#### 3.1. Lens Array Characterization

The focal length of biconcave X-ray lenses is calculated according to the basic optics formula [

22]:

where

R is the radius of curvature,

N the number of stacked biconcave lenses and 1-

$\delta $ the energy-dependent real part of the index of refraction of the material in the X-ray range. Here, each CRL consists of 20 crossed cylindrical lenses which leads to an effective total of 10 biconcave lenses for each direction. The radius

R is 20

$\mathsf{\mu}$m and

$\delta =5.65\times {10}^{-6}$ at 9 keV. The expected focal length therefore is 177 mm. Scanning the distance between the detector and the array (see

Figure 3a) yields the minimal width of the focal spots at (200 ± 15) mm which is in good agreement with the design value. Note that the real composition of the photoresist is unknown and therefore the exact value of

$\delta $ may vary slightly. Additionally, lens errors may add to the shift. The slight difference in spot size in

x- and

y-direction originates from different sizes of the source in horizontal and vertical direction. The visibility

V (

$V=({I}_{max}-{I}_{min})/({I}_{max}+{I}_{min})$ with

${I}_{max}$ being the maximum and

${I}_{min}$ the minimum intensity in the square zone of each spot) of the spots peaks at the same distance as the effective focal length.

The achieved visibility of 0.93 is considerably higher than typical values for grating interferometry [

42]. Even in white-beam illumination (centered at 15 keV), we still obtained a high visibility of 0.5.

Figure 3b shows three line plots in

x-direction of different spot rows (solid lines, the upper two are shifted for clarity) in monochromatic mode. All of them show high gain factors compared to the free beam intensity (dashed lines). Two points have to be noted. First, not only intensity within the horizontal line is concentrated into the spots, but also in the perpendicular direction. Therefore intensity appears not to be conserved (area below dashed versus solid lines). Second, the minima in

Figure 3b are not the global minima in each zone of the spot. The SHARX consists of crossed cylinders acting as 2D focusing lenses at the exact intersection point, while focusing works only in one direction at positions away from the cylinder cross points. The minima in

Figure 3b reflect such positions leading to an increased background intensity compared to areas of no intensity increase or even intensity decrease. The absolute visibility, however, was derived from analysing a square area around each spot. Nevertheless, the spot sizes and thus one major factor to spot shift sensitivity are dominated by two effects, the finite size of the synchrotron source and the aberrations of the cylinder lenses (spherical aberration and partly focusing regions due to the reduced fill factor of the cylinders).

#### 3.2. Reconstruction of The Phase Shift of a Diamond Lens

The SHARX can be used to analyze the changed wavefront of the X-ray beam in the exit plane of an X-ray optical element. For demonstration we analysed a CRL composed of 8 plano-concave parabolic lenses made from single-crystal diamond. Each unit lens has a design radius of curvature of 200

$\mathsf{\mu}$m at the vertex of the parabola and an entrance aperture of around 0.9 mm [

43,

44] leading to a focal length of about 2.5 m (at 8.5 keV).

Figure 4a clearly illustrates the effect produced by the CRL inserted in the X-ray beam: the almost rectilinear grid of X-ray spots formed by the SHARX (left part) is bent toward the central axis of the CRL (right part). Shift in meridional and sagittal planes of each beamlet are derived by comparing reference and distorted image. The resulting local refraction angle in the CRL exit plane is shown as a vector field in

Figure 4b. As expected, spots at the peripheral part of the CRL experience larger shifts than peaks located near the lens central axis. From these shifts we could easily reconstruct the X-ray phase shift using, for instance, the modified Southwell algorithm [

38] (see also

Section 2.3). The calculated phase shift (

Figure 4c) is in reasonable agreement with the prediction based on the design values.

To verify the accuracy of the performed phase front metrology, the deflection of the X-rays introduced by the diamond CRL was derived independently from a microtomography (CT) with the CT rotation axis coinciding with the CRL optical axis [

45]. The reconstructed distribution of material density was subjected to a segmentation procedure in order to outline the surfaces of the diamond. These in turn were used to calculate the thickness of diamond projected onto the CRL exit plane. Finally, this projected thickness was converted into X-ray deflection angles using the relation

$\beta (x,y)=\Delta T(x,y)\times {\delta}_{d}$, where

$\Delta T(x,y)$ is the finite difference approximation to the first derivative of the thickness and 1-

${\delta}_{d}$ the real part of the index of refraction of diamond. The result is shown as dashed line in

Figure 4d and compared to the SHARX deflection angles (red stars and black squares). Both angular shifts match well within a 4

$\mathsf{\mu}$rad confidence and relate to a CRL with effective radius of curvature at the apex of 205

$\mathsf{\mu}$m. The reproducibility of differential phase in the unperturbed area outside the diamond lens was derived to be 1.8 and 4

$\mathsf{\mu}$rad in horizontal and vertical direction, respectively. Smaller deviations can be detected by reducing the noise on the data by averaging (0.5

$\mathsf{\mu}$rad in the ablation run). A very good agreement between the measured beam deflection by the SHARX and the indirectly estimated deflection from the CT demonstrates that a precise phasefront metrology can be performed with the SHARX device.

2D representations of the differential phase and diffraction contrast in

x and

y-direction are shown in

Figure 4e. Phase-contrast is shown for both directions, reproducing the finding of phase front curvature. The diffraction patterns show no significant signal as expected from a smooth material. However, the edges display an elevated diffraction level. Here two effects appear: the abrupt beam displacement within the individual beamlets, leading to a high curvature of the wave front, and an enhanced roughness due to processing. The first effect is in line with the interdependence of diffraction and phase from objects sized within the spatial resolution [

46].

#### 3.3. In-Situ Imaging of Laser-Induced Cavitation Bubble

Pulsed laser ablation in liquid (PLAL) is employed to produce ligand-free nanoparticles in (e.g., aqueous) suspension [

47]. Due to its hierarchical processes spanning several length scales and the fast dynamics of structure formation the mechanisms and control of the process are still under investigation [

48]. Within the present nanosecond excitation it is known that ablated material is ejected from the target by phase explosion, while at the same time a plasma is ignited to modify particle formation and foster energy coupling into the water. The latter is easily visible as a millimeter-sized transient cavitation bubble [

41]. This bubble develops into a hemispherical shape of some 1.5 mm radius within a typical 150

$\mathsf{\mu}$s lifetime. It is known that this bubble contains a major part of the ejected mass from the target as nanoparticles [

41,

49] and the mutual interaction governs particle ripening [

50].

With the SHARX setup we could measure differential phase and diffraction in addition to the absorption. The bubble is imaged with the spatial resolution of the pitch of the SHARX (50

$\mathsf{\mu}$m). While the absorption and phase shift contain information about the bubble structure, the diffraction channel contains the scattering signal from structures in the nanometer up to the sub-micrometer scale as indicated by the sensitivity graph in

Figure 5. Due to the short exposure time an averaging over a number of (nominally reproducible) events is necessary for improving the signal-to-noise ratio.

Figure 5f shows an absorption contrast radiography without the SHARX for orientation [

40] over a large field of view. Due to the limited size of the SHARX array we sampled separately two different positions relative to the cavitation bubble, which were merged afterwards to cover the region of interest (see

Figure 5f).

Figure 5a–d shows the results at maximum bubble expansion for the different contrast channels. The differential phases in both directions clearly show the apex of the bubble, which coincides with the highest phase gradient. The directional sensitivity is unambiguous. A reconstruction of the absolute phase could be calculated as for the diamond lens (not shown here).

As known from investigations with Hartmann masks, the diffraction channel is referred to the scattering of spatially unresolvable structures [

14,

15,

32]. As explained in theory by Lynch et al. [

25] the sensitivity for different particle sizes is mainly dependent on the pitch of the spots, the X-ray energy and the distance between the sample and the detector. The sensitivity peak for the present setup is located at a structure size of 300 nm with a FWHM (full width at half maximum) from 110 to 1500 nm [

25,

26] as shown in the sensitivity curve in

Figure 5e. From earlier studies it is known that the main size fraction of particles produced by PLAL is smaller than the present range [

27,

51]. However, ripening processes produce large particles that enter the visibility interval. These larger agglomerates (>100 nm) will be visible in diffraction contrast with the present setup. A change in the setup geometry (lens pitch, distances) furthermore would enable to modify the sensitivity range.

The small observed signals in the diffraction channel in

Figure 5d (average of the two directional ones as no anisotropic scattering is expected) is of similar spatial distribution as the absorption signal. Therefore it is reasonable to consider a homogeneous filling of the bubble with scattering structures. On the other hand, some residual crosstalk from absorption or phase-contrast [

33,

52] may contribute to the signal and is subject of ongoing investigations. The high diffraction signal at the bubble boundary is again attributed to the crosstalk from an unresolvable sharp phase change [

46].

A signal decomposition as in [

52,

53] could not shed more light on this issue. These signals are visible by Fourier analysis as well [

18]. A further setup optimization and data analysis are required. It should be kept in mind that the integration time for the presented data only adds up to a total of 14 ms given the 33

$\mathsf{\mu}$s exposure of the individual frames.