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Coherent diffraction imaging

Coherent diffraction imaging (CDI) is a lensless computational imaging technique that reconstructs high-resolution, three-dimensional images of specimens—particularly noncrystalline materials and biological samples—from the intensity of their coherent diffraction patterns, bypassing the need for physical lenses or direct imaging optics.^[1] This method relies on illuminating isolated samples with coherent radiation sources, such as X-rays from synchrotrons or X-ray free-electron lasers (XFELs), and capturing the resulting speckle patterns with high-sensitivity detectors.^[2] By addressing the phase problem through iterative algorithms like hybrid input-output (HIO) or error reduction, CDI enables quantitative electron density mapping at nanoscale resolutions, typically 10–50 nm, making it ideal for radiation-sensitive structures that cannot be crystallized for traditional crystallography.^[1] The foundational concept of CDI traces back to David Sayre's 1952 proposal to recover images from oversampled diffraction data, but practical realization awaited advances in coherent sources and computing; the first experimental demonstration occurred in 1999 using soft X-rays on lithographic test patterns.^[1] Subsequent developments, including the application of XFELs in the 2000s, extended CDI to dynamic, single-shot imaging of fragile biological specimens like viruses and cells under cryogenic conditions to mitigate radiation damage.^[2] Variants such as Bragg CDI (for crystalline lattices) and ptychography (for extended samples via scanning illumination) have further broadened its scope, achieving resolutions down to ~2 nm in materials science applications.^[1] In biology and materials research, CDI has revolutionized the study of heterogeneous, unstained samples, enabling 3D visualization of yeast cells, bacterial nucleoids, and nanomaterials without preparative artifacts.^[2] Its non-destructive nature and compatibility with femtosecond pulses from XFELs allow probing ultrafast processes, such as protein dynamics, while ongoing refinements in phase-retrieval algorithms and detector technology continue to push resolution limits toward atomic scales.^[1]

Fundamentals

Imaging Process

Coherent diffraction imaging (CDI) is a lensless microscopy technique that employs a coherent beam to illuminate an isolated sample, capturing the far-field diffraction intensity pattern on a detector to encode structural information. In this approach, the absence of physical lenses eliminates optical aberrations, relying instead on the wave nature of the illuminating radiation to form a speckled diffraction pattern through interference of scattered waves. The recorded pattern represents the squared modulus of the Fourier transform of the sample's exit wave, providing a direct link between the object's density distribution and the measurable intensities. The imaging process commences with meticulous sample preparation to ensure isolation and stability, such as dispersing nanoparticles or nanocrystals in a vacuum or cryogenic environment to minimize background scattering. A coherent beam, typically generated from synchrotron or free-electron laser sources, is directed onto the sample, interacting primarily through elastic scattering where photons or particles are redirected without energy loss. This scattering produces a diffraction pattern in the far field, where the intensity distribution arises from constructive and destructive interference, forming a unique "fingerprint" of the sample's structure. The relationship is mathematically expressed as

I(\mathbf{q}) = \left| \mathcal{F} [\rho(\mathbf{r})] \right|^2,

where I(\mathbf{q}) is the measured intensity as a function of the scattering vector \mathbf{q}, \mathcal{F} denotes the Fourier transform, and \rho(\mathbf{r}) is the electron density (or equivalent contrast mechanism) of the sample. This far-field approximation holds when the detector is positioned sufficiently downstream, ensuring the pattern reflects the angular distribution of scattered amplitudes. The foundational idea of CDI traces to David Sayre's 1952 proposal on recovering images from oversampled diffraction data, with a 1980 extension outlining the potential for reconstructing images from coherent diffraction patterns of non-crystalline specimens using oversampling to enable phase recovery. The technique's experimental viability was demonstrated in 1999 by Jianwei Miao and collaborators, who successfully recorded and analyzed soft X-ray diffraction patterns from lithographic test patterns, marking the first lensless imaging of non-periodic objects at nanoscale resolution. Relative to conventional lens-based imaging, CDI circumvents limitations like spherical and chromatic aberrations in optics, offering the potential for atomic-scale resolution constrained solely by the illuminating wavelength and sample radiation tolerance, while excelling with aperiodic or heterogeneous samples that defy traditional crystallography. The resulting intensity pattern, however, captures only the amplitude information, posing the phase problem for full reconstruction.

Coherence Requirements

Coherent diffraction imaging (CDI) relies on the interference of scattered waves to form interpretable diffraction patterns, necessitating a highly coherent illuminating beam to preserve phase relationships across the sample. Coherence in CDI is characterized by transverse (spatial) and longitudinal (temporal) components. Transverse coherence describes the correlation of the electric field phases in the plane perpendicular to the beam propagation direction, while longitudinal coherence pertains to phase correlations along the propagation axis. The degree of coherence, denoted as \gamma, quantifies this correlation and is defined as \gamma = \frac{|\langle E_1 E_2^* \rangle|}{|E_1| |E_2|}, where E_1 and E_2 are the electric fields at two points, and the asterisk denotes the complex conjugate; \gamma ranges from 0 (incoherent) to 1 (fully coherent).^[3] For successful CDI, the transverse coherence length L_c must exceed the sample size D to ensure visibility of fine speckle structures in the diffraction pattern, with theoretical analyses indicating a requirement of L_c \approx 2D to satisfy Shannon sampling criteria for phase retrieval. Similarly, the longitudinal coherence length \delta = \lambda^2 / \Delta\lambda, where \lambda is the wavelength and \Delta\lambda the spectral bandwidth, should surpass the sample thickness to maintain phase stability throughout the interaction volume. Partial incoherence degrades the diffraction pattern by blurring speckles, which reduces contrast and effective resolution, ultimately lowering the oversampling ratio \sigma = D / \Delta—where \Delta is the detector pixel size projected onto the sample plane—and complicating reconstruction.^[4] Coherence is typically measured using Young's double-slit interferometry, where the visibility of interference fringes from two closely spaced slits provides an estimate of the transverse coherence length via the van Cittert-Zernike theorem.^[5] Early CDI experiments at synchrotrons, such as those in the late 1990s, required pinhole filtering to achieve sufficient transverse coherence by selecting a small portion of the partially coherent beam. In contrast, modern free-electron lasers (FELs) inherently deliver beams with high coherence due to their laser-like properties, enabling brighter and more robust CDI implementations without extensive filtering.

The Phase Problem

In coherent diffraction imaging (CDI), the phase problem originates from the fact that diffraction detectors capture only the intensity of the scattered radiation, corresponding to the squared magnitude of the Fourier transform of the object's scattering potential, |F[\rho(\mathbf{r})]|^2, while the phase, \arg(F[\rho(\mathbf{r})]), is irretrievably lost during measurement.^[6] This information loss stems from the nature of typical detectors, such as CCDs or pixel arrays, which are sensitive to photon counts but insensitive to the wave's phase shifts.^[6] Mathematically, the phase problem is formulated as recovering the complex-valued object \rho(\mathbf{r}) from the measured intensities I(\mathbf{q}) = |F[\rho(\mathbf{r})]|^2, where F denotes the Fourier transform and \mathbf{q} are the scattering momenta.^[6] Direct inversion via the inverse Fourier transform is impossible, as the mapping from \rho(\mathbf{r}) to |F[\rho(\mathbf{r})]|^2 is highly nonlinear and underdetermined, with infinitely many objects producing identical intensity patterns due to ambiguities like global phase shifts, translations, and conjugate flips.^[6] Uniqueness requires oversampling the diffraction pattern, defined by the ratio \sigma of the number of measured intensity pixels to the number of degrees of freedom in the real-space object; for one-dimensional cases, \sigma > 2 suffices under additional constraints. The phase problem was first recognized in the context of X-ray crystallography shortly after Max von Laue's 1912 discovery of X-ray diffraction by crystals, with William Henry Bragg explicitly noting in 1915 that phases of diffracted waves could not be directly measured, complicating structure determination from intensities alone.^[7] This challenge persisted as a central barrier in crystallographic methods throughout the 20th century and became equally foundational to CDI upon its emergence in the late 1990s, where non-crystalline specimens demanded phase recovery without periodic constraints.^[7] In CDI, the consequences of the phase problem are profound: without phase information, the measured diffraction pattern cannot be straightforwardly inverse-transformed to yield the real-space image of the object, necessitating computational reconstruction to estimate the lost phases.^[6] This indirect approach introduces sensitivity to noise and initial guesses, potentially leading to non-unique or artifact-prone reconstructions if insufficient constraints are applied.^[6] To address the underdetermined nature and achieve uniqueness, a key constraint in CDI is the assumption of finite object support, where the scattering potential \rho(\mathbf{r}) is nonzero only within a known bounded region.^[6] Bates' theorem provides a theoretical foundation, proving that in one dimension, with an oversampled Fourier intensity measurement (\sigma = 2) and strict support enforcement, the phase can be uniquely recovered up to trivial ambiguities such as translation and conjugate symmetry. Extensions of this result to two and three dimensions confirm similar uniqueness guarantees when oversampling exceeds twice the support area (or volume) in each spatial dimension, enabling reliable phase retrieval for isolated, finite objects. Coherent illumination in CDI facilitates this oversampling by producing speckle patterns with the required redundancy in the far-field diffraction.

Reconstruction Techniques

General Reconstruction Principles

Coherent diffraction imaging (CDI) addresses the phase problem by employing iterative projection methods that enforce physical and mathematical constraints on the estimated object to recover the phase information lost in intensity measurements. The core principle involves alternating projections between real space (object domain) and reciprocal space (diffraction domain), starting from an initial guess of the object's phase or density. In each iteration, the estimate is projected onto constraint sets that reflect known properties of the sample and the measurement, gradually refining the reconstruction until convergence is achieved. This framework, rooted in the oversampling phasing method, enables unique phase recovery under appropriate sampling conditions.^[1] Key constraints guide the projection process and ensure physical realism. The modulus constraint requires the calculated diffraction intensities to match the measured ones within noise limits, achieved by replacing the magnitudes of the Fourier transform with the square roots of the observed intensities while preserving the phases. The support constraint confines the object's density to a known finite boundary, typically determined a priori from low-resolution images or geometric assumptions, setting density to zero outside this region. Additionally, the positivity constraint enforces non-negative values for the electron density or amplitude in real space, reflecting the physical nature of scattering objects like materials or biological samples. These constraints collectively reduce the ambiguity in phase retrieval by narrowing the solution space.^[1]^[8] Successful reconstruction relies on the oversampling theorem, which stipulates that the diffraction pattern must be sampled at least twice the Nyquist rate in each dimension for uniqueness in two-dimensional phase recovery, corresponding to an oversampling ratio σ ≥ 2, where σ is the ratio of the total sampled pixels to the support pixels. This oversampling provides redundant data that introduces correlations allowing error reduction through iterative feedback between domains. In three dimensions, the requirement increases to σ ≥ 4 for discrete objects.^[9] Convergence of the iterative process is evaluated using metrics such as the R-factor, defined as

R = \frac{\sum |I_{\text{meas}} - I_{\text{calc}}|}{\sum I_{\text{meas}}}

where I_{\text{meas}} and I_{\text{calc}} are the measured and calculated intensities, respectively; values below 0.1 often indicate reliable reconstructions. The mean inner product between real-space estimates from independent runs serves as another metric to assess solution uniqueness, with values approaching 1 signifying minimal ambiguity. These metrics monitor progress and validate the final image against experimental data. Computationally, each iteration typically requires $10^6 to $10^9 operations, dominated by fast Fourier transforms and scaling with the square (in 2D) or cube (in 3D) of the image size, making efficient implementations essential for large datasets.^[10]^[11]

Iterative Phase Retrieval Algorithms

Iterative phase retrieval algorithms address the phase problem in coherent diffraction imaging by alternately enforcing constraints in the real and Fourier domains, such as the measured intensity modulus in reciprocal space and a finite support in real space. The error reduction (ER) algorithm serves as a foundational method, involving simple projection alternation between these domains to iteratively minimize discrepancies. It enforces the modulus constraint by replacing the magnitude of the current Fourier estimate with the measured intensities while preserving phases, then applies the support constraint by setting values outside the support to zero in real space; however, ER exhibits slow convergence, often requiring thousands of iterations, and is primarily used as a baseline or final polishing step. The hybrid input-output (HIO) algorithm, introduced by Fienup in 1982, improves upon ER by incorporating a feedback mechanism to escape local minima and accelerate convergence. In HIO, after applying the Fourier modulus constraint to obtain an output estimate g'(\mathbf{r}), the next input is updated as g_{n+1}(\mathbf{r}) = g'_n(\mathbf{r}) for points inside the support, while outside the support it uses g_{n+1}(\mathbf{r}) = g_n(\mathbf{r}) - \beta g'_n(\mathbf{r}), where \beta is a feedback parameter typically set to approximately 0.9 to balance stability and progress. This modification allows HIO to achieve recognizable reconstructions in 20-100 iterations, far outperforming ER in speed. In practice, reconstructions often begin with random initial phases applied to the square root of the measured intensities, followed by support refinement using techniques like the shrink-wrap algorithm, which iteratively tightens the support boundary based on the current density estimate to reduce artifacts. Typical runs involve 100-1000 iterations, alternating 10-30 HIO cycles with 5-10 ER steps for refinement. Convergence is assessed by monitoring the R-factor, defined as R = \sqrt{ \frac{ \sum | |F_\text{obs}|^2 - |F_\text{calc}|^2 | }{ \sum |F_\text{obs}|^2 } }, or the Fourier error metric, halting when stagnation occurs below $10^{-3}. Despite their effectiveness, these algorithms can stagnate in local minima, particularly with noisy diffraction data, leading to ambiguous or incorrect reconstructions. For well-oversampled patterns (oversampling ratio \geq 2 in each dimension), success rates reach approximately 80%, but performance drops for undersampled or highly noisy cases.

Advanced and Specialized Methods

Advanced and specialized methods in coherent diffraction imaging (CDI) have emerged in the 2010s and 2020s to address challenges in non-convex optimization, noise robustness, partial coherence, and dynamic scenarios, often building upon foundational iterative phase retrieval algorithms. These approaches incorporate proximal operators, sparsity priors, temporal constraints, machine learning, and wavefront modeling to enhance reconstruction reliability and push spatial resolutions beyond traditional limits. Such techniques are particularly valuable for complex datasets from synchrotron and free-electron laser sources, enabling atomic-scale imaging and real-time processing. The generalized proximal smoothing (GPS) algorithm, introduced in the late 2010s, employs proximal operators to tackle the non-convex phase retrieval problem in noisy CDI data. By smoothing the objective function and iteratively applying proximal mappings, GPS improves convergence and robustness compared to classical methods, achieving higher fidelity reconstructions even with low signal-to-noise ratios. This framework has been demonstrated to outperform hybrid input-output algorithms in simulations and experimental X-ray diffraction patterns, reducing artifacts in reconstructed images.^[12] Recent frameworks like SPRING, developed in 2025, leverage sparsity priors and memetic phase retrieval for reliable atomic structure recovery in single-particle CDI. SPRING processes X-ray free-electron laser diffraction data by combining global optimization with local refinements, exploiting the sparse nature of atomic densities to mitigate ambiguities in phase retrieval and yield high-resolution density maps. In benchmarks with simulated and experimental datasets, it consistently recovers structures with sub-angstrom accuracy, surpassing conventional algorithms in handling oversampled patterns. Similarly, the serialCDI method, proposed in 2025, exploits temporal continuity across serially recorded far-field diffraction patterns to reconstruct dynamic samples. By enforcing inter-frame constraints through adaptive region matching, serialCDI enables rapid imaging of time-varying objects, such as evolving nanostructures, with reduced computational overhead and improved temporal resolution in pump-probe experiments.^[13]^[14] Machine learning integrations, particularly neural networks, have advanced CDI by providing rapid phase initialization and denoising. Convolutional neural networks trained on simulated diffraction datasets can predict initial phase estimates from intensity patterns, accelerating convergence in iterative solvers and enabling real-time reconstructions during experiments. For instance, deep networks applied to imperfect or noisy patterns achieve sub-10% error in phase recovery, as shown in 2025 studies on synchrotron data, where they denoise patterns while preserving fine structural details without prior knowledge of the sample. These approaches are especially effective for electron-based CDI, where networks trained on diverse datasets handle aberrations and partial coherence.^[15]^[16]^[17] To handle partial coherence inherent in synchrotron sources, multi-mode decomposition algorithms model the illumination as a superposition of coherent modes, correcting for coherence degradation in CDI reconstructions. This technique decomposes the mutual coherence function into a finite number of transverse modes, allowing algorithms to propagate each mode separately and recombine them for accurate phase retrieval in partially coherent setups. Applied to undulator beamlines, such methods recover images with up to 20% improved contrast in experimental ptychographic CDI data, mitigating blurring from source emittance.^[18]^[19] High-numerical-aperture (high-NA) CDI methods, advanced in 2025, incorporate wavefront corrections to push resolutions toward sub-10 nm by addressing curvature and aberration effects in the Ewald sphere representation. These approaches model the diffracted field rigorously in reciprocal space, applying corrections for high-NA geometries to resolve fine features in extended samples. Demonstrated with X-ray data, they achieve resolutions below 10 nm half-period, enabling detailed imaging of nanoscale devices without lenses.^[20]

Radiation Sources and Implementations

X-ray Based CDI

X-ray based coherent diffraction imaging (CDI) utilizes synchrotron and free-electron laser (FEL) sources to generate coherent beams in the hard X-ray regime, typically with photon energies of 1-10 keV. Third- and fourth-generation synchrotrons, such as the Advanced Photon Source (APS) at Argonne National Laboratory and the European Synchrotron Radiation Facility (ESRF), provide stable, high-brilliance beams suitable for extended exposure experiments on non-biological samples.^[21]^[22] In contrast, FEL facilities like the Linac Coherent Light Source (LCLS) at SLAC and the European XFEL deliver ultrashort, intense pulses that enable single-shot imaging, particularly for radiation-sensitive specimens.^[21] These sources produce beams with sufficient transverse coherence lengths, on the order of micrometers to millimeters, to satisfy the requirements for CDI.^[21] The primary advantages of X-ray CDI stem from the penetrating nature of hard X-rays, which allow imaging of thick, bulk samples up to hundreds of micrometers without significant absorption or scattering artifacts that limit other modalities.^[23] Additionally, the technique offers elemental sensitivity through anomalous scattering near absorption edges, enabling contrast enhancement for specific atomic species in complex materials.^[24] Experimental setups typically involve focusing the beam to a spot size of 1-10 micrometers using Kirkpatrick-Baez (KB) mirrors, which provide high numerical aperture focusing while preserving coherence.^[25] Far-field diffraction patterns are recorded with area detectors positioned 1-10 meters from the sample to capture the speckle structure at small scattering angles.^[5] A key historical milestone was the first demonstration of three-dimensional (3D) X-ray CDI in 2002 at Stanford University, where a non-crystalline gold nanocrystal was reconstructed at 8 nm resolution using oversampling phase retrieval on a series of rotated diffraction patterns.^[26] More recently, advances at FELs have enabled CDI of frozen-hydrated whole cells, achieving around 40 nm resolution and revealing internal structures like organelles without labels.^[27] One persistent challenge in X-ray CDI is radiation damage, which can alter sample structure during exposure; this is mitigated at FELs through single-shot imaging with femtosecond-duration pulses, allowing diffraction data collection before significant atomic motion occurs.^[28]

Electron Based CDI

Electron-based coherent diffraction imaging (CDI) employs transmission electron microscopes (TEMs) with coherent electron beams accelerated to energies between 100 and 300 keV to generate diffraction patterns from nanoscale specimens. These setups leverage the high brightness and partial spatial coherence of field-emission guns in modern TEMs, enabling the illumination of isolated nanostructures with beam diameters on the order of 1-10 nm. Unlike X-ray CDI, electron CDI benefits from the electrons' strong interaction with matter, which enhances signal for low-Z elements, but limits penetration depth, necessitating ultrathin samples typically below 100 nm to minimize multiple scattering effects. A key advantage of electron CDI stems from the short de Broglie wavelength of the accelerated electrons, calculated as \lambda = \frac{h}{\sqrt{2mE}}, where h is Planck's constant, m the electron mass, and E the kinetic energy; for 200 keV electrons, this yields \lambda \approx 0.002 nm, supporting potential atomic-scale resolution without lens aberrations. This wavelength enables high scattering angles and fine structural detail, particularly for light atoms in materials like carbon-based nanostructures, where electron scattering cross-sections are significantly larger than for X-rays. Adaptations for cryogenic electron microscopy (cryo-EM) have been explored to extend CDI to frozen-hydrated biological samples, preserving native states while capturing diffraction from vitrified specimens at liquid nitrogen temperatures. The technique's milestones include the first demonstration in 2003, where a double-wall carbon nanotube was reconstructed at 0.1 nm resolution using iterative phase retrieval on nanodiffraction patterns from a 200 kV TEM.^[29] This work highlighted CDI's ability to resolve non-periodic structures by overcoming the phase problem through oversampling. More recent advances, such as in 2023, have integrated deep-learning convolutional neural networks to enhance phase retrieval in electron CDI, achieving sub-angstrom resolution on complex nanomaterials and improving reconstruction robustness against noise.^[17] These developments have enabled applications like 3D strain mapping in nanocrystals, revealing lattice distortions at the atomic level without destructive sectioning.^[30] Challenges in electron CDI primarily arise from dynamical scattering in specimens thicker than ~50 nm, which distorts intensity patterns and complicates phase retrieval, often requiring specialized thin preparation techniques like ultramicrotomy or focused ion beam milling. Additionally, achieving full beam coherence demands precise alignment and low-dose conditions to avoid sample damage, particularly for beam-sensitive materials. Despite these hurdles, electron CDI offers surface and volume sensitivity distinct from deeper-penetrating X-ray methods, positioning it as a complementary tool for high-resolution nanomaterial analysis.

Emerging Radiation Modalities

Recent advancements in coherent diffraction imaging (CDI) have been driven by the development of twisted X-rays, which carry orbital angular momentum (OAM) and are generated using helical undulators at synchrotron or free-electron laser facilities. These beams feature a helical wavefront that imparts quantized OAM to photons, enabling vortex diffraction patterns distinct from conventional Gaussian beams. In 2024, experiments demonstrated CDI with twisted X-rays, where the OAM facilitates selective imaging of chiral structures through mechanisms like helical dichroism, providing contrast based on the handedness of samples without relying on traditional absorption or phase differences. This approach breaks symmetry constraints in diffraction, allowing reconstruction of three-dimensional chiral features with enhanced sensitivity to molecular orientations.^[31]^[32] Extreme ultraviolet (EUV) and soft X-ray modalities have expanded CDI capabilities, particularly at seeded free-electron lasers (FELs) such as FERMI@Elettra, which produce fully coherent pulses in the 4-100 nm wavelength range. These sources offer high peak brightness and transverse coherence, ideal for lensless imaging of radiation-sensitive biological samples, where the "water window" (2.3-4.4 nm) enables high-contrast visualization of carbon-based structures against water with minimal absorption. Dedicated end-stations at FERMI support CDI experiments, achieving resolutions down to tens of nanometers for isolated biomolecules and cellular components by leveraging the short pulse durations to outrun damage.^[33] Prospects for fourth-generation synchrotron sources, such as diffraction-limited storage rings operational by 2025, promise significantly brighter and more coherent X-ray beams for CDI, with emittances reduced to the diffraction limit via multi-bend achromat lattices. These upgrades will increase flux by orders of magnitude while maintaining high coherence volumes, enabling CDI of weaker scatterers like dilute nanomaterials with improved signal-to-noise ratios and reduced exposure times. Additionally, coherent EUV sources from tabletop high-harmonic generation (HHG) lasers, driven by femtosecond infrared pulses, have realized compact CDI setups, demonstrating resolutions as fine as 22 nm for nanopatterns and potential for lab-scale biological imaging without large-scale facilities.^[34]^[35]^[36] Overall, these emerging modalities introduce new contrast mechanisms, such as OAM-induced dichroism, and enhance performance for weak or chiral scatterers, broadening CDI's applicability in materials and life sciences while adapting coherence requirements to shorter wavelengths and structured beams.^[31]

Applications and Extensions

Static Nanostructure Imaging

Coherent diffraction imaging (CDI) enables the label-free, high-resolution characterization of static nanostructures, including nanocrystals, nanoparticles, and defects in materials, by reconstructing their three-dimensional (3D) electron density from diffraction patterns. This approach has achieved nanoscale resolutions in 3D, allowing detailed visualization of morphology and internal features such as strain fields.^[37] These capabilities stem from the use of coherent X-ray or electron beams at synchrotron or free-electron laser sources, which provide the necessary flux for nanoscale imaging.^[37] The typical workflow for static nanostructure imaging involves positioning an isolated sample in a coherent beam to collect oversampled far-field diffraction patterns, often in Bragg geometry for crystalline targets to enhance sensitivity to lattice distortions. Phase retrieval is then performed using iterative algorithms constrained by a finite support function derived from the sample's estimated size and shape, yielding the 3D density distribution. For full 3D tomography, multiple diffraction patterns are acquired over a rotation series of the sample, enabling volumetric reconstruction without prior knowledge of the structure beyond basic constraints. This oversampling—typically by a factor of 2 or more relative to the Nyquist limit—ensures the uniqueness of the phase retrieval solution.^[37] A major advantage of CDI for static nanostructures is its lensless, nondestructive nature, which facilitates quantitative 3D mapping of electron density and strain in a single experiment, free from artifacts introduced by optical elements. Rotation-series tomography further supports comprehensive structural analysis, revealing defects and asymmetries in non-periodic samples. However, the technique demands well-isolated specimens to avoid signal overlap in diffraction space, and radiation dose constraints—often limited to 10^9-10^10 photons per pattern—restrict its use on beam-sensitive materials, necessitating cryogenic or ultrafast implementations in some cases.^[38] Early demonstrations in the 2000s highlighted CDI's potential through strain mapping in nanocrystals, where Bragg CDI reconstructed 3D displacement fields with nanoscale resolution, quantifying lattice distortions.^[39] More recent advances have pushed toward atomic resolution, as exemplified by deep-learning-enhanced electron CDI achieving sub-Ångström reconstructions of nanostructures in 2023, with applications extending to biomolecules for label-free structural determination.^[17]

In Situ and Dynamic Imaging

In situ coherent diffraction imaging (CDI) enables the study of materials under realistic operating conditions by incorporating environmental cells that simulate gas or liquid flows, allowing real-time observation of structural changes in catalysis and battery systems. These setups typically involve operando cells where nanocrystals or nanoparticles are exposed to reactive environments, such as hydrogen gas flow for catalytic reactions or electrolyte solutions for battery cycling, to capture evolving strain and morphology without ex situ preparation artifacts. For instance, Bragg CDI (BCDI) has been applied to track lattice strain in battery electrode materials like LNMO cathodes during charge-discharge cycles, revealing inhomogeneity dynamics under liquid electrolyte flow.^[40] Similarly, in catalysis, BCDI monitors nanoparticle faceting and defect formation under gas-phase conditions, providing insights into active site evolution during reactions like CO oxidation.^[40] Dynamic CDI extends these capabilities to time-resolved processes, particularly through single-shot imaging at X-ray free-electron lasers (XFELs), which capture ultrafast phenomena on picosecond to femtosecond timescales before sample damage occurs. Pump-probe schemes at XFELs, such as those using optical excitation followed by X-ray probing, have visualized non-equilibrium dynamics in nanocrystals, including phonon propagation and electronic state changes in materials like gold or vanadium dioxide. A milestone in this area was the demonstration of in situ CDI for tracking nanocrystal dynamics in 2018, where time-evolving complex exit waves were reconstructed with nanoscale resolution for evolving 3D structures.^[41] More recently, hybrid approaches combining CDI with X-ray photon correlation spectroscopy (XPCS-CDI) have enabled in situ tracking of Brownian motion in colloidal gold nanoparticles, bridging imaging and correlation timescales to quantify diffusion coefficients down to 200 nm particles in 2024.^[42] Serial CDI at synchrotrons further advances 4D (3D spatial + time) imaging of dynamic samples by employing fly-scan techniques, where continuous sample scanning collects overlapping diffraction patterns to exploit temporal continuity for robust reconstructions. This method, demonstrated in 2025, reconstructs evolving local structures in extended samples like polymer films or biological assemblies, achieving ~10 nm resolution over volumes spanning micrometers with acquisition times under 1 second per frame.^[14] In biological contexts, dynamic CDI variants have applicability to hydrated samples, as shown in zone-plate-based implementations that resolve fast structural changes on millisecond scales in 2025.^[43] These serial approaches leverage inter-frame constraints to enhance phase retrieval for non-stationary objects. Key challenges in in situ and dynamic CDI include motion blur from environmental flows or intrinsic sample dynamics, which degrades diffraction pattern quality and reconstruction fidelity. Mitigation strategies, such as stroboscopic pumping synchronized with X-ray pulses, temporally gate the illumination to freeze ultrafast motions, as applied in polaron tracking within organic semiconductors in 2024.^[44] Additionally, handling the massive data volumes from serial or time-series acquisitions requires advanced computational frameworks, often incorporating temporal regularization in iterative phase retrieval algorithms. Despite these hurdles, ongoing developments in XFEL and synchrotron upgrades continue to push the boundaries of temporal resolution and environmental compatibility.^[44]^[41]

Biological and Materials Science Uses

In biology, coherent diffraction imaging (CDI) has enabled the determination of virus and protein structures without staining or crystallization, leveraging the technique's ability to reconstruct high-resolution images from diffraction patterns of non-crystalline samples. A seminal demonstration involved imaging individual mimivirus particles using an X-ray free-electron laser (XFEL), achieving a full-period resolution of 32 nm in a single pulse, which revealed the virus's unstained, radiation-damage-free structure and internal features like the capsid and core.^[45] More recently, CDI at XFEL facilities has facilitated nanoscale imaging of cellular organelles, such as mitochondria, with resolutions around 32 nm, allowing visualization of whole, unstained organelles in their native-like states without sectioning.^[46] In materials science, CDI excels at analyzing defects in semiconductors by mapping lattice strains and distortions at the nanoscale, providing insights into material performance and failure mechanisms. For instance, Bragg CDI (BCDI) has been used to image defect structures in crystalline nanoparticles, quantifying strain fields with sub-nanometer sensitivity to inform semiconductor device optimization.^[47] Additionally, CDI captures phase transitions in alloys, such as the ordering processes in Fe-Al systems, yielding three-dimensional maps of evolving microstructures during thermal treatments that reveal domain formation and growth dynamics.^[48] CDI offers unique insights into radiation-tolerant imaging of hydrated biological samples, where XFEL pulses enable "diffraction before destruction" to preserve delicate, water-containing structures that would otherwise degrade under prolonged exposure.^[49] In materials, multi-energy CDI supports 3D chemical mapping by exploiting energy-dependent scattering contrasts near absorption edges, as demonstrated in hyperspectral imaging of quantum materials to differentiate elemental compositions and electronic states.^[50] Recent advances include CDI applications to drug delivery nanoparticles, where XFEL imaging of individual polymeric capsules in solution has resolved their internal architectures and loading distributions at tens-of-nanometers resolution, aiding design for targeted therapies.^[51] In optoelectronics, CDI quantifies strain in quantum dots, mapping heterostrain distributions in nanowires that influence emission properties and device efficiency.^[52] CDI's interdisciplinary impact lies in bridging cryo-electron microscopy (cryo-EM) and X-ray tomography through hybrid approaches, combining CDI's label-free, quantitative density mapping with cryo-EM's high-resolution details for comprehensive cellular imaging.^[53]

Ptychography

Ptychography is a scanning variant of coherent diffraction imaging (CDI) designed to image extended samples by acquiring a series of diffraction patterns from overlapping illuminations. In this technique, a focused coherent beam, such as X-rays or electrons, is raster-scanned across the specimen, illuminating localized regions with substantial overlap—typically 60% or more—between adjacent positions to ensure data redundancy. For each scan position, the far-field diffraction intensity is recorded using a detector, forming a dataset that encodes the sample's complex transmission function. Unlike single-shot CDI, ptychography exploits the shared information from overlapping regions to iteratively reconstruct both the object's amplitude and phase, as well as the illumination probe's profile, through phase retrieval algorithms that enforce consistency across the measurements. This simultaneous recovery of probe and object mitigates errors from unknown beam characteristics, enabling quantitative imaging without prior knowledge of the illumination. A primary advantage of ptychography over standard CDI lies in its ability to handle extended or non-isolated samples, which pose challenges in traditional CDI due to phase ambiguities and the need for finite support constraints. The overlapping illuminations provide multiple views of the same specimen features, creating a robust constraint that improves convergence and reduces artifacts, allowing imaging of large fields of view up to micrometers while maintaining nanoscale resolution. Additionally, ptychography demonstrates enhanced robustness to partial coherence in the incident beam, as the algorithm can incorporate coherence models to correct for degradation in image quality, outperforming lens-based methods that are more sensitive to source imperfections. These properties make it particularly suitable for real-world experimental conditions at synchrotron or laboratory sources.^[54]^[55]^[56] Reconstruction in ptychography relies on iterative algorithms that propagate estimates of the object and probe between real and reciprocal space. The extended ptychographic iterative engine (ePIE) is a widely adopted method, extending the original piecewise error reduction approach by updating the probe function alongside the object through gradient-based adjustments that minimize discrepancies between measured and simulated diffraction patterns. Complementary techniques, such as the hybrid input-output (HIO) algorithm adapted for ptychography, incorporate feedback mechanisms to escape local minima, often combined with ePIE in a sequential manner for superior convergence on noisy data. These algorithms typically converge in tens to hundreds of iterations, depending on overlap and dataset size, yielding high-fidelity phase-contrast images.^[57] In applications, ptychography excels in high-resolution scanning at synchrotron facilities, where it has achieved spatial resolutions of 5-10 nm for imaging nanostructures in materials like semiconductors and biological tissues, surpassing the diffraction limits of conventional optics. For volumetric reconstruction, ptychographic tomography integrates scanning diffraction data with sample rotation, enabling 3D imaging with isotropic nanometer resolution, as demonstrated in studies of porous materials and cellular structures. Recent developments include fast computational schemes, such as partial Fourier transform-accelerated ePIE variants introduced in 2024, which reduce reconstruction times for large-scale datasets by focusing on low-frequency components initially, facilitating near-real-time processing. In 2025, advances like uncertainty-aware Fourier ptychography have further enhanced imaging stability for live cells. Integration with standard CDI has also advanced hybrid imaging modalities, such as ptychographic Fresnel CDI, which combines near-field diffraction with scanning to extend applicability to weakly scattering or dynamic samples.^[58]^[59]^[60]^[61]

Holographic Methods

Holographic methods in coherent diffraction imaging (CDI) extend the technique by incorporating a reference wave to enable direct phase recovery through interference patterns, bypassing the need for iterative algorithms typically required in pure CDI. In Fourier transform holography (FTH), a coherent reference beam—often generated by a pinhole, grating, or structured mask embedded near the sample—interferes with the diffracted light from the object, producing a hologram that encodes both amplitude and phase information. Reconstruction is achieved by performing a Fourier transform on the recorded intensity pattern, yielding the complex-valued exit wave of the sample in a single step. This approach, first demonstrated with X-rays to achieve submicrometer resolution on gold nanostructures as small as 60 nm, combines the lensless nature of CDI with the simplicity of classical holography.^[62]^[63] A key variant is near-field (Fresnel) holography, which operates in the propagation-based regime where the detector is placed close to the sample, capturing inline interference patterns akin to Gabor holography. Here, the reference wave is implicitly provided by the undiffracted portion of the incident beam, allowing phase retrieval through Fresnel propagation modeling without explicit external references. Recent advances have enabled such methods without relying on wavefront priors, improving robustness for experimental setups with imperfect illumination, as shown in simulations and reconstructions of complex scatterers. These techniques maintain the non-iterative essence of holography while adapting to near-field geometries for enhanced signal collection.^[64]^[65] The primary advantages of holographic methods include their non-iterative reconstruction, which accelerates processing and enables single-shot imaging suitable for dynamic processes, such as ultrafast structural changes in materials. For instance, tabletop extreme ultraviolet (EUV) sources have facilitated FTH imaging of nanostructures with resolutions approaching 50 nm in a single exposure, demonstrating feasibility outside large-scale facilities. In 2025, hybrid approaches like Fourier holo-ptychographic microscopy have combined holography with ptychography to improve resolution and field of view. However, these methods suffer limitations: the reference structure occupies space in the beam path, reducing the signal-to-noise ratio from the object and typically yielding lower resolution compared to optimized iterative CDI, often by a factor of 2–3 due to twin-image artifacts and reference-induced noise. To mitigate this, hybrid approaches combine holographic data with CDI for validation, enhancing phase accuracy in applications like nanostructure metrology. Coherence requirements remain comparable to standard CDI, emphasizing partial spatial coherence for stable interference.^[66]^[67]^[68]^[69]

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