Transmission electron microscopy (TEM), both with conventional collimated illumination and with a focused probe in scanning mode, is a very powerful tool for studying materials at the nanoscale. The ability to resolve atomic structure in materials allows for detailed investigation of how properties at the atomic level impact macroscopic behavior and properties of materials Haider et al., 1998. In this work, we focus on scanning transmission electron microscopy (STEM) Crewe et al., 1970, although references to high-resolution TEM (HRTEM) will be made throughout.

The most commonly used imaging method in STEM is annular dark field (ADF), which integrates the intensity on a detector covering an annular range outside the directly transmitted beam – meaning it detects scattered electrons. This method gives easily interpretable images where the detected intensity is roughly proportional to $Z^2$, where $Z$ is the atomic number Pennycook & Jesson, 1991. However, the collection efficiency of ADF is poor and the $Z^2$ dependence of the intensity complicates imaging of lighter elements together with heavier ones Klenov & Stemmer, 2006. A solution is to place a detector in the bright field (BF) region and collect the direct beam and its interference with scattered beams. However this complicates the interpretation of the images, necessitating computational techniques to extract useful information from the measured data.

Phase Retrieval Methods¶

When a converged electron probe passes through a thin sample, the electron wavefunction is modulated in both phase and amplitude. Most of the sample information we are typically interested in is encoded in the phase of the electron wavefunction. However, this information is “lost” during intensity measurements using physical detectors – giving rise to the so-called “phase problem” Drenth et al., 1975Fienup, 1982. This problem is especially apparent in weakly-scattering samples, such as soft or biological materials, which exhibit negligible amplitude contrast. To remedy this, a number of phase-retrieval techniques have been developed to computationally reconstruct this phase information from intensity measurements.

The simplest such technique is the integrated center of mass (iCOM) method Dekkers & De Lang, 1974Lazić et al., 2016, which recovers the projected sample phase by integrating an estimate of the scattered probe COM. Direct ptychographic methods, such as single side-band (SSB) Pennycook et al., 2015Yang et al., 2016, Wigner distribution deconvolution (WDD) Rodenburg & Bates, 1992Yang et al., 2017, optimum bright field (OBF) Ooe et al., 2021Ooe et al., 2024, and tilt-corrected bright field (tcBF) Yu et al., 2024Varnavides et al., 2024, retrieve the phase more explicitly using the interference information in overlapping regions between unscattered and scattered beams. The phase information can also be retrieved by solving the inverse scattering problem of how the probe and sample interact to produce the observed intensity measurements. Solving this iteratively, in a family of techniques under the umbrella of iterative ptychography Rodenburg & Maiden, 2019, provides self-consistent estimates for both the probe and the sample.

Article Outline¶

Multiple different methods may be used to extract information from such detector datasets, each with its own usecases and limitations. In this work, we investigate the efficacy of STEM phase retrieval methods to transfer phase information, as a function of the imaging system properties and experimental parameters, using a combination of analytical and numerical analyses and interactive widgets.

The manuscript is structured as follows: First, we introduce the theory of image formation in STEM experiments following Hammel & Rose (1995), highlighting the limiting components for the contrast transfer of information (CTF) – namely the probe aperture and the aberrations surface. We then introduce analytical expressions for the CTF for each reconstruction method using pixelated detectors, before investigating how robust the CTF is to acquisitions with few-pixel, segmented detectors. Motivated by our observations, we propose a direct ptychographic algorithm which combines tcBF and SSB, to achieve upsampled reconstructions without missing frequencies (CTF zero-crossings). Finally, we consider the effect of Poisson-limited detectors and finite electron fluence, by introducing the spectral signal-to-noise (SSNR) formalism for statistically evaluating the transfer of information.