A Comparative Analysis of STEM Phase Retrieval Techniques: Evaluating Transfer of Information and Dose Efficiency

Reconstructing the phase information of weakly-scattering samples using intensity measurements is a longstanding problem in many imaging and diffraction fields, including electron microscopy [1, 2]. Recent hardware and algorithmic developments have renewed interest in using a set of converged probe diffraction intensities, that is 4D-STEM measurements [3], for dose-efficient phase retrieval in STEM. These include integrated center-of-mass (iCOM) imaging [4], tilt-corrected bright-field (tcBF) STEM [5], as well as direct ptychography, such as single-side band (SSB) and Wigner distribution deconvolution (WDD) [6-7], and iterative ptychography [8]. These STEM phase retrieval techniques all have their strengths and weaknesses, as well as distinct experimental acquisition and computational reconstruction requirements. iCOM imaging is the most computationally inexpensive technique yet suffers from low-spatial frequency artifacts and scan-step limited resolution. tcBF relaxes scan sampling requirements and provides robust estimates of the aberration surface yet exhibits “Thon-like” oscillations and contrast reversals characteristic of HRTEM. Direct ptychography techniques improve the fidelity of iCOM imaging and are robust at low electron doses yet require precise knowledge of the often-unknown incoming illumination. Iterative ptychography offers “super” resolution beyond the numerical aperture and can solve for the unknown illumination yet is very computationally expensive and slow at converging low spatial frequencies. Figure 1 illustrates the above observations graphically by formulating the contrast transfer function (CTF) as the outcome of convolution with the converged probe. The aperture autocorrelation function, which can be understood geometrically as the area of the double-overlap region of shifted apertures, acts as an envelope function for the CTF and is modulated by the probe aberration function [9]. While iCOM and tcBF should be performed in-focus and out-of-focus respectively, SSB illustrates strong signal under both conditions – provided the aberration surface is known. While the CTF is an important tool in assessing the performance of imaging techniques, representing the maximum usable signal in an ideal “infinite-dose” measurement, it fails to capture the inevitable effect of finite-dose at realistic detectors and is thus of limited utility to STEM practitioners. As two simple failure modes, notice how iCOM is predicted to obtain the zero-frequency component with high fidelity, which is inaccurate since, without a reference wave, we can only measure relative phase changes. Similarly, the iterative ptychography CTF is predicted to be unity, which we know from experimental observations of “Thon-like” oscillations in biological reconstructions over vitreous ice to be erroneous (Fig. 2) [10]. To investigate this, we perform numerical simulations on white-noise objects, that is samples with random phase and constant Fourier amplitude, for various electron doses repeatedly and compute the spectral signal-to-noise ratio (SSNR) [11]. The choice of a white-noise object implies this is trivially related to the square-root of the recently proposed detective quantum efficiency metric without the need for a reference reconstruction [12]. Figure 2 summarizes the results and highlights how for iCOM, tcBF, and SSB, the SSNR is, as expected, independent of electron dose, and can in-fact be modeled analytically. The SSNR for iterative ptychography however, illustrates curious behavior: At low electron doses, it asymptotically approaches the SSB SSNR while at higher electron doses, it starts to “fill-in” high-frequency information. The analytical and numerical results are compared against experimental tcBF and iterative ptychography reconstructions of biological crystals at large defocus [10]. The SSNR results are placed in the context of how practitioners can choose between different STEM phase retrieval techniques and design acquisition parameters. Moreover, we investigate the robustness of information transfer for each of these techniques using segmented detectors and discuss their promise for sub-second phase retrieval. Finally, we discuss how we can boost low spatial frequency convergence by using STEM ptychographic holography, where one uses a diffraction grating to scan multiple beams across the sample while keeping one of them over vacuum to act as a reference [13]. Analytical models for STEM phase retrieval contrast transfer functions (CTFs), highlighting the convolutional role of the converged probe and demonstrating different techniques are best suited for in-focus / defocused acquisitions respectively. Numerical and analytical results of the spectral signal to noise ratio (SSNR) for STEM phase retrieval techniques. The derived functional forms for tcBF and ptychography are compared against experimental power spectra of biological crystals.

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