Single-shot, coherent, pop-out 3D metrology

The paper addresses the challenge of rapid, large-scale 3D reconstruction of thin, extended specimens at nanometer resolution. Traditional tomography (X-ray or electron) needs a tilt series, which is slow, limited in temporal resolution, and often infeasible for extended, high-aspect-ratio samples due to occlusion and absorption at high tilts. Coherent imaging approaches such as ptychographic laminography also require multiple tilts and large-scale facilities. The authors posit that with strong priors—specifically, homogeneous, amorphous materials—3D metrology from a single 2D image could be feasible. Building on holography and multislice beam propagation, they propose a single-shot method that exploits bright-field TEM images to encode depth (via Thon ring spacing in the power spectrum and an effective CTF defocus tied to the specimen’s center-of-mass depth) and thickness (via energy-filtered absorption contrast). This promises rapid, accessible 3D nanometrology suited to dynamics, process control, and extended surfaces, without rotation or destructive preparation.

The work situates itself amid advances in tomography (electron and X-ray) that, while powerful, are limited by acquisition time, missing-cone issues, and infrastructure requirements. Machine learning has enabled reduced-angle tomography through strong priors, suggesting potential for fewer views. Holography (Gabor) encodes depth in phase and has benefited from digital detectors and computation, as has the multislice formalism for modeling multiple scattering. Lensless digital holography has been used to infer 3D positions of many particles, and compressive holography can reconstruct sparse 3D densities from single holograms, but dense specimens pose ill-posed inverse problems. Prior phase retrieval and depth sectioning methods typically require through-focal series. The authors leverage these developments to propose a single-shot approach specialized to homogeneous, amorphous materials, using the dominance of the single-slice sum in the power spectrum and energy-filtered BF-TEM.

Concept and forward model: Using the multislice formalism under a kinematic/weak phase linearization, the squared amplitude of the Fourier transform of a BF-TEM image can be decomposed into a dominant single-slice sum B(k) and a double-sum interference term I(k). For amorphous, random, uncorrelated slices of comparable scattering density, I(k) averages out, leaving an effective contrast transfer function (CTF) modulated by an envelope that includes specimen thickness and partial coherence. Critically, the effective defocus is offset by half the specimen thickness, i.e., centered at the specimen’s center of scattering mass. CTF fitting for depth (defocus): Depth is inferred patch-wise from Thon-ring spacings in the azimuthally averaged power spectrum by semi-empirical CTF fitting with a Gaussian-like envelope and noise model. The aberration function includes spherical aberration Cs and defocus Δf (including astigmatism when needed). The fitted defocus yields the local center-of-mass depth relative to the zero-defocus plane. Thickness from absorption (energy filtering): With an energy filter to exclude inelastically scattered electrons, local thickness T(x,y) is computed via a log-ratio Beer–Lambert relation using a flat-field I0(x,y) and specimen image I1(x,y), T = Imfp ln(I0/I1). The inelastic mean free path Imfp is calibrated using a region of known thickness in the same material and imaging conditions or via a calibration specimen. Patch-wise reconstruction and stitching: The image is partitioned into overlapping patches for CTF fitting (depth) and smaller windows for thickness averaging. Running-window voting with visitation weights discards patches with high fit error (e.g., near sharp edges), producing robust maps of defocus Δf(x,y) and thickness T(x,y). A 3D volume is then reconstructed by popping out material symmetrically about the local center-of-mass depth for each pixel: voxels within [Δf − T/2, Δf + T/2] along z are filled. Overlaps are blended via weighted averages and gaps are inpainted via nearest neighbors. Anisometry from voxel scaling and effective magnification variation with depth can be corrected when significant (>~10%). Dual-layer extension: For two amorphous layers, the power spectrum can be modeled as an incoherent sum of two single-layer CTFs with separate envelopes and amplitudes. Fitting this additive model yields separate layer defocuses (depths). Layer thicknesses can be deduced if one layer’s thickness is known or, with sufficient dose and patch size, from amplitude ratios to estimate relative thickness. Resolution, sampling, and calibration: The xy resolution is limited by the patch size W used for CTF fitting; z precision improves with larger patches and higher dose but trades off xy resolution. A sinc-like envelope imposes a node at λ k^2 T = 1, setting a lower bound on resolvable spatial frequency and hence a transverse resolution limit dependent on the maximum sample thickness Tmax. Practical criteria include ensuring at least three prominent CTF maxima within the first envelope node and ≥8 frequency samples between maxima. Dose studies show z error decreases markedly up to ~100 e Å−2 exit dose. Patch size must increase with thickness to sample Thon rings adequately. Acquisition and processing flow: The process includes calibrating Cs, detector pixel size, magnification range given sample thickness, dose budget, flat-field capture, energy filtering, ensuring ≥3 visible CTF rings across the field, determining defocus range and patch size, then imaging (optionally scanned and stitched), followed by pop-out reconstruction (defocus map, thickness map, 3D model). Optimization uses Levenberg–Marquardt fitting (SciPy), with coarse-to-fine bounds and optional 2D (with astigmatism) or 1D fits depending on astigmatism significance. Experimental and simulation specifics: Simulations used 200 keV electrons, Cs = 1.2 mm, detector pixel size 6 μm. Experiments used a JEOL 2200 TEM with a DE16 direct electron detector and omega energy filter (20 eV zero-loss window). Reconstructions were visualized in TomViz. Example pop-out reconstructions used patch sizes corresponding to 20–40 nm voxel sizes, and total doses of ~2000–2500 e Å−2.

- Introduced single-shot pop-out 3D metrology: a coherent BF-TEM method that simultaneously infers local depth (via CTF defocus/Thon rings) and thickness (via energy-filtered absorption) from a single image of amorphous, homogeneous specimens. - Effective CTF defocus is centered at the specimen’s depth center-of-mass, enabling depth inference; interference cross-terms are negligible for amorphous random slices, validating an effective single-layer CTF model. - Demonstrated 3D reconstructions from single energy-filtered BF-TEM images: • Nanochannel etched in amorphous SiN: reconstruction reveals top-side etch and flat bottom surface; xy half-period resolution ~30 nm (patch-size limited), total dose ~2000 e Å−2. • Nanopit etched through SiN: reconstruction (40 nm voxel) reveals petal-like rim features and debris on top surface and a hidden double-conical cavity widening toward the bottom; total dose ~2500 e Å−2. - Resolution and parameter dependencies (simulations): • Thickness: Thicker samples strengthen Thon rings (positive for depth fits) but reduce exit dose due to inelastic scattering (negative via noise); at modest thicknesses the positive effect dominates. • Patch size: Larger patches improve z precision (better frequency sampling) but reduce xy resolution; minimum patch size increases with Tmax to satisfy sampling before the envelope’s first node. • Dose: With 20 nm patches, z-depth errors drop sharply with exit dose up to ~100 e Å−2; ~5 nm z accuracy achievable for SiN features thinner than ~0.5 Imfp (~66.5 nm for SiNx) at ~100 e Å−2 exit dose. - Dual-layer extension validated numerically: • With known thickness prior for one layer, additive-CTF fitting accurately recovers depths and thicknesses (e.g., 25 nm top membrane and bottom wedge varying 2–50 nm). • With sufficient dose (~20k e Å−2) and large patches (e.g., 400×400 px), relative layer thicknesses can be inferred from fitted amplitude ratios even without priors. - For specimens thinner than 40 nm, sub-10 nm 3D resolution is feasible at ~100 e Å−2; for ~200 nm-thick SiN, demonstrated ~30 nm resolution. Single-shot acquisitions (2–6 s exposures, no tilting) support rapid, large-area metrology.

The study demonstrates that, under the prior of amorphous homogeneous material, the BF-TEM image’s Fourier magnitude functions equivalently as a hologram whose Thon-ring spacings encode local depth relative to focus, while energy-filtered absorption provides local thickness. Combining these measurements in overlapping patches enables a 3D “pop-out” reconstruction from a single image, addressing the challenge of fast 3D metrology for extended, thin samples where tomography is impractical. The approach complements tomography: it is not designed to recover full 3D density distributions of general heterogeneous or crystalline samples, but excels in rapid, minimally invasive 3D assessment over large fields of view, potentially at millisecond timescales with fast detectors. Simulations clarify the trade-offs among thickness, patch size, and dose, providing practical criteria to tune acquisition for target xy and z resolutions. Numerical dual-layer tests indicate scalability to multilayered amorphous structures using an additive-CTF model. Experimentally, reconstructions reveal features (e.g., flat bottoms, hidden double-conical cavities) not inferable from thickness alone, underscoring the importance of depth encoding in the CTF. The method’s general wave-optics basis suggests applicability to coherent bright-field imaging with electrons, X-rays, or visible light, particularly in near-field in-line holographic geometries with appropriate filtering and aberration calibration.

Pop-out 3D metrology provides a computational wave-optics framework for single-shot 3D reconstruction of amorphous, homogeneous-density specimens using coherent bright-field imaging. By fitting Thon-ring CTFs to infer local center-of-mass depth and using energy-filtered absorption to measure thickness, it reconstructs 3D structures rapidly without sample tilting or destructive preparation. Experiments demonstrate ~30–40 nm resolution on ~200 nm SiN and simulations indicate sub-10 nm resolution for ≤40 nm specimens at ~100 e Å−2. The approach generalizes to dual-layer structures using an additive CTF model and is compatible with standard TEM hardware (with energy filtering and direct detectors). With automation, it can extend to micrometer-scale fields of view and probe dynamic processes at high temporal resolution. Future work includes determining practical multilayer limits, refining models for partially crystalline materials and higher-order aberrations, optimizing dose/patch-size trade-offs, extending to other coherent modalities (X-ray/optical in-line DHM), and integrating high-speed detectors (kHz to ultrafast) for real-time 3D dynamics.

- Assumes amorphous, homogeneous-density specimens; performance degrades with increasing crystallinity due to non-negligible cross-terms and correlated scattering. - Does not recover full 3D density as tomography does; under-constrained for general heterogeneous samples. - Requires energy filtering to exclude inelastic electrons; otherwise thickness estimates are biased and CTF contrast reduced. - Accurate calibration needed: inelastic mean free path (sample- and condition-specific), aberrations (Cs, astigmatism), detector response linearity. - Resolution trade-offs: z precision improves with larger patches and higher dose but at the cost of xy resolution and potential specimen dose limits. - Envelope (from thickness, coherence, motion, drift) suppresses Thon rings beyond the first node, limiting usable spatial frequencies, especially for thick samples. - Effective magnification varies with depth for very thick or multi-layer samples; may require voxel scaling corrections when changes are non-negligible. - Dual-layer/multilayer separability depends on distinct Thon-ring patterns, adequate dose, and patch size; practical upper limit on layers depends on signal and optical thickness.