The 2D approximation quickly breaks down in reflection ptychography

TL;DR

This paper develops a 3D model for reflection ptychography, revealing that the common 2D approximation quickly becomes invalid with increasing sample thickness, especially near Bragg minima.

Contribution

It derives explicit thickness criteria for 2D model validity in reflection ptychography and demonstrates how incorporating depth-dependent propagation improves reconstructions.

Findings

2D approximation validity is limited to very thin samples in reflection geometries.

Conventional 2D reconstructions show thickness-dependent artifacts near Bragg minima.

Depth-dependent forward modeling enables accurate thickness recovery.

Abstract

Ptychographic reconstructions in reflection geometries are commonly interpreted with the same two-dimensional thin-sample model used in transmission, yet the validity of this approximation has not been established. We develop a three-dimensional weak-scattering description of reflection ptychography and derive explicit thickness criteria for when a two-dimensional model remains accurate. Because the sampled axial spatial frequency range is dominated by the rotation of the Ewald sphere rather than its curvature, reflection geometries impose far stricter thin-sample conditions than transmission geometries. The allowable thickness is reduced by one to two orders of magnitude for a representative extreme ultraviolet geometry, depending on the tolerance for appearance of artifacts. Simulations verify that conventional two-dimensional reconstructions may exhibit the thickness-dependent artifacts as predicted by the theory, with particularly strong distortions near specular Bragg minima. We further show that incorporating the correct depth-dependent propagation into the forward model resolves these distortions and enables recovery of sample thickness. These results establish practical validity limits for two-dimensional reflection ptychography and identify a path toward quantitative depth-sensitive reconstructions at all geometries.