Abstract

Host–guest chemistry in microporous crystals underpins key applications in catalysis 1 , separation 2 and sensing 3 , yet the unambiguous identification of guest species within host frameworks remains a central challenge. Recent advances in low-dose phase-contrast electron microscopy have enabled real-space imaging of guest moieties and transformed this field 4 , 5 . Here, however, we show that standard implementations of these techniques can generate artefactual in-pore contrast owing to non-ideal contrast transfer, thus compromising the reliability of guest identification. We address this limitation with a reconstruction method based on Gaussian-apodized single-sideband electron ptychography, which suppresses such artefacts and yields chemically interpretable phase images. We apply this method to an important catalytic system, achieving unbiased identification of guest metal–oxo clusters within microporous zeolite hosts. Our results are supported by complementary diffraction and spectroscopy. Beyond revealing previously unreported framework-confined metal active sites in zeolites, this method is expected to enable high-fidelity visualization of a broad range of local inhomogeneities across crystalline porous materials.

Main

Host–guest chemistry originally described supramolecular complexes 6 and was later extended to other material systems, particularly microporous materials such as zeolites and metal–organic frameworks. Precise identification of guest species within host matrices is essential for understanding how host structures govern guest binding, transport and transformation but remains challenging, largely because of the limitations of existing characterization methods. Spectroscopic techniques can probe host–guest interactions but typically lack spatial resolution 7 , 8 . Diffraction methods determine the average structure of crystalline host–guest composites 9 , yet they are insensitive to spatial inhomogeneities arising from non-uniform guest distributions.

Electron microscopy enables real-space imaging of non-periodic structures at atomic resolution, making it particularly powerful for identifying guest species within host materials. However, achieving high-resolution images while preserving structural fidelity has long been hindered by beam damage and the weak contrast of guest species 10 . Notably, recent advances in low-dose phase-contrast imaging—such as low-dose high-resolution transmission electron microscopy (HRTEM) 11 , integrated differential phase-contrast (iDPC) scanning transmission electron microscopy (STEM) 12 , optimum bright-field (OBF)-STEM 13 and electron ptychography 14 —have transformed the field by offering substantially greater dose efficiency than conventional single-channel intensity-integrating STEM 15 . These methods have been employed to visualize adsorbed or immobilized guest species in zeolites 16 , 17 and metal–organic frameworks 18 . Nevertheless, the phase-contrast formation mechanisms that govern the capabilities and limitations of these techniques—and their impact on the achievable image fidelity—have not always been carefully considered. As a result, assigning guest species solely from in-pore image contrast can be unreliable, particularly without rigorous cross-validation using complementary characterization methods.

Herein, we show that widely used phase-contrast imaging methods can produce artefactual contrast, particularly within pores, thereby hindering reliable identification of guest species because of their band-pass–shaped phase-contrast transfer functions (pCTFs). We further develop a method of electron ptychographic reconstruction that overcomes this limitation. When we applied it to an important catalytic system, it faithfully identified cobalt clusters within zeolite pores in previously unrecognized forms.

Artefacts from band-pass phase-contrast transfer

Phase-contrast electron microscopy generates image contrast from specimen-induced phase shifts arising from interference between the transmitted and diffracted beams 19 . The conversion of these phase shifts into measurable contrast is modulated in reciprocal space by a spatial-frequency-dependent transfer function (often referred to as a pCTF) 20 , which is determined by the microscope optics and the imaging modality (and, where applicable, the reconstruction algorithm). Equivalently, in real space, the phase-contrast image corresponds to the convolution of the specimen-induced phase with the associated point spread function (PSF).

In HRTEM mode with parallel illumination, the interpretation of image contrast is often not trivial because the pCTF oscillates between positive and negative values over a broad range of spatial frequencies 21 . These oscillations cause contrast reversals and can generate artefactual contrast, which is particularly pronounced in empty voids, channels or pores, as reported previously 22 . This limitation of HRTEM is well recognized.

In STEM modes with convergent illumination, pCTFs typically show only weak or no oscillations between positive and negative values, making artefact-generation mechanisms less evident—and, therefore, less widely appreciated—than in HRTEM. Nevertheless, many phase-contrast STEM modalities, including bright-field, annular bright-field, DPC- and OBF-STEM as well as single-sideband ptychography (SSB-Ptycho), exhibit band-pass transfer characteristics, as described by their pCTFs 23 . Such band-pass transfer behaviour corresponds in real space to a PSF with pronounced sidelobes, which can manifest as negative haloes around atoms and depresses their surrounding phase. This effect has been noted in attempts to probe interatomic charge transfer in two-dimensional materials 24 . When imaging nanoporous materials, the same effect can delocalize and redistribute a strong contrast from pore walls into the pore interior so that it generates a spurious in-pore contrast, an issue that has largely been overlooked.

To illustrate this point, we use SSB-Ptycho as a representative phase-contrast STEM approach and examine how its standard implementation influences the identification of guest species within zeolite pores, with ZSM-5 (an MFI-type zeolite) as a model system. The working principle of SSB-Ptycho has been described in detail elsewhere 25 and is summarized here. Briefly, SSB-Ptycho starts from a four-dimensional STEM (4D-STEM) dataset consisting of convergent beam electron diffraction (CBED) patterns recorded at each probe position (Fig. 1a). After Fourier transforming the dataset with respect to probe position, the signal at each spatial frequency comprises a central zero-order disc that partially overlaps with displaced sideband discs (Fig. 1b). Integrating over only the double-overlap regions yields the object spectrum, and an inverse Fourier transform then produces a complex real-space image from which both amplitude and phase (proportional to the projected potential) maps are obtained. The pCTF magnitude of SSB-Ptycho is proportional to the area of the double-overlap region as a function of spatial frequency 26 , giving a characteristic band-pass profile (Supplementary Fig. 1).

a, Schematic of the 4D-STEM data acquisition. A focused probe Ψ in is raster-scanned across the specimen with transmission function φ, generating an exit wave Ψ out . At each probe position, a pixelated detector records the CBED patterns intensity I D = |Ψ D | 2 , yielding a four-dimensional dataset. b, Reconstruction workflow and comparison with standard SSB-Ptycho. The four-dimensional dataset is Fourier transformed with respect to probe position to form G(K, Q). The signal in the double-overlap regions is integrated to recover the object spectrum. In standard SSB, this integration corresponds to a band-pass transfer. In GASSB, Gaussian apodization is applied to the double-overlap regions so as to reshape the transfer function into a Gaussian form. An inverse fast Fourier transform (FFT −1 ) then yields the reconstructed transmission function in real space. The mathematical symbols used in the figure are defined in the Methods, where further details of the reconstruction procedures are also provided.

SSB-Ptycho phase images reconstructed from simulated 4D-STEM datasets acquired along the [010] zone axis of ZSM-5 exhibit pronounced, conformal contrast within the 10-membered-ring (10-MR), 6-MR and 5-MR channels, even though these channel spaces are empty (they contain no guest species in the simulation) (Extended Data Fig. 1). These artefactual contrasts persist upon varying the off-zone-axis angles (Supplementary Fig. 2). We attribute these in-channel artefactual contrasts to the band-pass–shaped pCTFs of SSB-Ptycho, which yield PSFs with oscillatory sidelobes that appear as circular ripples in simulated phase images of a single carbon atom (Extended Data Fig. 2). When convolved with the ZSM-5 projected potential, sidelobe contributions from neighbouring atomic columns overlap within the pores and give rise to spurious in-pore contrast. This effect becomes more pronounced as the convergence semi-angle (α) decreases (Extended Data Fig. 2c). In the presence of aberrations, such as a coma, the artefacts can become eccentric and non-conformal to the pore geometry, making them easy to misinterpret as in-pore guest species, and they can obscure the contrast of genuine guests (Extended Data Fig. 2d). Applying a band-pass filter to the simulated projected potential of ZSM-5 produces a similar in-pore contrast, corresponding to the well-known ‘ringing’ artefact 27 , which represents another manifestation of the same underlying mechanism (Supplementary Fig. 3). These observations indicate that phase-contrast imaging methods with band-pass–shaped pCTFs have inherent limitations for reliably probing host–guest chemistry in porous materials.

Gaussian-apodized SSB ptychography

To eliminate artefacts arising from PSF sidelobes, we developed a 4D-STEM electron ptychography method coupled with Gaussian apodization. This approach, termed Gaussian-apodized SSB ptychography (GASSB-Ptycho), reshapes the pCTF towards a Gaussian profile by applying spatial-frequency-dependent weighting to the double-overlap regions (Fig. 1). A detailed description is provided in Methods. Because the Gaussian function preserves its functional form under a Fourier transformation, a Gaussian-shaped pCTF corresponds to a Gaussian PSF and intrinsically suppresses sidelobe formation. A simulation-based comparison of SSB- and GASSB-Ptycho images of a single carbon atom provides direct validation of this effect (Extended Data Fig. 3).

We first evaluated the performance of GASSB-Ptycho using simulated, aberration-free 4D-STEM datasets acquired along the [010] zone axis of pristine ZSM-5, at both infinite and experimental electron doses (approximately 5,700e − Å − 2 ). The reconstruction shows clean 10-MR, 6-MR and 5-MR channels and agrees well with the simulated electrostatic potential, in clear contrast to a standard SSB-Ptycho reconstruction, which displays pronounced in-pore artefacts (Extended Data Fig. 1).

We subsequently applied GASSB-Ptycho to experimental datasets acquired from calcined ZSM-5 without guest species and compared the results with standard SSB-Ptycho. To mitigate the influence of residual aberrations in the experimental data, we incorporated the singular-value-decomposition matrix inversion algorithm 28 into the reconstruction process (Extended Data Fig. 4). In standard SSB-Ptycho reconstructions, the artefactual contrast appears in the 10-MR channels as bright spots that can be readily misinterpreted as guest species, such as metal clusters (Fig. 2a). Similar artefacts are also observed when the same dataset is reconstructed using OBF- and iDPC-STEM, two methods widely used for zeolite imaging (Supplementary Fig. 4). By comparison, GASSB-Ptycho yields a reconstruction with clean, artefact-free channels (Fig. 2b).

a, b, Phase images of ZSM-5 reconstructed from the same 4D-STEM dataset acquired along the [010] zone axis using standard SSB-Ptycho (a) and GASSB-Ptycho (b). Scale bars, 2 nm. The corresponding projected structural model is shown on the right. c, d, Phase images of MIL-101(Cr) reconstructed from the same 4D-STEM dataset acquired along the [101] zone axis using standard SSB-Ptycho (c) and GASSB-Ptycho (d). Scale bars, 5 nm. The corresponding projected structural model is shown on the right. Insets, false-colour images to enhance the visibility of the weak contrast. e–h, FFT patterns of the reconstructions in a (e), b (f), c (g) and d (h). The dashed lines highlight the contrast transfer properties across spatial frequencies.

Artefacts arising from band-pass phase-contrast transfer have a particularly strong impact on identifying structural features associated with small pores formed by many closely spaced atoms. For example, Fig. 2c shows an SSB-Ptycho reconstruction from experimental 4D-STEM data of the metal–organic framework MIL-101(Cr) acquired along the [101] zone axis, in which the supertetrahedra—formed by terephthalate linkers and Cr 3 O trimers—are barely discernible. By contrast, the GASSB-Ptycho reconstruction clearly resolves the Cr 3 O trimers and the supertetrahedra they form (Fig. 2d). The Gaussian-type contrast transfer modulation achieved by GASSB-Ptycho is more clearly reflected in the fast Fourier transform (FFT) patterns of the reconstructed images (Fig. 2e–h).

The systematic comparisons above, based on both simulated and experimental data, demonstrate that GASSB-Ptycho effectively suppresses artefacts arising from PSF sidelobes, which are prevalent in reconstructions obtained with conventional phase-contrast imaging methods. This capability makes GASSB-Ptycho well suited for probing host–guest chemistry in nanoporous materials, for which the faithful identification of guest species is essential and spurious in-pore contrast can lead to severe interference and misassignment.

Identifying metal–oxo active sites in zeolite

Cobalt-incorporated ZSM-5 (Co–ZSM-5) has shown outstanding catalytic performance in diverse reactions, such as selective oxidation 29 and dehydroaromatization 30 . However, the precise chemical nature and spatial distribution of the Co–oxo active sites remain elusive, largely because methods capable of probing them directly with high fidelity are still lacking. To investigate this important catalytic system using the GASSB-Ptycho approach developed here, we synthesized a Co–ZSM-5 catalyst with a Co loading of 0.967 wt% through a well-established ion-exchange method 31 .

Elemental mapping with energy-dispersive X-ray spectroscopy confirmed the incorporation and widespread distribution of Co throughout the ZSM-5 crystals (Supplementary Fig. 5). Powder X-ray diffraction of Co–ZSM-5 exhibited attenuated low-angle reflections compared with pristine ZSM-5 (Supplementary Fig. 6), indicative of the presence of guest species within the micropores that modulate the local electron density 32 . Raman and infrared spectroscopy indicate that the confined Co species are Co–oxo motifs terminated by –OH groups (Fig. 3c,d), consistent with previous reports 33 , 34 .

a, Reciprocal lattice of Co–ZSM-5 reconstructed by PEDT. b, Difference electrostatic potential maps of Co–ZSM-5 referenced to pristine ZSM-5, viewed along the [010], [100] and [001] zone axes with the ZSM-5 framework overlaid. Red and black ellipses denote circumferentially distributed and radially distributed extra potential, respectively. c, Raman spectra of ZSM-5 and Co–ZSM-5, with other bands at 1,082 cm −1 and 1,197 cm −1 in Co–ZSM-5 assigned to Co–O–Si asymmetric stretching and framework-coupled vibrations associated with Co species, respectively. d, Infrared spectra of ZSM-5 and Co–ZSM-5 measured at 500 °C. The bands centred at 3,532 cm −1 and 3,724 cm −1 are typically assigned to H-bonded and isolate silanol groups, respectively. The other band at 3,680 cm −1 in Co–ZSM-5 is attributed to Co–OH groups. e–g, GASSB-Ptycho (e), SSB-Ptycho (f) and annular dark-field STEM (g) images of Co–ZSM-5 reconstructed from the same 4D-STEM dataset. Scale bars, 2 nm. Boxed regions are shown magnified on the right. h, Line profiles extracted along the arrows in e–g (blue, orange and green, respectively), showing contrast within the zeolite channels. Black and purple triangles mark channels with and without cobalt species, respectively. a.u., arbitrary units.

Precession-enhanced electron diffraction tomography (PEDT) was performed to obtain averaged structural information on the Co–oxo species within ZSM-5 (Fig. 3a). Datasets collected from five crystals (Supplementary Fig. 7 and Supplementary Tables 1 and 2) were merged to generate a difference electrostatic potential map, referenced to calcined pristine ZSM-5 with empty channels. The map clearly reveals further electrostatic potential within the 10-MR channels along the [010] direction (Fig. 3b).

Notably, the signal is offset from the channel centre and can be divided into two merged components: (1) a circumferentially distributed component along the T 1 , T 2 and T 5 sites and (2) a radially distributed component extending from the T 2 site towards the channel centre. This off-centre positioning of the positively charged metal–oxo species reflects specific interactions with negatively charged [AlO 4 ] − sites in the ZSM-5 framework, which mirrors the Al-siting preference 5 .

Although PEDT confirmed the presence of Co–oxo species within the micropores and provides insight into their preferred locations, the resulting averaged structural information cannot resolve their heterogeneity or determine their exact chemical forms. In this context, GASSB-Ptycho is better suited for such characterization, given its high spatial resolution and proven ability to suppress in-pore artefacts, as demonstrated above. The reconstruction reveals the coexistence of guest-occupied and empty channels in Co–ZSM-5, ruling out the possibility that the observed in-channel contrast arises from artefacts (Fig. 3e). By comparison, an annular dark-field image from the same 4D-STEM dataset can hardly resolve the guest species because of the high noise level (Fig. 3g), and the standard SSB-Ptycho reconstruction shows spurious contrast in all channels, which interferes with the faithful identification of the genuine Co–oxo clusters (Fig. 3f). The differences among the three imaging methods are more clearly reflected in the line profiles in Fig. 3h.

We statistically analysed the preferential locations of Co–oxo species within the 10-MR channels and the fraction of empty channels, based on GASSB-Ptycho-reconstructed images. The site preference follows the order T 2 > T 1 ≈ T 5 > T 3 > T 6 ≈ empty channel (Extended Data Fig. 5), consistent with the PEDT results.

We examined the GASSB-Ptycho-reconstructed images, focusing on ultrathin regions at the crystal edge (Supplementary Fig. 8), where the overlapping of several guest species along the projection direction is minimized. In these regions, two distinct contrast motifs are observed in the 10-MR channels: an off-centre ‘single-dot’ feature and an off-centre ‘twin-dot’ feature, which we assign to mononuclear and binuclear Co–oxo species, respectively. Figure 4a shows representative image motifs corresponding to mononuclear and binuclear Co–oxo clusters and their superpositions, together with the optimized structural models and simulated images. The combined contribution of these motifs is consistent with the extra in-channel electrostatic potential inferred from PEDT (Fig. 3b).

a, Identified configurations of cobalt species in ZSM-5 channels: (I) empty channel, (II) mononuclear Co species, (III)–(V) binuclear Co species, and (VI)–(VII) binuclear Co species superimposed with another mononuclear Co species or another binuclear species. Upper two rows, structural models, shown from different views. Third row, simulated GASSB-Ptycho image. Fourth row, experimental GASSB-Ptycho image. Dashed circular and elliptical outlines highlight single-dot and twin-dot features, respectively. b, Normalized Co K-edge XANES spectra of Co foil, CoO, Co 2 O 3 , and Co–ZSM-5. Inset, magnified view of the Co K-edge XANES spectra in the 7,712–7,723 eV region. c, Magnetic exchange schemes (right) and temperature-dependent magnetization measured from 2 K to 300 K (left). Inset, Curie–Weiss fitting over 50–300 K. C denotes the Curie constant. d, Positive-ion TOF-SIMS of Co–ZSM-5. Insets, representative fragment peaks assigned to mononuclear and binuclear Co species. e, Wavelet-transform EXAFS maps of Co foil, CoO and Co–ZSM-5 in R–K space. R denotes the apparent radial distance from the absorbing atom, and K denotes the magnitude of the photoelectron wave vector. FC, field-cooled; ZFC, zero-field-cooled. Scale bar, 1 nm.

Time-of-flight secondary-ion mass spectrometry (TOF-SIMS) detected Co-related secondary ions during sputtering of Co–ZSM-5 that are absent in pristine ZSM-5 (Fig. 4d and Extended Data Fig. 6). In the low m/z region (m/z < 100), signals at m/z = 59, 76 and 93 are attributed to Co + , CoOH + and Co(OH) 2 + species, respectively. In the higher m/z region (m/z > 100), signals at m/z = 103, 150 and 151 are attributed to Co–O–Si + , Co 2 O 2 + and Co 2 O(OH) + species, respectively. These observations confirm the coexistence of mononuclear and binuclear Co–oxo species confined within ZSM-5 and provide insight into their possible structural motifs. The coordination environment of Co was further probed by Co K-edge Fourier transform extended X-ray absorption fine structure (EXAFS) spectra, using Co foil and CoO as reference samples (Supplementary Fig. 9). The spectra indicate a tetra-coordinated first-shell environment with a Co–O distance of 2.02 Å. In addition, the wavelet-transform EXAFS map shows an intense maximum at approximately 3.15 Å −1 , which we assigned to the first-shell Co–O coordination, and a weaker lobe at approximately 6.9 Å −1 that matches the Co–O–Co feature in the CoO reference (Fig. 4e). This Co–O–Co contribution supports the presence of binuclear Co–oxo species.

X-ray absorption near-edge structure (XANES) spectra and X-ray photoelectron spectroscopy indicate the coexistence of Co 2+ and Co 3+ valence states in the Co–oxo guest species (Fig. 4b and Supplementary Fig. 10). We also used temperature-dependent magnetization measurements to probe the valence configuration of the binuclear Co–oxo species, noting that homovalent Co 2+ –O–Co 2+ and Co 3+ –O–Co 3+ pairs are typically antiferromagnetically coupled through superexchange, whereas heterovalent Co 2+ –O–Co 3+ pairs are ferromagnetically coupled through double exchange 35 , 36 . Fitting the high-temperature (over 50 K) inverse susceptibility χ, which was derived from zero-field-cooled and field-cooled magnetization data with the Curie–Weiss law, yielded a positive Curie–Weiss temperature Θ = 8.26, indicating ferromagnetic coupling and, thus, the presence of Co 2+ –O–Co 3+ pairs in these guest species (Fig. 4c). These results collectively support the heterovalent nature of the binuclear Co–oxo species.

Taken together, we elucidate the molecular structures of the Co–oxo guest species in ZSM-5 by combining GASSB-Ptycho imaging with complementary diffraction and spectroscopic analyses. These tetra-coordinated Co–oxo guest species bind to negatively charged [–Si–O–Al–O–] moieties in the framework and can be broadly grouped into two classes (based on the experimental findings and chemical plausibility): (1) mononuclear species, including bidentate (–O) 2 Co 3+ (OH) 2 and (–O) 2 Co 2+ (OH)(H 2 O) and (2) binuclear species, including bidentate (–O) 2 Co 3+ (μ 2 –O) 2 Co 2+ (H 2 O) 2 and tetradentate (–O) 2 (OH)Co 3+ (μ 2 –O)Co 2+ (OH)(O–) 2 .

Summary

We have demonstrated that widely used phase-contrast STEM imaging modalities can generate spurious in-pore contrast when applied to porous materials, so we developed an electron-ptychography-based remedy termed GASSB-Ptycho. This approach produces molecular-level structural fingerprints of guest moieties within host frameworks with high resolution and fidelity. Using GASSB-Ptycho, we unambiguously identified the coexistence of mononuclear and binuclear Co–oxo species within the micropores of ZSM-5. These assignments, corroborated by complementary diffraction and spectroscopic measurements, provide unprecedented insight into the industrially important Co–ZSM-5 catalytic system. More broadly, GASSB-Ptycho is readily extendable to a wide range of beam-sensitive materials, making it a general strategy for probing complex host–guest chemistry.

Methods

4D-STEM data acquisition

4D-STEM experiments were carried out on a Thermo Fisher Scientific Spectra 300 microscope operated at 300 keV and equipped with a 128 × 128 px electron microscope pixel array detector. Before each experiment, aberration was corrected up to third order using a standard gold sample. After the specimen of interest had been located, the crystal zone axis was identified and aligned in transmission electron microscopy mode using an established low-dose procedure to minimize dose consumption 11 . The microscope was subsequently switched to STEM mode for data acquisition with convergence semi-angles of around 10 mrad. The camera length was set so that the CBED pattern occupied approximately two-thirds of the detector. The scan step size and dwell time were 0.376 Å and 1 ms, respectively.

Calculations and simulations

Structural optimization was performed using the CP2K package 37 . Core electrons were treated using Goedecker–Teter–Hutter pseudopotentials 38 , and valence electrons were expanded in the DZVP-MOLOPT-SR basis set 39 . An auxiliary plane-wave cutoff of 450 Ry was applied to the electronic density to ensure the convergence of total energies and forces. The geometry optimization was carried out using the conjugate gradient algorithm, with all atoms allowed to relax fully. Convergence criteria were set as follows: maximum force below 0.0045 Hartree/Bohr, maximum geometry change below 0.005 Å, root-mean-square displacement below 0.0035 Å and root-mean-square force below 0.003 Hartree/Bohr.

4D-STEM data simulations were carried out using the multislice method as implemented in the open-source package Dr. Probe 40 . The accelerating voltage and convergence semi-angle were chosen to match the experimental conditions, with partial source coherence included in the simulations. The ZSM-5 model was constructed with a thickness of 6 nm along the [010] direction, corresponding to approximately three unit cells, consistent with the thickness of the experimental specimen as determined by position-averaged CBED 41 (Supplementary Fig. 8).

Data processing and reconstruction

GASSB-Ptycho inherits the reconstruction principles and part of the processes from SSB-Ptycho. These shared components are briefly outlined here as the background for implementing GASSB-Ptycho.

In 4D-STEM, at each probe position R, a diffraction pattern I D (K, R) is recorded on the pixelated detector, which can be expressed as:

Here K denotes the scattering vectors, A(K) is the aperture function, Ψ(K) is the Fourier transform of the specimen transmission function and otimes denotes the convolution operation. Taking the Fourier transform of I D (K, R) with respect to R results in:

Here Q represents the image spatial frequency, and the asterisk indicates the complex conjugate. The Fourier transform of I(K, R) with respect to the probe position R is defined as:

Under the weak phase object approximation, we have:

Ψ s (Q) denotes the scattered component of Ψ(Q) and can be derived by integrating over the double disc overlaps. δ(Q) is the Dirac delta function. The first term in G(K, Q) represents the zero-order beam, and the remaining two terms correspond to the overlap between the zero-order beam and the −Q and +Q beams, respectively.

The general workflow of SSB- and GASSB-Ptycho is illustrated in Fig. 1 and the processing steps are summarized as follows.

Common steps (steps 1–5)

1. A complete 4D-STEM dataset, I D (K, R), is acquired using a pixelated detector.

2. I D (K, R) is Fourier transformed with respect to R to obtain G(K, Q).

3. Lens aberrations are corrected using the singular-value-decomposition matrix inversion method 28 , yielding the aberration-corrected dataset G a (K, Q).

4. For each Q, the pixel count N p (Q) is determined from the double-overlap region between A(K)and A*(K + Q). The signal is summed over K within these regions to obtain Ψ s (Q).

5. The averaged spectrum Ψ s-avg (Q) is then obtained by dividing Ψ s (Q) by N p (Q).

SSB-Ptycho reconstruction

6. Ψ s-avg (Q) is multiplied by N p (Q) to obtain Ψ s (Q), where N p (Q) is a two-dimensional band-pass function.

7. An inverse Fourier transform of Ψ s (Q) yields the real-space complex transmission function φ s (R).

8. The phase of φ s (R) is plotted over all probe positions in the STEM scan to obtain the SSB-Ptycho reconstruction.

GASSB-Ptycho reconstruction

6. Ψ s-avg (Q) is multiplied by a two-dimensional Gaussian apodization function to obtain Ψ′ s (Q).

7. An inverse Fourier transform of Ψ′ s (Q) yields the real-space complex transmission function φ′ s (R).

8. The phase of φ′ s (R) is plotted over all probe positions in the STEM scan to obtain the SSB-Ptycho reconstruction.

The form of the apodization function

The two-dimensional Gaussian function is used as an apodization function and is given by:

This form represents a two-dimensional Gaussian function where the variables Q x and Q y are both scaled by the factor α. The function describes a Gaussian distribution in two dimensions with circular symmetry and centred at the origin (0,0).

The property of the apodization function

The Fourier transform of a Gaussian function preserves its Gaussian form:

So,

which, by using a Gaussian function as the apodization function, consequently avoids the generation of ringing artefacts. Here mathcalF denotes the Fourier transform operation, gaus and Gaus represent Gaussian functions in real space and reciprocal space, respectively.

The window of the Gaussian apodization function

The window half-width determines the specific spatial frequency cutoff, thereby influencing the image resolution. In GASSB-Ptycho, this half-width is typically set to 2α (α denotes the convergence semi-angle), corresponding to the contrast transfer limit in direct STEM imaging.

Steps in the SSB-Ptycho reconstruction

In the SSB-Ptycho reconstruction procedure described above, steps 5 and 6 can be omitted because their net effect is null, leaving the result unchanged from step 4, namely, Ψ s (Q). In practice, SSB-Ptycho proceeds directly from step 4 to step 7. Steps 5 and 6 are included here solely to align the SSB-Ptycho workflow with that of GASSB-Ptycho for ease of comparison and to make the band-pass nature of conventional SSB-Ptycho reconstruction more explicit.