Combined tilt- and focal series scanning transmission electron microscopy is a recently developed method to obtain nanoscale three-dimensional (3D) information of thin specimens. In this study, we formulate the forward projection in this acquisition scheme as a linear operator and prove that it is a generalization of the Ray transform for parallel illumination. We analytically derive the corresponding backprojection operator as the adjoint of the forward projection. We further demonstrate that the matched backprojection operator drastically improves the convergence rate of iterative 3D reconstruction compared to the case where a backprojection based on heuristic weighting is used. In addition, we show that the 3D reconstruction is of better quality.

Aberration-corrected scanning transmission electron microscopes (STEMs) provide sub-Angstrom lateral resolution; however, the large convergence angle greatly reduces the depth of field. For microscopes with a small depth of field, information outside of the focal plane quickly becomes blurred and less defined. It may not be possible to image some samples entirely in focus. Extended depth-of-field techniques, however, allow a single image, with all areas in focus, to be extracted from a series of images focused at a range of depths. In recent years, a variety of algorithmic approaches have been employed for bright-field optical microscopy. Here, we demonstrate that some established optical microscopy methods can also be applied to extend the ∼6 nm depth of focus of a 100 kV 5th-order aberration-corrected STEM (αmax = 33 mrad) to image Pt-Co nanoparticles on a thick vulcanized carbon support. These techniques allow us to automatically obtain a single image with all the particles in focus as well as a complimentary topography map.