Phase-Vortex Removal for Quantitative X-Ray Nanotomography with Near-Field Ptychography

X-ray ptychotomography in the near-field regime is a promising technique for high-resolution, quantitative, and nondestructive investigations in materials science, paleontology, and biomedicine. X-ray near-field ptychography has been previously demonstrated in projection and tomography mode, but the quantitativeness of the reconstructed data has never been discussed in detail. Here, we use measurements of a sample made of aluminum and nickel microparticles to evaluate the quantitativeness of the volumetric mass-density data. Moreover, we propose an algorithm (VortRem) for the removal of phase vortexes, a type of artifact that frequently occurs in holographic methods. VortRem and the results presented here may be fundamental for extending the applicability of this emerging technique to quantitative three-dimensional characterization studies of light as well as dense samples down to the nanoscale.

Figure 1

A schematic representation of the setup used for near-field ptychography. (a) The scanning pattern used for each projection. (b) A hologram of $\mathrm{Al}$ and $\mathrm{Ni}$ particles in the glass capillary. The horizontal and vertical stripes in the image are caused by wave-front variation produced by the multilayer-coated Kirkpatrick-Baez mirrors. (c)–(f) The same sample detail at different scan positions [regions of interest (ROIs) of $300\times 300$ pixels], where the displacement of the sample image with respect to the background features can be appreciated. The nonhomogeneous background combined with the scanning procedure provides the necessary information to reconstruct the transmission function of the sample and the illuminating wave front.

Figure 2

The (a) phase (in radians) and (b) amplitude (in arbitrary units) of the illumination function retrieved with the near-field ptychographic algorithm. The features of the mirrors used for focusing the beam are clearly visible in both images.

Figure 3

(a) The phase part of the transmission function of the sample after 500 iterations of the DM algorithm. (b),(c) Enlargements of areas containing vortices.

Figure 4

The numerical curl of the phase map shown in Fig. 3, as computed with Eq. (1).

Figure 5

The low-pass filtered version of the numerical curl. Neighboring vortices of opposite charge are seen to mostly cancel each other, so that after thresholding, a single vortex, centered in (c), is left to be corrected.

Figure 6

Vortex cancellation using a naive antivortex (a) produces strong gradients in the areas known to be uniform (b).

Figure 7

(a) An antivortex map corrected using the absorption part of the reconstruction. (b) No global gradient or offset is introduced in the highly transmissive area of the sample.

Figure 8

(a) The phase projection of $\mathrm{Al}$ and $\mathrm{Ni}$ particles in a capillary obtained with the near-field ptychographic and VortRem algorithms. The profile taken along the dashed line can be seen superimposed on the figure. The dashed-square area at the bottom left of the image is used to determine the phase sensitivity. (b) A sagittal slice of the reconstructed value, with an axial slice through the same volume as an inset. The scale bar in (a) and (b) is the same. (c) A combined histogram of the density values measured in the full volume. The result of the fit of the histogram peaks with squared-Lorentzian functions is used to determine the refractive-index and mass-density values of the materials of the sample. The blue vertical lines represent the tabulated values for pure materials.

Figure 9

(a)–(c) Segmentation of the microparticles, color coded according to their size, measured here by means of the equivalent diameter. The top row [i.e., panels (b) and (d)] shows the rendering and measurements of the $\mathrm{Al}$ particles, while the bottom row [i.e., panels (c) and (e)] shows the corresponding data for the $\mathrm{Ni}$ particles. (d),(e) Scatter plots illustrating the values of the $\delta$ measurements of the segmented particles as a function of their volume. The measurements for the full particles are plotted with blue diamonds, while the measurements for the same particles where 10 pixels have been removed from their edges are plotted with orange diamonds.