Distribution of unresolvable anisotropic microstructures revealed in visibility-contrast images using x-ray Talbot interferometry

X-ray Talbot interferometry has been widely used as a technique for x-ray phase imaging and tomography. We propose a method using this interferometry for mapping distribution of parameters characterizing anisotropic microstructures, which are typically of the order of $\mu \mathrm{m}$ in size and cannot be resolved by the imaging system, in a sample. The method uses reduction in fringe visibility, which is caused by such unresolvable microstructures, in moiré images obtained using an interferometer. We applied the method to a chloroprene rubber sponge sample, which exhibited uniaxial anisotropy of reduced visibility. We measured the dependencies of reduced visibility on both the Talbot order and the orientation of the sample and obtained maps of three parameters and their anisotropies that characterize the unresolvable anisotropic microstructures in the sample. The maps indicated that the anisotropy of the sample’s visibility contrast mainly originated from the anisotropy of the microstructure elements’ average size. Our method directly provides structural information on unresolvable microstructures in real space, which is only accessible through the ultra-small-angle x-ray scattering measurements in reciprocal space, and is expected to be broadly applied to material, biological, and medical sciences.

Figure 1

Schematic illustration of experimental setup of x-ray Talbot interferometry

Figure 2

Schematic views of samples (a) and images obtained using x-ray Talbot interferometry at $\Phi =0^{\circ }$ [(b) absorption, (c) differential-phase, and (d) visibility-contrast images]. In upper figures, cylinder axis of sample is parallel to optical axis, while in lower figures, it is perpendicular to optical axis. Talbot order $p$ was fixed at 0.5.

Figure 3

Dependencies of visibility-contrast image on orientations of two samples in Fig. 2. (a) $\Phi =0^{\circ }$, (b) $\Phi ={45}^{\circ }$, and (c) $\Phi ={90}^{\circ }$.

Figure 4

(a) $-\ln \left(V/V_0\right)$ image at $\Phi =0^{\circ }$ for lower sample in Fig. 3 ($p=0.5$). (b) Dependence of $-\ln \left(V/V_0\right)$ on $\Phi$ for a pixel in square area in (a) ($p=0.5$; open circles: experimental results; solid curve: best-fit sine function). (c) Color representation of anisotropy of $-\ln \left(V/V_0\right)$ in square area in (a) (color: ${\Phi }_{\min }$; brightness: amplitude of best-fit sine function). (d) Map of normalized visibility averaged over $\Phi$ in same area.

Figure 5

(a) Experimental results of $pd$ dependence of $-\ln \left(V/V_0\right)$ for a pixel in square area in Fig. 4a (crosses: $\Phi ={15}^{\circ }$; filled circles: $\Phi ={60}^{\circ }$; triangles: $\Phi ={105}^{\circ }$). Solid, dashed, and dotted curves are best-fit curves obtained by least-squares fittings using model of Eq. (7). Dependencies of $\xi$, $H$, and ${\sigma }^2$ on $\Phi$ for the pixel are shown in (b), (c), and (d) (open circles) with best-fit sine functions (solid curves).

Figure 6

Color representation of anisotropies of (a) $\xi$, (b) $H$, and (c) ${\sigma }^2$ in square area in Fig. 4a [brightness scales: 0 $\sim$ 3 (a); 0 $\sim$0.25 (b); 0 $\sim$ 1.5 (c)]. Amplitudes of best-fit sine curves for 100 arbitrarily chosen pixels are plotted against ${\Phi }_{\max }$ for (d) $\xi$, (e) $H$, and (f) ${\sigma }^2$.

Figure 7

(a) Map of ${\sigma }^2$ averaged over $\Phi$ and (b) absorption image ($-\ln T$ image, where $T$ is transmittance of sample) in square area of image in Fig. 4a.

Figure 8

Normalized autocorrelation function $\gamma$ defined by Eq. (3) for a pixel.

Figure 9

Absorption tomograms of CR sponge used in our experiment. (a) Schematic view of sample; (b) and (c) are section images along planes A and B.