Single-Shot Coherent Diffractive Imaging

• Single-shot CDI is a lensless imaging technique that captures a single diffraction pattern per ultrafast pulse, requiring robust phase retrieval to reconstruct complex structures.
• Experimental implementations range from XFEL to laboratory HHG setups, achieving nanometer-scale resolution and enabling 3D imaging of nanocrystals and nanoparticles.
• Advanced inversion strategies, including symmetry enforcement, convex optimization, and neural-field methods, improve reconstruction accuracy and expand imaging capabilities.

Single-shot coherent diffractive imaging (CDI) is a lensless imaging technique in which a coherent short-wavelength pulse scatters from an individual object and the resulting far-field diffraction pattern is used to recover real-space structure, without lenses and without repeated exposures of the same specimen. In the single-shot regime, one pulse produces one diffraction pattern for one particle, and the structure is inferred entirely from that one pattern. At X-ray free-electron lasers (XFELs), this regime is tied to “diffraction-before-destruction,” because the sample is destroyed after a single exposure; in laboratory extreme-ultraviolet implementations, it likewise enables imaging of isolated particles in free flight from one pulse (Jr et al., 2017). The method has been demonstrated for single-shot 3D structure determination of individual nanocrystals using symmetry and Ewald-sphere curvature (Xu et al., 2013), for quantitative 3D imaging of Au/Pd core-shell nanoparticles with elemental specificity (Jr et al., 2017), for wide-angle 3D morphology retrieval of isolated faceted nanostructures (Colombo et al., 2022), and for time-resolved tracking of laser-driven electronic modifications in helium nanodroplets (Schäfer-Zimmermann et al., 27 Aug 2025).

1. Physical basis and the meaning of “single-shot”

CDI reconstructs the real-space electron density or complex object transmission from the intensity of the scattered field. Under the kinematic approximation, the scattering amplitude is the Fourier transform of the electron density,

$$F(\mathbf{q}) = \int \rho(\mathbf{r})\, e^{-i\,\mathbf{q}\cdot\mathbf{r}}\, d\mathbf{r},$$

and the detector records only

$I(\mathbf{q}) = |F(\mathbf{q})|^2.$

The phase of $F(\mathbf{q})$ is therefore lost and must be recovered computationally by phase retrieval (Jr et al., 2017). In a transmission formulation, the same relationship is often written as

$$F(\mathbf{q}) = \mathcal{F}\{O(\mathbf{r})\}, \qquad I(\mathbf{q}) \propto |F(\mathbf{q})|^2,$$

with $O(\mathbf{r})$ encoding the complex refractive response of the object (Schäfer-Zimmermann et al., 27 Aug 2025).

The single-shot qualifier means that only one diffraction pattern is available for a given object. In XFEL imaging, each pulse is so intense that the sample is destroyed after one exposure, so there is no possibility of collecting multiple views of the same particle (Jr et al., 2017). In this sense, single-shot CDI is not merely a data-acquisition mode but an inversion regime in which phase retrieval must succeed from one noisy, incomplete, and object-specific diffraction pattern (Latychevskaia et al., 2011).

A central enabling condition is oversampling: the diffraction pattern must be sampled more finely than the Nyquist limit set by object size, providing redundancy for numerical phase recovery (Jr et al., 2017). However, several later developments explicitly seek to reduce or bypass the practical burden of strong object-domain priors. These include holographically informed CDI (Latychevskaia et al., 2011), randomly coded masks (Seaberg et al., 2015), sparsity-based sub-wavelength single-shot imaging (Szameit et al., 2011), and untrained neural-field phase retrieval that eliminates handcrafted object-specific constraints (Hu et al., 10 Dec 2025).

A common misconception is that single-shot CDI is synonymous with static structural imaging. That is not sustained by recent work. The helium-nanodroplet pump–probe study shows that single-shot diffraction can also encode transient changes in the complex refractive index and thereby track ultrafast electronic dynamics, not only geometry (Schäfer-Zimmermann et al., 27 Aug 2025).

2. Experimental realizations and source modalities

Single-shot CDI has been implemented across distinct source classes. At SACLA, single-shot diffraction patterns of high-index faceted gold nanocrystals were recorded with $5.4\ \mathrm{keV}$, $\sim 10\ \mathrm{fs}$ XFEL pulses, focused to a $\sim 1.5\ \mu\mathrm{m}$ spot, yielding the first experimental single-shot 3D structure determination of individual nanocrystals at $\sim 5.5\ \mathrm{nm}$ resolution (Xu et al., 2013). In a later SACLA experiment on Au/Pd core-shell nanoparticles, the beam parameters were $6\ \mathrm{keV}$, $I(\mathbf{q}) = |F(\mathbf{q})|^2.$0, $I(\mathbf{q}) = |F(\mathbf{q})|^2.$1, and $I(\mathbf{q}) = |F(\mathbf{q})|^2.$2 photons per pulse, focused by Kirkpatrick–Baez mirrors to a $I(\mathbf{q}) = |F(\mathbf{q})|^2.$3 spot; 39,151 diffraction patterns were acquired, 34 high-quality single-shot patterns were selected, and quantitative 3D reconstructions were obtained with $I(\mathbf{q}) = |F(\mathbf{q})|^2.$4 resolution (Jr et al., 2017).

Single-shot CDI is not restricted to XFELs. A laboratory HHG implementation used harmonics of a 792 nm Ti:sapphire laser, with dominant contributions from the 13th and 15th harmonics, to record bright wide-angle scattering patterns of isolated helium nanodroplets. The geometry corresponded to $I(\mathbf{q}) = |F(\mathbf{q})|^2.$5, implying $I(\mathbf{q}) = |F(\mathbf{q})|^2.$6, and enabled shape classification of individual droplets in free flight (Rupp et al., 2016). This established that single-shot gas-phase nanoscopy with lab-based short-wavelength pulses is feasible (Rupp et al., 2016).

The source class and geometry strongly condition what information is encoded in one pattern. Wide-angle XUV and soft-X-ray scattering can carry 3D morphology and optical-property information in a single image, but only if the forward model accounts for wide-angle propagation and strong refractive effects; the propagation multi-slice Fourier transform method was introduced precisely to address this simulation problem (Tuemmler et al., 30 Jul 2025). In soft-X-ray wide-angle imaging of isolated silver nanoparticles at 243 eV, detector coverage up to $I(\mathbf{q}) = |F(\mathbf{q})|^2.$7 and $I(\mathbf{q}) = |F(\mathbf{q})|^2.$8 were used to reconstruct arbitrary convex polyhedral morphologies from single diffraction patterns (Colombo et al., 2022).

The experimental meaning of single-shot is therefore source-dependent but structurally consistent: one coherent pulse, one object, one diffraction record. What changes across implementations is the balance among wavelength, bandwidth, angular coverage, probe structure, and the amount of prior information injected into reconstruction.

3. Recovering three-dimensional information from one pattern

The central technical problem in single-shot CDI is that one detector image is two-dimensional, whereas the target is generally three-dimensional. Several distinct strategies have been used to lift one pattern into a 3D structural inference.

In the gold nanocrystal and Au/Pd nanocube experiments, the decisive ingredients were Ewald-sphere curvature and object symmetry. Because elastic scattering satisfies $I(\mathbf{q}) = |F(\mathbf{q})|^2.$9, the sampled reciprocal-space points lie on the Ewald sphere rather than on a flat plane. For nearly cubic particles with octahedral symmetry, the measured Fourier magnitudes on this curved manifold can be replicated by all 48 octahedral symmetry operations, substantially filling 3D reciprocal space from a single exposure (Jr et al., 2017). In the earlier trisoctahedral nanocrystal work, crystallographic point group $F(\mathbf{q})$0 symmetry was likewise used to generate 48 symmetry-related spherical diffraction patterns; self-common arcs verified symmetry and orientation, and a 3D Cartesian diffraction volume was assembled from a single shot (Xu et al., 2013).

For the Au/Pd core-shell nanoparticles, the interpolation of Fourier magnitudes onto a 3D grid was performed with

$F(\mathbf{q})$1

with a spherical interpolation kernel of radius $F(\mathbf{q})$2 voxels (Jr et al., 2017). Orientation refinement exploited the fact that most nanocubes lay flat on the Si$F(\mathbf{q})$3N$F(\mathbf{q})$4 membrane, yielding angular precision of $F(\mathbf{q})$5 (Jr et al., 2017).

A different route is wide-angle model-based reconstruction without imposing specific particle symmetries. For isolated silver nanostructures, the object was parameterized as a convex polyhedron with uniform refractive index, and the diffraction pattern was matched by forward simulation rather than voxel-based model-free phase retrieval. The shape was defined by $F(\mathbf{q})$6 planes, and a memetic optimization combining differential evolution and Nelder–Mead simplex minimized

$F(\mathbf{q})$7

This enabled retrieval not only of highly symmetric motifs but also of imperfect shapes and agglomerates that were not accessible previously (Colombo et al., 2022).

A plausible implication is that “single-shot 3D CDI” is not a single inversion strategy but a family of inference problems. Symmetry-based Ewald-sphere lifting, convex-polyhedron forward fitting, and structured-illumination single-shot ptychography each solve different versions of the same dimensionality deficit under different physical assumptions (Levitan et al., 2024).

4. Phase retrieval, inversion strategies, and computational formulations

The classical reconstruction loop in CDI alternates between reciprocal space and object space, enforcing measured Fourier magnitudes in one domain and support, positivity, or related priors in the other. In the Au/Pd nanoparticle work, phase retrieval was performed with the Oversampling Smoothness (OSS) algorithm. For each single-shot 3D Fourier grid, 1000 independent runs were executed, each with 1000 OSS iterations; positivity and loose-support constraints were used in real space, measured $F(\mathbf{q})$8 values were imposed in reciprocal space, and ten progressive low-pass filters promoted smooth convergence (Jr et al., 2017). Agreement between measured and reconstructed Fourier magnitudes was monitored by

$F(\mathbf{q})$9

The top 10% of runs, ranked by $$F(\mathbf{q}) = \mathcal{F}\{O(\mathbf{r})\}, \qquad I(\mathbf{q}) \propto |F(\mathbf{q})|^2,$$0-factor, were averaged to produce the final reconstruction (Jr et al., 2017).

In the earlier gold nanocrystal study, OSS was also used, combined with symmetry enforcement every 30 iterations. One hundred independent reconstructions were run per diffraction volume, and the best solutions were used to form tight supports and final averages (Xu et al., 2013). This illustrates a recurrent feature of single-shot CDI: even when only one diffraction pattern exists, the inversion is typically ensemble-based at the algorithmic level, with many random starts used to stabilize phase retrieval.

Alternative detector-domain constraints have been proposed to improve convergence from a single noisy pattern. A holography-inspired Fourier-domain update,

$$F(\mathbf{q}) = \mathcal{F}\{O(\mathbf{r})\}, \qquad I(\mathbf{q}) \propto |F(\mathbf{q})|^2,$$1

and its stabilized variant

$$F(\mathbf{q}) = \mathcal{F}\{O(\mathbf{r})\}, \qquad I(\mathbf{q}) \propto |F(\mathbf{q})|^2,$$2

were introduced as fast alternatives to hard modulus replacement, yielding low-resolution reconstructions first and then refining them to high resolution without requiring a support mask (Latychevskaia et al., 2011). The paper explicitly frames this as advantageous for single-shot CDI, where only one diffraction pattern is available and support information may be poor (Latychevskaia et al., 2011).

Other strategies move information into the forward operator rather than into object-space constraints. Randomly coded masks generate multiple coded diffraction patterns

$$F(\mathbf{q}) = \mathcal{F}\{O(\mathbf{r})\}, \qquad I(\mathbf{q}) \propto |F(\mathbf{q})|^2,$$3

allowing mask constraints to replace conventional support priors; experimentally, 3–4 masks were sufficient for stable, unique reconstructions in the tested geometry (Seaberg et al., 2015). Holographic coherent diffraction imaging derives phase from the Fourier transform of an inline hologram, using the relation

$$F(\mathbf{q}) = \mathcal{F}\{O(\mathbf{r})\}, \qquad I(\mathbf{q}) \propto |F(\mathbf{q})|^2,$$4

so that oversampling requirements can be relaxed (Latychevskaia et al., 2011). Sparsity-based single-shot sub-wavelength CDI instead minimizes the number of degrees of freedom in a known basis and was demonstrated experimentally for 100 nm features with 30 nm resolution under 532 nm illumination (Szameit et al., 2011).

A recent extension replaces handcrafted support and positivity constraints by untrained coordinate-based neural fields and a physics-consistent forward model. In that formulation, the measured intensity is

$$F(\mathbf{q}) = \mathcal{F}\{O(\mathbf{r})\}, \qquad I(\mathbf{q}) \propto |F(\mathbf{q})|^2,$$5

while the learned reconstruction is parameterized as a neural field $$F(\mathbf{q}) = \mathcal{F}\{O(\mathbf{r})\}, \qquad I(\mathbf{q}) \propto |F(\mathbf{q})|^2,$$6 and optimized through

$$F(\mathbf{q}) = \mathcal{F}\{O(\mathbf{r})\}, \qquad I(\mathbf{q}) \propto |F(\mathbf{q})|^2,$$7

with data-fidelity and perceptual losses (Hu et al., 10 Dec 2025). This suggests a shift from explicit constraint engineering to implicit priors embedded in representation and forward physics.

5. Quantitative observables: resolution, composition, and dynamic electronic response

Single-shot CDI is often associated with qualitative morphology, but several implementations are explicitly quantitative. In the Au/Pd core-shell nanoparticle study, model fitting to the reconstructed electron densities yielded an Au core size of $$F(\mathbf{q}) = \mathcal{F}\{O(\mathbf{r})\}, \qquad I(\mathbf{q}) \propto |F(\mathbf{q})|^2,$$8, a Pd shell thickness of $$F(\mathbf{q}) = \mathcal{F}\{O(\mathbf{r})\}, \qquad I(\mathbf{q}) \propto |F(\mathbf{q})|^2,$$9, and an average intensity ratio

$O(\mathbf{r})$0

in close agreement with the tabulated scattering factor ratio

$O(\mathbf{r})$1

The agreement within 2% was used to claim elemental specificity in the reconstructed 3D density (Jr et al., 2017).

Spatial resolution in that work was determined by Fourier shell correlation,

$O(\mathbf{r})$2

with the resolution defined at $O(\mathbf{r})$3. The average FSC between every pair of the 34 reconstructions gave a 3D resolution of $O(\mathbf{r})$4 (Jr et al., 2017). In the earlier gold nanocrystal work, 15 FSC curves between six independent reconstructions yielded $O(\mathbf{r})$5 resolution (Xu et al., 2013).

In wide-angle silver-nanoparticle imaging, resolution was characterized differently because the reconstruction is parametric rather than voxel-based. Stability analysis based on repeated reconstructions yielded transition widths at boundaries that were mostly well below the theoretical real-space limits of $O(\mathbf{r})$6 transversely and $O(\mathbf{r})$7 along the beam direction, with higher uncertainty at facets perpendicular to the beam and in agglomerate contact regions (Colombo et al., 2022).

Single-shot CDI can also quantify transient optical-property changes rather than static structure. In helium nanodroplets pumped by an NIR field and probed with ultrashort XUV pulses, the small-angle coherent diffraction signal dropped by up to about 30% when pump and probe overlapped in time (Schäfer-Zimmermann et al., 27 Aug 2025). The interpretation was a transient increase in XUV absorption caused by laser-induced modification of the electronic structure. The effective refractive index was written as

$O(\mathbf{r})$8

and the simulations showed that at the energy of the 13th harmonic, $O(\mathbf{r})$9 increased by about an order of magnitude while $5.4\ \mathrm{keV}$0 remained large and negative (Schäfer-Zimmermann et al., 27 Aug 2025). The final modeled detector signal reproduced the observed roughly symmetric $5.4\ \mathrm{keV}$1 intensity drop centered near zero delay (Schäfer-Zimmermann et al., 27 Aug 2025).

A related theoretical development argues that at strong resonant XUV fields the standard linear-response description of CDI can break down. A density-matrix-based scattering model predicts Rabi cycling, saturation, and inelastic sidebands in the diffraction signal from helium nanodroplets, leading to the proposal of “quantum coherent diffractive imaging (QCDI)” as a distinct strong-field regime (Kruse et al., 2020). This suggests that the “object” in single-shot CDI need not remain a static, linearly scattering density even during one pulse.

6. Limits, misconceptions, and trajectories of development

The most important limitation in symmetry-assisted single-shot 3D CDI is the symmetry requirement itself. In the Au/Pd study, 48 octahedral symmetry operations were essential to fill reciprocal space; without such symmetry, one Ewald-sphere cut would provide too little information for robust 3D phase retrieval (Jr et al., 2017). The method is therefore best suited to highly symmetric objects such as nanocubes, octahedra, icosahedra, and certain viruses (Jr et al., 2017). The earlier gold nanocrystal study makes the same point in a different form: single-shot 3D structure determination became possible because the nanocrystals belonged to point group $5.4\ \mathrm{keV}$2 (Xu et al., 2013).

A second limitation is that wide-angle one-pattern inversion is highly model-dependent when symmetry is absent. Convex-polyhedron fitting substantially broadens the accessible morphology class, but it still assumes convexity, faceting, and uniform internal refractive index. Non-convexity, internal density gradients, and contact regions in agglomerates remain difficult (Colombo et al., 2022). A plausible implication is that the information content of one diffraction pattern is often sufficient only after aggressive regularization by geometry, symmetry, sparsity, or learned representation.

A third limitation concerns bandwidth and non-monochromatic probes. Conventional CDI assumes quasi-monochromatic illumination, but isolated attosecond pulses are extremely broadband. A numerical monochromatization scheme based on regularized inversion of a matrix depending only on the diffracted spectrum was introduced to enable coherent diffraction imaging with broadband isolated attosecond sources, because standard CDI fails for the $5.4\ \mathrm{keV}$3 bandwidth typical of attosecond pulses (Huijts et al., 2019).

Another misconception is that overlap is indispensable for quantitative coherent imaging of extended samples. Recent work in structured-illumination single-shot ptychography reframed a grid of overlapping beams as a single complex illumination function and achieved a resolution 3.5 times finer than the numerical-aperture-based limit imposed by traditional single-shot ptychography algorithms (Levitan et al., 2024). A subsequent overlap-free formulation in Fresnel CDI geometry reported overlap-free single-shot reconstruction with amplitude SSIM 0.904, compared with 0.968 for overlap-constrained reconstruction, at approximately $5.4\ \mathrm{keV}$4 the per-GPU throughput of LSQ-ML at matched $5.4\ \mathrm{keV}$5 resolution (Hoidn et al., 24 Feb 2026). These results do not eliminate the ill-posedness of single-shot CDI, but they shift the source of redundancy from scan overlap to probe structure and learned physics.

More broadly, recent developments point in two directions. One is improved forward modeling for wide-angle and strongly refractive scattering, where pMSFT provides qualified guidance for choosing between SAXS, Born, MSFT, Hare, and pMSFT according to wavelength, refractive index, and angle (Tuemmler et al., 30 Jul 2025). The other is reconstruction frameworks that relax handcrafted constraints, such as physics-guided neural fields for static and dynamic CDI (Hu et al., 10 Dec 2025). Together, these suggest that the future of single-shot CDI lies less in a universal inversion formula than in tighter coupling of source physics, probe design, scattering simulation, and representation learning.

Single-shot CDI therefore occupies a distinctive position within coherent imaging. It is simultaneously a destructive-imaging strategy, a wide-angle inverse-scattering problem, and, increasingly, a test bed for reconstruction methods that must operate with minimal redundancy. Its major achievements to date—single-shot 3D nanocrystal reconstruction (Xu et al., 2013), element-specific 3D core-shell imaging (Jr et al., 2017), wide-angle retrieval of imperfect faceted nanostructures (Colombo et al., 2022), laboratory HHG single-particle imaging (Rupp et al., 2016), and femtosecond tracking of transient opacity (Schäfer-Zimmermann et al., 27 Aug 2025)—show that one diffraction pattern can encode far more than a projected silhouette, provided the geometry, priors, and forward model are commensurate with the underlying physics.