 24 bunches separated by 153.4 ns at the Advanced Photon Source (APS), only provides a single frame of data in the ICF and HED regime. As an example, a typical limiting resolution for XPCI at a synchrotron is ∼1.5 μm. Material flow, such as shocks, traveling faster than 1.5 µm/100 ps–15 km/s will have substantial motion blur. To estimate a useful limiting pressure for dynamic XPCI experiments at a synchrotron, pressure in the strong shock limit for a γ = 5/3 ideal gas material scales as $P≈2/γ+1ρ0Us2$⁠, where ρ₀ is the initial material density and U_s is the shock speed in the material. For a polymer with ρ₀ = 1 g/cm³ and setting U_s = 15 km/s yields an upper limit pressure ∼170 GPa (1.7 Mbar) where motion blur begins to severely degrade the resolution. Single frames per experiment in this pressure regime also requires highly reproducible drive and target parameters on each experiment for systematic research, thus dynamic XPCI at synchrotrons in the HED regime (≥100 GPa) is not practical.

Dynamic XPCI at synchrotrons has been used to examine brittle fracture and material strength,^42–46 compression of porous materials^47,48 and engineered lattice materials,^50–51 shock-induced void collapse,^52,53 shock-induced compaction of granular materials,^54,55 equation of state,^55–57 and electrical discharges and exploding wires.^58–60 One notable example of shock compaction of granular material is shown in Fig. 8, where a Cu impactor strikes a sample containing packed glass spheres, launching a shock wave traveling through the granular material, followed later by compaction behind the shock.⁵⁵ In the low pressure HED regime, exploding wires and pulsed-power loads imaged with XPCI show details of the fast wire explosion followed by shocks driven in the surrounding medium. Figure 9 shows an XPCI image frame of an exploding Cu wire array, submerged in water.⁶⁰ In addition to the details of the exploding wires, a shock wave launched in the surrounding water is observed. The electron bunch structure of synchrotrons provides a near ideal time-scale for obtaining multi-frame movies of dynamic material and pulsed discharge experiments via XPCI imaging. However, as noted previously, the typical synchrotron x-ray pulse duration of ∼100 ps and spatial resolution of 1.5 µm, together with the 10s of ns spacing between bunches is not well suited to multi-frame XPCI imaging of the dynamics in HED and ICF experiments exceeding ∼100 GPa pressure, which is the main focus for this Review article.

X-ray free electron lasers (XFELs) provide the ultimate in spatial resolution (10–100 nm), time resolution (∼10-fs), and flux per pulse (∼10¹² photons/pulse) for experiments in the ICF and HED regime, with the flexible tuning of x-rays between 5 and 25 keV. Using a similar argument as above in Sec. IV B, dynamic imaging of material flow above ∼10 nm/10 fs ∼ 1000 km/s will be limited by motion blur. This occurs for pressures greater than ∼1000 TPa (10 Gbar), which enables dynamic XPCI imaging over a wide range of ICF and HED phenomena, limited only by drive pressure currently available at XFELs. The repetition rate of most XFELs is in the range of ∼10 Hz–MHz, and this typically only allows for single-pulse XPCI in high pressure (>Mbar) regimes, requiring carefully reproducible laser drive and target (sample) conditions in the experiment in order to accurately follow the dynamic evolution. Recent experiments^61,62 have begun to explore new multi-pulse capabilities for XFELs, using 4–8 x-ray pulses, separated at the ∼300 ps–1 ns time scale, enabled by ultrafast x-ray imaging detectors,⁶³ albeit with spatial resolution limited to ∼1-µm by the detector and geometric magnification constraints, not yet realizing the spatial resolution potential for XFELs, which will be discussed later in Sec. V.

Single pulse XPCI experiments at XFELs have imaged optical laser-driven shocks into various materials such as single crystal diamond,⁶⁴ silicon,^6,65 germanium,⁶⁶ aerogel foam,⁶⁷ and liquid water microjets.⁶⁸ For the shocked solid experiments,^65–67 XPCI directly measured shock compression, and used phase retrieval or forward model to obtain detailed structure near the shock and laser ablation fronts. In the liquid microjet experiment,⁶⁸ a pump-probe technique was used to shock the thin water jet with an intense optical laser, and probe the dynamics of the jet breakup using XPCI. An example XPCI image from single crystal silicon⁶⁵ shocked to ∼25 GPa is shown in Fig. 10. In contrast to current dynamic experiments using laser-plasma x-ray sources and synchrotrons, the signal-to-noise ratio, spatial coherence, and detector resolution are sufficient to observe Fresnel diffraction fringes at the shock front and laser ablation front. Forward modeling of XPCI was used to infer shock compression and spatial scale-lengths in this experiment.⁶⁵ 

Increasingly complex dynamics in the HED regime have been examined using single-pulse XPCI, such as Rayleigh–Taylor driven turbulence,⁶⁹ laser-driven shock cavitation in liquid water,⁷⁰ and shock-driven void collapse in the HED regime.^71,72 In the research by Rigon et al. in Ref. 69, hydrodynamic turbulence in a foam, driven by a Rayleigh–Taylor unstable foil expanding into the foam, was imaged with 600-nm resolution using XPCI and a novel single-crystal LiF detector,⁷³ that acted as an image plate, but with ∼1 mm² field of view, dynamic range ∼10⁶, and high detector spatial resolution ∼1 μm. Figure 11 shows an example XPCI image probed with 7 keV x-rays from their experiment using a laser-driven thin plastic foil modulated at a 40-µm wavelength to seed Rayleigh–Taylor instability. As the unstable foil expands into the 0.1 g/cm³ polymer foam, a transition to hydrodynamic turbulence is observed later in time, characterized by small scale density fluctuations that are easily observed with XPCI.

Figure 12 shows recent experimental results of strong-shock driven void collapse, led by Pandolfi, Gleason and co-workers,^71,72 where a several-ns optical laser drives a shock at ∼200 GPa into an epoxy target⁷² with an embedded ∼1-µm thick SiO₂ hollow spherical shell, 20–40 μm diameter, and the evolution of the collapsing void is followed in nearly identical laser and target conditions using XPCI at 18 keV, with ∼400-nm spatial resolution. Figure 12(a) shows the initial unshocked void, and the shock has propagated midway through the void in Fig. 12(b). Fresnel diffraction at the steep shock front is easily observed. The turbulent structure at the top of Fig. 12(b) is the optical laser ablation region. In Fig. 12(c), the shock has passed completely through the void, and the initial void position is shown as the yellow dashed circle, measured on a static XPCI image for this shot. A supersonic jet is observed in front of the shock after void collapse, as expected from radiation hydrodynamic simulations,⁷¹ and the remaining SiO₂ shell material is entrained in the moving shock. The fine speckle structure in these raw, un-normalized images is due to small defects in the Be compound refractive lenses (CRL) and will be removed using advanced algorithms^68,74,75 in the ongoing analysis. Though these recent efforts were single shot (i.e., one target per image per shock transit) additional efforts are being pioneered for high resolution imaging sequences of void collapse to multi-Mbar conditions with multiple time slices per shock transit in the same target, which leverage gated-detector technologies⁶³ and XFEL pulse trains.^61,62

Coherent x-ray diffractive imaging (CXDI) has been used in dynamic pump-probe experiments,^76,77 where the detector is in the far-field, and the experiments use an intense short-pulse optical laser as the pump, and the sample is probed using x-rays measured with CXDI. In the pioneering experiment by Barty and co-workers,⁷⁶ an intense short-pulse optical laser is used to ablate a silicon nitride membrane with a nano-etched pattern. CXDI with soft x-rays at 13.5 nm was used to measure the evolving ablation region. Iterative phase retrieval was used to reconstruct the images with 50-nm resolution within a few μm field of view. Finally, CXDI was recently used at 5.5 keV by Ihm and co-workers in pump-probe experiments to observe melt in single Au nanospheres heated by an intense short-pulse laser.⁷⁷ Time-resolved diffraction patterns were measured, and iterative phase retrieval was used to reconstruct the image during the melt phase, with 10-nm resolution. Figure 13(a) shows the measured CXDI diffraction patterns at various times with respect to the optical pump laser pulse, and the reconstructed images of the melting Au nanosphere are shown in Fig. 13(b). The potential of CDXI has yet to be fully explored in the much higher pressure and temperature regimes of HED and ICF, and will be discussed further in Sec. V.

In addition to being extremely sensitive to small defects, cracks, weak shocks, and other density perturbations, allowing one to readily visualize these features in the near-field (N_F ≫ 1) and Fresnel (N_F ∼ 1) regimes, quantitative areal density maps can be extracted, in principle, via phase retrieval, similar to absorption-based radiography. However, generalized phase retrieval is a well-known challenge, and is a non-linear ill-posed problem. Traditional iterative projection algorithms for phase retrieval, such as Gerchberg–Saxton⁷⁸ and Fienup-type,⁷⁹ use numerical free-space propagation of the complex wave field (Fresnel or Fraunhofer propagators, depending on N_F) between the object and detector plane, imposing the measured intensity and other constraints at the detector plane, and finite support and other physical constraints at the object plane, refining the retrieved phase at the object plane after 100s–1000s of iterations. One example of a physical constraint at the object plane can be found in Ref. 81. Compressed sensing uses prior knowledge about the object or its physics constraints, and has been successfully used^44,70 to better inform the forward model in phase retrieval. Phase retrieval for objects with phase jumps exceeding 2π in the holographic or Fresnel regime is notoriously difficult⁸¹ using traditional Gerchberg–Saxton or Fienup iterative algorithms, especially in the presence of noise,⁷⁹ which is often the case for Fresnel-regime dynamic XPCI images reviewed in this article. A complete review of the numerous phase retrieval approaches is beyond the scope of this article, but this challenge presents an opportunity for the dynamic XPCI community.

For quantitative phase retrieval, it is necessary for the XPCI images to be photometrically accurate, including background subtraction, and normalization for spatial variations in the x-ray beam in the absence of an object (so-called white field or flat field image), as well as sensitivity variations in the detector. At synchrotrons, besides noise associated with finite photon detection statistics, pulse-to-pulse x-ray intensity variations of ∼5% RMS is typical⁸² and can affect how one normalizes for white field since dynamic image may have been taken with a different x-ray pulse intensity.⁶⁸ Maximum output energy for XFELs occurs in the self-amplified spontaneous emission (SASE) mode, which is inherently stochastic in nature and can produce ∼10% variations in space, ∼10% in pulse energy, and ∼0.2% in x-ray wavelength.⁸³ While essentially monochromatic, since XPCI with XFELs typically uses compound-refractive lenses to focus the x-ray beam, the ∼0.2% wavelength variation pulse to pulse can cause enough change in the x-ray focus location due to chromatic aberration to make a noticeable change to magnification that must be accurately accounted for.^68,75

The highest spatial resolution detectors typically use a single crystal scintillator optically coupled to a CCD camera and can produce <1 µm resolution at the detector plane.⁸⁴ However, scintillators have a finite decay time, which can produce an after-image on subsequent frames, limiting the inter-frame time for a given scintillator.⁸² Development of single-crystal LiF detectors, with spatial resolution, dynamic range, and field of view is promising for some single shot applications,⁷³ but not for the general multi-frame need. Ultra-high speed direct-detection x-ray (UXI) CMOS cameras,⁶³ with ns-scale interframe time, show some promise for multi-frame capability,^61,62 but their current 25-µm pixel sizes are not well-matched to XPCI in the near-field or Fresnel regime at XFELs unless their pixel sizes can be made much smaller. For an XFEL using a CRL to focus the x-ray beam before the object and produce large geometric magnification at some large distance past the object, the effective spatial resolution and field-of-view are intimately connected. One can only get high resolution by focusing the XFEL closer to the object, thus further limiting the field of view. At some point, due to the intense XFEL beam, concerns arise about heating the sample in a multi-pulse setting, where the first pulse and subsequent pulses may heat the target significantly and affect the dynamic behavior of the experiment. However, this challenge presents an opportunity for use of the current UXI with a multi-pulse XFEL. At a substantially high pressure, the plasma will be hot enough that heating by the x-ray probe is negligible, and the opportunity to do multi-frame CXDI with a small field of view can be explored in the high pressure HED regime.

Both synchrotron and XFEL dynamic XPCI experiments yield very large datasets during a given experimental beam time. XPCI data processing and phase retrieval can be challenging often for just a single image frame, not to mention a dataset with hundreds to thousands of images. This presents a great opportunity for data science. Experiments that do single shot pump-probe XPCI imaging often have relatively simple targets, such as liquid microjets, thin films, etc., in order to create reproducible experimental conditions shot-to-shot. The challenge of producing more complex targets for HED and ICF experiments^69,72 that number in the hundreds and are nearly identical presents an opportunity for advanced and additive manufacturing.

Finally, XPCI at large laser facilities using incoherent laser-plasma x-ray sources often suffer from source brightness, spatial coherence, and spatial resolution. Substantial development and progress in short-pulse driven betatron sources in laser plasmas may eventually provide a sufficiently bright, coherent x-ray source to overcome this issue.^85,86 Another phase imaging technique called Talbot-Lau interferometer or deflectometer, which is sensitive to phase gradients, also shows much promise for use of incoherent laser-plasma x-ray sources in dynamic experiments.^88–91 Talbot-Lau uses x-ray gratings with specific periodicity and locations to effectively break up the incoherent x-ray source into multiple spatially coherent beamlets, which are then detected as they deflect past a final grating. Since an incoherent source can be used, it can be placed closer to the object to be imaged, producing higher flux at the detector. However, precise alignment of the optics as well as possible damage due to shrapnel and debris from the laser-plasma remain concerns.^88–91 

The author would like to acknowledge useful discussions with Andrew Leong, John Barber, Cindy Bolme, Kyle Ramos, and Brian Jensen (LANL), Nino Landen (LLNL), Richard Sandberg (BYU), and Arianna Gleason and Bob Nagler (SLAC). This paper was dedicated to the late Jeff Koch (deceased) at LLNL, who had many enlightening discussions with the author on this subject, and who tirelessly worked to help implement this technique at large laser facilities such as the National Ignition Facility. This work was supported by the Laboratory Directed Research and Development program of Los Alamos National Laboratory under Project Nos 20180100DR and 20210717ER, NA-10 Defense Programs, NA-115 Advanced Manufacturing Development Program, and the LANL Inertial Confinement Fusion Program. Los Alamos National Laboratory was operated by Triad National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy under Contract No. 89233218CNA000001.

The author have no conflicts to disclose.

David S. Montgomery: Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.