Recent work has revived interest in the scattering matrix formulation of electron scattering in transmission electron microscopy as a stepping stone toward atomic-resolution structure determination in the presence of multiple scattering. We discuss ways of visualizing the scattering matrix that make its properties clear. Through a simulation-based case study incorporating shot noise, we shown how regularizing on this continuity enables the scattering matrix to be reconstructed from 4D scanning transmission electron microscopy (STEM) measurements from a single defocus value. Intriguingly, for crystalline samples, this process also yields the sample thickness to nanometer accuracy with no a priori knowledge about the sample structure. The reconstruction quality is gauged by using the reconstructed scattering matrix to simulate STEM images at defocus values different from that of the data from which it was reconstructed.

Three-dimensional phase contrast imaging of multiply-scattering samples in X-ray and electron microscopy is extremely challenging, due to small numerical apertures, the unavailability of wavefront shaping optics, and the highly nonlinear inversion required from intensity-only measurements. In this work, we present a new algorithm using the scattering matrix formalism to solve the scattering from a non-crystalline medium from scanning diffraction measurements, and recover the illumination aberrations. Our method will enable 3D imaging and materials characterization at high resolution for a wide range of materials.

Three-dimensional phase-contrast imaging of multiply-scattering samples in x-ray and electron microscopy is challenging due to small numerical apertures, the unavailability of wave front shaping optics, and the highly nonlinear inversion required from intensity-only measurements. In this work, we present an algorithm using the scattering matrix formalism to solve the scattering from a noncrystalline medium from scanning diffraction measurements and simultaneously recover the illumination aberrations. We demonstrate our method experimentally in a scanning transmission electron microscope, recovering the scattering matrix of a heterogeneous sample with two layers of multiwall carbon nanotubes filled with TaTe2 core-shell structures, spaced 10nm apart in the axial direction. Our work enables phase contrast imaging and materials characterization in multiply-scattering samples at high resolution for a wide range of materials.

Increasing interest in three-dimensional nanostructures adds impetus to electron microscopy techniques capable of imaging at or below the nanoscale in three dimensions. We present a reconstruction algorithm that takes as input a focal series of four-dimensional scanning transmission electron microscopy (4D-STEM) data. We apply the approach to a lead iridate, PbIrO, and yttrium-stabilized zirconia, YZrO, heterostructure from data acquired with the specimen in a single plan-view orientation, with the epitaxial layers stacked along the beam direction. We demonstrate that Pb-Ir atomic columns are visible in the uppermost layers of the reconstructed volume. We compare this approach to the alternative techniques of depth sectioning using differential phase contrast scanning transmission electron microscopy (DPC-STEM) and multislice ptychographic reconstruction.

Increasing interest in three-dimensional nanostructures adds impetus to electron microscopy techniques capable of imaging at or below the nanoscale in three dimensions. We present a reconstruction algorithm that takes as input a focal series of four-dimensional scanning transmission electron microscopy (4D-STEM) data. We apply the approach to a lead iridate, Pb$_2$Ir$_2$O$_7$, and yttrium-stabilized zirconia,Y$_{0.095}$Zr$_{0.905}$O$_2$ , heterostructure from data acquired with the specimen in a single plan-view orientation, with the epitaxial layers stacked along the beam direction. We demonstrate that Pb-Ir atomic columns are visible in the uppermost layers of the reconstructed volume. We compare this approach to the alternative techniques of depth sectioning using differential phase contrast scanning transmission electron microscopy (DPC-STEM) and multislice ptychographic reconstruction.