Versatile 3D reconstruction framework for hard X-ray grazing incidence imaging of nanostructures

TL;DR

The paper tackles the challenge of 3D nanoscale reconstruction from grazing-incidence coherent imaging, where traditional DWBA-based projection methods fail to capture strong multiple scattering. It introduces a multislice forward model for ptychography that propagates the wavefield through perpendicular slices, enabling depth-resolved phase retrieval across arbitrary incidence and rotation geometries. The approach is implemented in the open-source PyGRAPES package and validated on experimental GISAXS data and diverse simulations, including rotated samples, nanoparticle growth, and multilayer stacks. The results demonstrate high lateral resolution and meaningful depth information for surface and near-surface structures, with broad implications for detailed surface characterization and materials science.

Abstract

Coherent imaging techniques such as ptychography offer powerful capabilities for 3D resolution of nanoscale structures. By application in grazing incidence, such techniques may achieve exceptional surface sensitivity as demonstrated by grazing incidence small angle scattering. This requires however an extension of the conventional analysis based on the Distorted Wave Born Approximation which is typically limited to stratified-layer models and statistical descriptions of in-plane structures. The prevailing implementations of reconstruction algorithms for ptychography based on the projection approximation fails to capture the significant multiple scattering that occurs in grazing incidence. We present a ptychographic reconstruction framework that replaces the single-scattering model with a multislice wave-propagation formalism tailored to grazing incidence. The framework supports simultaneous phase retrieval and reconstruction, and can incorporate multiple incidence angles, multiple rotation angles, and flexible experimental geometries into a single inversion. Reconstructions can be initialized from a random guess without strong structural priors, enabling the recovery of complex surface and near-surface nanostructures. This reconstruction framework is applied to both experimental and simulated datasets, demonstrating its versatility.

Figures (11)

• Figure 1: In the projection approximation (left), the sample’s transmission function is projected onto a single plane. In multislicing (right), the sample is divided into several slices perpendicular to the optical axis ($z$), and each of these slices is projected onto a separate plane. The wave field is then propagated from plane to plane, allowing for propagation effects to arise within the sample volume and for multiple interactions of the wave field with the sample.
• Figure 2: The grazing incidence geometry is simulated by aligning the optical axis parallel to the substrate surface, such that the slices are perpendicular to the substrate. The angle of incidence is then applied to the incident wave field in the form of a constant phase gradient. Reflection and multiple scattering arise naturally. Here $\theta_i$ is the angle of incidence, while $\psi_j$ refers to the $j_{th}$ wave in the simulation. $\Delta z$ is the slice thickness.
• Figure 3: a) A typical experimental setup for grazing incidence X-ray ptychography. Black circles indicate overlapping scan positions which are acquired as the sample is translated along the surface. b) a schematic illustrating the acquisition of multiple incidence angles used in a single reconstruction. c) Schematic illustrating another imaging mode akin to laminography, where different scans are acquired as the sample is rotated about its surface normal. Note that in b) and c) only a single beam footprint is shown for clarity.
• Figure 4: Reflectivity curve of W/Si multilayer at $\lambda = 1.54Å$. Kinematic curve calculated as in Nielsen Nielsen2011. Dynamical curve calculated using slab-model approach using the Abeles matrix method, as per Vignaud and Gibaud (2019)Vignaud2019REFLEX
• Figure 5: Top: a specific varying 2D structure used as the basis for both multislice and the projection approximation. The structure is entirely Au in the first case, and entirely Si in the second case, with an incidence angle of $\theta_i=0.4^\circ$ Middle: the output intensities, normalized, of the far-field exit wave, simulated via single slice and multislice with Au, showing reasonably good correlation. Bottom: The output intensities, normalized, of the far-field exit wave, simulated via single slice and multislice with Si, where multiple scattering becomes significant.