Talbot-Lau x-ray deflectometer: Refraction-based HEDP imaging diagnostic

When probing weakly absorbing materials with 1–100 keV x rays, refraction effects are dominant. For low-Z, the real part of the index of refraction n = 1 − δ + iβ is much larger than the imaginary part (Fig. 1). Thus, x-ray phase contrast imaging can provide far better contrast than attenuation radiography techniques in High Energy Density (HED) plasma experiments within this energy range.¹ 

Talbot-Lau X-ray Deflectometry (TXD) is a refraction-based electron density gradient diagnostic for highly dense matter.² TXD implementations in high-intensity laser and pulsed-power environments will be presented. Optimization of x-ray backlighter emission and spatial resolution will be discussed in terms of detector and backlighter. Future developments of TXD techniques will be discussed.

Talbot-Lau (TL) interferometry is commonly used at synchrotron facilities or in conjunction with medical x-ray tubes for biological imaging or materials science.^3,4 This phase-contrast imaging technique can deliver electron density gradient information in plasmas by measuring x-ray refraction angles using the following equation:

where α is the angular refraction, λ is the probing beam wavelength, r_e is the electron radius, and N_e is the electron density.

A probing x-ray beam will be deviated due to the changes in the index of refraction within a sample. Hence, highly dense objects can be diagnosed by measuring the x-ray refraction angle.^2,5,6 It should be noted that Talbot-Lau interferometry integrates along the direction of the probing beam, similar to standard x-ray imaging diagnostics. Hence, TL interferometry measures line integrated electron density gradient information directly.

TL interferometry uses micro-periodic gratings (source, phase, and analyzer) to measure small angular deviations of a beam transversing an object (Fig. 2). The source grating g₀ ensures x-ray coherence by generating an array of micro-sources.^7–9 TL interferometry works with incoherent and polychromatic sources, as those available in high-intensity laser or pulsed-power environments.

Constructive image formation is achieved when g₀ = g₂ (L/D), where g_i is the period of the grating and L and D are shown in Fig. 2. The phase grating g₁ is typically a π-shift grating, splitting the beam into ±1 orders, set at the Talbot distance¹⁰ d_T = m g₁²/8 λ, where m is the (odd) order of the interference pattern. It is important to note that each Talbot order has a specific interferometer contrast curve (Fig. 3) so that larger orders have a narrower energy bandwidth due to a narrower contrast peak centered at the interferometer design energy. Considering Eq. (1), when solving for electron density, a narrower energy bandwidth translates to a more accurate measurement. Additionally, Talbot orders translate to a larger interferometer length. In a magnifying geometry, d_T scales with Talbot magnification M_T = (D + L)/L. The analyzer grating g₂ turns refraction angle changes into intensity changes at the detector plane.

Talbot-Lau interferometry was proposed recently as a HED plasma diagnostic.^5,11 By providing refraction-based images, TL imaging can characterize highly dense regions in plasmas by measuring electron density gradients through refraction angle detection. Post-processing can provide electron density maps with high spatial resolution. TL interferometry can identify regions of hydrodynamic instabilities, for example, not observable in conventional radiography due to the lack of attenuation contrast.¹ 

Talbot-Lau X-ray Deflectometry (TXD)^6,12,13 has been developed, where the analyzer is rotated a small angle (ϴ ≈ sin ϴ), producing a moiré pattern with period P_M = (g₂/2) sin(ϴ/2). The refraction angle is given by α ≈ W_eff F, where W_eff is the effective angular resolution of the interferometer and F is the fringe shift. The effective angular resolution is given by¹⁴ (a) g₀/p when the object is located between g₀ and g₁ and p is the distance between the object and g₀ or (b) g₂/q when the object is located in between g₁ and g₂ and q is the distance between the object and g₂, as shown in Fig. 2. The fringe shift is measured through Fourier techniques, separating the phase shift from attenuation, thus obtaining simultaneous information from a single moiré image.² 

The TL interferometers had periods of 2.4, 3.85, and 12 μm for g₀, g₁, and g₂, respectively (as described in Ref. 15), and were set in m = 1, 3, 5, and 7 with M_T = 6 for all configurations. Figure 3 shows the interferometer contrast curve along with 8 keV emission expected from Cu laser–target interactions and x-pinch discharges. Fringe contrast is a measure of the interferometer quality. In TXD, contrast is defined as the ratio between the intensity amplitude and the intensity mean of the signal. It is worth noting that the energy bandwidth is proportional to 1/m; hence, larger Talbot orders increase the accuracy of electron density retrieval, as per Eq. (1).

TXD has been benchmarked on the Multi-Tera Watt (MTW), LULI2000 (Laboratoire pour l'Utilisation des Lasers Intenses), and Omega EP lasers.^15–17 MTW experiments demonstrated grating survival and moiré fringe formation irradiating Cu foils at 45° with I ∼ 5 × 10¹⁶ W/cm² (<30 J, 8 ps). The electron density of a solid static object was retrieved with ∼92% accuracy and spatial resolution of 50 m.¹⁵ Moiré images were recorded in LULI2000 experiments (2 × 10¹⁴ W/cm²) with <60 J, 10 ps laser pulses. Image plates showed lower contrast than x-ray CCD images recorded on MTW (7% compared to <25%, respectively).¹⁶ A resolution of 24 μm was measured for 25 μm diameter Cu wire targets,¹⁸ motivating follow-up MTW experiments using wire and Cu sphere targets of 20 μm and 14–25 μm diameter, achieving a spatial resolution of <25 μm.¹⁶ 

In recent studies, “bookend” targets¹⁹ composed of a Cu wire wedged between two CH foils were tested alongside Cu wires, spheres, and edge-on planar foils. Laser spot size was also changed for each target to optimize fringe contrast by varying the laser intensity (I ∼ 8–30 × 10¹⁵ W/cm²). A future publication will address x-ray backlighter optimization through laser intensity and target type variation. Cu grid images (Fig. 4) were used to measure spatial resolution in the vertical and horizontal directions, that is, parallel and perpendicular to the grating bars, respectively. Spatial resolutions (at the object plane) of ∼5 to 10 μm FWHM were measured for edge-on foil, wire, sphere, and bookend targets. Considering pixel size, the measurements were restricted to 3–5 μm, which varied depending on detector binning and object magnification.

Fringe contrast loss was observed (<22%) and caused by high-energy emission from wire, sphere, and bookend targets due to hot electron generation,¹⁷ which has been observed in experiments^20,21 with laser intensities >10¹⁸ W/cm2. A lower fringe contrast of ∼5% was measured with image plates (Fig. 5) as they have lower gain at <10 keV and higher gain at >20 keV, unlike x-ray CCD detectors.

A TXD diagnostic has been implemented on Omega EP. Laser pulses of 50–150 J, 10 ps (1–3 × 10¹⁷ W/cm⁻²) were used to drive Cu foil, wire, and bookend targets.¹⁶ To identify the optimal target type, a HOPG spectrometer was used to measure Cu K-shell emission and the interferometer contrast was determined from moiré images. A spatial resolution of >60 μm was measured for Cu foils at 45° in the first Omega EP campaign [Fig. 6(a)]. ¹⁷ In a new Omega EP campaign, a spatial resolution of >10 μm was measured for Cu foil targets at normal laser incidence [Fig. 6(b)], that is, edge-on orientation with respect to TXD line-of-sight.

As in MTW experiments, an improved spatial resolution for edge-on Cu foil backlighter targets was accompanied by additional high-energy emission. In 8 keV Talbot-Lau X-ray Deflectometry, x-ray emission outside the contrast curve (Fig. 3) will contribute to moiré image intensity base levels, thus diminishing contrast. On Omega EP, a fringe contrast of 3%–10% was measured for edge-on Cu foils, compared to 11%–17% measured for Cu foils at 45°. While Cu bookend targets were expected to deliver better spatial resolution based on MTW data, they emitted a higher level of high-energy photons, which wiped out the moiré fringe contrast in the images recorded on Omega EP using an x-ray CCD detector due to low contrast thus making fringe shift measurements more challenging.

The ablation front of a 3.0 × 3.0 mm² × 125 μm CH foil irradiated by a 160 J, 1 ns laser pulse with a spot size of 120 μm was probed through TXD. An x-ray refraction angle map was retrieved, for the first time, from the moiré image recorded. Figure 7(a) shows a highly dense ablation front close to the foil (left side), above the TXD detection limit given by critical density (10²⁵ cm⁻³). While moiré fringe contrast was lower than expected, phase retrieval was performed by tracking fringes individually, allowing for refraction angle mapping, as shown in Fig. 7(b).

Based on these results, a new TXD variation has been designed for MTW and Omega EP. A monochromatic TXD diagnostic will use a multilayer mirror to select 8 keV emission, improving interferometer contrast by filtering out emission outside interferometer design energy. Systematic studies will optimize 8 keV emission considering the spatial resolution and detector spectral response by varying the target type and laser intensity (spot size and energy).

Table I summarizes results for all laser-produced x-ray backlighter targets.

TXD diagnostics have also been implemented in a pulsed-power environment. For example, the study of the plasma density outside of an imploding liner in MagLIF-relevant experiments can be a straightforward and useful application. An ∼350 kA, 350 ns Marx-bank pulsed-power generator, Llampudeñ,²² produced x-ray backlighting from copper x-pinch emission, allowing for electron density mapping of a low-Z object through TXD.¹³ Gratings did not show any damage due to the intense magnetic field or heating induced by plasma radiation. Nevertheless, these experiments show the need for using a disposable protective filter in front of the source grating to avoid damages induced by debris from both electrodes and the plasma object.

While the x-pinch load was not properly optimized (photon fluence and source size), moiré fringe patterns with contrast up to 14% were obtained. Considering that x-ray CCDs are prone to EMPs in pulsed-power environments, a Carestream D-speed x-ray film was used as the detector. It should be noted that the Bio-Max equivalent x-ray film²³ is expected to deliver higher contrast and should be tested in future studies. In these experiments, the electron density of a static object was measured with <16% error.¹³ Refraction angle mapping was limited by source size. These studies demonstrated the potential of TXD as an electron density diagnostic in the pulsed-power experiment.

Follow-up experiments were carried out on GenASIS,²⁷ an ∼200 kA, 150 ns Linear Transformer Driver (LTD) generator, where a new TXD diagnostic platform was implemented. X-ray backlighting was produced by driving Cu wire,²⁸ hybrid,²⁹ and laser-cut³⁰ x-pinch loads. Emission was analyzed and optimized by addressing TXD backlighter requirements:¹⁶ spectrum, spatial and temporal resolution, emission timing, number of sources, and reproducibility.

It was found that the Moiré fringe contrast recorded with image plates was 0.5% for 2 × 25 μm Cu wire x-pinches, 1.4% for single 25 μm Cu wire hybrid x-pinches, and 7.3% for 25 μm thick Cu foil laser-cut x-pinches. This is compared to the 27% moiré fringe contrast measured with an x-ray CCD using a Cu rotating anode medical x-ray tube [Fig. 8(a)]. The much lower contrast is justified by the fact that x-ray films have higher gain at 8 keV than image plates.^24–26 A detailed analysis of these results will be published in a separate manuscript.

When compared to wire and hybrid x-pinches [Fig. 8(b)], laser-cut x-pinches [Fig. 8(c)] were found to be the optimal x-pinch configuration for TXD. Laser-cut x-pinches were found to have high reproducibility and improved spatial resolution, with source sizes measured at ≤5 μm, limited by diagnostic resolution. Their pulse duration of ∼1 ns FWHM was also limited by diagnostic capabilities. Laser-cut x-pinches delivered up to 10⁶ W peak power (∼10⁸ W/cm²) near 8 keV, matching the optimal emission for the Talbot-Lau interferometer contrast (Fig. 3). The x-ray backlighting properties of the three x-pinch configurations can be found in Ref. 31.

The electron density of a Be sheet of 0.5 × 2.5 × 5.0 mm³ was retrieved from Moiré images. The Be sheet was set at a small angle so that edges produce a constant fringe shift, while the center of the object produces no shift. Conversely, attenuation varies at the edges and is constant at the center. Following Eq. (1) and considering that one fringe shift is equivalent to the system W_eff (199 µrad), an electron density of 4.39 ± 1.43 × 10²³ cm⁻³ was retrieved for the hybrid x-pinch [Fig. 8(b)], representing 11.1% underestimation of the tabulated value. Laser-cut x-pinches [Fig. 8(c)] delivered an electron density of 5.13 ± 2.48 × 10²³ cm⁻³, representing an overestimation of the tabulated value by 3.8%, well within the measurement uncertainty.

Plasma loads were imaged through TXD for the first time using laser-cut x-pinch backlighting.³¹ These results will be presented and discussed thoroughly elsewhere. Experimental images matched XWFP simulations,³² demonstrating that TXD can be a powerful x-ray refraction-based diagnostic for dense Z-pinch loads.

The development of TXD diagnostics for HEDP experiments has been presented. Grating survival and moiré fringe formation have been demonstrated in high-intensity laser and pulsed-power environments. Refraction angle mapping and electron density retrieval were achieved with Cu backlighter targets. Thin foils, wires, and spheres were irradiated by 10–150 J, 8–30 ps laser pulses to produce K_α emission in the MTW, LULI2000, and Omega EP laser systems. Additionally, Llampudkeñ (∼350 kA, 350 ns) and GenASIS (∼200 kA, 150 ns) pulsed-power generators used Cu wire x-pinch loads. GenASIS was also used to drive novel configurations: hybrid and laser-cut x-pinches. X-ray backlighting was provided by Cu K_α emission (8 keV). Electron density profiles were retrieved achieving 4%–11% accuracy when compared to tabulated values. The first x-ray images of plasma objects recorded through TXD were obtained. The ablation front of a laser-irradiated CH foil was imaged through TXD for the first time on OMEGA EP for densities >10²³ cm⁻³, and a refraction angle map was obtained from post-processing. X-ray backlighting for TXD was optimized considering targets and detectors. It was shown that targets can improve spatial resolution, while detector can optimize moiré fringe contrast, thus achieving accurate plasma characterization through TXD.

This work was supported by the NNSA (Grant Nos. NLUF DE-NA0002955 and HEDLP DE-NA0003882 and DE-NA0003842), the DoE Office of Science (FES) [Grant No. DE-SC0020005, LaserNetUS (Omega Laser Facility)], and Fondecyt (Grant No. 1171412). The authors thank LLE science and engineering staff for their support throughout the years as well as students and professors from Pontificia Universidad Catolica de Chile, University of Michigan, École Polytechnique, and Johns Hopkins University. A.C. and V.B. acknowledge the Conseil Régional Aquitaine (project INTALAX), and the Agence Nationale de la Recherche (ANR-10-IDEX-03-02, ANR-15-CE30-0011), for their funding support.

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