X-ray lensless microscopy from undersampled diffraction intensities

X-ray coherent diffraction imaging including ptychography provides the nanoscale resolved three-dimensional description of matter. The combination of these approaches to the Bragg geometry case arouses a strong interest for its capability to provide information about strain state in crystals. Among the existing approaches, ptychography is particularly appealing because it allows the investigation of extended or weakly scattering samples. Coherent diffraction imaging approaches, based on redundancy in the collected diffraction intensity data set, are highly time consuming and rely on state-of-the-art mechanical setups, both being strong limitations for a general application. We show here that these can be overcome by regularization-based inversion algorithms introducing a priori structural knowledge. This method, which can be generalized to other wavelengths or beam sources, opens new possibilities for the imaging of radiation-sensitive specimens or very large samples.

Figure 1

The different bases involved in Bragg CDI. The sample related basis is ($\mathbf{x},\mathbf{y},\mathbf{z}$); ${\theta }_B$ denotes the Bragg angle corresponding to the Bragg vector ${\mathbf{G}}_{hkl}$. (Top) The measurement basis is $({\mathbf{q}}_1,{\mathbf{q}}_2,{\mathbf{q}}_3)$, with the first two directions along the area detector and the last one being probed as the sample is tilted along the rocking curve. (Bottom) The conjugate of this basis is $({\mathbf{r}}_1,{\mathbf{r}}_2,{\mathbf{r}}_3)$. The blue frame $W_{\mathbf{r}}$ is the smallest numerical window which still fully contains the exit field, represented in dark green.

Figure 2

Numerical sample. (a) Three-dimensional view of the numerical object. The axis black lines are 0.4 $\mu$m long. (b${}_1$),(b${}_2$) Modulus of the test object shown in the $(y,z)$ and $(x,y)$ planes, indicated in color in (a) (in arbitrary units). (b${}_3$) Phase (in radians) shown in the $(x,y)$ plane. (b${}_4$),(b${}_5$) Diffraction patterns in $({\mathbf{q}}_1,{\mathbf{q}}_2)$ at ${\mathbf{q}}_3=-2\delta q_3$ and ${\mathbf{q}}_3=0$, respectively. (c) Photon shot noise corrupted diffraction intensity patterns taken along the rocking curve with the beam centered between the lines (in photons). ${\mathcal{S}}_{+}$ is the full data set while ${\mathcal{S}}_{-}$ extracts one over four diffraction patterns from ${\mathcal{S}}_{+}$. The arrow length represents 50 $\mu$m${}^{-1}$.

Figure 3

Regularization on numerical data. (a) Same as Fig. 2b for the reconstructed object obtained from the ${\mathcal{S}}_{+}$ data set. (b) Same as (a) for the reconstructed object obtained from the undersampled data set ${\mathcal{S}}_{-}$. (c) Same as (b) introducing the regularization based on the support of the sample.

Figure 4

Regularization on undersampled experimental data. (a) The illumination function $P$ in the focal plane. The brightness represents $\left|P\right|$ on a logarithmic scale, whereas the color represents its phase. (b) A scanning electron microscopy top view of the two Si lines. The green and pink dots represent two beam positions. (c) and (d) Measured diffraction pattern at the center of the rocking curve, with beam at the green-dot and pink-dot positions, respectively. The arrow length is 30 $\mu$m${}^{-1}$. (e) Three-dimensional isosurface of $\left|\rho \right|$ retrieved from the under-sampled data set using regularization. The axis black lines are 0.4 $\mu$m long. (f) Two-dimensional cross section showing the retrieved displacement field component.