Multislice Electron Ptychography

• Multislice Electron Ptychography is a phase-retrieval method that uses a multislice forward model to reconstruct depth-dependent 3D electrostatic potentials from 4D-STEM data.
• It overcomes conventional STEM limitations by providing quantitative, phase-sensitive 3D imaging with sub-Ångström lateral resolution and enhanced depth sectioning via tilt coupling or tomography.
• Recent advances in inverse algorithms, sampling designs, and deep generative priors have improved reconstruction speed, noise robustness, and axial resolution for thicker, beam-sensitive materials.

Multislice electron ptychography (MEP) is a phase-retrieval approach in scanning transmission electron microscopy that combines 4D-STEM measurements with a multislice forward model of multiple elastic scattering to reconstruct depth-dependent electrostatic potential, rather than a single thin projection. In this formulation, a thick specimen is represented as a stack of slices, and the incident probe is successively transmitted and Fresnel-propagated through them; the inverse problem is then solved from position- and momentum-resolved diffraction data. Across recent work, MEP is described as enabling quantitative, phase-sensitive, three-dimensional imaging at sub-Ångström lateral resolution, with depth information recovered from a single scan in favorable cases and further extended by tilt coupling or tomography when higher axial resolving power is required (Chen et al., 2021, Gilgenbach et al., 2023, Ribet et al., 2024).

1. Conceptual scope and distinction from conventional STEM

MEP addresses a regime in which conventional imaging modes are limited by projection, contrast, dose, and multiple-scattering effects. For beam-sensitive upconverting core-shell nanoparticles, conventional imaging in scanning transmission electron microscopy was reported to have sufficient resolution to probe atomic structure, yet contrast, dose, and projection limitations made those modes insufficient for fully characterizing the structures; the cited study therefore used multislice ptychography to recover depth-dependent information under a low electron dose (Ribet et al., 2024). In thick specimens more generally, MEP is formulated as an explicit solution of the multiple-scattering problem, in contrast to single-slice ptychography or conventional projection imaging (Gilgenbach et al., 2023, Chen et al., 2021).

A common misconception is that ptychography is necessarily a two-dimensional technique. The literature instead distinguishes several levels of three-dimensionality. Single-orientation MEP can recover separate slice phases and therefore provide optical sectioning along the beam direction, with no tilt series or tomography required for certain classes of 3D localization problems (Chen et al., 2021). At the same time, when the target is a larger reconstructed volume beyond the depth-of-field limit, ptychographic multi-slice electron tomography has been used to reconstruct tilt-series 4D-STEM measurements, experimentally yielding atomic resolution three-dimensional phase-contrast imaging in a volume surpassing the depth of field limit (Romanov et al., 2023). Small-angle tilt coupling represents an intermediate strategy: it couples only a few projections at small tilt angles and improves depth resolution by more than threefold to the sub-nanometer scale (Dong et al., 2024).

This places MEP at the intersection of computational imaging, inverse scattering, and atomic-resolution 3D metrology. The technique is now used not only for structural imaging, but also for strain mapping, point-defect localization, interface metrology, chemical-order analysis, and light-element sensitivity in systems where projection methods are inadequate (Zhu et al., 2024, Li et al., 25 Jul 2025, Karapetyan et al., 9 Jul 2025).

2. Multislice forward scattering model

The core physical model divides the specimen into thin slices normal to the beam. A representative formulation writes the incident probe mode at scan position $x_j$ as

$\psi_{0,k}(r)=P_k(r-x_j),$

the transmission through slice $n$ as

$T_n(r)=\exp[i\sigma V_n(r)],$

and the slice-to-slice propagation as

$\psi_{n+1}(r)=P(\Delta z_n)[T_n(r)\psi_{n,k}(r)].$

After the final slice, the detector intensity is

$$I_{\mathrm{model},j}(q)=\sum_k|\mathcal F\{\psi_{\mathrm{out},k}(r)\}(q)|^2.$$

This notation emphasizes three ingredients that recur throughout the literature: a complex probe, slice transmission functions derived from projected electrostatic potentials, and Fresnel propagation between slices (Gilgenbach et al., 20 May 2025).

Equivalent formulations appear across studies with different sign conventions and symbols. One widely used propagator form is

$$\hat P(k_x,k_y)=\exp[-\,i\pi\lambda\Delta z(k_x^2+k_y^2)],$$

with the real-space action implemented by forward FFT, multiplication by the propagator, and inverse FFT (Chen et al., 2021, KP et al., 2024). In this framework, the multislice recursion

$\psi_{j+1}(r)=P(\Delta z)\,[t_j(r)\psi_j(r)]$

is the basic object-to-wave mapping, while the scan converts it into a 4D-STEM dataset by recording a full diffraction pattern at each probe position (Zhu et al., 2024).

The same formalism accommodates thick-specimen effects that are central to MEP. Thermal vibrations can be included through frozen phonon multislice, and finite source size through mode decomposition or mixed-state probes (Gilgenbach et al., 2023). Mixed-state maximum-likelihood solvers have been used with $M=12$ incoherent modes in a $300\,\mathrm{keV}$ experiment on PrScO$\psi_{0,k}(r)=P_k(r-x_j),$0, while other experimental reconstructions have used 6, 8, or 10 probe modes to capture partial coherence (Chen et al., 2021, KP et al., 2024, Kim et al., 2024, Huang et al., 8 Sep 2025). This suggests that partial coherence is not a peripheral correction but a standard component of state-of-the-art MEP implementations.

3. Inverse reconstruction algorithms and sampling design

The inverse problem is to recover the object slices, and often the probe, from diffraction intensities. ePIE-style updates, least-squares maximum-likelihood solvers, and gradient-based autodifferentiation frameworks all appear in the recent literature. In a standard reconstruction loop, one forward-propagates the probe through all slices, replaces the detector-plane amplitude with the measured amplitude, inverse-transforms to the exit plane, and back-propagates the resulting difference wave through the slice stack to update object and probe (Zhu et al., 2024, KP et al., 2024). Gradient-based maximum-likelihood solvers compute derivatives by back-propagating the error term through the multislice forward model, and one study additionally mixed low-frequency content between neighboring slices to prevent low-frequency ambiguities when $\psi_{0,k}(r)=P_k(r-x_j),$1 depth of focus (Chen et al., 2021).

Recent software work has broadened the algorithmic landscape. The open-source Python package phaser provides a unified interface to both conventional and gradient descent based ptychographic algorithms, including ePIE-style and LSQML multislice solvers, as well as JAX- and PyTorch-based autodifferentiation engines. Reconstructions are specified in a declarative format and can be run from a command line, Jupyter notebook, or web interface; NumPy, CuPy, and JAX are supported as computational backends. With the JAX backend, a $\psi_{0,k}(r)=P_k(r-x_j),$2 improvement in iteration speed over fold_slice was reported on benchmark problems, and the gradient descent solver was reported to achieve improved reconstruction quality (Gilgenbach et al., 20 May 2025).

Robustness depends strongly on sampling. A dedicated study defined physically informed metrics including fundamental ptychographic sampling, areal oversampling, probe sampling, and Ronchigram magnification. The central empirical conclusion was that $\psi_{0,k}(r)=P_k(r-x_j),$3 px/Å is required for stable reconstructions, that a hard cutoff at $\psi_{0,k}(r)=P_k(r-x_j),$4 px/Å separates converged from non-converged reconstructions over all convergence angles and detector samplings, and that FPS $\psi_{0,k}(r)=P_k(r-x_j),$5 and $\psi_{0,k}(r)=P_k(r-x_j),$6 are necessary but not sufficient (Gilgenbach et al., 2023). Under these principles, experimental multislice reconstructions were achieved at a scan step of $\psi_{0,k}(r)=P_k(r-x_j),$7 Å/px, with $\psi_{0,k}(r)=P_k(r-x_j),$8 and $\psi_{0,k}(r)=P_k(r-x_j),$9, enabling sub-Å resolution over fields of view $n$0 nm (Gilgenbach et al., 2023).

The computational burden remains substantial. For one thick-object implementation, a full iteration over $n$1 scan positions and $n$2 slices was described as costing $n$3 operations, with reconstruction times ranging from hours to days on a GPU cluster depending on slice number, detector sampling, and scan size (Yang et al., 25 Feb 2025). This has motivated efforts toward better solver design, better initialization, and stronger priors.

4. Resolution, depth sectioning, and contrast mechanisms

One of the defining claims of MEP is that it can reach the atomic-resolution limits set by lattice vibrations. In a $n$4 PrScO$n$5 experiment, the recovered electrostatic-potential widths implied residual instrumental blurring under $n$6 pm, with the remaining spot size dominated by Debye-Waller broadening rather than the imaging system (Chen et al., 2021). A later thick-object study combined energy filtering and extended local-orbital ptychography (eLOP) to reconstruct silicon as thick as $n$7 nm and SrTiO$n$8 up to $n$9 nm, reporting an information limit of $T_n(r)=\exp[i\sigma V_n(r)],$0 pm and atomic position precision of $T_n(r)=\exp[i\sigma V_n(r)],$1 pm in the $T_n(r)=\exp[i\sigma V_n(r)],$2 nm Si dataset (Yang et al., 25 Feb 2025).

Depth sectioning is more nuanced. In the same PrScO$T_n(r)=\exp[i\sigma V_n(r)],$3 work, the experimental depth resolution estimated from a surface-edge method was $T_n(r)=\exp[i\sigma V_n(r)],$4 nm, while simulated single interstitial dopants in a $T_n(r)=\exp[i\sigma V_n(r)],$5 nm matrix yielded FW80M values of $T_n(r)=\exp[i\sigma V_n(r)],$6 nm for Pr, $T_n(r)=\exp[i\sigma V_n(r)],$7 nm for Sc, and $T_n(r)=\exp[i\sigma V_n(r)],$8 nm for O at dose $T_n(r)=\exp[i\sigma V_n(r)],$9 e/Å$\psi_{n+1}(r)=P(\Delta z_n)[T_n(r)\psi_{n,k}(r)].$0 (Chen et al., 2021). In a later study on interstitial atoms in a garnet oxide heterostructure, single atoms were detected and located in three dimensions with depth resolution better than $\psi_{n+1}(r)=P(\Delta z_n)[T_n(r)\psi_{n,k}(r)].$1 nm, together with deep-sub-Ångström lateral resolution; simulations further indicated that atomic-layer depth resolution should be possible using strongly divergent electron probe illumination (Chen et al., 2024). Small-angle tilt coupling pushed this further: conventional MEP on a Ca–Co film gave $\psi_{n+1}(r)=P(\Delta z_n)[T_n(r)\psi_{n,k}(r)].$2 nm depth resolution, whereas TCMEP with $\psi_{n+1}(r)=P(\Delta z_n)[T_n(r)\psi_{n,k}(r)].$3 yielded $\psi_{n+1}(r)=P(\Delta z_n)[T_n(r)\psi_{n,k}(r)].$4–$\psi_{n+1}(r)=P(\Delta z_n)[T_n(r)\psi_{n,k}(r)].$5 nm under experimental dose, and $\psi_{n+1}(r)=P(\Delta z_n)[T_n(r)\psi_{n,k}(r)].$6 nm at a bilayer SrTiO$\psi_{n+1}(r)=P(\Delta z_n)[T_n(r)\psi_{n,k}(r)].$7 interface (Dong et al., 2024).

Contrast formation is also not reducible to a single monotonic atomic-number law. A 2025 analysis showed that ptychographic $\psi_{n+1}(r)=P(\Delta z_n)[T_n(r)\psi_{n,k}(r)].$8-dependence is highly dependent on the integrated area of an atom column. For small integration radius $\psi_{n+1}(r)=P(\Delta z_n)[T_n(r)\psi_{n,k}(r)].$9, the reconstructed per-column signal follows a nearly monotonic power law, $$I_{\mathrm{model},j}(q)=\sum_k|\mathcal F\{\psi_{\mathrm{out},k}(r)\}(q)|^2.$$0 with $$I_{\mathrm{model},j}(q)=\sum_k|\mathcal F\{\psi_{\mathrm{out},k}(r)\}(q)|^2.$$1. At larger $$I_{\mathrm{model},j}(q)=\sum_k|\mathcal F\{\psi_{\mathrm{out},k}(r)\}(q)|^2.$$2, the behavior becomes strongly non-monotonic and follows trends in orbital shell structure. The reconstructed projected potential aligned with the transmission function with an overall deviation of only $$I_{\mathrm{model},j}(q)=\sum_k|\mathcal F\{\psi_{\mathrm{out},k}(r)\}(q)|^2.$$3, and the non-monotonic regime was shown to accentuate contrast between adjacent elements such as Cu and Zn in $$I_{\mathrm{model},j}(q)=\sum_k|\mathcal F\{\psi_{\mathrm{out},k}(r)\}(q)|^2.$$4-CuZn, even for $$I_{\mathrm{model},j}(q)=\sum_k|\mathcal F\{\psi_{\mathrm{out},k}(r)\}(q)|^2.$$5 nm thick samples (Denzer et al., 11 Jun 2025). This suggests that elemental sensitivity in MEP is partly an analysis choice, not solely a property of the forward model.

5. Scientific applications and demonstrated use cases

MEP has been used to extract three-dimensional strain and defect information from single 4D-STEM scans in systems where dose or projection constraints are severe. In photon upconverting core-shell nanoparticles, low-dose multislice ptychography reconstructed both a defect-free particle and a particle with a large break in the side, and mapped the distribution of strain in 3D by computing distortion fields from high-resolution potential images of each slice. The defect-free nanoparticle showed twisting of the shell, while the broken particle revealed the 3D position of the crack, the core, and dislocations (Ribet et al., 2024).

In ferroic oxides, MEP has been used to resolve depth-dependent order parameters and chemically frustrated boundaries. A chemical anti-phase boundary in Pb$$I_{\mathrm{model},j}(q)=\sum_k|\mathcal F\{\psi_{\mathrm{out},k}(r)\}(q)|^2.$$6MgWO$$I_{\mathrm{model},j}(q)=\sum_k|\mathcal F\{\psi_{\mathrm{out},k}(r)\}(q)|^2.$$7 was found to be inclined along the electron beam direction with a finite width of chemical intermixing, while nearby regions exhibited antiferroelectric-like displacements that contrasted with the predominantly paraelectric matrix (Zhu et al., 2024). In sodium niobate, multislice electron ptychography separated the ferroelectric interior from the relaxed surface structure, identified the soft phonon mode and related structural distortions with picometer precision, and showed that the polar distortion is dominated by anion displacements relative to the niobium sublattice (KP et al., 2024). In the relaxor ferroelectric PMN-0.32PT, direct comparison with molecular dynamics simulations linked the polar structure to unit-cell-level charge imbalance induced by chemical disorder, and showed that compressive strain enhances out-of-plane correlations and ferroelectric-like order without affecting the in-plane relaxor-like structure (Zhu et al., 2024).

The method has also become a tool for semiconductor and defect metrology. In implanted 4H-SiC, reconstructed volumes along the implantation direction quantified damage over depths up to $$I_{\mathrm{model},j}(q)=\sum_k|\mathcal F\{\psi_{\mathrm{out},k}(r)\}(q)|^2.$$8 nm and showed that silicon vacancies can be identified beyond that depth and used to measure local strain (Kim et al., 2024). A separate SiC case study evaluated sensitivity to silicon vacancies, antisite defects, and substitutional transition-metal dopants, concluding that isolated point defects can be located within a unit cell along the sample depth and that electron energy, dose, defocus, and convergence semi-angle govern defect contrast (Bhat et al., 2024). In gate-all-around transistors, MEP uncovered distortions and defects at the gate-oxide/channel interface, found that the silicon in the $$I_{\mathrm{model},j}(q)=\sum_k|\mathcal F\{\psi_{\mathrm{out},k}(r)\}(q)|^2.$$9-nm-thick channel gradually relaxes away from the interfaces so that only $$\hat P(k_x,k_y)=\exp[-\,i\pi\lambda\Delta z(k_x^2+k_y^2)],$$0 of the atoms remain in a bulk-like structure, and measured different atomic-scale roughness profiles at the top and bottom interfaces (Karapetyan et al., 9 Jul 2025).

Light-element sensitivity is another notable application domain. In multi-principal-element alloy hydrides, MEP resolved heterogeneous hydrogen distributions, quantified hydrogen-induced lattice displacements with picometer precision, and provided depth-sectioned hydrogen maps; the same study argued that MEP overcomes the weak-scattering and mobility constraints that limit conventional electron microscopy for hydrogen (Li et al., 25 Jul 2025).

6. Limitations, controversies, and active methodological directions

MEP is computationally intensive and experimentally sensitive. Large 4D datasets, iterative multislice propagation, partial coherence, detector geometry, and scan instabilities all contribute to reconstruction difficulty (KP et al., 2024, Gilgenbach et al., 2023). Residual defocus, astigmatism, source-size broadening, inelastic scattering, sample-thickness variation, and inter-slice cross-talk have all been identified as practical limits on phase sensitivity and depth resolution (Zhu et al., 2024, Kim et al., 2024). One zeolite study stated that accuracy of reconstructed defocus and thickness maps is limited by dose and convergence semi-angle, even when surface-adaptive updates are introduced (Zhang et al., 24 Apr 2025).

Another common misunderstanding is that single-scan MEP and electron tomography are interchangeable. The literature instead presents them as complementary. Single-orientation MEP can provide true depth sectioning and single-dopant localization, but the finite convergence angle sets a missing-wedge-like limit on axial transfer (Chen et al., 2021, Chen et al., 2024). Small-angle tilt coupling improves that missing wedge without a full tilt series (Dong et al., 2024), whereas ptychographic multi-slice electron tomography extends 3D imaging beyond the depth-of-field limit by reconstructing tilt-series 4D-STEM measurements (Romanov et al., 2023). A plausible implication is that the choice between single-scan MEP, few-tilt coupling, and full tomography is fundamentally application-dependent rather than hierarchical.

Current methodological directions are aimed at thickness, adaptivity, and priors. Energy filtering and eLOP extended accurate ptychographic reconstruction to Si as thick as $$\hat P(k_x,k_y)=\exp[-\,i\pi\lambda\Delta z(k_x^2+k_y^2)],$$1 nm, approximately three times larger than the usual thickness threshold for conventional multislice electron ptychography (Yang et al., 25 Feb 2025). Surface-adaptive electron ptychography for zeolites optimized probe defocus and slice thickness during reconstruction to resolve surface morphology, thickness variations, and atomic structure simultaneously (Zhang et al., 24 Apr 2025). On the algorithmic side, MEP-Diffusion introduced a diffusion model as a generative prior integrated via Diffusion Posterior Sampling and reported a $$\hat P(k_x,k_y)=\exp[-\,i\pi\lambda\Delta z(k_x^2+k_y^2)],$$2 improvement in SSIM over existing methods on full 3D volumes (Belardi et al., 23 Jul 2025). A separate deep generative prior framework parameterized both probe and sample within an automatic-differentiation mixed-state multislice forward model and reported greater noise robustness, faster convergence, improved depth regularization, and minimal user-specified regularization (McCray et al., 11 Nov 2025).

Taken together, these developments define MEP not as a single algorithm, but as a family of multislice inverse-imaging methods. The stable elements are the slice-wise forward model, 4D-STEM data redundancy, and joint recovery of probe and object; the active research frontier lies in how best to regularize, accelerate, and physically constrain that inverse problem for ever thicker, noisier, more heterogeneous, and more beam-sensitive materials systems.