Information in 4D-STEM: Where it is, and How to Use it

TL;DR

This work extends Rose's contrast transfer formalism to 4D-STEM by decomposing information channels into coherent phase, coherent amplitude, and incoherent amplitude terms, and by separating the phase transfer into symmetric (tcBF) and antisymmetric (tcDPC) components. By coherently combining these channels into acBF, the method achieves continuous, nonzero phase transfer up to the 2$\alpha$ limit within the bright-field disk, offering a fast direct-ptychography–like reconstruction under the WPOA. It also uncovers a coherent amplitude transfer (ACTF) and introduces tilt-corrected dark-field (tcDF) for depth-sensitive information from a single dataset, enabling one-shot depth sectioning and richer information content than traditional WPOA-based approaches. Together, these results provide a unified, analytically tractable framework that generalizes phase-contrast theory to 4D-STEM and informs both direct and iterative ptychography under various coherence and thickness conditions, with practical implications for low-dose, high-resolution imaging of light- and heavy-element specimens.

Abstract

Contrast transfer mechanisms for electron scattering have been extensively studied in transmission electron microscopy. Here we revisit H. Rose's generalized contrast formalism from scattering theory to understand where information is encoded in four-dimensional scanning transmission electron microscopy (4D-STEM) data, and consequently identify new imaging modes that can also serve as crude but fast approximations to ptychography. We show that tilt correction and summation of the symmetric and antisymmetric scattering components within the bright-field disk -- corresponding to tilt-corrected bright field (tcBF) and tilt-corrected differential phase contrast (tcDPC) respectively -- enables aberration-corrected, bright-field phase contrast imaging (acBF) that makes maximal use of the 4D-STEM information under the weak phase object approximation (WPOA). Beyond the WPOA, we identify the contrast transfer from the interference between inelastic/plural scattering electrons, which show up as quadratic terms, and show that under overfocus conditions, contrast can be further enhanced at selected frequencies, similar to phase-contrast TEM imaging. There is also usable information encoded in the dark field region which we demonstrate by constructing a tilt-corrected dark-field image (tcDF) that sums up the incoherent scattering components and holds promise for depth sectioning of strong scatterers. This framework generalizes phase contrast theory in conventional/scanning transmission electron microscopy to 4D-STEM and provides analytical models and insights into full-field iterative ptychography, which blindly exploits all above contrast mechanisms.

Figures (12)

• Figure 1: CTEM/STEM reciprocity. (a) Scattering schematic of incident electrons at incident angles $\bm{\theta}$ for the STEM. Scattered electrons at angle $\bm{\theta}$ are recorded by a pixelated detector. (b) Reciprocity of CTEM and STEM. Defocus leads to a shift in images formed with tilted illumination in CTEM or from off-axial BF detectors in STEM.
• Figure 2: PCTF as a 4D function. (a,b) Projections of the disk overlap regions of the PCTF in the doubly reciprocal space, indicating the double overlap and triple overlap regimes of contrast transfer. (a) When viewed at constant spatial frequency $\boldsymbol{\omega}$, the PCTF shows where information transfer is located in the diffraction plane. (b) When viewed at constant detector position $\boldsymbol{\Theta}$, the PCTF shows the spectral response of an image formed from a single position in the diffraction plane. (c) Schematic showing the real space shifts for the image formed from each detector pixel. Correction of the shifts before combining the images recover the coherent contrast transfer. In tcDF imaging, the shifts are constant within each area of uniform color.
• Figure 3: (a) Phase contrast transfer function (PCTF) and (b) Detective quantum efficiency (DQE) of tcBF, tcDPC and acBF at 300 kV, 30 mrad convergence semi-angle. The tcDPC curve in (a) shows the imaginary part of its PCTF. The tcDPC envelope is equal to the PCTF of in-focus tcDPC. acBF fills in the difference between tcBF and tcDPC, which yields constantly non-zero contrast transfer.
• Figure 4: Imaging of a simulated single C atom at 300 keV and 30 mrad convergence semi-angle under different methods, (a) axial BF, (b) tcBF, (c) tcBF with post-summation CTF correction (sign flipping), (d) acBF. (e-h) Corresponding fast Fourier transform (FFT) power spectra of (ad). The dashed circles on the FFTs indicate the spatial frequencies corresponding to $1 \alpha$ and $2 \alpha$. acBF demonstrates superior information transfer with no information loss at all frequencies up to $2 \alpha$.
• Figure 5: Imaging of a dataset (taken from li2025atomically) of a 48-nm-thick metal-organic framework acquired at a dose of $\sim 100 \mathrm{e-} / \text{Å}^2$ reconstructed using different 4D-STEM phase contrast methods, (a) axial BF, (b) tcBF, (c) acBF, (d) multislice ptychography, along with their respective Fourier transforms (e-h). tcBF and acBF have been upsampled by 2. A zoom-in of the NU-1000 structure shows the Zr clusters are well resolved by both acBF and ptychography, and the Fourier transforms $(\mathrm{g}-\mathrm{h})$ indicates a resolution of better than $2 \text{Å}$. The dashed circles on the FFTs indicate the spatial frequencies corresponding to $1 \alpha$ and $2 \alpha$.