Multislice Electron Ptychography

• Multislice electron ptychography is a computational imaging technique that reconstructs atomic-scale 3D electrostatic potentials by explicitly modeling multiple elastic scattering.
• The method employs iterative phase retrieval algorithms integrated with multislice forward propagation to achieve deep-sub-angstrom resolution even in thick specimens.
• Innovations like eLOP and energy-filtered 4D-STEM extend its applicability to specimens >40 nm, enabling quantitative atomic-number discrimination and compositional imaging.

Multislice electron ptychography is a computational imaging technique that enables quantitative recovery of three-dimensional (3D) electrostatic potential distributions in crystalline and complex materials, reaching deep-sub-angstrom lateral resolution and sub-picometer precision in atomic coordinates—capabilities previously limited by specimen thickness and multiple electron scattering. By explicitly modeling multiple elastic scattering via the multislice formalism and integrating robust phase retrieval algorithms, this approach extends 4D scanning transmission electron microscopy (4D-STEM) into a true volumetric probe for structural, chemical, and defect characterization at the atomic scale (Yang et al., 25 Feb 2025, Denzer et al., 11 Jun 2025, Zhang et al., 24 Apr 2025, Gilgenbach et al., 20 May 2025).

1. Multislice Forward Propagation Framework

In multislice electron ptychography, a specimen of total thickness $L$ is divided into $N$ thin slices (thickness $\Delta z = L/N$), each described by a 2D projected electrostatic potential $V_j(x,y)$. The total transmission through slice $j$ is

$T_j(x,y) = \exp\left[ -i \sigma V_j(x,y) \right]$

with $\sigma = 2\pi m e / (\lambda h^2)$ the electron–matter interaction constant, $m$ the electron mass, $e$ the elementary charge, $\lambda$ the relativistic electron wavelength, and $h$ Planck’s constant (Yang et al., 25 Feb 2025, Denzer et al., 11 Jun 2025). Free-space propagation between slices is implemented in Fourier space as

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

yielding the recursive update: $$\psi_{j+1}(x,y) = \mathcal{F}^{-1}\!\Bigl\{ P(k_x,k_y)\, \mathcal{F}[T_j(x,y) \psi_j(x,y)] \Bigr\}$$ where $\mathcal{F}$ denotes the 2D Fourier transform. The exit wave after the final slice is propagated to the detector, and its squared modulus produces the simulated far-field diffraction intensities.

Crucially, this formalism naturally incorporates multiple (dynamical) scattering, enabling inversion of thick specimens where single-slice or projection models fail (Yang et al., 25 Feb 2025). The continuous 3D potential $V(x,y,z)$ is faithfully represented when $\Delta z$ is sufficiently small to avoid phase wrapping and loss of axial information.

2. Ptychographic Phase Retrieval and Reconstruction Engines

The multislice model is embedded within iterative ptychographic phase retrieval algorithms such as the extended Ptychographical Iterative Engine (ePIE), difference-map (DM), and least-squares maximum-likelihood (LSQ-ML) solvers (Gilgenbach et al., 20 May 2025). These engines operate as follows:

1. Forward projection: Advance the current probe estimate through the object slices to obtain simulated exit waves and diffraction patterns at each scan position.
2. Data constraint: Replace the simulated amplitude with the square root of the measured intensity, while retaining the computed phase, at the detector plane.
3. Backpropagation: Propagate the difference signal (“error wave”) backward through the multislice stack, updating both the object slices and probe parameters.
4. Gradient updates: Use step-size-controlled updates and, if desired, regularization (total-variation, Tikhonov, or atomic-prior) to guide convergence.

Mixed-state (incoherent) probe models, in which several independent probe modes and weights are recovered simultaneously, are essential to account for partial spatial and temporal coherence of the illumination (Yang et al., 25 Feb 2025, Gilgenbach et al., 20 May 2025). The modular “phaser” framework provides a unified, multi-backend Python environment supporting all such solvers, with JAX-enabled autodifferentiation delivering up to 6× speedups over conventional MATLAB or CuPy engines (Gilgenbach et al., 20 May 2025).

3. Overcoming Depth and Thickness Limitations

Historically, multislice electron ptychography was restricted to specimens thinner than ∼20–40 nm due to ill-conditioning introduced by dynamical scattering and probe aberrations (Yang et al., 25 Feb 2025). Two major innovations have overcome this limit:

• Extended Local-Orbital Ptychography (eLOP): The probe function is parameterized as a position-dependent coherent mode, with aberration coefficients fitted as smooth functions of scan position via low-order Zernike polynomials or physically motivated bases. This corrects for slowly varying defocus, astigmatism, and higher-order aberration fields that otherwise degrade atomic resolution in thick (>40 nm) specimens. Simultaneous gradient-backpropagation allows recovery of both object and probe parameters (Yang et al., 25 Feb 2025).
• Energy-filtered 4D-STEM: Post-specimen zero-loss energy filtering is implemented to reject inelastically scattered electrons, which contribute a diffuse low-frequency background and erode high-spatial-frequency phase signal. Energy-filtered reconstructions extend the achievable thickness for high-fidelity 3D recovery—eLOP combined with filtering enables robust imaging in Si up to 85 nm (Yang et al., 25 Feb 2025).

Sustained positional precision below 0.5 pm and information transfer to 16–18 pm are achieved in 60–85 nm thick SrTiO₃ and Si crystals, far exceeding the 40 nm threshold of conventional probe models (Yang et al., 25 Feb 2025).

4. Atomic Number Contrast and Quantitative Compositional Imaging

Multislice electron ptychography offers direct access to the projected atomic potentials of individual columns, enabling quantitative discrimination of atomic number (Z) contrast (Denzer et al., 11 Jun 2025). The integrated potential of an atomic column,

$C_Z(R) = \iint_{| r | < R} \hat V_Z(r) \, d^2r$

exhibits two regimes:

• For small integration radii (∼0.2 Å), $C_Z \propto Z^{0.66}$: monotonic growth with Z,
• For larger radii (∼0.9 Å), non-monotonic oscillations appear, tracking electron shell closures (e.g., minima at Z=10,18,30).

Contrast-to-noise ratio (CNR) analysis demonstrates that with optimal integration radius (R≈0.8–1.0 Å), single atomic-number discrimination (ΔZ=1, e.g., Zn versus Cu) is feasible in ∼20 nm thick samples, outperforming ADF- or iDPC-STEM by at least 2–3× in CNR at comparable dose and source size. Precise control of integration radius, source coherence, and electron dose are critical for Z-contrast optimization (Denzer et al., 11 Jun 2025).

5. Surface Morphology, Thickness Variation, and Large-FOV Imaging

Surface-adaptive electron ptychography (SAEP) generalizes the multislice formalism by treating slice thicknesses and probe defocus as free, continuously optimizable parameters during reconstruction. The algorithm employs a chain-rule gradient calculation to recover smooth variations in surface undulation, particle thickness, and atomic structure across large fields of view (Zhang et al., 24 Apr 2025). In zeolite nanoparticles, this enables simultaneous mapping of atomic-scale channel structure, surface corrugation (up to 30 nm), and local thickness variation (40 nm), at sub-nanometer accuracy.

The SAEP and related frameworks integrate adaptive regularization, physical parameter optimization (defocus, vacuum thickness), and robust gradient-based updates—essential for stability and accuracy in highly inhomogeneous, beam-sensitive, or porous materials.

6. Software, Regularization, and Computational Strategies

Implementation of multislice ptychography at scale requires GPU acceleration, modular solver architectures, and sophisticated data management. The "phaser" package supports command-line, web-based, and Jupyter interfaces, handling modular hooks for forward models, solvers, noise, and regularization. Analytical and autodifferentiation-based solvers are supported, allowing the integration of custom priors (e.g., total variation, sparsity), novel network-based regularizers, and user-defined physical constraints (Gilgenbach et al., 20 May 2025).

Typical best practices include:

• Areal oversampling $S_a = \pi r_p^2/s^2 \geq 10$, where $r_p$ is probe radius and $s$ scan step,
• Ronchigram magnification $M_r \geq 1$ px/Å, with large enough convergence angle and sufficiently fine detector sampling,
• Mixed-state probe with 6–10 modes to compensate partial coherence,
• Depth-wise or slice-wise regularization to suppress unphysical cross-talk or noise,
• Slice thickness $\sim 1$ nm for volumetric resolution and computational efficiency (Zhang et al., 24 Apr 2025, Gilgenbach et al., 2023).

High memory (tens of GB GPU RAM), efficient FFT implementations, and JIT compilation (JAX) are crucial for practical inversion on large fields of view.

7. Outlook and Implications

By inverting dynamical scattering with adaptive probe modeling and energy filtering, multislice electron ptychography enables deep-sub-ångström and sub-picometer precision volumetric imaging in specimens up to ∼85 nm thick. This lifts restrictions on sample preparation, permitting 3D mapping of strain fields, dislocation cores, charge distributions, and atomic species under near−native conditions (Yang et al., 25 Feb 2025). The technique is broadly applicable in physics, chemistry, and semiconductor engineering, including for materials that cannot be artificially thinned without altering their intrinsic properties. Its modular computational frameworks promise extensibility to generative priors, new physical constraints, and automation for routine, high-precision 3D characterization of complex solids (Yang et al., 25 Feb 2025, Zhang et al., 24 Apr 2025, Gilgenbach et al., 20 May 2025).