Advances in Electron Ptychography:
Sub-Ångstrom 3D Resolution, Automated Tomography
and Transverse Quantum State Characterization

Invited talk · IMC21 · September 2026 · Liverpool, UK

Prof. Dr. Philipp Pelz

FAU Erlangen-Nürnberg

2026-09-01

Outline

Part 1 · Sub-Ångstrom 3D Resolution

Simulation reaches 10⁸ atoms. Can measurement follow — atom by atom, in 3D?

1 · Sub-Ångstrom 3D

2 · Fused FF-STEM

3 · Transverse state

Ptychographic tomography: put the scattering physics in the model

Depth information is encoded in the illumination angles — for large convergence angles, a single scan already carries 3D information.

Forward model — multislice, paraxial, time-independent: \[\mathcal{M}_{\psi}(\mathbf{V}) = \prod_{k=0}^T (\mathcal{P}^{\Delta z} T^{\mathbf{V}_k})\psi\]

Note

Atomic 3D resolution from a 4D-STEM tilt series — with multiple scattering modelled explicitly, not approximated away

First unknown 3D atomic structure solved by phase-contrast tomography

1. 4D-STEM tilt series
2. mixed-state ptychography at every tilt
3. joint linear tomography and alignment
4. sub-pixel atomic peak tracing
5. 3D atomic-structure determination

Volume size: (6 nm)³

Note

Elliptic double-wall CNT resolved in 3D; a novel ZrTe₂ phase, confirmed stable by DFT

Pelz et al. (2023)

End-to-end reconstruction alleviates the missing wedge

Fully end-to-end MSPT differentiates through the whole chain — affine resampling of the volume, z-resampling, batch-cropping, mixed-state multislice, far-field propagation, and gradient backpropagation through all of it.

Light and heavy atoms recovered over a 90° tilt range

Note

Physical priors recover the missing wedge → near-isotropic 3D resolution (Nyquist 0.82 Å), at 12× virtual sub-sampling

Romanov et al. (2024), You et al. (2024)

The payoff: atomic resolution in all three directions

scale bar: 2 nm

scale bar: 2 nm

Note

Unknown 3D structure, solved — the specimen unknown is addressed. On to the rest.

You et al. (2024)

Chemistry, and the cost of scale

Ptychography delivers weak chemical contrast. Spectroscopic STEM provides complementary contrast. Existing forward models however are expensive and do not scale well.

Core-loss EELS maps cannot be read off directly

• Atomic-resolution EELS elemental maps are not projections: the focused probe undergoes strong dynamical scattering
• Quantitative interpretation needs full quantum-mechanical simulation of the probe before and after the core-loss transition
• So the forward model becomes the bottleneck for inversion

Simultaneous five-edge atomic-resolution maps across a LaAlO₃/SrTiO₃ interface

Note

Core-loss chemistry is the richest — and one of the more expensive — signals to simulate

Pelz (2026)

Bi-partitioned PRISM-EELS: and it scales

Probe leg (\(\mathcal{S}_1\)) and detector leg (\(\mathcal{S}_2\)), each on a sparse set of parent beams, reconstructed on a small window at the ionized atom

Bi-partition both scattering matrices → the per-scan exit propagation disappears

Note

LaAlO₃/SrTiO₃, five core-loss edges at once, 32 041 probe positions: ~89 s, 2.4 GB on one GPU — >50× lower memory

Pelz (2026)

Building on Ophus 2017, Brown 2019, Pelz 2021

Part 2 · Fused Full-Field STEM

One scan already illuminates several detectors. Why throw any of them away?

1 · Sub-Ångstrom 3D

2 · Fused FF-STEM

3 · Transverse state

Sensor fusion: gap-free contrast in near-real time

Fused Full-Field STEM (FF-STEM) on Gd₂O₃ nanohelices

• Direct ptychography → coherent high-frequency phase
• Tilt-corrected dark-field → low-frequency amplitude contrast
• fused in Fourier space by Wiener-type weighting with SSNR-based weights

Note

Closes the low↔︎high spatial-frequency gap → one aberration-free image. Analytic, non-iterative: < 1 s on a single GPU.

You, Varnavides, et al. (2026)

Fusion works on real, low-dose data

Experimental FF-STEM vs. parallax across detectors: AuNPs and Gd₂O₃ nanohelices (Timepix4, 1211 e⁻/Ų)

Note

Robust across samples and detectors

You, Varnavides, et al. (2026)

Part 3 · Transverse Quantum State Characterization

Many structured-illumination proposals assume a coherent state. What transverse quantum state does a programmable MEMS phase plate actually deliver?

1 · Sub-Ångstrom 3D

2 · Fused FF-STEM

3 · Transverse state

Motivation: a wealth of proposals for programmable electron optics

Mixed-state ptychography measures the whole illumination

Dominant probe mode in the aperture plane, ℓ = −17 … +17. Hue = phase, brightness = amplitude. The winding number is directly countable; the dark wedge is the needle-band shadow.

Note

21 charge states, from standard 4D-STEM using ptychography
Two stages: BF-disk shift estimation with a vortex ramp, then a mixed-state iterative solver.

MEMS device: CNR Modena (V. Grillo). You, Rosi, et al. (2026)

The device is linear and calibratable

From the OAM spectrum of the reconstructed probe density matrix: \[\langle \ell \rangle = \sum_\ell \ell\, P_\ell, \qquad P_\ell = \rho_{\ell\ell}\]

Note

16.8 ħ per applied volt across ℓ = −17 … +17.
programmed charge maps onto the delivered charge by stable gain

You, Rosi, et al. (2026)

But the delivered beam is not one wave

Note

At ℓ = +17 the dominant mode carries only 42 % of the power. Purity falls and entropy rises: the higher the charge you program, the more mixed the beam you get.

You, Rosi, et al. (2026)

Partial coherence, measured in both planes

Aperture-plane coherence envelope |γ(Δk)|

Note

The aperture-plane coherence width collapses by ~3× from ℓ = 0 to |ℓ| = 17, while the real-space envelope stays near the source-size limit — the mixedness lives in the aperture, exactly where a vortex phase plate acts.

You, Rosi, et al. (2026)

Dose budget to reach given resolution

• We derive a generalized weak-phase SSNR for mixed probes: \(\Gamma_\rho(q,k) = J(k-q,k) - J(k,k+q)\), giving \(\mathrm{SSNR}_\rho = \sqrt{2L_{\mathrm{up}}}\,D_{\mathrm{coh}}\). → DQE → partial coherence-aware dose budget

Note

Dose rule: below \(q \approx 0.5\,q_{\mathrm{probe}}\) the vortex is the cheaper probe; above it, pay for purity or use a round aperture.

You, Rosi, et al. (2026)

Reduces to Nellist–Rodenburg, Bennemann and Varnavides in the appropriate limits

Acknowledgements

thank you

Shengbo YouFAU

Sihan ShaoFAU

Nikita PalatkinFAU

Mingjian WuFAU

Tadahiro YokosawaFAU

Erdmann SpieckerFAU

Andreas HutzlerHI ERN

Lucía MoralesHI ERN

Georgios VarnavidesTU Delft

Sagar KhavnekarThermo Fisher

Darya ChernikovaISTA

Baixu ZhuIndiana University

Ricardo EgoavilThermo Fisher

Stefano VespucciThermo Fisher

Dileep KrishnanThermo Fisher

Florian K. M. SchurISTA

Paolo RosiCNR Nanoscience

Colin OphusStanford

Vincenzo GrilloCNR Modena

Daniel StroppaDSTL

M. MeffertDECTRIS

Xingchen YeIndiana University

Supported by the ERC Starting Grant HyperScaleEM

Summary

• Sub-Ångstrom 3D — ptychographic electron tomography reaches sub-Ångström, near-isotropic 3D beyond the depth-of-focus limit
• core-loss EELS simulations at scale (Pelz 2026) — BiP-PRISM allows core-loss EELS + 4D-STEM past one million atoms on one GPU
• FF-STEM (You, Varnavides, et al. 2026) — Wiener sensor fusion gives gap-free, aberration-free contrast in < 1 s, robust at low dose
• Transverse quantum state characterization (You, Rosi, et al. 2026) — mixed-state ptychography can now be used to characterize the transverse quantum state of a programmable MEMS phase plate

References 📚

Pelz, Philipp M. 2026. The BiP-PRISM Algorithm for Fast and Scalable Core-Loss STEM-EELS Simulations. https://arxiv.org/abs/2607.00756.

Pelz, Philipp M., Sinéad M. Griffin, Scott Stonemeyer, et al. 2023. “Solving Complex Nanostructures with Ptychographic Atomic Electron Tomography.” Nature Communications 14 (11): 7906. https://doi.org/10.1038/s41467-023-43634-z.

Romanov, Andrey, Min Gee Cho, Mary Cooper Scott, and Philipp Pelz. 2024. “Multi-Slice Electron Ptychographic Tomography for Three-Dimensional Phase-Contrast Microscopy Beyond the Depth of Focus Limits.” Journal of Physics: Materials 8 (1): 015005. https://doi.org/10.1088/2515-7639/ad9ad2.

You, Shengbo, Andrey Romanov, and Philipp M Pelz. 2024. “Near-Isotropic Sub-Ångstrom 3d Resolution Phase Contrast Imaging Achieved by End-to-End Ptychographic Electron Tomography.” Physica Scripta 100 (1): 015404. https://doi.org/10.1088/1402-4896/ad9a1a.

You, Shengbo, Paolo Rosi, Enzo Rotunno, et al. 2026. Transverse Quantum-State Characterization of Programmable Electron Optics. https://arxiv.org/abs/2608.05749.

You, Shengbo, Georgios Varnavides, Sagar Khavnekar, et al. 2026. “Gap-Free Information Transfer in 4D-STEM via Fusion of Complementary Scattering Channels.” Advanced Science 13: e76620. https://doi.org/10.1002/advs.76620.

Backup · OAM channel decomposition

Azimuthal-harmonic spectra

OAM density matrix

Adjacent-OAM coherence

Note

Median adjacent-OAM coherence 0.30: the delivered state is largely an incoherent OAM mixture, not a coherent fractional superposition

Backup · Channel purity and mode powers

Power in the dominant OAM channel

Stacked fractional mode powers

Backup · SSNR and DQE in full

SSNR(q)

Coherence envelope D_coh(q)

DQE(q)

Backup · Sample-plane coherence and mode gallery

Sample-plane coherence envelope |γ(Δx)|

Dominant mode, sample plane, ℓ = −17 … +17

Backup · Controls and object self-consistency

Round-beam controls, aperture plane

Retracting the device recovers a 98 % pure mode; inserting it at ℓ = 0 costs a few percent.

Object phase reconstructed with each probe — same specimen, every charge state

The specimen is recovered consistently across the whole charge series, which is what licenses comparing probes.