Ptychography at all wavelengths

Abstract

Ptychography is a computational imaging technique that operates across multiple wavelength regimes, from electron (picometres) to X-ray (~0.1 nm), extreme ultraviolet (~10 nm), and visible light (micrometres). By reconstructing both amplitude and phase from diffraction patterns, ptychography enables high-resolution, quantitative imaging without conventional limitations imposed by lens-based optics. Ptychography has enabled advances across a range of scales: achieving record-breaking atomic resolution with electron microscopy, becoming an indispensable tool at X-ray synchrotron facilities worldwide, and overcoming the trade-offs between resolution and field-of-view in optical imaging. This Primer provides a unified treatment of ptychography across these wavelength regimes. First, we discuss theoretical foundations, reconstruction algorithms, experimental considerations and wavelength-specific challenges. We then give examples of raw and processed data from various configurations and wavelengths. Next, we highlight key applications of ptychography in life sciences, materials science and industry. We also discuss data standards, open-source software implementations, and best practices for ensuring reproducibility across different wavelength regimes. Finally, we consider limitations and future opportunities for ptychography. Together with accompanying datasets and code implementations, this Primer aims to serve newcomers and experienced practitioners in the field, facilitating broader adoption of ptychography across different disciplines.

1. Introduction

Ptychography is a computational imaging approach that operates across a wide wavelength range, offering excellent resolution, field of view, and quantitative contrast. It combines diffraction measurements with computational reconstruction to overcome the limitations of conventional lens-based systems, transforming a hardware design problem into a computational one that can be solved post-measurement. The technique was originally conceived as a solution to the challenge of recovering the phase of a wavefield, which carries crucial information about how a wave has been delayed by interactions with a specimen (Box 1). By computationally recovering this otherwise inaccessible phase, ptychography can reveal structures and properties of a specimen without using conventional lens-based optics.

Box 1 |. The phase problem.

Amplitude and phase information is required for the complete wavefield characterization of electromagnetic and matter waves. Amplitude represents the strength of the wave, while phase encodes wavefront information about the propagation direction. Conventional detectors such as image sensor and photographic film measure intensity (the squared amplitude) but cannot capture the phase information. This ‘phase problem’ was first recognized in the field of crystallography through the analysis of X-ray diffraction patterns.

Electromagnetic waves such as visible light, extreme ultraviolet and X-rays experience phase shifts primarily through variations in refractive index $n\left(x\right)$ or differences in material thickness $t$ (Panel a). These differences create wavefront distortions $\mathrm{\Delta }\phi$ following the relationship $\mathrm{\Delta }\phi =(2\pi /\lambda )\left(n\right(x)-1)t$, where $\lambda$ is the wavelength. Fundamentally, $n\left(x\right)$ originates from bound electrons oscillating in response to electromagnetic fields and reradiating waves that interfere with the incident beam.

Electron waves interact with matter through atomic electrostatic potentials $V\left(x\right)$. The resulting phase shift follows $\mathrm{\Delta }\phi =\sigma V\left(x\right)t$, where $\sigma$ is an interaction constant (panel b). Unlike photons, electrons probe the entire Coulomb field of an atom; thus, electron waves are sensitive to atomic number, chemical bonding, and local electromagnetic fields. Other matter waves, such as neutrons and ions, interact via their own specific mechanisms (such as, nuclear forces and magnetic moments), each offering unique contrast capabilities.

Ptychographic imaging spans nearly nine orders of magnitude in length scale, from picometre-scale atomic dimensions to metre-scale macroscopic structures. At atomic scales, electron ptychography leverages extreme phase sensitivity to precisely map electrostatic potentials at sub-angstrom resolution. In the nanometre and micrometre regimes, X-ray ptychography excels at visualizing internal electron density variations while extreme ultraviolet (EUV) implementations provide surface-sensitive imaging, which is important for semiconductor metrology. At visible wavelengths, optical ptychography overcomes the traditional trade-off between resolution and field of view, enabling high-throughput quantitative phase imaging for biological and materials applications.

The development of ptychography spans decades of methodological innovation. Initially proposed in 1969 (ref.¹), it remained largely conceptual until advances in computation and detector technology enabled practical implementations. The theoretical foundation was strengthened by parallel developments of iterative phase retrieval, beginning with the Gerchberg–Saxton algorithm in 1972 (ref.²) and refined by the hybrid input–output method in the 1980s (ref.^3,4). Early ptychographic implementations relied on direct closed-form inversion methods, with the Wigner-distribution deconvolution approach being developed in 1992 (ref.⁵). Subsequently, experimental demonstrations achieved double-resolution imaging in 1993 (ref.⁶), and surpassed the resolution barrier of conventional electron microscopy in 1995 (ref.⁷). A key breakthrough came in 2004 when iterative phase retrieval was integrated into ptychography^8,9, bringing the technique into its modern form. Since then, each wavelength regime has cultivated specialized strategies while preserving the shared foundation of iterative phase retrieval.

A typical experimental setup of ptychography illuminates an object $O(x,y)$ with a coherent probe beam $P(x,y)$ and uses a detector at distance $d$ to record the resulting diffraction pattern (Fig. 1a). In optical, EUV, X-ray systems, $d$ is the physical propagation distance from object to detector. In electron microscopy, $d$ corresponds to the camera length, an effective propagation distance controlled by post-specimen lenses rather than the actual physical separation.

Fig. 1 |. Principle and implementation of conventional ptychography.

a, Experimental setup where a confined probe beam $P(x,y)$ illuminates an object $O(x,y)$. The object translates to positions $(x_i,y_i)$ with overlapping illumination regions, and the resulting wavefields propagate a distance $d$ to the far-field detector plane for intensity acquisition. This far-field propagation can be modelled by a Fourier transform that converts real-space coordinates $(x,y)$ to reciprocal-space coordinates $\left(k_x,k_y\right)$. Other key parameters include convergence angle $\theta$, detector pixel size $ps$, and bright-field radius $r_{bf}$ measured in pixels. Right panels illustrate the sampling relationship, where $N\times \mathrm{\Delta }x$ defines the field of view in real space and $2\pi /\mathrm{\Delta }x$ sets the reciprocalspace extent. $N$, number of pixels in the x direction; $\mathrm{\Delta }x$, object pixel size in real space; $\mathrm{\Delta }{k}_x$, reciprocal-space pixel size. b, The diffraction pattern captured at different scan positions where adjacent diffraction patterns share similar features owing to overlapping illumination. The red dashed box indicates the pattern from position ($x_i,y_i$) indicated in part a. c, The complex object reconstructed from the ptychogram in b. The red dashed circles mark the regions illuminated in each scan position as the object is translated following the trajectory shown by the arrows in the top right. Inset, reconstructed probe beam in which brightness indicates amplitude ($A$) and hue indicates phase ($\phi$), as defined by the colour wheel.

The operation of ptychography relies on three fundamental concepts: wavefield propagation, specimen translation, and overlapping measurements. The first concept, wavefield propagation, is essential for encoding phase information into measurable intensity variations. As the wavefield travels from the object to the detector plane, it undergoes a Fourier transform that converts real-space information into reciprocal-space diffraction patterns. Without this propagation, the detector would simply record exit wave intensities, losing all phase information.

The second concept, specimen translation, differentiates ptychography from single-shot coherent diffraction imaging¹⁰, which reconstructs objects from individual patterns using a priori constraints such as finite support boundaries³, but struggles with extended or complex specimens. Ptychography addresses this limitation by translating the object to different positions $(x_i,y_i)$ in real space or shifting the object spectrum in reciprocal space by changing the illumination angle. This translation process in real or reciprocal space produces multiple diffraction patterns, forming a ptychogram (Fig. 1b). This strategy maintains only two fundamental unknowns—$O(x,y)$ and $P(x,y)$—while generating multiple measurements. It transforms what would otherwise be an underdetermined problem into a well-conditioned inverse problem, enabling reliable convergence even under challenging conditions such as partial coherence, detector noise, and positioning errors.

The third concept, overlapping illumination regions between adjacent positions enables consistent phase recovery across the entire field of view. Typically adjacent measurements overlap by 50% or more^11,12, which increases measurement diversity while only moderately expanding the number of unknowns. Conventional scanning systems use minimal overlap solely for image stitching. Conversely, the extensive overlap in ptychography measurements is used to break the mathematical ambiguity between $O(x,y)$ and $P(x,y)$ that cannot be resolved from intensity measurements alone. When the same regions of the specimen are illuminated by different portions of the probe, the resulting diffraction patterns contain shared information (Fig. 1b). Reconstruction algorithms use this redundancy to recover the object’s complex image and the illumination probe beam (Fig. 1c).

This Primer provides a unified treatment of ptychography across wavelength regimes. First, we discuss the diverse experimental configurations, computational reconstruction algorithms, and key performance characteristics. Second, we provide examples of raw and reconstructed data in different wavelength regimes, demonstrating how diffraction patterns are transformed into high-resolution complex-valued images at various length scales. Next, we present key applications of ptychography in life sciences, materials characterization, and industrial metrology. We then examine reproducibility by considering existing data standards and software tools. Finally, we outline limitations and future opportunities for ptychography. Open-source datasets and code are provided for hands-on exploration (Supplementary Note). Readers seeking specialized topics such as its historical development, detailed mathematical derivations, and domain-specific applications should consult other related resources^13–19.

2. Experimentation

The success of ptychography relies on the careful co-design of experimental data acquisition and computational reconstruction. This section explores the diverse experimental configurations for data acquisition; the key components of the computational framework, including the iterative reconstruction algorithms, the system calibration methods, and the advanced models needed to account for complex sample interactions; and wavelength-specific considerations.

2.1. Experimental configurations

Conventional and Bragg ptychography.

Conventional ptychography (left, Fig. 2a) translates an object $O$ across a localized probe $P$ while recording far-field diffraction patterns^8,9. The captured ptychogram $I_i\left(k\right)$ can be written as

$$I_i\left(k\right)={\left|{ℱ}_{r\to k}\left\{P\left(r\right)\cdot O\left(r-r_i\right)\right\}\right|}^2,$$ (1)

where $r=(x,y)$ and $k=\left(k_x,k_y\right)$ are the real and reciprocal space coordinates, respectively. ${ℱ}_{r\to k}$ denotes the Fourier transform for modeling far-field light propagation from real space to reciprocal space. The scan positions $r_i=\left(x_i,y_i\right)$ create overlapping illuminated regions, generating the diversity necessary for robust phase retrieval. This imaging model assumes thin specimens where probe–object interactions can be approximated with a multiplication. This approximation holds when the sample thickness $T\le 5.2\mathrm{\Delta }{w}^2/\lambda$, where $\mathrm{\Delta }w$ is the smallest resolvable linewidth in the reconstruction and $\lambda$ is illumination wavelength²⁰. Multi-slice modeling or tomographic approaches can be used to measure specimens with thickness beyond this limit to account for multiple scattering effects^20–25.

Fig. 2|. Experimental configurations for ptychography.

a, Left, in conventional ptychography a specimen is translated across a localized probe beam and far-field diffraction patterns are recorded in reciprocal space. Right, ptychography can be performed in a reflection mode where the probe beam reflects off the sample surface and the scattered light is collected by a detector positioned at an angle. Bragg ptychography is a specific reflection mode operating at the Bragg condition, which can measure lattice strain and defects in crystalline materials. b, Near-field ptychography uses extended, structured illumination through a diffuser, recording near-field diffraction patterns in the Fresnel regime. c, Fourier ptychography uses a lens system to access reciprocal space. Angle-varied illumination (left) shifts different portions of the object’s Fourier spectrum into the lens aperture, while camera translation (right) directly captures different Fourier spectral regions. d, Coded ptychography (left) operates in the near-field and synthetic aperture ptychography (right) operates in the far-field, both using a coded surface on an image sensor as an effective probe. Translating this coded sensor provides measurement diversity equivalent to object scanning in conventional ptychography. $ℱ$: Fourier transform; Prop: free-space propagation.

The object pixel size in real space $\mathrm{\Delta }x$ links digital pixels to physical dimensions. Determining $\mathrm{\Delta }x$ enables mapping between physical scan positions $x_i$ and their corresponding pixel coordinates ($x,y$) in the reconstruction, ensuring that the overlapping regions between measurements are properly registered. For configurations with farfield propagation, $\mathrm{\Delta }x=(\lambda \cdot d)/(N\cdot ps)$, where $N$ is the total pixel count along the $x$ direction, and $ps$ is the detector pixel size. For electron ptychography, it becomes $\mathrm{\Delta }x=\left(\lambda \cdot r_{bf}\right)/(N\cdot \theta )$, where $r_{bf}$ is bright-field disc radius (in pixels) and $\theta$ is the convergence angle. These sampling relationships show how propagation distances and detector pixel size impact spatial sampling of the object. Detailed derivations and typical values of $\mathrm{\Delta }x$ across wavelength regimes are provided in the Supplementary Information.

Bragg ptychography²⁶ and reflection ptychography^27–29 both operate in reflection geometry but serve different purposes (Fig. 2a, right). Bragg ptychography specifically targets crystalline materials, requiring precise angular alignment to satisfy Bragg diffraction conditions²⁶. It adopts the imaging model of Eq. (1) but captures diffraction from specific crystal planes, yielding complex-valued reconstructions in which phase encodes lattice distortions along the Bragg vector. In contrast, general reflection ptychography operates at arbitrary incident angles without Bragg constraints, enabling surface topography mapping, multilayer characterization, and imaging of non-crystalline reflective structures^27–29.

Near-field ptychography.

Near-field ptychography uses an extended, structured beam for sample illumination (Fig. 2b). The detector is placed close to the specimen to record near-field diffraction patterns³⁰. The near-field condition can be quantified using the Fresnel number, $a^2/(\lambda \cdot d)$, where $a$ denotes the illuminated area size. Nearfield ptychography operates with a Fresnel number ≫ 1, while far-field conditions require < 1. The imaging model can be written as:

$$I_i\left(r\right)={\left|\left\{P\left(r\right)\cdot O\left(r-r_i\right)\right\}⊛ps{f}_f\left(d\right)\right|}^2,$$ (2)

where $⊛$ denotes the convolution operation and $ps{f}_f\left(d\right)$ denotes the free-space propagation kernel for a distance $d$. This convolution operation implements the free space propagation of the wavefield $P\left(r\right)\cdot O\left(r-r_i\right)$ to the detector plane. In practical implementations, this convolution can be performed via Fourier-space multiplication using the angular spectrum method. The near-field approach offers several advantages over conventional far-field ptychography: it can achieve large field-of-view coverage with fewer measurements; the requirements of the focusing optics are relaxed as near-field implementations use extended beams instead of tightly focused probes; the distributed illumination reduces dose rates; and the resulting intensity patterns reduce dynamic range demands on the detector³⁰. In the optical regime, high laser intensities permit continuous scanning without motion blur and small sensor pixels allow minimal sample-to-detector distances that maximize field coverage, making near-field ptychography particularly suited for high-throughput quantitative phase imaging³¹.

Fourier ptychography.

Fourier ptychography (FP) adapts ptychographic principles to lens-based microscopy³². Instead of scanning a probe across the sample, FP uses a microscope with angle-varied illumination to achieve similar measurement diversity (Fig. 2c, left). The technique works by sequentially illuminating the object $O\left(r\right)$ with plane waves $e^{i{k}_{i}r}$ at wavevector $k_i$, creating exit wavefields $O\left(r\right)\cdot e^{i{k}_{i}r}$. The microscope objective performs a Fourier transform on each exit wavefield, producing shifted spectra $\hat{O}\left(k-k_i\right)={ℱ}_{r\to k}\left\{O\left(r\right)\cdot e^{i{k}_{i}r}\right\}$ in reciprocal space. The pupil function $Pupil\left(k\right)$ of the objective acts as a bandpass filter, capturing only a portion of each shifted spectrum in reciprocal space. A tube lens then performs a second Fourier transform, yielding the detected intensity:

$$I_i\left(r\right)={\left|{ℱ}_{k\to r}\left\{Pupil\left(k\right)\cdot \hat{O}\left(k-k_i\right)\right\}\right|}^2,$$ (3)

Here the pupil function acts as the probe in reciprocal space and the illumination wavevector $k_i$ provides the necessary translational shifts of the object spectrum. Since its introduction in 2013 (ref.³²), FP has evolved into a distinct field, catalyzing major advances in computational imaging^33–35.

The resolution of conventional microscopy is limited by the numerical aperture (NA) of the objective $N{A}_{\mathit{objective}}=\mathrm{s}\mathrm{i}\mathrm{n}\left(\alpha \right)$, where $\alpha$ is the maximum collection angle. Under plane-wave illumination, the smallest resolvable linewidth equals $\lambda /\left(2\cdot N{A}_{\mathit{objective}}\right)$, with higher NA objectives resolving finer object details. FP overcomes this NA limitation by using oblique illumination angles that shift high-resolution information into the collection cone of the objective. The maximum incident angle $\beta$ contributes an additional $N{A}_{\mathit{illumination}}=\mathrm{s}\mathrm{i}\mathrm{n}\left(\beta \right)$, creating a synthetic aperture $N{A}_{\mathit{synthetic}}=N{A}_{\mathit{objective}}+N{A}_{\mathit{illumination}}$. For example, FP with $\beta ={53}^{\circ }$ enables a 0.1-NA objective (resolving ~5λ linewidth) to resolve 0.56λ linewidth, achieving 9× resolution improvement while retaining the large field of view of the low-NA optics. This principle of using angled illumination to encode high-frequency information is conceptually similar to structured illumination microscopy^36–38, in which sinusoidal patterns create moiré fringes that can be observed by NA-limited objectives. In FP, this concept is extended by combining sequential plane-wave illumination with iterative phase retrieval, enhancing the resolution while recovering quantitative phase. The computational framework used to recover complex wavefield from FP measurements can also be adapted for structured illumination microscopy reconstruction, enabling simultaneous recovery of non-uniform intensity patterns and the object profile^39–41.

FP can be extended to imaging metre-scale or larger scenes by translating a camera system that consists of an image sensor with attached photographic lens (Fig. 2c, right)^42–45. During image formation, coherent light from remote objects undergoes a Fourier transform as it propagates to the camera system. The photographic lens in the camera system then performs a second Fourier transform to form the image. The lens aperture acts as a circular window that restricts which spatial frequencies in reciprocal space can pass through to the sensor, analogous to the pupil function in microscope setups. Instead of varying the illumination angle, this implementation physically translates the camera system in reciprocal space to capture different portions of the object’s Fourier spectrum. The resolution limit imposed by the physical dimension of the lens aperture can thus be overcome by combining multiple images taken at different positions. Importantly, this macroscopic FP handles objects with arbitrary thickness by modulating the exit wavefield through camera translation rather than varying illumination angles. The recovered wavefield at the lens aperture plane can be computationally propagated to any position along the optical axis, enabling three-dimensional scene recovery without needing to model how light interacts with the objects.

Coded and synthetic aperture ptychography.

Coded ptychography^46,47 (Fig. 2d, left) and synthetic aperture ptychography⁴⁸ (Fig. 2d, right) use coded sensors for image acquisition. While coded ptychography operates in the near-field regime for on-chip imaging, synthetic aperture ptychography works in the far-field to synthesize an aperture larger than the physical detector size. In both methods, an extended uniform beam illuminates the specimen and the diffracted wavefield propagates to a structured element fabricated on the image sensor. This element, termed coded surface, can be a diffusing mask⁴⁶, a disorder-engineered surface⁴⁷, or a layer of fixed blood cells^49,50. The coded surface must be thin at wavelength scale while providing strong scattering to modulate high spatial frequency content into detectable signals. The choice typically depends on fabrication considerations: blood cells offer the simplest preparation process, engineered surfaces provide optimal performance but require specialized fabrication processes, and diffusers offer a middle ground with moderate fabrication complexity. After being modulated by the coded surface, the wavefield propagates a short distance $d$ to the pixel array for intensity detection. The imaging model can be written as

$$I_i\left(r-r_i\right)={\left|\left\{P\left(r-r_i\right)\cdot W\left(r\right)\right\}⊛ps{f}_f\left(d\right)\right|}^2,$$ (4)

where $W\left(r\right)$ denotes the diffracted wavefield of the specimen at the coded surface plane, $P\left(r\right)$ is the transmission profile of the coded surface, which acts as the effective probe beam. Unlike conventional ptychography in which the specimen moves, here the coded sensor itself translates to position $r_i$, enabling in-situ imaging of vibration-sensitive specimens. Reconstruction recovers $W\left(r\right)$ at the coded surface plane, which can be computationally propagated to any axial plane post-measurement, providing an extended depth of field without needing to model the probe–object interaction. In synthetic aperture mode⁴⁸ (Fig. 2d, right), far-field sensor translation synthesizes an effective aperture larger than the detector size, simultaneously improving resolution and expanding field of view.

2.2. Computational framework

Iterative reconstruction.

The forward imaging models in Eqs. (1)–(4) describe how the wavefield propagates to the detector plane and forms the observed diffraction patterns. Ptychographic reconstruction addresses the inverse problem: recovering the object $O$ and probe beam $P$ from measured intensities. Gradient-based algorithms minimize a cost function $ℒ$ that quantifies the mismatch between measured intensities $J_i$ and estimated intensities $I_i$ based on the current estimates of $O$ and $P$:

$$ℒ\left(P,O\right)=\sum _i\sum _{k}d\left[w\left(I_i\left(k\right)\right),w\left(J_i\left(k\right)\right)\right],$$ (5)

where summation extends over all scan positions (indexed by $i$) and all detector pixels (indexed by $k$). For each scan position $i$ and each detector pixel $k$, a distance metric $d[·,·]$ measures the discrepancy between the weighted measured intensity $w\left(J_i\right)$ and the weighted estimated intensity $w\left(I_i\right)$, where $I_i$ is calculated using the current estimates of object and probe and the associated forward models. Regularization terms can be added to Eq. (5) to stabilize reconstruction in illuminated regions where signal-to-noise is poor, or to incorporate prior knowledge such as smoothness constraints or sparsity⁵¹. The object and probe estimates are iteratively updated following the negative gradient direction to minimize $ℒ:O^{\mathrm{update}}=O-{\eta }_o\cdot \partial ℒ/\partial O$ and $P^{\mathrm{update}}=P-{\eta }_P\cdot \partial ℒ/\partial P$, where ${\eta }_o$ and ${\eta }_P$ are the respective step sizes.

The various reconstruction strategies differ in whether they process scan positions sequentially or globally and in how they choose step sizes (Table 1). Sequential methods process one scan position at a time, offering fast initial convergence with minimal memory requirements. Examples include the foundational ptychographical iterative engine (PIE)⁹, its variants (extended PIE⁵², three-dimensional PIE²¹, regularized PIE⁵³, weighted average of sequential projections⁵⁴), and sequential quasi-Newton method⁵⁵. Among these, the ePIE algorithm⁵² has become particularly popular owing to its predetermined step size, which eliminates the need for parameter tuning. Global methods process multiple scan positions simultaneously, providing increased robustness at higher computational cost. Examples include conjugate gradient descent⁵⁶, global quasi-Newton method⁵⁷, and the least-squares solver¹⁴. These approaches differ in how they determine step size: conjugate gradient method performs a line search to find the optimal step size, quasi-Newton methods use second-order approximation to predict the step size, and least-squares solvers calculate the optimal step size for each scan position analytically.

Table 1 |.

Gradient-based ptychographic reconstruction

Strategy	Loss function	Object update rulea
Sequentialb	${(J_i^{\gamma }-{\left|{\psi }_i\right|}^{2\gamma })}^2$	$O_i^{\text{update}}=O_i-{\eta }_o\gamma P^{\mathrm{*}}{ℱ}^{-1}\left({\left|{\psi }_i\right|}^{2\gamma }-J_i^{\gamma }\right)\frac{{\psi }_i}{{\left|{\psi }_i\right|}^{2(1-\gamma )}}$
Global or batch updatec	${\sum }_i{(J_i^{\gamma }-{\left|{\psi }_i\right|}^{2\gamma })}^2$	$O^{\text{update}}=O-{\eta }_o\gamma P^{\mathrm{*}}{ℱ}^{-1}{\sum }_i\left({\left|{\psi }_i\right|}^{2\gamma }-J_i^{\gamma }\right)\frac{{\psi }_i}{{\left|{\psi }_i\right|}^{2(1-\gamma )}}$
Feature-domain	${\left.{\sum }_i\left|\nabla J_i^{\gamma }-\nabla \right|{\psi }_i\right|}^{2\gamma }\mid$	$\begin{array}{c}{O}^{\text{update}}=O-{\eta }_o\gamma P^{\mathrm{*}}{ℱ}^{-1}\sum _i{\nabla }^T\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\nabla {\left|{\psi }_i\right|}^{2\gamma }\\ -\nabla J_i^{\gamma })\frac{{\psi }_i}{{\left|{\psi }_i\right|}^{2(1-\gamma )}}\end{array}$
Poisson statistics	${\sum }_i\{{\left|{\psi }_i\right|}^2-J_i\mathrm{l}\mathrm{o}\mathrm{g}{\left|{\psi }_i\right|}^2\}$	$O^{\text{update}}=O-{\eta }_o{P}^{\mathrm{*}}{ℱ}^{-1}{\sum }_i\left(1-\frac{J_i}{{\left|{\psi }_i\right|}^2}\right){\psi }_i$

$J_i$, measured intensity; $O_i=O\left(r-r_i\right)$, object at scan position $r_i;{\psi }_i=ℱ\left\{P\cdot O_i\right\}$, exit wavefield propagating to the detector plane; $\eta$, step size for; $ℱ$, Fourier transform; *, complex conjugation; $\nabla$ and ${\nabla }^T$, finite difference approximations to the gradient and the negative divergence (the adjoint of the gradient), respectively; $\gamma$, 0.5 for amplitude and 1 for intensity; sign(·), signum function that returns +1, 0, or −1 based on the sign of its argument.

^aThe update rules for the illumination probe beam can be obtained by exchanging $O$ and $P$. For each iteration, the probe beam $P$ needs to be re-centred to remove any global translation ambiguity and prevent drift accumulation.

^bFor sequential update with $\gamma =0.5$ and ${\eta }_o=1/|P{|}_{\mathit{max}}^2$, the algorithm reduces to the extended ptychographical iterative engine (ePIE) approach⁵², where $|P{|}_{\mathit{max}}$ is the maximum amplitude of the probe function. Alternatively, setting $\gamma =0.5$ with ${\eta }_o=|P|/[|P{|}_{\mathit{max}}\cdot (|P{|}^2+ϵ)]$ yields the quasi-Newton method⁵⁵, where $ϵ$ is a small constant for numerical stability.

^cFor global or batch update, the step size for each scan position can be determined analytically¹⁴.

Gradient-based methods can also be adapted for specific experimental conditions by modifying the distance metric $d[·,·]$ and weighting function $w(\cdot )$ (Table 1). The choice of $d[·,·]$ reflects assumptions about noise statistics: a squared error metric assumes Gaussian noise and is suitable for high-flux conditions, whereas a negative loglikelihood metric implements Poisson statistics for photon-limited imaging where shot noise dominates. The weighting function $w(\cdot )$ offers another dimension of flexibility. Standard implementations use identity weighting (where the weighting function returns its input unchanged) to operate directly on intensities or amplitudes. Conversely, feature-domain reconstruction^58–60 uses gradient operators $\left(w\right(\cdot )=\nabla (\cdot \left)\right)$ to operate on image gradients rather than raw intensities, providing robustness against systematic errors and artifacts in FP.

Projection-based algorithms offer alternative reconstruction strategies. The Gerchberg–Saxton algorithm², difference map^61–63, relaxed averaged alternating reflections⁶⁴, and semi-implicit relaxed Douglas–Rachford⁶⁵ methods alternate between constraint sets rather than following explicit gradients. These methods seek fixed points of projection operators rather than minima of cost functions, enabling them to escape local minima. However, they can exhibit oscillatory convergence behaviour and struggle to incorporate regularization priors. The alternating direction methods of multipliers^66,67 represent a hybrid approach that combines gradient descent for smooth terms with projections for handling constraints, offering robustness against corrupted data while maintaining the flexibility to incorporate regularization priors. However, optimal performance of this hybrid approach often requires careful tuning of additional hyperparameters that control how the optimization is split across constraints.

System calibration.

The accurate characterization of experimental parameters is important for ptychography experiments. Calibration measurements are performed before data acquisition to characterize the probe and system parameters, while computational corrections are applied post-measurement to refine these parameters. Preacquisition calibration measurements involve capturing ptychograms of known calibration objects and reconstructing both the object and probe from these measurements. The recovered probe function is then used as the initial estimate for subsequent sample measurements, ensuring faster convergence and high quality reconstructions. Ideal calibration objects include weakly scattering specimens with phase variations within $0-2\pi$, including blood smears for visible light⁵⁰ and silicon nitride membranes with nanoparticles for shorter wavelengths⁶⁸. Probe stability depends primarily on the ptychographic configuration rather than wavelength. This stability determines how frequently the system must be recalibrated. Conventional and Bragg ptychography require frequent recalibration owing to their sensitivity to illumination conditions and object position. Conversely, Fourier and coded ptychography have inherently stable effective probes (fixed pupil function of the optical system and invariant transmission profile of the coded surface, respectively) that do not vary over time.

Other system parameters also require careful calibration. In conventional and Bragg ptychography, determining the focus position of the illumination beam is crucial for object placement and initial probe beam estimation. For FP, the light-emitting diode (LED) array position must be calibrated to determine the illumination angles; this calibration can be achieved by monitoring the brightfield-to-darkfield transition region^35,50 or through coded sensor measurements⁶⁹. In coded ptychography, it is important to precisely know the distance between the coded surface and the detector array to enable accurate propagation calculations⁵⁰.

Post-acquisition refinement complements the pre-acquisition calibration by computationally correcting the residual errors. System parameters can be updated alongside the object and probe during iterative optimization. This post-acquisition optimization can correct for sample positioning errors^70–73, angle misalignment^{57,69,74–76}, aberrations^57,77–80, intensity fluctuations⁸¹, focusing^81,82, vignetting effect⁸³, and other experimental imperfections⁸⁴.

Advanced modeling and specialized reconstructions.

Although iterative algorithms form the foundation of ptychographic reconstruction, real-world experimental conditions often require more sophisticated modeling approaches (Figure 3). 3D reconstruction through ptycho-tomography is one of the most straightforward extensions of ptychography (Fig. 3a, left). It rotates the specimen around a single axis perpendicular to the beam propagation direction, typically through hundreds of angular positions over 180° or 360° range. At each rotation angle, a full ptychogram is acquired to reconstruct 2D projections, which are then combined via standard tomographic algorithms to recover the 3D complex refractive index distribution⁸⁵. For planar samples that are unsuitable for conventional tomography, ptycho-laminography tilts the rotation axis relative to the beam (Fig. 3a, right)⁸⁶. Ptychotomography and laminography are most commonly implemented with X-rays due to X-rays’ penetration depth, enabling non-destructive 3D characterization of materials and biological specimens with nanoscale resolution and high contrast^85–90.

Fig. 3|. Advanced ptychography strategies.

a, Left, ptycho-tomography uses lateral scanning with sample rotation at different angles $\omega$ to achieve 3D reconstruction. Right, ptycho-laminography is similar to ptycho-tomography but uses a tilted rotation axis. b, Mixed-state modeling decomposes measurements into multiple incoherent modes, obtaining multi-state representations of the probe $P_{\lambda }$ and object $O_{\lambda ,i}$, to handle partial coherence. For the object, amplitude shown in grayscale at top left and phase shown in red colour at bottom right. c, Multi-slice modeling is used to measure thick samples by representing them as sequential axial slices. Ptychogram can be captured by translating the sample or by changing the angle of illumination. d, Orthogonal probe relaxation decomposes a varying probe $P$ into orthogonal modes $\left(U_1,U_2,\dots U_k\right)$ to handle illumination instabilities. e, Left, super-resolution extrapolation iteratively extends beyond the detector’s measured region (red dashed box). During each iteration, the central region replaces amplitude with measured values while preserving phase, whereas the extrapolated region outside the red box keeps both amplitude and phase from the previous iteration, gradually building up high-frequency information. Right: subpixel modeling computationally divides each physical pixel into multiple virtual sub-pixels to improve sampling density. f, Fourier ptychographic diffraction tomography is used to measure thick samples by varying the angle of illumination to access different regions of the 3D scattering potential spectrum, with sampling constrained by Ewald spheres. $ℱ$: Fourier transform. g, Neural-field representations parameterize the object and probe as continuous functions using multilayer perceptron instead of discrete pixel arrays to enable reconstruction at arbitrary spatial positions.

Partial coherence of illumination is unavoidable in experiments due to source limitations, temporal fluctuations, and mechanical instabilities. For example, mechanical vibrations and thermal drifts cause the illumination to interact with slightly different regions of the specimen throughout the acquisition, creating an effect equivalent to multiple shifted probe positions being incoherently superimposed. This degrades the effective coherence of the measurements, resulting in blurred features and reduced contrast. Mixed-state ptychography addresses partial coherence by decomposing the illumination probe beam into multiple mutually incoherent modes (Fig. 3b)^91,92. This approach is useful for synchrotron experiments with limited spatial coherence⁹¹, optical implementations with LED illumination⁹³, and fly-scan operation involving specimen movement during acquisition^89,94–97. Mixed-state modeling also addresses the partial coherence caused by spectral multiplexing^98–102, where each wavelength component of the light source creates a mutually incoherent mode that the algorithm aims to separate.

Optically thick specimens violate the thin-object assumption implicit in the object–probe interaction model of Eq. (1). Multi-slice modeling (Fig. 3c) addresses this thickness limitation by representing the object as sequential thin sections with wave propagation between layers^{20–24,103–108}. Two implementations exist: conventional scanning in which the sample is translated in real-space^{20–24,106–108}, and angle-varied illumination that probes stationary samples from different directions^{103–105,109}. This multi-slice model captures multiple scattering effects within thick samples. It is crucial for biological imaging with visible light^{21–24,103–105} and also benefits atomic-scale electron imaging of relatively thick specimens^106–108.

Probe instabilities arise during measurements from source fluctuations, drift, or intentional modulation. When reconstruction algorithms assume a fixed probe but the actual probe varies, the resulting model mismatch produces artifacts and degrades image quality. To address this limitation, orthogonal probe relaxation^97,110 decomposes the probe into orthogonal basis functions that capture the primary variation modes, relaxing the fixed-probe assumption (Fig. 3d).

The resolution of a standard reconstruction is limited by the physical aperture of the detector, which restricts the maximum spatial frequency that can be captured. The super-resolution strategy overcomes this limitation by extrapolating diffraction patterns to a larger virtual detector¹¹¹. The algorithm (Fig. 3e, left) updates only the central measured pixels with recorded intensities while leaving the extrapolated region unchanged, allowing spatial frequency information to build up beyond the physical detector boundaries. Additionally, subpixel reconstruction treats each detector pixel as containing multiple virtual sub-pixels, computationally increasing the effective sampling density to capture finer spatial details (Fig. 3e, right)^{46,47,112,113}. Resolution and signal-to-noise ratio can also be enhanced by optimizing the spatial-frequency spectrum of the illumination itself; for instance, by partially clipping a focused beam near its focus¹¹⁴.

Conventional FP assumes thin specimens where angle-varied illumination simply translates the 2D object spectrum into the lens aperture’s passband. For thick specimens, however, oblique illumination fundamentally alters the scattered spectrum rather than simply translating it, causing this assumption to break down. Fourier ptychographic diffraction tomography addresses this limitation by modeling the 3D object as a volumetric scattering potential function. Each illumination angle samples the 3D reciprocal space along a curved Ewald sphere surface rather than a planar section. By strategically selecting illumination angles to provide 3D reciprocal space coverage, the measurements can be then synthesized to reconstruct the 3D complex refractive index distribution, enabling optical sectioning without requiring mechanical rotation of the sample (Fig. 3f)^25,115.

Beyond traditional pixel-based representations, neural-field approaches¹¹⁶ parameterize objects and probes as continuous functions using implicit neural networks (Fig. 3g)¹¹⁷. This representation maps spatial coordinates to complex values, providing memory-efficient storage for large-scale reconstructions while handling irregular sampling. Neural network frameworks are also used in automatic differentiation schemes to compute gradients for complex forward models without requiring analytical derivations^118–127. Automatic differentiation also allows regularization terms to be easily incorporated into the cost function^125–127.

2.3. Wavelength specific considerations

The implementation of ptychography across different wavelength regimes requires careful consideration of physical factors that impact experimental design and reconstruction approaches (Table 2). While conventional and Fourier ptychography can be implemented across all wavelength regimes, certain configurations are more prevalent in specific domains. Bragg ptychography is primarily used with X-rays for crystallographic applications, while coded and synthetic aperture approaches are currently implemented in optical systems but could be adapted to X-ray and EUV regimes.

Table 2.

Wavelength considerations for ptychography

	Optical	EUV	X-ray	Electron
Typical wavelength	Visible: 400–700 nm; DUV: 100–300 nm; Near infrared: ~1.5 pm	~13.5 nm	Soft X-ray: 3 nm; Hard X-ray: 0.1 nm	4.2 pm @ 80 keV
Resolution (resolvable linewidth)	Lensless: ~300 nm47; Lens-based: ~130 nm197	~18 nm141	~4 nm (hard X-ray record)89	<0.5 Å (ref. 133,136,202)
Resolution / wavelength	Lensless: ~0.6×; Lens-based: ~0.3×	~1–5×	~40–400×	~20–200×
Source property	Laser: excellent coherence & flux; LED: moderate coherence, low flux	HHG: high coherence & moderate flux; Synchrotron & free electron laser: variable coherence & high flux	HHG (soft X-ray only): high coherence & moderate flux; Synchrotron & free electron laser: variable coherence & high flux	Field emission guns: high coherence and sufficient flux for rapid data collection
Detector & sampling	Standard detectors; Larger pixels acceptable for far-field operations; Small pixels critical for near-field operations47	Specialized detectors (vacuum-compatible); High dynamic range	Specialized detectors (vacuum-compatible for soft X-ray); Photon-counting capability; High dynamic range	Direct electron detectors (e.g., Dectris Arina131, Timerpix3132); 4D-STEM requires fast recording
Sample preparation	Standard microscope slides (with or w/o coverslips); Live samples in culture medium	Thin support membranes; Vacuum compatible	Thin membranes / TEM grids; Cryo-cooling for bio-samples; Vacuum for soft X-ray	Ultrathin sections on TEM grids; Conductive coating; Cryo-cooling for bio-samples; Vacuum compatible
Contrast mechanisms	Refractive index104,105,115,185; Absorption32,47,97; Birefringence198–201; Intrinsic molecular contrast with DUV light60	Refractive index via quantitative phase151; Absorption edges (element-specific contrast)166	Electron density85; Absorption edges (spectro-ptycho)152; Strain (Bragg ptycho)162; Magnetization vector159,160; Crystal orientation161	Electrostatic potential (strong phase contrast)135; Absorption (inelastic scattering)203; Magnetic fields (electron beam deflection)134,204
Key challenges	Computationally demanding; Large data volume for certain implementations	Limited source flux; Samples must be vacuum compatible; Efficient optics required; Need to maintain stability and avoid contamination	Beam damage; Limited beamtime at synchrotrons; Limited detector speed and dynamic range; Large data volume	Severe beam damage; Strict thickness constraints; Extreme stability needed; High data rates and large volume needed

EUV, extreme ultraviolet; DUV, deep ultraviolet; NIR, near infrared; TEM, transmission electron microscopy; STEM, scanning transmission electron microscope; HHG, high-harmonic generation.

Optical ptychography encompasses a broad range of wavelengths, from visible light ($\lambda \approx 400-700nm$), extending to deep-ultraviolet (DUV) $(\lambda \approx 190-300nm)$ and infrared $(\lambda \approx 1-10\mu m)$. Most implementations modify existing optical platforms such as light microscopes and camera systems, typically employing lens-based FP or combinations with lensless near-field approaches. For detection, complementary metal-oxide-semiconductor (CMOS) and charge-coupled device (CCD) sensors are commonly employed. These detectors provide high pixel counts (multi-megapixel) with micron-scale pixel dimensions, providing adequate sampling of diffraction patterns. Data acquisition generally ranges from ~10 to a few hundreds of diffraction patterns, with high photon counts and minimal radiation damage concerns. The availability of highly coherent laser sources, accurate positioning stages, and well-characterized optical elements create favorable conditions for robust reconstruction. Sample preparation uses standard microscope slides or Petri dishes, and live specimens can be imaged without vacuum chambers or special atmospheric requirements. The physical contrast mechanism arises from variations in refractive index and absorption. Beyond visible light, DUV ptychography offers enhanced resolution and intrinsic molecular contrast for biological specimens⁶⁰. Infrared ptychography enables chemical-specific imaging through vibrational spectroscopic contrast mechanisms, which is valuable for material characterization and bio-applications⁸⁰.

EUV ptychography ($\lambda \approx 10-100nm$) occupies an intermediate regime between optical and X-ray implementations, achieving wavelength-limited resolution comparable to soft X-ray methods while maintaining the accessibility of table-top systems. Synchrotron beamlines and high-harmonic generation (HHG) systems are often used as the EUV source. EUV is strongly absorbed in air; therefore, the entire beam path must be enclosed in a vacuum chamber. Detector requirements include vacuum compatibility and high dynamic range, with pixel sizes typically at the 10-μm scale. Samples must also be vacuum-compatible, limiting live specimen imaging but enabling detailed inspection of semiconductor devices and mask structures. The contrast arises from variations in complex refractive index, with high sensitivity to surface topography owing to the limited penetration depth. This wavelength range is important for semiconductor metrology and lithography inspection for defect detection.

At the wavelengths of soft X-ray ptychography ($\lambda \approx 0.5-10nm$), zone plate optics are commonly used for focusing but suffer from aberrations and limited efficiency; therefore, ptychographic reconstruction is valuable for computationally enhancing the imaging performance. In the ‘water window’ (2.3–4.4 nm), water becomes relatively transparent while carbon and nitrogen remain highly absorbing^128,129, enabling high-contrast imaging of hydrated specimens without staining. The need for a vacuum during operation introduces practical challenges including pump-down delays and setup constraints, but dedicated beamlines at synchrotron facilities provide optimized experimental configurations. The contrast mechanism offers excellent sensitivity to electron density variations and elemental composition, enabling detailed structural and chemical mapping of specimens.

Hard X-ray ptychography ($\lambda \approx 0.1nm$) stands out from the other wavelength regimes for the large difference between the achievable resolution and wavelength—often several hundred times. A comparable setup at optical wavelengths to a typical hard X-ray experiment would involve using a camera with 30 cm² pixels to image a sample positioned at least 30 km away with 200 μm resolution, which is more akin to astronomy than microscopy. Nevertheless, the penetrative power of hard X-rays enables one of the most common applications of ptychography: 3D volumetric imaging of materials and biological specimens via ptycho-tomography^85,87,89,90. Beamlines at synchrotron facilities provide highly coherent X-ray beams, with specialized photon-counting detectors that offer excellent sensitivity and dynamic range. Sample preparation is relatively flexible owing to the high penetration depth, with mounting options ranging from thin support membranes to capillaries. Cryo-cooling is often employed to mitigate damage to radiation-sensitive biological materials during measurement^88,130. The image contrast arises directly from variations in electron density, offering sensitivity to internal structures without requiring destructive sectioning.

Electron ptychography ($\lambda \approx 1-10pm$) is one of the most challenging regimes to implement. Even small instrument instabilities and sample drift can affect atomic-resolution measurements. Unlike other methods that translate samples, electron ptychography scans the beam to avoid drift-induced blurring at atomic resolution. High resolution electron ptychography operates in transmission, requiring specimens to be electron transparent, typically ranging from tens of nanometres down to a single atom in thickness. Such thin samples can move or vibrate if not sufficiently well supported, causing motion blur and incoherence by introducing a defocus spread. Phonons also cause atomic vibrations even for a well-supported sample, limiting the ultimate resolution. Thicker samples offer mechanical stability but introduce inelastic scattering that degrades signal-to-noise ratios. Additionally, the electron beam itself causes sample damage through knock-on displacement and radiolysis, while also attracting hydrocarbon contamination. Despite these obstacles, beam scanning with direct electron detectors enables rapid data collection up to megahertz rates. The contrast mechanism, based on electrostatic potential interactions, provides sensitivity to light elements and subtle atomic structure variations. Early implementations of electron ptychography relied on the use of cameras to image fluorescent screens (scintillators) that converted electrons to light. This indirect detection introduced spatial blurring, limited dynamic range, and severely restricted acquisition speeds. Direct electron detection cameras have eliminated this intermediate conversion, enabling single-electron sensitivity, million-to-one dynamic range, and megahertz scan rates^131,132. Such advances in camera technology combined with reconstruction algorithms have further improved the resolution, sensitivity, and clarity that can be achieved^{107,133–137}.

3. Results

This section presents representative raw data and reconstructions from optical, EUV, X-ray, and electron implementations. We demonstrate how each wavelength regime transforms diffraction patterns into quantitative amplitude and phase images, achieving resolutions that exceed conventional hardware limitations through computational reconstruction.

3.1. Optical ptychography

Conventional ptychography produces raw diffraction patterns that manifest as intricate speckle structures in reciprocal space (Fig. 4a, top)¹¹¹. These seemingly random intensity distributions encode the specimen’s spatial frequency content through coherent interference. Iterative phase retrieval algorithms extract the complex-valued object from these patterns, with specialized techniques like super-resolution extending capabilities beyond detector limits. For example, using a 675 nm laser probe and collecting 400 diffraction patterns with ~70% overlap, super-resolution extrapolation increased the resolution of a conventional ptychography reconstruction by 3 times relative to the NA of the detector, enabling the 1.23 μm linewidth to be resolved (Fig. 4a, bottom)¹¹¹. This demonstrates how ptychographic redundancy enables computational techniques to extend the resolution of a measurement beyond hardware constraints.

Fig. 4|. Data acquisition and reconstruction in optical ptychography.

a, Top, raw diffraction pattern in reciprocal space ($k_x,k_y$) obtained with a conventional ptychography configuration (shown in inset). Bottom, corresponding reconstructed image obtained with super-resolution extrapolation (inset shows the zoomed-in view resolving the 1.23 μm linewidth). b, Top, low-resolution raw image in real space $(x,y)$ obtained with Fourier ptychography using an LED array (shown in inset) for object illumination and captured with a 2×, 0.08-NA objective lens. NA, numerical aperture. Bottom, reconstructed image, which is obtained by synthesizing 137 raw images captured at different incident angles, achieves an effective NA of 0.5 (inset shows the zoomed-in view resolving the 780 nm linewidth). c, Top, the raw low-resolution, large field-of-view coded diffraction pattern obtained from coded ptychography (configuration shown in inset). Bottom, the recovered high-resolution gigapixel image obtained from a series of raw images (inset shows the zoomed-in view resolving the 308 nm linewidth). Panel a adapted with permission from ref. ¹¹¹. Panel b adapted with permission from ref. ³². Panel c adapted with permission from ref. ⁴⁷.

FP uses a low-NA lens to invert the acquisition paradigm of conventional ptychography, capturing real-space raw images under angle-varied LED illumination (Fig. 4b, top)³². During reconstruction, the algorithm synthesizes a larger effective NA, increasing the resolution while retaining the large field of view of low-NA optics (Fig. 4b, bottom)³². Therefore, FP can acquire tissue-level context and cellular resolution across gigapixel-scale images¹³⁸. Additionally, the data acquisition speed of FP is limited only by camera frame rate and LED switching time, making it well suited for high-throughput microscopy.

Coded ptychography uses a lensless configuration in which coded surfaces are permanently fabricated on the image sensors (Fig. 4c, top)⁴⁷. The captured raw image exhibits deterministic modulation that encodes high-frequency object information into detectable signals. The deterministic modulation pattern serves as a stable, characterized probe that remains invariant across measurements, enabling gigapixel-scale reconstructions to maintain consistent resolution across the entire field of view^139,140. This approach has achieved one of the highest NA demonstrated without lenses, producing a reconstruction from 450 diffraction measurements that resolves the 308-nm linewidth using 1.85-μm pixels⁴⁷ (Fig. 4c, bottom). The resolution of coded ptychography with fixed plane wave illumination is fundamentally limited by NA = 1, corresponding to the maximum collection angle of 90 degrees in air; this maximum possible NA allows for resolving a ~250 nm linewidth with green light.

All three implementations surpass the constraints of the physical optics, demonstrating that the performance limits are defined by computational processing rather than detector resolution or lens quality.

3.2. EUV ptychography

HHG sources can generate coherent EUV radiation through nonlinear frequency conversion in noble gases. In tabletop HHG setups (Fig. 5a), infrared laser pulses interact with gas jets to produce high harmonics. The system first removes the fundamental infrared beam (EUV-IR separation), yielding a polychrome EUV spectrum spanning 70–90 eV (wavelengths of 13–18 nm) as shown in the figure inset. Further spectral filtering via reflective mirrors then selects specific harmonics to create a quasi-monochrome beam required for ptychographic imaging. EUV ptychography can be performed in transmission or reflection mode. Transmission-mode measurements are often used for membrane-supported nanostructures or ultrathin (<100 nm) specimens. The raw image from such measurements is a far-field diffraction pattern, characterized by a bright central region with high-angle scattering features that encode the fine structural details of the sample (Fig. 5b, top)¹⁴¹. Reconstruction algorithms use a series of such patterns collected at different scan positions to recover the complex object, which is visualised by representing the quantitative amplitude with brightness and phase with hue. The recovered image from transmission-mode EUV measurements of a Siemens star target resolves the 18 nm linewidth (Fig. 5b, bottom)¹⁴¹. Reflection-mode EUV ptychography is often used for opaque samples such as semiconductor devices, lithography masks, and metallic structures. The raw diffraction pattern, after being corrected for the 45° angle of incidence, contains the far-field scattering information from the surface of the sample (Fig. 5c, top). The angle correction process remaps the detector intensities to the object’s reciprocal space, enabling reconstruction at arbitrary angles of incidence in reflection mode. The final recovered image displays the surface topography of the sample, where phase (hue) is related to feature height and amplitude (brightness) corresponds to material reflectivity. Such measurements have been used to reveal the surface topography of titanium nanostructures on silicon (Fig. 5c, bottom)²⁷.

Fig. 5 |. EUV ptychography with tabletop high harmonic generation.

a, Schematic of a tabletop extreme ultraviolet (EUV) ptychography system in transmission (top) and reflection mode (bottom), which uses a gas jet for high harmonic generation (HHG). The system first separates the generated EUV harmonics from the fundamental infrared beam (EUV-IR separation), producing the polychrome spectrum shown in the inset. Further spectral filtering then selects specific harmonics to create a EUV monochrome beam for imaging. b, Top, raw diffraction pattern in reciprocal space ($k_x,k_y$) from a transmission-mode measurement of a Siemens star target, (data from ref. ²⁰⁵). Bottom, recovered complex wavefield in real space ($x,y$) demonstrating quantitative amplitude ($A$) and phase ($\phi$) reconstruction, represented by brightness and hue, respectively. c, Top, raw diffraction data from a reflection-mode measurement of titanium nanostructures on silicon. Bottom, corresponding phase-retrieved surface topography with nanometer-scale surface relief. Panel b adapted with permission from ref. ¹⁴¹. Panel c adapted with permission from ref. ²⁷.

3.3. X-ray ptychography

X-ray ptychography leverages its penetrating power and short wavelengths to achieve nanoscale resolution in thick specimens. Kirkpatrick–Baez mirrors are used to focus X-rays from synchrotron beamlines into various ptychographic configurations (Fig. 6a). Conventional X-ray ptychography generates far-field diffraction patterns with characteristic radial intensity decay from a bright central region (Fig. 6b, top). Blurring effects in the raw data arise from the partial coherence of the synchrotron source. Consequently single-state reconstruction, which assumes that the source is fully coherent, yields artifacts and poor contrast. By decomposing the beam into multiple incoherent modes, mixed-state modeling improves the reconstruction quality, enabling measurements of a resolution test pattern to resolve individual pillars with sharp boundaries (Fig. 6b, bottom)⁹¹. This coherence correction is essential for ptychographic implementations where perfect coherence is unattainable.

Fig. 6 |. Data acquisition and reconstruction in X-ray ptychography.

a, Schematic of X-ray ptychography using a synchrotron beamline where Kirkpatrick–Baez mirrors focus the X-rays into conventional, Bragg, near-field and Fourier ptychography configurations. b, Top, a single far-field diffraction pattern in reciprocal space ($k_x,k_y$) from conventional X-ray ptychography measurements of nanofabricated gold pillars. Bottom, a standard single-state reconstruction (top left) in real space $(x,y)$ and a superior mixed-state result (bottom right), which computationally corrects for the partial coherence of the synchrotron beam. Amplitude ($A$) is represented by brightness and phase ($\phi$) by hue. c, Top, a diffraction pattern collected at a specific Bragg reflection from a helium-implanted tungsten crystal (data from ref. ¹⁴²). Bottom, the recovered phase map, which reveals periodic fringes that correspond to lattice distortions and strain fields. d, Top, the raw image obtained from a near-field X-ray ptychography measurement of a Siemens star pattern, where an upstream diffuser produces a complex speckle pattern for sample illumination. Bottom, the reconstructed phase image resolving the ~100 nm linewidth. e, Top, a raw X-ray Fourier ptychography image of an integrated circuit captured using a zone-plate lens. Bottom, the recovered phase image, which has resolution beyond the diffraction limit of the zone-plate lens. Panel b adapted with permission from ref. ⁹¹. Panel c bottom adapted with permission from ref. ¹⁴². Panel d adapted with permission from ref. ³⁰. Panel e adapted with permission from ref. ¹⁴³.

Bragg configurations of X-ray ptychography operate at crystallographic reflections to probe lattice distortions. For such measurements of a helium-implanted tungsten crystal, the captured raw diffraction pattern (Fig. 6c, top) encodes information about the local strain field and the reconstruction yields a phase map representing the projection of the displacement field along the Bragg vector (Fig. 6c, bottom)¹⁴². Strain maps can then be derived from the spatial derivatives of the phase, revealing nanoscale lattice distortions. Similarly, grazing-incidence ptychography²⁸ leverages reflection geometry at shallow angles (<1°) for total external reflection. This configuration provides surface sensitivity with sub-nanometre height precision while maintaining millimetre-scale field coverage across the footprint of the elongated beam. Therefore, this approach enables simultaneous topographical and chemical characterization of surfaces over large areas, advantageous for wafer inspection and thin film characterization.

In near-field ptychography, a diffuser is inserted into the beam path to create a structured illumination wavefront. The raw data exhibit holographic fringes that are characteristic of Fresnel diffraction (Fig. 6d, top). Unlike the localized illumination used in conventional ptychography, this approach illuminates extended sample areas, trading spatial resolution for increased field coverage per scan position. A reconstruction from near-field ptychography measurements simultaneously retrieved the Siemens star test pattern and the structured probe function, achieving 100–200 nm resolution over a relatively large field of view of 65 × 65 μm² (Fig. 6d, bottom)³⁰. The extended illumination also provides sufficient information diversity across the entire field such that the image can be reconstructed from just 16 diffraction patterns. Whereas conventional ptychography might require hundreds of positions to cover the same area with a focused probe.

X-ray FP, which adapts the aperture synthesis concept to X-ray microscopy, captures a series of raw images in real space by scanning the illumination angle or translating the objective lens (Fig. 6e, top). Each raw frame samples a limited portion of reciprocal space owing to the modest NA of X-ray optics. Ptychographic reconstruction synthesizes these images to increase the resolution beyond the physical diffraction limit of the objective. For example, X-ray FP achieved ~47 nm resolution by synthesizing an effective NA of ~0.0018 from a physical zone plate objective with a NA of ~0.001, representing a ~2× increase of the objective’s NA (Fig. 6e, bottom)¹⁴³.

3.4. Electron ptychography

Electron ptychography exploits elastic electron scattering and modern detector technology to achieve atomicresolution imaging. The technique is often performed inside a scanning transmission electron microscope (STEM). An electron probe is systematically scanned across the specimen using scanning coils, just as in conventional STEM operation. However, unlike conventional STEM in which the probe is focused to the smallest spot at the specimen plane, electron ptychography often uses a defocused probe to increase the illumination area (Fig. 7a). At each scan position, a pixelated camera records the complete far-field diffraction pattern, preserving all scattered electron intensities. This scanning process generates a four-dimensional dataset with two real-space dimensions ($x,y$) from the probe positions and two reciprocal-space dimensions ($k_x,k_y$) from the diffraction patterns. In contrast, conventional STEM integrates electrons over specific angular ranges using monolithic detectors, discarding the detailed diffraction information.

Fig. 7 |. Data acquisition and reconstruction in electron ptychography.

a, Schematic of a typical electron ptychography setup inside a scanning transmission electron microscope. A scanning defocused probe is used for sample illumination and the resulting diffraction patterns are captured by a detector in reciprocal space. b, Top, the raw diffraction pattern of gold nanoparticles from a dark-field electron ptychography measurement in reciprocal space ($k_x,k_y$), displayed in log-scale intensity to reveal weak dark-field signals alongside the intense bright-field disk. Bottom, the recovered quantitative phase image, which resolves gold atomic columns with 0.236 nm spacing in real space $(x,y)$. c, Top, the raw data from an atomic-resolution electron ptychography measurement of monolayer MoS₂ sample, showing the bright-field disk and diffraction extending to high scattering angles (data from ref. ²⁰⁶). Bottom, the reconstruction yields a real-space phase image with sub-angstrom resolution, clearly resolving the atomic lattice. Panel b adapted with permission from ref. ¹⁴⁴. Panel c adapted with permission from ref. ¹³³.

Dark-field electron ptychography can achieve atomic resolution by processing both the central bright-field disk and high-angle scattered electrons outside the disk. Raw diffraction patterns require log-scale visualization to reveal the weak dark-field signals alongside the intense central disk (Fig. 7b, top). As a demonstration, gold nanoparticles were imaged in a modified scanning electron microscope (SEM) at 30 keV using a standard CCD detector without an aberration corrector¹⁴⁴. Processing both bright-field and dark-field signals achieved ~0.24 nm resolution, a five-fold improvement over the 1.2-nm limit of the SEM. It successfully resolved individual gold atomic columns with 0.236 nm spacing (Fig. 7b, bottom). These results demonstrate that electron ptychography can extract atomic information even with a microscope that would normally capture much lower resolution images¹⁴⁴.

While the CCD detector in early demonstrations could capture both bright-field and dark-field signals, its limited dynamic range often led to saturation of the bright-field disk or poor signal in the dark-field regions. Achieving maximum resolution requires not only aberration-corrected STEM with large probe convergence angles but also detectors with sufficient dynamic range to capture the full signal without saturation. Modern direct electron detectors meet these requirements through pixel array that achieves single-electron sensitivity with ~1,000,000:1 dynamic range. Raw diffraction patterns of monolayer MoS₂ acquired in an aberration-corrected STEM using such detectors reveal a prominent bright-field disk surrounded by higher-order diffraction extending to large scattering angles (Fig. 7c, top)¹³³. The reconstruction from this data achieved 0.039 nm resolution, ~2.5 times beyond the conventional STEM limit operating at 80 keV (Fig. 7c, bottom)¹³³. This improvement arises from capturing the full dynamic range of scattered electrons without saturation. When combined with multislice modeling to account for dynamic scattering, electron ptychography achieves ~20 pm resolution, approaching the fundamental limit imposed by atomic thermal vibrations^107,136.

The intelligent use of the full diffraction information in electron ptychography enables higher dose efficiency than conventional STEM modes that integrate electron signals over specific regions. Conventional annular dark field imaging in STEM integrates scattered electrons to produce Z-contrast images where brightness scales with atomic number (Z), making it widely used in materials science¹⁴⁵. However, annular dark field suffers from poor dose efficiency and limited sensitivity to light elements. Electron ptychography addresses these issues, achieving dose efficiency and sensitivity comparable to transmission electron microscopy (TEM) while retaining interpretability: atomic positions in ptychographic phase images correspond to actual locations, unlike planewave TEM where contrast reversals and lens aberrations can produce misleading artifacts¹⁴⁶. Moreover, ptychography can be performed simultaneously with annular dark-field imaging in a focused probe configuration¹⁴⁷. This dualmode imaging combines the chemical sensitivity of Z-contrast imaging with the phase sensitivity of ptychography, enabling the detection of charge transfer and bonding variations that are invisible to Z-contrast alone¹³⁷.

4. Applications

This section presents representative applications of ptychography in life sciences, materials characterization, and industrial metrology, demonstrating how its quantitative contrast capabilities and resolution enhancements address imaging challenges across length scales.

4.1. Life sciences

Optical ptychography has transformed biomedical imaging through high-throughput, label-free implementations. For example, coded ptychography at visible wavelengths has been used for gigapixel phase imaging to monitor live cell cultures across centimetre-scale fields (Fig. 8a)⁵⁰. The recovered phase maintains subcellular resolution and provides information related to cellular dry mass, enabling label-free tracking of cell division, growth, and drug responses across extended periods. By visualizing bacterial growth with picogram sensitivity¹³⁹, coded ptychography reduces antibiotic response tracking from days to hours. This acceleration is critical for treating infections such as sepsis. The technique can also differentiate cells and crystals in blood and urine samples^46,48, providing label-free alternatives to flow cytometry^47,49.

Fig. 8 |. Life science applications.

a, Gigapixel lensless phase imaging of live human embryonic kidney 293 (HEK 293) cells via optical coded ptychography. Insets, zoomed-in views on the phase ($\phi$) images at the same time as the large image and 1 hour and 2 hours later, showing cell division as indicated by the white arrows. b, The recovered phase image from a synthetic aperture ptycho-endoscopy measurement of an ex vivo mouse colon sample using a handheld lensless fiber bundle tip (inset), which achieves super-resolution imaging of colonic crypts at the optical regime. Brightness indicates amplitude ($A$) and hue indicates phase ($\phi$). c, 3D refractive index distributions of a single HeLa cell at the x–z, x–y and y–z planes, reconstructed using Fourier ptychographic diffraction tomography at visible wavelengths. d, Extreme ultraviolet (EUV) ptychographic imaging of fixed mouse hippocampal neurons, which reveals sub-100-nm neural structures without causing radiation damage. e, The reconstructed phase (top left) and amplitude (bottom right) from a soft X-ray ptychography measurement of a mouse fibroblast cell doped with CoFe₂O₄ nanoparticles, enabling visualization of nanoparticle clusters through intrinsic contrast (appearing as dark spots in amplitude). f, An X-ray ptychographic image of frozen-hydrated algae overlayed with a coloured fluorescence microscopy image of the same sample, revealing sub-20-nm structural details alongside elemental distributions. g, Ptychographic X-ray computed tomography of a mouse femur bone showing quantitative electron density mapping. The golden regions represent bone matrix while the dark branching regions are osteocyte lacunae and their connecting network, revealing bone microstructures within intact bone tissue. h, 3D reconstruction of rotavirus double-layered particle obtained with electron ptychography. Panel a adapted with permission from ref. ⁵⁰. Panel b adapted with permission from ref. ⁹⁷. Panel c adapted with permission from ref. ¹¹⁵. Panel d adapted with permission from ref. ¹⁵¹. Panel e adapted with permission from ref. ¹⁵². Panel f adapted with permission from ref. ¹⁵⁶. Panel g adapted with permission from ref. ⁸⁵. Panel h adapted with permission from ref. ¹⁵⁷.

For tissue-level analysis, lensless ptychographic scanners and lens-based FP platforms acquire gigapixel images at visible wavelengths^59,140,148, providing solutions for digital pathology applications. DUV FP exploits the intrinsic absorption properties of biomolecules at 265 nm to reveal cellular structures that cannot be observed with conventional light microscope⁶⁰. For ultrathin specimens, plasmon-enhanced ptychography uses metamaterial substrates to increase visibility 10-fold over standard light microscope, imaging unstained tissue sections as thin as 70 nm¹⁴⁹. Endoscopic applications extend ptychography to clinical settings. Synthetic aperture ptycho-endoscopy uses 532 nm laser light for illumination and a lensless fiber bundle tip for diffraction data acquisition⁹⁷. The lensless fiber bundle tip serves as the effective coded sensor, with densely packed fiber cores modulating the diffracted wavefield. Handheld motion introduces translational shifts that facilitate the acquisition of a ptychogram. The reconstruction process synthesizes a virtual aperture that extends beyond the fiber bundle’s physical dimensions, achieving super-resolution imaging similar to synthetic aperture ptychography. Despite challenges from handheld operation and fiber bending, the technique can visualize colonic crypts on the surface of opaque ex-vivo mouse colon tissue imaged in reflection mode (Fig. 8b). The achieved resolution of 548 nm surpasses that of conventional fiber endoscopy and is sufficient to resolve sub-cellular details needed for optical biopsy.

Fourier ptychographic diffraction tomography at visible wavelengths reconstructs 3D refractive index distributions throughout cells and tissues. For example, 3D imaging of HeLa cells (~10 μm) revealed nucleoli and sub-cellular structures with distinct refractive index values throughout the cellular volume (Fig. 8c)¹¹⁵. Other multislice and tomographic ptychography approaches^{22,23,25,103–105,109} similarly enable computational optical sectioning for 3D microscopy, offering a label-free alternative to confocal microscopy. Additionally, ptychographic optical coherence tomography has enabled volumetric imaging of mouse brain tissue with depth-resolved contrast and speckle-free reconstruction¹⁵⁰.

Moving to shorter wavelengths, EUV ptychography enabled imaging of mouse hippocampal neurons at sub 100 nm without observable radiation damage (Fig. 8d)¹⁵¹. The phase contrast reveals delicate neural structures that would be invisible in conventional amplitude imaging. Additionally, the water window in the soft X-ray regime can add elemental specificity for cellular imaging. For example, soft X-ray ptychography demonstrated iron-specific contrast between iron-containing nanoparticles and cellular structures in mouse fibroblasts, enabling visualization of nanoparticle clusters through intrinsic contrast (Fig. 8e)¹⁵². Such applications extend to freeze-dried bacterial cells¹⁵³, resin-embedded plant tissues¹⁵⁴, and frozen-hydrated yeasts¹⁵⁵.

Correlative imaging approaches combine the strengths of multiple modalities. For example, high-resolution ptychographic measurements combined with fluorescence molecular localization can provide insight on complex biological systems. The synergy between soft X-ray ptychography and fluorescence microscopy enabled measurements of frozen-hydrated algae to achieve sub-20 nm structural resolution while mapping elemental distributions (Fig. 8f)¹⁵⁶.

The penetrating power of hard X-rays enables 3D imaging of organ-scale specimens. Hard X-ray ptychotomography reconstructed a ~30-μm thick mouse femur bone with quantitative electron density mapping, revealing hierarchical bone structure from the organ level down to microscopic features (Fig. 8g)⁸⁵. This capability also extends to soft tissues. For example, hard X-ray cryo-ptychographic tomography visualized myelinated axons and subcellular features in mouse brainstem tissue at ~100 nm resolution through nearly 100 μm of material without heavy metal staining⁸⁸.

The high dose efficiency of electron ptychography makes it suitable for imaging radiation-sensitive specimens. 3D reconstruction of rotavirus particles achieved 1.86 nm resolution while preserving capsid structures under ultralow electron exposure (Fig. 8h)¹⁵⁷. Ptychographic single particle analysis further achieved a resolution of 5.8 Å for apoferritin protein complex with high symmetry (ref.¹⁵⁸). These low-dose measurements can determine the structure of specimens that would be destroyed by conventional STEM measurements.

4.2. Materials sciences

The ability of ptychography to quantitatively map strain fields, electromagnetic properties, and 3D structures across multiple length scales make it suitable for materials characterization. The technique can reveal structure–property relationships that were previously inaccessible using conventional methods.

Soft X-ray vector ptycho-tomography reconstructs magnetization vectors throughout sample volume by measuring differential absorption between polarized X-rays at multiple sample orientations. This technique maps complex magnetization structures in ferromagnetic systems, including vortices and domain walls¹⁵⁹. Building on this principle, the technique has been applied to image topological magnetic monopoles in nickel-infiltrated ferromagnetic meta-lattices (Fig. 9a), providing direct 3D visualization of these quasiparticles¹⁶⁰. These capabilities offer insights into topological magnetic structures relevant for magnetic storage and quantum device applications.

Fig. 9 |. Materials science and industrial applications.

a, Soft X-ray vector ptycho-tomography image of topological magnetic monopoles (TMMs) in a nickel-infiltrated ferromagnetic meta-lattice. The red and blue dots indicate TMMs and anti-TMMs, respectively. The red and blue surfaces represent virtual TMMs and virtual anti-TMMs on the magnetic void surfaces, respectively. Inset, magnetization vector field (white arrows) showing the hedgehog spin texture around a TMM (red dot). b, X-ray linear dichroic orientation tomography of polycrystalline V₂O₅. Colours indicate the in-plane orientation of the V₂O₅ crystallographic c-axis, with hues mapping to different angular orientations. This 3D visualization reveals multiple grains with distinct orientations that are invisible in conventional electron density imaging. c, 3D Bragg projection ptychography measurements of embedded SiGe (eSiGe) stressor crystals in silicon on insulator (SOI) structures at two cut-planes, revealing the amplitude $A$ (top), which is linked to material density, and lattice distortions (bottom), which is linked to the reconstructed phase $\phi$. d, Reconstructed phase images from multi-slice electron ptychography of PrScO₃ at different depths, showing atomic-resolution reconstruction of the crystal structure. e, The skyrmion lattice in FeGe imaged using Lorentz electron ptychography. Brightness indicates the amplitude, and hue indicates the direction of the magnetic field. Inset, magnified view showing the magnetic field vectors with arrows representing both the amplitude and direction of the magnetic field. f, Super-resolution image of an integrated circuit obtained with reflected Fourier ptychography at visible wavelengths. g, 3D extreme ultraviolet reflectometry image of a patterned silicon wafer, mapping chemical composition (Si₃N₄ and SiO₂) and As-dopant distributions. h, 3D burst ptychography image of electron density ${\rho }_{ED}$ in a processor circuit, achieving 4 nm isotropic resolution. Panel a adapted with permission from ref. ¹⁶⁰. Panel b adapted with permission from ref. ¹⁶¹. Panel c adapted with permission from ref. ¹⁶². Panel d adapted with permission from ref. ¹⁰⁷. Panel e adapted with permission from ref. ¹³⁴. Panel f adapted with permission from ref. ¹⁶⁵. Panel g adapted with permission from ref. ¹⁶⁶. Panel h adapted with permission from ref. ⁸⁹.

X-ray linear dichroic orientation tomography extends ptychographic capabilities to the characterization of polycrystalline materials. The method exploits the anisotropic X-ray absorption of a sample to determine the crystal orientation at each voxel, providing insight on structure–property relationships in functional materials. Applied to V₂O₅ catalysts (Fig. 9b), this technique maps crystallographic orientations throughout the volume, revealing 3D grain boundaries, defects, and texture¹⁶¹.

3D Bragg projection ptychography maps strain fields in semiconductor devices by reconstructing lattice distortions from single-angle measurements. Unlike conventional techniques requiring multiple crystal orientations, this approach provides 3D strain information from measurements at one orientation angle, reducing acquisition time and radiation dose. Applied to embedded SiGe stressor crystals within silicon-on-insulator structures, it revealed complex strain fields critical for device performance (Fig. 9c)¹⁶². Such results provide insight into how engineered strain enhances carrier mobility in modern electronics.

At the atomic scale, multislice electron ptychography has achieved resolution approaching the fundamental physical limits. For example, measurements of PrScO₃ crystals resolved 59 pm Pr–Pr separations and light oxygen atoms (Fig. 9d)¹⁰⁷. It achieved 23 pm resolution, which is primarily limited by atomic thermal vibrations rather than instrumentation. Advances in local-orbital ptychography, which perform orbital-based modeling to separate atomic contributions, have further pushed the resolution limit to 14 pm while enabling element-specific phase decomposition¹³⁶. These developments enable direct observation of structure–property relationships in complex materials, where electronic and magnetic behaviour is primarily determined by subtle positional changes of light elements.

Lorentz electron ptychography maps magnetic structures by detecting beam deflections from magnetic fields, providing direct visualization of magnetic textures beyond conventional diffraction limits¹³⁴. This approach was used to image skyrmion lattices in FeGe, revealing internal structures including cores, boundaries, and dislocations (Fig. 9e). The high dose efficiency of the technique enables dynamic studies under varying magnetic fields and temperatures, essential for developing skyrmion-based memory devices. The ability to directly visualize the evolution of the magnetic phase can also guide the development of spintronic devices.

4.3. Industrial applications

The key industrial applications of ptychography are semiconductor manufacturing, quality control, and optical characterization. Reflection-mode FP^163,164 at visible wavelengths achieves super-resolution imaging beyond physical optics alone permits. For example, a 10×, 0.28-NA objective lens achieved a synthetic NA of 1.06 through angle-varied illumination, enabling inspection of patterned features on silicon wafers for rapid defect detection (Fig. 9f)¹⁶⁵. This reflection configuration excels at inspecting opaque materials, revealing surface defects, particles, and pattern irregularities that are critical to device yield (the percentage of functional chips per wafer). As the semiconductor industry transitions to EUV lithography, reflection-mode FP can be implemented at EUV wavelengths to inspect EUV masks and wafers.

Reflection-mode EUV imaging can non-destructively characterize multilayer semiconductor structures. By recovering phase images at different incident angles, this technique produces 3D maps of the chemical composition and dopant distributions in patterned silicon wafers (Fig. 9g)¹⁶⁶, revealing buried layers and interfaces inaccessible to surface inspection. Such depth-resolved characterization is essential for characterizing gate-all-around transistors, buried power rails, and other 3D structures in advanced nodes.

Hard X-ray burst ptychography further enhances resolution for non-destructive analysis. This technique captures multiple low-exposure frames at each scan position, then computationally aligns them to correct beam pointing instabilities. Applied to 7-nm node processors, it achieved 4-nm isotropic resolution throughout the entire processor volumes (Fig. 9h)⁸⁹. Burst ptychography accelerates the acquisition speed by 170-fold compared to conventional ptychography, transforming failure analysis from a weeks-long bottleneck to an overnight procedure, enabling rapid iterations in process development.

Beyond semiconductor applications, ptychography can address broader quality control and optical metrology challenges in manufacturing. Various optical implementations have been used to characterize wavefront aberrations⁷⁹, metasurfaces^167,168, optical lenses⁴⁷, spatial light modulators¹⁶⁹, and sophisticated optical components¹⁷⁰, which are difficult to measure with conventional metrology techniques. Additionally, EUV and X-ray implementations can characterize diffractive optical elements with nanometre-scale precision^171–173 and perform high-resolution wavefront sensing for advanced optical systems¹⁰¹. Ptychographic principles also extend to macroscopic imaging challenges. Snapshot FP using camera arrays enables single-shot synthetic aperture imaging^44,174, which can simultaneously capture multiple perspectives to enhance resolution beyond the limits of individual cameras. As manufacturing increasingly demands high resolution and large area coverage, such multi-aperture FP systems could bridge the scale gap in macroscopic inspection. Applications include quality control of display panels, turbine blades, and engine parts. These components present unique inspection challenges: conventional microscopy requires time-consuming scanning to cover such large areas, while single-camera systems lack the resolution to detect microscopic defects across metre-scale surfaces.

5. Reproducibility and data deposition

Ptychography spans multiple wavelength regimes, each with distinct experimental configurations and reconstruction algorithms. This diversity has led to different data management approaches across communities; however, common standards are emerging.

Hierarchical Data Format 5 (HDF5)¹⁷⁵ is the standard for ptychographic data storage for many X-ray and electron ptychography groups. HDF5 efficiently handles large multidimensional arrays while preserving rich metadata. In contrast, EUV and optical ptychography communities have maintained more diverse and flexible approaches to data storage. For example, optical implementations use various formats including proprietary software files (MATLAB), common image formats, programming language-specific data arrays (Python, Julia), and custom binary structures, depending on the computational tools employed^{13,35,50,57,58}. Despite format differences, all communities recognize the importance of preserving essential metadata, including illumination wavelength, detector specifications, probe characterization, pixel sizes, sample-to-detector distances, and scan position coordinates. Ongoing standardization efforts focus on establishing these minimal metadata requirements to ensure that datasets remain interpretable and reconstructable regardless of the specific format chosen, while respecting the practical constraints and workflows of each wavelength regime.

The use of standardized algorithms and proper version control in reconstruction software is also needed to ensure reproducibility. Established platforms such as PtyPy¹⁷⁶, PyNX¹⁷⁷, PtyGer¹⁷⁸, PtychoShelves¹⁷⁹, and PtyLab¹³ offer extensive functionality supporting various experimental geometries and advanced reconstruction algorithms. While these platforms implement similar core algorithms, they use different data formats and parameter conventions, potentially affecting reproducibility when switching between packages. Each platform has distinct strengths. For example, PtyPy provides an abstract representation of physical experiments facilitating algorithm portability. Its modular architecture can handle challenging datasets ranging from high-resolution X-ray ptychography at synchrotron sources to complex electron ptychography experiments with severe partial coherence issues. PyNX, PtyGer, and PtychoShelves are optimized for high-throughput analysis at synchrotron facilities but require careful configuration of reconstruction parameters, graphics processing unit memory management, and beamline-specific settings. PtyLab unifies conventional and FP workflows across multiple programming languages, allowing flexible conversion between conventional and FP datasets. More specialized tools have also emerged for particular applications: phaser¹⁸⁰ and ptychoscopy¹⁸¹ for electron ptychography, SHARP⁶⁴ for X-ray synchrotron data processing, and various reconstruction protocols for optical ptychography^{13,35,50,54,57,58}.

The sophistication of software packages often presents a steep learning curve for newcomers, requiring familiarity with complex parameter files, and careful management of software dependencies. For example, the external libraries and specific software versions are needed for the code to function properly. Version conflicts and missing dependencies can prevent software from running. The diversity of experimental configurations and reconstruction strategies across wavelength regimes often obscures the underlying unified principles. To address these accessibility challenges, the simplified implementations in the Supplementary Note demonstrate the core ptychographic concepts across wavelength regimes with minimal complexity for educational purposes. Establishing robust practices for data sharing, software documentation, and reproducible workflows remains essential for maximizing the influence of the technique.

6. Limitations and optimizations

Radiation damage remains a fundamental constraint when using ionizing radiation, particularly for biological specimens and soft materials. In electron ptychography, knock-on damage from elastic scattering directly displaces atoms and inelastic interactions cause radiolysis and specimen heating. Similarly, soft X-rays induce photoionization and hard X-rays induce structural changes through absorbed dose. These structural changes during measurement cause reconstruction artifacts as the imaging model assumes a static object. Reconstruction strategies address radiation damage through different approaches. For example, mixed-state algorithms achieve sub-angstrom resolution at substantially lower doses than conventional methods¹³⁵, orthogonal probe relaxation accounts for specimen and probe changes during measurement to prevent artifacts¹¹⁰, and machine learning approaches enhance reconstruction quality from incomplete, low-dose datasets^182–184.

Reconstruction quality is constrained by sample modeling assumptions. The multiplication between the object and probe beam assumes that specimens are thin. This approximation becomes problematic in conventional ptychography and FP when imaging optically thick samples. Although multi-slice or ptycho-tomography approaches can model thicker specimens, they are computationally complex and have large memory requirements. Additionally, in FP the phase transfer characteristics make it difficult to recover slowly varying phase with multiple 2π wraps, such as those found in agar plates, bacterial colonies, crystals, and cytology smears. This limitation can be partially addressed by incorporating coded detection elements that enhance phase sensitivity through structured modulation¹⁸⁵.

The requirement that the illumination is coherent impacts the experimental design and achievable resolution. Insufficient coherence causes interference fringes to blur, limiting the achievable resolution. The development of fourth-generation synchrotron sources and cold field emission electron guns provide enhanced spatial coherence for X-ray and electron ptychography, respectively, while HHG offers naturally coherent EUV sources. Nevertheless, residual partial coherence in all systems necessitates mixed-state algorithms that decompose illumination into incoherent modes⁹¹.

Detector technology influences ptychographic performance across all implementations, with ideal detectors meeting multiple simultaneous requirements. High dynamic range captures both the intense un-scattered beam and weak high-angle scattering signals that differ by many orders of magnitude. Single-photon or single-electron sensitivity is essential for dose-efficient imaging of radiation-sensitive specimens. High acquisition rates are needed for rapid data acquisition to minimize drift and capture dynamic processes, while small pixel sizes ensure adequate sampling of fine diffraction features. Direct electron detectors now achieve million-to-one dynamic range with up to megahertz acquisition rates^131,132. Additionally, photon-counting X-ray detectors with energy discrimination enable modalities such as spectroscopic ptychography¹⁸⁶. For optical implementations, event-based sensors that record only pixel changes¹⁸⁷ could reduce data volumes and increase acquisition speeds.

The need for high data acquisition speed and throughput present interconnected challenges that limit practical applications. Conventional ptychography must illuminate small portions of the object sequentially to build up large fields of view. Mechanical scanning stages introduce positioning errors and settling time delays. This serial acquisition creates fundamental throughput bottlenecks. Optimizations include the use of a fly-scan mode to eliminate settling delays^94–96,188, strategies such as sparse sampling, adaptive acquisition, and multiplexed measurements to reduce the number of acquisitions^{51,55,189–192}, and on-the-fly reconstructions that minimize processing delays and approach real-time reconstructions^193,194. However, these optimizations cannot fully overcome the fundamental speed limitations of sequential scanning and often require specialized hardware such as synchronized motion controllers, graphics processing unit clusters, or compromise image quality through undersampling.

Computational requirements scale with data volume and desired resolution, creating barriers to commercialization. Unlike conventional microscopy methods, which provide immediate visual feedback, reconstruction times for multi-slice or tomographic approaches used to handle thick specimens can extend from hours to days even with graphics card acceleration¹¹⁵, creating a usability gap that hinders commercial adoption. For ptychographic systems aimed at clinical or industrial applications, the inability to provide real-time imaging during specimen positioning or focus adjustment presents a fundamental workflow challenge. Neural network approaches can reduce the reconstruction time while maintaining comparable quality^{117,182,194–196}.

7. Outlook

The evolution of ptychography from a specialized crystallographic method to a versatile imaging modality demonstrates how computational approaches can transcend hardware limitations. From a technological innovation perspective, the development of ptychography can be understood through three complementary aspects: configurations (for example, conventional, Bragg, Fourier and coded), modelling (for example, ptychotomography, multi-slice, mixed state and neural field representation), and wavelengths (optical, EUV, X-ray and electron). The unexplored combinations within this multidimensional innovation space offer opportunities for future developments beyond incremental improvements to existing methods. It is likely that future developments will increasingly draw from the collective knowledge base, with innovations in one wavelength regime rapidly propagating to others, accelerating progress across the field. With increasing commercial adoption, educational integration, and technological convergence, the next phase of ptychography development is likely to produce innovations that fundamentally change how the world is imaged and understood from atomic to macroscopic scales.

Cross-wavelength adaptations of imaging configurations present immediate opportunities for innovation. Synthetic aperture ptychography could be adapted to X-ray and EUV regimes to enable extended field-of-view imaging for semiconductor metrology and materials characterization. For instance, translating coded X-ray detectors could synthesize apertures substantially larger than the physical size of the detector, potentially achieving nanometre-scale resolution across large fields of view. FP, which has been demonstrated at optical wavelengths, could improve EUV mask inspection by synthetically expanding the NA of existing inspection tools, potentially achieving sub-10 nm resolution. Coded ptychography also offers cross-wavelength opportunities. Integrating disorder-engineered surfaces directly onto EUV-sensitive detectors could enable compact, lensless inspection systems for in-line semiconductor metrology, eliminating the need for expensive EUV optics while enabling inspection at the same EUV lithography wavelength. In the DUV regime, coded ptychography could be combined with slide scanning and handling systems to create high-throughput cellular analysis platforms that exploit intrinsic protein and nucleic acid absorption for label-free chemical imaging.

Advanced modelling would also benefit from cross-wavelength developments. Neural field representations developed for optical ptychography could address the massive data volumes in 4D-STEM, providing memory-efficient storage and accelerating the reconstruction of electron ptychographic datasets. Automatic differentiation frameworks used in optical and X-ray ptychography could accelerate the development of complex forward models for multi-wavelength EUV ptychography, where simultaneous reconstruction at multiple harmonics could provide element-specific contrast. Mixed-state algorithms essential for handling partial coherence in synchrotrons could also be used to model the extended source in LED-based implementations of coded ptychography.

The commercialization of ptychography will require that the cost, usability, and workflow integration are addressed alongside technical performance. While ptychography offers performance advantages including higher throughput than conventional microscopes, better contrast than standard phase-contrast imaging, and unique capabilities such as 3D reconstruction from 2D measurements, widespread adoption still faces challenges. These include computational requirements necessitating graphics processing unit infrastructure, reconstruction times incompatible with real-time workflows, and software complexity requiring specialized expertise. FP is under commercial development for digital pathology and industrial inspection. Additionally, coded ptychography on a chip is under development for point-of-care applications such as rapid urinalysis, antibiotic sensitivity testing, and cytopathology screening. Success will depend on developing turnkey systems with automated operation and hardware acceleration that maintain performance advantages while achieving costs competitive with conventional robotic microscopy platforms.

From an education perspective, ptychography has emerged as an effective teaching tool for Fourier optics and computational imaging concepts. Open-source implementations and online tutorials enable students to build functional systems with existing classroom resources⁵⁰. This accessibility allows hands-on learning of phase retrieval, spatial frequency concepts, and the relationship between measurement and computation. The educational value creates a positive feedback loop: accessible implementations attract new researchers who drive further innovation.

Glossary

Wavefield

The complete description of a wave containing both amplitude and phase information

Real space

The physical coordinate system ($x,y$) in which specimens exist

Reciprocal space

Also known as Fourier space. The spatial frequency domain ($k_x,k_y$) in which far-field diffraction patterns form

Ptychogram

a complete dataset of diffraction patterns captured at different scan positions

Wavevector

A vector pointing in the propagation direction of the wave. To observe fine details, waves must scatter at large angles, creating large transverse wavevector components

Pupil function

The aperture function of an optical system, which limits what spatial frequencies pass through. In Fourier ptychography, it acts as the effective ‘probe’ by scanning through reciprocal space

Bragg diffraction condition

The specific angle and wavelength combination at which crystal lattice reflections interfere constructively, enabling selective imaging of crystal orientations and strain

Coherence

The degree to which different parts of a wavefield maintain fixed phase relationships

Cost function

A mathematical metric quantifying the discrepancy between the measured and estimated diffraction patterns that algorithms minimize through iterative refinement

Ewald sphere

A spherical surface in reciprocal space that determines which spatial frequencies of an 3D object can be measured for a given illumination angle

Strain field

The spatial distribution of lattice distortions in crystalline materials, which affects properties such as carrier mobility in semiconductors

Dose efficiency

The information extracted per unit of radiation damage, with ptychography achieving 10–100× improvement over conventional methods by using all of the scattered photons or electrons

Data availability

The experimental datasets that accompany the code implementations in the Supplementary Information are available at https://doi.org/10.6084/m9.figshare.29205677.