Extended Local-Orbital Ptychography (eLOP)

• Extended local-orbital ptychography is a high-resolution imaging method that uses a local-orbital basis and multislice propagation to analyze thick specimens.
• It achieves resolution limits of 16–18 pm by incorporating spatially varying aberration corrections and energy filtering to enhance signal fidelity.
• The technique provides sub-picometer atomic positioning and robust structural insights for materials like silicon and SrTiO3 in advanced electron microscopy.

Extended local-orbital ptychography (eLOP) is an advanced computational imaging methodology developed to enable high-precision, high-resolution structural analysis of comparatively thick samples in transmission electron microscopy (TEM) and scanning transmission electron microscopy (STEM), particularly within the context of 4D-STEM experiments. By leveraging a local-orbital expansion for both probe and object, and incorporating explicit spatial variation of optical aberrations at each scan position, eLOP achieves ptychographic reconstructions for samples substantially thicker than those accessible by conventional multislice electron ptychography. This technique attains information limits down to 16–18 pm and sub-picometer atomic position precision, providing new opportunities for investigating intrinsic structural and physical phenomena in a broad array of materials systems (Yang et al., 25 Feb 2025).

1. Theoretical Formalism: Multislice Propagation and Scattering Model

In the eLOP approach, the electromagnetic (electron) wave propagation through a sample is modeled using the multislice algorithm. The initial probe wave $\Psi_j^{(0)}(r)$ illuminates slice $n=0$ of a thick object at scan position $s_j$. The projected potential $V_n(r)$ defines the scattering at each slice $n$ of thickness $\Delta z$, with electron-sample interaction parameter $\sigma$. Propagation through a slice is formalized as:

$$\Psi_j^{(n+1)}(r) = P_n(r) \cdot [\Psi_j^{(n)}(r) \otimes h(\Delta z)]$$

where $P_n(r) = \exp[-i\sigma V_n(r)]$ is the transmission function, and $h(\Delta z)$ is the Fresnel propagator. The exit-wave after $n=0$0 slices, $n=0$1, yields the experimental observable—the far-field diffraction pattern:

$n=0$2

with $n=0$3 denoting the 2D Fourier transform. This formalism accommodates multiple scattering effects crucial for thick specimens.

2. Local-Orbital Expansion and Aberration Parameterization

A key innovation in eLOP is the use of a minimal local-orbital basis for both the probe and object. The probe at position $n=0$4 is expressed as:

$n=0$5

where $n=0$6 are local basis functions, typically chosen from low-order Zernike polynomials (to capture phase aberrations) and localized wavelets. Similarly, the object transmission in each slice $n=0$7 is written as:

$n=0$8

with $n=0$9 being compactly supported orbitals. The use of a limited set of basis functions enforces spatial smoothness and allows for position-dependent probe variation without excessive parameter proliferation.

Aberration retrieval is formalized by parameterizing the probe’s phase at each scan position in terms of a sum over Zernike polynomials:

$s_j$0

The coefficients $s_j$1 are allowed to vary smoothly across the scan, explicitly modeling spatial aberration variations that are otherwise neglected in conventional ptychography pipelines (CPP), where a single, stationary probe is assumed.

3. Energy Filtering and Data Acquisition

To suppress the deleterious effects of inelastically scattered electrons, 4D-STEM datasets are acquired using a post-column energy filter with a 20 eV zero-loss slit. This replaces $s_j$2 by the zero-loss intensity, significantly improving signal-to-noise ratio, especially in thick sample regions where inelastic scattering would otherwise obscure high-frequency information. The theoretical model itself is unchanged by filtering, but reconstruction fidelity is substantially enhanced.

Experimental setups employ aberration-corrected TEM/STEM instruments at 300 keV (e.g., FEI Titan with XFEG and $s_j$3 corrector), a pixelated 4D-STEM camera (e.g., $s_j$4 pixels, padded to $s_j$5), convergence semiangle of $s_j$6 mrad, and scan step $s_j$7 Å. Thickness calibration is performed via low-loss EELS, yielding an inelastic mean free path $s_j$8 nm for SrTiO$s_j$9 (Yang et al., 25 Feb 2025).

4. eLOP Reconstruction Algorithm

The eLOP reconstruction involves the following algorithmic workflow:

• Initialization: Begin with a uniform object guess (e.g., $V_n(r)$0), initial probe basis, and $V_n(r)$1. Optionally, estimate thickness by low-loss EELS or as a variable in the loss.
• Iteration: For each step $V_n(r)$2:
  • Forward-propagate current probe, object, and aberrations to simulate $V_n(r)$3.
  • Compute Fourier-domain error:

  $V_n(r)$4

  and back-propagate $V_n(r)$5. - Update object coefficients $V_n(r)$6 using local-orbital projections:

  $V_n(r)$7 - Update probe-orbital coefficients and aberration maps by projecting the loss gradient onto the corresponding basis elements:

  $V_n(r)$8 - Optionally refine estimated sample thickness. - Enforce smoothness with Laplacian penalty or low-pass filter on $V_n(r)$9. - Monitor convergence via relative loss change ($n$0) or probe/object updates.

The process is iterated until convergence criteria are met.

5. Information Limit, Precision, and Quantitative Metrics

eLOP achieves a significantly enhanced information limit and atomic-positioning precision relative to CPP. The information limit is determined from the radially averaged magnitude of the Fourier transform of the final phase image ("diffractogram") and defined by the 0.143-FRC criterion. For an 85 nm silicon sample, this corresponds to $n$1, i.e., 18 pm. For 60 nm SrTiO$n$2, the information limit is 16 pm. Atomic position precision is estimated by measuring column-to-column distances across all slices and computing the standard deviation; for Si, $n$3 pm.

Comparison with CPP is summarized below:

	Thickness Threshold (nm)	Information Limit (pm)	Atomic Position Precision (pm)
CPP	20–40	23–39	≈1–3 (lattice distortion risk)
eLOP	60 (SrTiO$n$4 unfiltered), 85 (Si filtered)	16–18	≈0.4

eLOP’s ability to explicitly model spatially varying aberration prevents lattice distortion and shape irregularities common in high-thickness CPP reconstructions (Yang et al., 25 Feb 2025).

6. Applications, Limitations, and Outlook

Accurate eLOP ptychographic reconstructions in thick objects provide high-fidelity structural data critical to elucidating intrinsic solid-state phenomena relevant for physics, chemistry, materials science, and semiconductor device engineering. Notable sample types imaged include FIB-prepared Si (85 nm thick) and SrTiO$n$5 (30–60 nm wedges) oriented along [001].

Limitations of eLOP include increased computational demands (increased memory and CPU/GPU resources for local-orbital expansions and aberration mapping), reliance on energy-filtered data without fully dynamical inelastic-scattering treatment, and the challenge of accurate sample-thickness calibration in real specimens.

Future directions may involve incorporation of explicit inelastic-scattering models in the multislice framework, adaptively learned basis sets for improved efficiency or accuracy, 3D ptychographic reconstructions (thickness mapping), and real-time GPU-based implementations to facilitate rapid feedback during experiments (Yang et al., 25 Feb 2025).

7. Comparative and Experimental Data

Simulated and experimental results illustrate the capabilities and robustness of eLOP:

• Simulations of a 60 nm test object demonstrate eLOP’s ability to recover position-dependent probe phase variation and reduce atomic distortion compared to CPP.
• Experimental phase images of SrTiO$n$6 at increasing thickness show that above $n$7 40 nm, CPP reconstructions collapse, while eLOP maintains high fidelity up to 60 nm.
• Quantitative data for both SrTiO$n$8 and Si confirm the resolution and atomic precision achievements described above.

The convergence and accuracy of eLOP across much greater thickness regimes markedly extend the practical utility of ptychographic imaging for dense, three-dimensional materials.