2. IMPROVED MONOCHROMATIC X-RAY BACKLIGHTING TECHNIQUE

One of the two LULI2000 beams was used to generate the X-ray source needed to perform the sample radiograph. This backlight beam was 1 ns long, smoothed with a random phase plate and focused on a 15 µm thick planar foil of vanadium with a 100 µm diameter to reach an intensity of about 10¹⁵ W/cm². The other beam was used to drive a shock on a plastic sliver. During this experiment, the X-ray radiography using a spherically bent crystal allowed us to determine the shocked plastic mass density as described in Section 4. The crystal set-up is presented in Figure 1. This quartz (11–20 orientation with 2D abut 4.9 Å) spherically bent crystal with a 15 × 50 mm size and a radius of R = 150 mm was set, according to Bragg's law, for the central wavelength λ₀ ~ 2.384 Å (Bragg angle of θ = 76.7°) in order to select the resonance line of Vanadium He_α at 2.382 Å and its intercombination line at 2.393 Å. It was aligned to perform an image with a magnification M (M ~ 10) of the object on an X-ray charge-coupled device. The large incidence angle on the crystal causes angular aberrations, which generate two focusing distances: the tangential and sagittal planes. To reduce this aberration in the tangential direction, the object was set at a distance a = 80 mm, the source at c = 100 mm from the crystal, and the detector was placed very close to the tangential focus, i.e., at a distance b = 825 mm. In our scheme, the X-ray source was inside the Rowland circle in order to collect more photons. Consequently, for a given spectral line, the imaged zone on the object is smaller. In such configuration, there is a compromise between intensity and field of view.

Fig. 1. (Color online) Improved backlighting scheme with a spherically bent crystal. Configuration in the tangential plane. Only formation of two points of the object on the image plane are considered in this drawing.

In order to measure the spatial resolution, tests have been performed using a 400 lpi grid in the object plane as shown in Figure 1. The obtained image is presented in Figure 2. We can notice, in background, the spectral intensity distribution of the source: the line on the left is the intercombination line, and the other one is the resonance line. Considering first the resonance line in Figure 2, the vertical grid wires are clearly visible in the foreground, but the horizontal ones are not so well resolved. It means that we demonstrate that a high resolution in the meridional plane, less than 10 µm, in a large field of view of about 800 µm was reached in the case of using resonance He_α line of Vanadium as a backlighter source. In the sagittal plane, the resolution was worse but still high enough with 25 µm. As the position of the detector was not optimized for obtaining images in different wavelengths of backlighter, the spatial resolution in the case of using intercombination line was not so high with about 16 µm in meridional plane, but better than for resonance line backlighting with abut 20 µm in sagittal plane. To simply sum up, we have a 2D map resolution that is the result of the image plan position in comparison with the sagittal and meridional object plan at a specific Bragg angle.

Fig. 2. (Color online) Monochromatic image of a 400 lpi gold grid and traces along the different directions of such image, obtained in the backlighting scheme of Figure 1 configuration. The distance crystal-object is 80 mm.

An additional test was done where the distance between object and crystal was not a = 80 mm like in the first experimental test, but a = 83 mm (Fig. 3b). The comparison of the images in Figures 3a and 3b clearly demonstrates how sensitive the spatial resolution of the obtained images depends on the position of the object. For example, in the case of object position 80 mm and He_α backlighting line, the spatial resolution is 10 µm in the meridional and 25 µm in the sagittal directions, but for the case of 83 mm object position, the spatial resolution in both direction is equal and on the order of 20 µm. One may summarize, that the key point, which we demonstrated thanks to these test grid images, is the possibility to obtain high spatial resolution in a specific direction even at enough big Bragg angles instead at small Bragg angles used previously in another experiments.

Fig. 3. (Color online) Monochromatic images of a 400 lpi gold grid. The crystal-object distance is (a) a = 80 mm and (b) a = 83 mm. The hole in the grid refers to the target chamber center.

3. GEOMETRICAL INTERPRETATION

To interpret the results obtained and presented in Figures 2 and 3, we developed a ray tracing code to calculate the resolution in the image plane. Let us consider one point A in the object plane which creates a spot C in the image plane (cf. Fig. 1). Its size δ must be determined in order to calculate the spatial resolution. The point A is illuminated by the source, which size is L = 100 µm and which is situated at a distance of 20 mm from the object. Thus, the corresponding beam divergence is about 5 × 10⁻³ rad. This beam illuminates the crystal on an approximately disc shaped area B with a Δ = 400 µm diameter. After the reflection on the crystal, this X-ray beam focuses at two different positions: the tangential focus f _t and the sagittal focus f _s. This implies a beam divergence that differs in both directions: ω_t = Δ/f _t and ω_s = Δ/f _s.

In the detector plane, the diameter of the spot C, corresponding to A in the object plane, is δ_t = ω_t |b − f _t| in the tangential direction and δ _s = ω_s |b–f _s| in the sagittal direction (our detector is placed between the two foci). The resolutions along both directions are then , where M _t and M _s are the magnifications. The foci and magnifications can then be calculated:

Confusion is possible between the magnifications of the object and of the source. As we are considering the resolution in the object's plane, we gave the magnifications of the object. The same remark applies to the foci. In our scheme, this gives: r _t < 1 µm and r _s ~ 33 µm. According to the experimental picture, the sagittal resolution is reasonably explained, but the tangential resolution is not as good as 1 µm. This is mainly due to the crystal imperfections, but the resolution also depends on the position on the image. Indeed, our calculation gives the resolution only for the image center. Nevertheless, we successfully developed a ray tracing program to compute the resolution everywhere on the image.

The source spectrum has not been used to calculate the final resolution because the V He_α X-ray lines are spectrally wide enough: the corresponding area on the crystal is larger than Δ. Consequently, in our scheme, the spectral width has almost no effect on the resolution, but it controls the field of view. The spectral width has to be taken into account for a source diameter of about 500 µm.

The above discussion shows that it is possible to favor one direction against the other by moving the position either of the object or the detector. For instance, if the detector is close to the tangential focus, the resolution will be better in the tangential direction. This behavior was already observed in Section 2 with a = 83 mm (Fig. 3b) where the detector was half-way between the tangential and sagittal foci, giving equal resolutions in both directions. As shown on Figure 4 where only a small part of the grid is simulated, our ray-tracing program could reproduce correctly the experimental images. The sagittal and tangential resolutions correspond to the experimental data for the two different positions of the object. Indeed, the a = 80 mm case shows clearly that the horizontal resolution is favored, whereas the a = 83 mm case denotes almost equal resolutions in both directions. The program takes into account a Gaussian-shaped X-ray source in order to get a valid resolution. The spectral shape of the source is supposed to be a rectangular function of the wavelength, with a 0.006 Å width. Although the program reproduces a very similar behavior of the experimental data, the calculated resolutions are slightly better, especially a resolution better than 5 µm is predicted. This could be explained by a limited resolution due to the opacity of the grid and the shape of the wires, but another reason is probably the imperfections of the spherically bent crystal that is not taken into account in our program. A next step would be to simulate these imperfections in order to explain the limit in resolution that has also been reported in different experiments, from 2 to 10 µm depending on the crystal quality. We did implement the effect of the rocking curve, but it does not modify significantly the result. Other parameters, like a non-specula reflection, can be added, but they rely on unknown characteristics of the crystal.

Fig. 4. Results of ray tracing modeling of image formation for a 400 lpi grid object. The crystal-object distance is (a) a = 80 mm and (b) a = 83 mm.

At the same time, our modeling demonstrated that future experiments should be very accurately designed and alignment of crystal, object, source and detector must be very accurate. Indeed, our modeling and experimental results clearly show that spatial resolution is changed dramatically during measurements in the two rather closed configurations, a = 80 mm and 83 mm. Another important point, which should be considered for efficient design of such type backlighting experiments, is the role of backlighting source size for obtained spatial resolution. Indeed, Figure 5 shows the difference of the spatial resolution of image with two source sizes. One can see that, with a smaller source size (~50 µm) for our backlighting system, a significantly better resolution can be achieved. This gives a good indication on the maximum source size that should be used to optimize resolution, knowing that the crystal's imperfections cannot allow a resolution better than a few microns. Summarizing this part, we underline that the key point we want to demonstrate, thanks to this test grid images, is the possibility to obtain high spatial resolution in a specific direction even at incidence angles larger than 10° contrary to previous experiments.

Fig. 5. The same modeling as in Figure 3b, but the source size is (a) L = 50 µm and (b) L = 500 µm.

4. DENSITY MEASUREMENT APPLICATION AND RESULTS

One important application of X-ray sources is to radiograph and image warm dense matter with the aim to achieve a direct measurement of the mass density (Rosmej et al., Reference Rosmej, Lee, Riley, Meyer-ter-Vehn, Krenz, Tschentscher, Tauschwitz, Tauschwitz, Lisitsa and Faenov2007). The backlighting scheme discussed above was used to diagnose dense plasma. In particular, radiography of shock wave propagation in solid target was performed in order to determine the mass density. Indeed, shock-wave-equations of state experiments require two parameters to be measured, like the shock and fluid velocities in order to extract the thermodynamic properties of the material. For low-Z material, X-ray radiography has been used to determine both these velocities (Cauble et al., Reference Cauble, Perry, Bach, Budil, Hammel, Collins, Gold, Dunn, Celliers and Da Silva1998; Collins et al., Reference Collins, Da Silva, Celliers, Gold, Foord, Wallace, Ng, Weber, Budil and Cauble1998). However, the development of direct probing techniques to determine another shock parameter, such as density, would allow more precise absolute equations of state determinations and would represent a real break-through in the field. Several attempts have been made on plastic in the past (Hammel et al., Reference Hammel, Griswold, Landen, Perry, Remington, Miller, Peyser and Kilkenny1993, Reference Hammel, Kilkenny, Munro, Remington, Kornblum, Perry, Phillion and Wallace1994; Ravasio et al., Reference Ravasio, Koenig, Le Pape, Benuzzi-Mounaix, Park, Cecchetti, Patel, Schiavi, Ozaki, Mackinnon, Loupias, Batani, Boehly, Borghesi, Dezulian, Henry, Notley, Bandyopadhyay, Clarke and Vinci2008), using long-pulse laser-plasma X-ray sources. But in these experiments point projection radiography was performed. With our scheme, it is possible to reach a higher spatial resolution (10 µm) in the shock wave direction. Moreover, as the X-ray image is monochromatic, the mass density measurement of shocked plastic can be deduced with limited error bars (Benuzzi-Mounaix, Reference Benuzzi-Mounaix, Koenig, Ravasio, Vinci, Ozaki, Rabec le Gloahec, Loupias, Huser, Henry, Bouquet, Michaut, Hicks, MacKinnon, Patel, Park, Le Pape, Boehly, Borghesi, Cecchetti, Notley, Clark, Bandyopadhyay, Atzeni, Schiavi, Aglitskiy, Faenov, Pikuz, Batani, Dezulian and Tanaka2006, Reference Benuzzi-Mounaix, Loupias, Koenig, Ravasio, Ozaki, Rabec le Gloahec, Vinci, Aglitskiy, Faenov, Pikuz and Boehly2008). These rear side diagnostics provided the shock velocity and the temperature. The general set-up of the experiment using the backlighting scheme described in this paper is presented in Figure 6a. It was performed on the LULI2000, employing one of the kJ beam to drive a shock into a plastic sliver, and another to generate the X-ray source. The laser was first focused on an ablator-pusher foil to generate a strong shock, which propagates through the plastic sliver. To cross check the density measurement from the X-ray radiography, usual shock diagnostics such as VISAR and self optical pyrometry (SOP) (Fig. 6a) were also implemented. A typical image obtained of the shock into the plastic sliver is shown in Figure 6b while the deduced compression across the shock front (using Abel inversion) is presented in Figure 6c. A compression (ρ/ρ₀) around 2.85 was obtained in good agreement with compression estimated using hydrodynamic simulations and data deduced from the rear side diagnostics (Benuzzi-Mounaix, Reference Benuzzi-Mounaix, Loupias, Koenig, Ravasio, Ozaki, Rabec le Gloahec, Vinci, Aglitskiy, Faenov, Pikuz and Boehly2008). To achieve a high level of accuracy in the mass density measurement, a well-known reflected spectrum range as well as a good spatial resolution is required.

Fig. 6. (Color online) (a) The target is composed by a pusher, a three layer target (10 µm CH-10 µm Al – 10 µm CH) and a plastic sliver (415 µm in the radiographic direction). Self optical pyrometry and VISAR were also implemented. (b) X-ray shadowgraphy of the shock propagation in CH. (c) Compression given by Abel inversion of the shock front. The angles are larger than 10°, in contrary to previous experiments.