Fig. 3: Finite-element distorted-wave Born approximation (FE-DWBA) approach for coherent surface scattering.

a The sketch shows the top and side views of a highly heterogeneous sample and the corresponding FEA grid pattern of the substrate-supported sample (not to scale). In the top view, the shading in yellow indicates the X-ray footprint, while the gray-scale shading shows the lateral electron density variation in the surface pattern. In the side view, the shadings highlight the lateral cells in the y-direction. b The four scattering contributions describe the full DWBA terms in two adjacent lateral cells. The transmission and reflection coefficients have also been indexed at each lateral interface. c CSSI pattern calculated with the FE-DWBA algorithm. The simulated scattering pattern reproduces the low exit angle speckles and the intense interference fringes in the dashed boxes, which reflect the nature of the multibeam or dynamical scattering mechanism, revealing the pattern thickness. d Quantitative comparison between the FE-DWBA simulation—the line scan at the center of the arc (equivalent to \({q}_{x}=0\)) along the dashed black line from \({\alpha }_{f}=0\) (horizon) to 0.3°, slightly above the silicon substrate critical angle (0.22°). The intensity oscillation, the signature of the dynamical scattering, below and near the critical angles of the substrate and the sample, matches well with the experimental data. e Scattering intensity line scan along the dashed white line containing the information on the form factor of the thin-film surface pattern along the length of the bar. The oscillation period corresponds to the bar length of 70.0 µm precisely. Therefore, this highlighted region in the single-view scattering image taken at an incident angle of 0.6° captures the 3D structural information of the sample. f 3D schematic of a more complex surface sample (shown not to scale), a petal-like pattern with a thickness of 55 nm. g Experimental data collected at 0.6° incident angle (right panel) and simulation by using the FE-DWBA method (left panel). The signature of the pattern thickness—the enhanced intensity modulations at the low exit angle is evident.