Table-top extreme ultraviolet grazing-reflection ptychography with full-pose self-calibration

As semiconductor manufacturing progresses toward nanometer-scale features, there is an escalating demand for advanced metrology that can address critical dimensions and overlay errors¹. Short-wavelength techniques such as extreme ultraviolet (EUV) imaging offer distinct advantages²⁻⁶, including nanometer resolution, deep penetration, and nondestructive inspection, making them suitable for semiconductor metrology. Furthermore, the development of tabletop EUV sources based on high harmonic generation provides high brightness and coherence, effectively facilitating laboratory-scale imaging⁷⁻⁹.

However, progress in EUV-based metrology has been hindered in recent years by several fundamental challenges. A primary obstacle is the scarcity of high-quality, high numerical aperture (NA) optics suitable for this spectral band. Within the EUV regime (approximately 10–100 nm), all materials strongly absorb radiation, necessitating the use of mirrors instead of lenses. The fabrication of these EUV mirrors demands specially designed multilayer coatings and exceptional surface smoothness, with height variations significantly smaller than the wavelength. These rigid requirements render EUV mirrors extremely difficult and costly to manufacture¹⁰⁻¹².

Ptychography has opened new avenues for lensless imaging¹³⁻¹⁵. Unlike conventional approaches that rely on complex optical systems, ptychography does not require any optical elements between the sample and detector; instead, it reconstructs the sample image directly from diffraction patterns. By leveraging redundant information from spatially overlapped scanning, ptychography recovers the illumination probe and image nonisolated specimens¹⁶⁻¹⁹. The NA in this technique is determined by the diffraction distance and detector size, and therefore, it approaches the theoretical limit²⁰. Thus far, ptychography in transmission geometry has proven to be an effective tool for high-resolution imaging in the EUV waveband, and several studies demonstrated its utilization in the analysis of lithographic components^{14, 21, 22} and biological samples^{23, 24}, and in the multiwavelength characterization of EUV optics^{25, 26}. However, the application of EUV ptychography in reflection configurations remains limited.

In practice, EUV reflection ptychography can be implemented across three distinct ranges of reflection angles: small angles (<10°)²⁷⁻²⁹, which are commonly used for photomask inspection in lithography; moderate angles (<60°)^{30, 31}, typically employed to explore the fundamental principles of reflective diffraction imaging and seldom applied in practical inspection tasks; and large reflection angles (~70°)³²⁻³⁵, which corresponds to grazing-reflection geometry. Within this grazing reflection range, most materials are reflective, and the phase response shows significant angular sensitivity, making it suitable for tasks such as defect analysis and compositional analysis in semiconductor inspection. Further, reflective diffraction imaging at other wavelengths uses grazing reflection systems^{34, 36}.

However, such a grazing-reflection configuration poses significant challenges to system alignment and calibration for ptychography. Moreover, the limited power of tabletop EUV sources results in weak reflected and diffracted signals. Consequently, the effect of system misalignment is exacerbated by a low signal-to-noise ratio (SNR), which significantly degrades imaging quality. This issue becomes pronounced for larger numerical apertures, wherein the high-frequency signal content is sparse. Transmission ptychography involves calibrating the diffraction distance and scan positions, which is a subject thoroughly addressed in many studies³⁷⁻⁴⁰. However, calibration methods for reflection ptychography remain relatively underdeveloped.

Although some efforts have been made to correct system misalignment⁴¹⁻⁴³, they are constrained by the oversimplified assumption that all parameters are correct, except for one or two corrections. Maintaining this assumption requires using highly precise hardware alignment solutions, which undoubtedly increase system cost and complexity and hinder its general applicability in practical scenarios. Further, this assumption remains unrealistic even with highly sophisticated hardware alignments because of inevitable discrepancies between idealized models and real physical systems. Consequently, improvements in image quality remain limited. Thus far, there is no comprehensive and holistic solution that can perform the integrated calibration of full-pose parameters in reflection ptychography systems, and the need for such a solution is genuine and urgent.

In this study, we report a grazing-reflection ptychography system operating on a 17.9 nm table-top EUV source that features full-pose self-calibration capability. The full-pose calibration is based on an advanced optimization framework that includes comprehensive system modeling, feature domain phase retrieval, and automatic differentiation. The system is used to image a customized sample and a real chip, and the proposed self-calibration algorithm demonstrates an unprecedented improvement in image quality. Our method addresses a fundamental challenge in reflection ptychography and significantly advances its practical application. This study has inspired progress in diffraction-based imaging techniques across broader regimes, including visible light, X-ray, and electron microscopy.

In transmission ptychography, the camera is placed parallel to the sample to capture diffraction signals in reciprocal space after the illumination probe interacts with the specimen. These signals are distributed in the Ewald sphere. Under paraxial approximation with a low numerical aperture, the pixel array on the detection plane can sample the reciprocal space diffraction signal uniformly. In the EUV reflection ptychography system (Fig. 1a), the detector is positioned perpendicular to the outgoing light but tilted relative to the sample. This configuration leads to the nonuniform sampling of the reciprocal space signal on the Ewald sphere. Diffraction patterns appear significantly distorted compared to those under normal-incidence illumination. In conventional approaches, the degraded imaging performance in reflection ptychography is largely attributed to the insufficient correction of this distortion.

Fig. 1  Principle of full-pose self-calibration for table-top EUV reflection ptychography. a Schematic of table-top EUV reflection ptychography system with grazing illumination. Misalignment attributed to scan position, diffraction distance, illumination pose, and detection pose. b Flow work of proposed method. The optical encoding block represents a physical experimental setup, whereas the digital forward modeling block simulates how system parameters are combined to generate predicted diffraction patterns. Following loss evaluation using a feature-domain (FD) loss function along with regularization terms, the gradients of each variable are computed via auto-differentiation and back-propagated. Finally, variables are updated using an optimizer.

To address this issue, we describe the mapping relationship $ {\bf{T}} $ between the diffraction pattern distributed on the Ewald sphere and tilted detector plane as a function of the pose parameters of the reflection system, including diffraction distance $ d $, three-dimensional illumination angle $ {\boldsymbol{\theta}} = \left [\theta_{x},\theta_{y},\theta_{z} \right ]^{\top} $, and three-dimensional detection rotation angle $ {\boldsymbol{\gamma}} = \left [\gamma_{x},\gamma_{y},\gamma_{z} \right ]^{\top} $. A detailed derivation of the mapping operator $ {\bf{T}} $ is provided in the Materials and Methods section. As depicted by the digital forward modeling block in Fig. 1b, we further integrate this relationship into the forward model of ptychography.

where $ {\bf{y}} _{j} $, $ {\bf{I}} _{j} $, $ {\bf{A}} $, $ {\bf{P}} $, $ {\bf{x}} $, $ {\bf{Q}}_{j} $, $ {\bf{r}}_j $, and $ \Delta {\bf{r}}_j= [\Delta x_{j}, \Delta y_{j} ]^{\top} $ represent the $ j $th diffraction pattern in total $ N $ frames, virtual pattern on the Ewald sphere, propagation operator, probe function, sample function, translation operator, two-dimensional estimated scan position, and position error, respectively. The scan position error is integrated using the Fourier shift theorem⁴⁴. After modeling the full-pose integrated forward propagation, we use the feature domain (FD) loss^{45, 46} for data fidelity assessment between the measurement $ {\bf{y}} _{j} $ and prediction $ \mathbf{\tilde{y}} $.

where $ \nabla = \left [ \nabla_{x}, \nabla_{y}\right ]^{\top } $ represents the first-order differentiation operator. Further, $ {\left \| \cdot \right \|}_{Huber} $ represents the Huber norm⁴⁷, which is a hybrid loss function combining $ L_1 $ and $ L_2 $ components to balance robustness against outliers with optimization efficiency. Compared to the element-wise loss functions such as the mean squared error (MSE) used in traditional phase retrieval, the FD loss is less sensitive to model degeneration.

Prior knowledge can be incorporated through regularization terms to enhance reconstruction stability, including support and denoising constraints. The complete loss function is expressed as $ {\cal{L}}_{full}={\cal{L}}_{data}+\sum\nolimits_{i}w_{i}{\cal{L}}_{reg}^{i} $, where $ w_{i} $ represents the weighting coefficient of the $ i $th regularization term. The complete workflow of the proposed method is shown in Fig. 1b. The optical encoding block represents the physical measurement process in the actual system, whereas the digital forward modeling block implements computational mapping from variables to the predictions. The loss function evaluates and backpropagates through a computational graph for accumulating the gradients of each variable using automatic differentiation. Materials and Methods present more details about the implementation of the experimental algorithm, including the sampling conditions, regularization, and optimizer settings.

We conducted a series of investigations into method performance and system behavior after completing the forward modeling and inverse problem design. The following investigations provide profound algorithmic and physical insights into reflection ptychography, which offers essential guidance for robust algorithm designs and optimized experimental configurations:

I. We validate the effectiveness of the FD loss function used in ptychography. Simulations are performed under different illumination doses and source coherence conditions. The detailed implementation is provided in Section S1. As shown in Figs. S1 and S2, the FD-Huber loss successfully recovered images under moderate noise levels and coherence degradation, outperforming both MSE and FD-$ L_{1} $ loss. This conclusion is corroborated by experimental results shown in Fig. S3.

II. In Section S2, we explore the sensitivity of image reconstruction to different pose parameters under varying reflection angle settings with simulation results presented in Fig. S4. We find that misalignment in all parameters can degrade reconstruction quality. The sensitivity to misalignments in $ \theta_x $, $ \theta_y $, and $ \Delta {\bf{r}}_j $ increases significantly as the geometry approaches grazing reflection. This sensitivity analysis explains why grazing-reflection ptychography yields poorer image reconstruction quality compared to normal-incidence or small-angle illumination and underscores the necessity of full-pose self-calibration.

III. We identify coupling relationships among the parameters, which are discussed in detail in Section S3. Parameter coupling indicates that the error of one parameter can be compensated by adjusting other parameters. This effect is most pronounced between the pose parameters and scan positions. Consequently, we introduce a segmented updating optimization strategy to decouple parameters during self-calibration.

IV. We discuss the illumination-dose dependence of full-pose self-calibration. The simulation results in Section S4 indicate that full-pose self-calibration is effective when the illumination dose reaches or exceeds $ 10^{7} $ photons.

Our reflection ptychography system uses a table-top 17.9 nm EUV source based on high-order harmonic generation (HHG). Detailed information is provided in Section S7. The focused 17.9 nm beam illuminates the sample at a grazing angle, and a scientific complementary metal-oxide-semiconductor (sCMOS) camera (Dhyana XF95, 2,048 × 2,048, 11 µm pixel size) was placed perpendicular to the reflected beam. Incidence angles relative to the normal and sample-camera distances of the sample surface were 70° and 50 mm in the design, respectively. Sample scanning was driven by a two-dimensional slip-stick piezo stage (LS35x.Lab, MultiFields Technology).

In this experiment, we employed a Fermat spiral-scanning strategy with 400 positions⁴⁸ and scanning overlap ratio of $ \sim 82\% $. At each position, four patterns were recorded with increasing exposure times to enable high dynamic range image fusion. Data recording at each position took approximately 1 s, whereas stage movement and stabilization required approximately 1.2 s. Full data acquisition was completed within approximately 15 min.

The pseudocolor imaging results of a customized wafer sample (silicon substrate with a gold pattern; scanning electron microscope (SEM) image provided in Fig. S9) and corresponding probe functions are presented in Fig. 2a, b. The probe inherently carries a linear phase gradient along the horizontal direction because of the highly tilted illumination. However, in our forward model (mapping operator T), this linear phase term is equivalently represented as a translation of the wavevector coordinates of the Ewald sphere. Consequently, the reconstructed probe wavefront does not explicitly contain this phase component. The pixel size of the reconstructed image is 43.5 nm × 70.0 nm at a numerical aperture (NA) of 0.19 × 0.095. Table 1 lists posture parameter values under their default settings ($ d = 50 $ mm, $ \theta_{y} = 70{}^{\circ} $, and rest = 0) and after calibration using different strategies. Fig. 2c1, c5 show the overall outline of the Siemens star pattern reconstructed with full-pose, partial-pose, and no-pose calibrations. The outer diameter of the Siemens star pattern is 80 $ \mu $m, and the pattern includes five groups of radial gratings from the outside with a minimum line width of 30 nm.

Fig. 2  Imaging results of the customized wafer sample. a Full-field pseudocolor image of the reconstructed sample. b Pseudocolor image of the reconstructed probe. Amplitude (A) and phase information ($ \varphi $) correspond to the brightness and hue in the pseudo-color image, respectively. c1–c5 are magnified Siemens star patterns reconstructed with different calibration strategies; d1–d5 are amplitude profiles and the measured diameter along $ {\rm{D}}_{1} $ and $ {\rm{D}}_{2} $; e1–e5 and f1–f5 show magnified images of ROI1 and ROI2 in c1–c5; and amplitude profiles 1–3 are plotted in g–i. j1–j5 show magnified maze patterns (marked by purple dashed box in a) reconstructed with different strategies.

The outer diameters are measured in the horizontal (D1) and vertical (D2) directions for images reconstructed using different strategies (Fig. 2d1, d5). The reconstructed patterns exhibit severe compression in the horizontal direction when the scan position error ($ \Delta{\bf{r}}_{j} $) is calibrated without pose parameters or when no calibration is performed. This perspective distortion stems from inaccurate $ \theta{y} $ estimations, leading to incorrect sampling intervals in the reciprocal space. In contrast, calibrating pose parameters while neglecting $ \Delta{\bf{r}}_{j} $ partially alleviates the distortion. The full-pose strategy yields the smallest errors, with both D1 and D2 deviations below 1$ \% $, outperforming other strategies including the one in which $ d $ is fixed.

Fig. 2e1, e5 display the central region (ROI1) of the Siemens star pattern. Images reconstructed without full-pose parameters show artifacts and reduce contrast near the center (Fig. 2e2, e5). The corresponding amplitude profiles (No. 1 in Fig. 2g) and (Fig. 2h) confirm this observation. Figs. S9b and d in Section S6 provide a direct visual comparison of Siemens star patterns under an SEM and full-pose EUV pattern, which shows high consistency between two modalities. For quantitative comparison, we calculate the Tamura coefficient of the gradient (TOG), which is a metric that characterizes the clarity of the complex-field image within each ROI⁴⁹. TOG values align with the visual assessment, which confirms that full-pose reconstruction yield the sharpest results. However, grating features are not fully resolved even in full-pose reconstruction, and the horizontal gratings appear clearer than the vertical gratings. This discrepancy can be attributed to the anisotropic nature of the NA of the system. Profile No. 1 in Fig. 2h indicates that the full width at half-maximum (FWHM) of the measured minimum grating is 82.6 nm, which is approximately equal to 1.9 pixel sizes. Images of ROI2, which contain a larger line pattern with punctate defects, are shown in Fig. 2f1, f5. As demonstrated by the profiles in Fig. 2i, only the full-pose strategy successfully resolved the defects.

Fig. 2j1, j5 show enlarged views of the maze pattern, and a direct comparison with the SEM image is shown in Figs. S9 c and e. With full-pose self-calibration, the EUV image clearly resolved both the horizontal and vertical lines while presenting a flat background, thereby indicating high consistency with the SEM image. In contrast, all other strategies were severely contaminated by artifacts.

In EUV reflection ptychography, material-specific surface characterization is enabled by rich information in the complex-field imaging mode. The amplitude of the complex value corresponds to the reflectance at each pixel, whereas its phase angle reflects the phase change induced by the interface phase shift governed by Fresnel relations and surface topography of the sample. The quality of sample characterization depends on both the image reconstruction quality and precision of the system parameters.

We used the phase of the reconstructed results to segment the gold patterns and silicon substrate and calculate the height using the specific method provided in Section S5. Fig. 3a, b show the surface topography of the regions of number "1" and "2" under the full-pose and no-pose conditions, respectively. These regions correspond to the yellow dashed areas in Fig. 2a. The height map in Fig. 3a exhibits sharp well-defined edges, indicating clear segmentation between the silicon substrate and gold pattern. In contrast, Fig. 3b shows significant glitch artifacts and pattern adhesion, which can be attributed to inaccurate component segmentation caused by poorer image quality. In addition, height histograms are provided to the right of each height map. Both histograms display two distinct peaks corresponding to the heights of the gold pattern and silicon substrate. The distance between the two peaks represent the average height difference between the substrate and pattern, which are 37.53 and 36.86 nm for the full-pose and no-pose methods, respectively. Compared to the no-pose result, the peaks in the full-pose histogram are sharper, indicating clearer material differentiation. Fig. 3c shows the atomic force microscopy (AFM) measurement results and height distribution of the same sample region, with a peak-to-peak distance of 37.87 nm between the silicon and gold peaks. This height is closer to that measured by the full-pose method. Overall, we demonstrate that full-pose correction is critically important for surface topography characterization.

Fig. 3  Surface topography characterization. a, b Height maps in the area marked by a dotted square in Fig. 2a measured by EUV reflection ptychography with and without full-pose calibration. c Height maps measured by atomic force microscopy (AFM). Height histograms are provided to the right of each height map.

This system was used to inspect chip samples to demonstrate its practical utility. The sample height variation (within 100 nm) was well within the vertical measurement range of our EUV reflection ptychography system, making it ideally suited for characterization. An SEM image of the chip sample is presented in Fig. S10 in Section S6. The system posture varies because the reinstallation of the clamping mechanism is required for sample replacement, and system hardware such as the camera is reinstalled between the two experiments. Consequently, the system parameters are reconstructed synchronously with the image during the reconstruction process. Parameters with a full-pose calibration and their default values (no pose) are listed in Table 2. Full-field pseudocolor images of the chip sample and illumination probe are shown in Fig. 4a with full-pose calibration, and the non-pose reconstructed image is shown in panel b.

Fig. 4  Inspection on the chip sample. a Full-field pseudocolor result of the chip sample with full-pose calibration and probe image displayed in the bottom-right corner. b Full-field pseudocolor result with default parameters (no-pose calibration). c, d and e, f show magnified views of regions ROI 1 and ROI 2, respectively, reconstructed using full-pose calibration and default parameters (no-pose calibration). The SEM image of the grating area is used as the background of profile 1 in c for reference.

Edge regions exhibited reduced brightness and sharpness compared to that that the center because of the low detection intensity at the FOV periphery. However, the overall structure remained discernible. We magnified the two regions of interest (ROIs) as shown in Fig. 4c, f. Images in Fig. 4c, e correspond to results obtained using the full-pose strategy, whereas those in Fig. 4d, f correspond to results from the default parameter set (no pose). In ROI1, the chip substrate, grating structures, and contaminants are represented by blue, green, and purple regions, respectively, indicating significant phase variations among these features. In the no-pose result, the boundaries of the purple contaminant regions are noticeably ambiguous, and the green grating structures in the lower right appear blurred. A comparison of Fig. S10b, c shows that the contamination area in the EUV image with full-pose self-calibration is highly consistent with that in the SEM image. Magnified view in the dashed box highlights the details of the grating structure. Profile 1 shows an amplitude distribution perpendicular to the grating orientation. In the full-pose result, the grating exhibits high contrast (0.49) with the smallest amplitude change corresponding to one pixel (43.5 nm). Fig. S10b shows an actual grating linewidth of 42.7 nm. The grating area of the SEM image is cropped and used as the background for Profile 1 in Fig. 4c for reference. Although sampling aliasing causes some local contrast loss and slight shifts in peak-valley positions, the EUV ptychography profile accurately matches the grating structure observed in the SEM image. This confirms that the system resolution (half-pitch) reaches a diffraction limit of 43.5 nm. In contrast, the profile at the same location in the no-pose condition shows a significantly lower contrast (0.13).

We measured the distances between adjacent grating regions (marked by solid yellow lines) and obtained values of 2.22 and 2.01 µm using the full-pose and no-pose methods, respectively. Compared with the ground truth of 2.26 µm provided by SEM, the relative errors were 1.9% and 7.7%, respectively. ROI2 was located at the edge of the FOV, where the number of reflected photons was significantly lower than that of ROI1. In Fig. 4f, scratches and grating structures are overwhelmed by noise, whereas in Fig. 4e, calibrated parameters significantly improve the SNR, which makes both scratches and grating clearly distinguishable. We conducted a full field image quality assessment of the reconstructed image of the chip to demonstrate the improvement achieved through full pose self calibration. Each image was evenly divided into 200 sub regions using a 10 × 20 grid, as indicated by the dashed lines. The TOG was calculated for every sub region except for those in the outermost border. The resulting TOG values are presented as heatmaps in Fig. 5a, b. Visually, the TOG values of the full pose reconstruction are significantly higher across the entire field of view compared to those of the no pose reconstruction. The statistical values of the segmented full-field TOG such as the mean are shown in the figures. This blockwise full field evaluation effectively demonstrates the enhancement in image quality caused by full pose self calibration. Our method with full-pose self-calibration enhances the overall SNR of imaging and achieves the best resolution reported to date. Further, the higher-order diffraction signals from the chip sample are stronger than those from the Siemens star sample. The resolution in diffraction imaging depends not only on the NA of the system, but also critically on the SNR of the diffraction signals.

Fig. 5  Block-wise full-field image evaluation of the chip sample. a and b show the block-wise TOG map corresponding to the reconstructed results with full-pose and no-pose calibration, respectively.

Unambiguous and unique results are critical for industrial applications, necessitating self-calibration methods that deliver consistent calibration performances across varying conditions. Tests were performed on customized wafer sample data across nine independent runs for the evaluation. Convergence curvatures for different pose parameters are plotted in Fig. 6b, while panel Fig. 6a describes system components that respond to the poses. For each run, initial estimates for $ d $ and $ \theta_{y} $ were selected randomly from the ranges of 48–50 mm and 70–73 degrees, respectively. In general systems, random initialization is not performed, and the other parameters default to zero.

Fig. 6  Consistency test of full-pose self-calibration. a System components corresponding to different pose parameters. b1–b4 Convergence of full-pose parameters in nine runs of the algorithm with random initialized $ d $ and $ \theta_{y} $. b1 Convergence curvature of the illumination pose parameters $ \theta_{x}, \theta_{y} $, and $ \theta_{z} $ b2 Corrected position error $ \Delta{x} $ and $ \Delta{y} $ of 400 positions. The gray bars indicate the mean value and pink bars represent the corresponding variation. b3 Convergence curvature of diffraction distance $ d $. b4 Convergence curvature of detection pose parameters $ \gamma_{x} $, $ \gamma_{y} $, and $ \gamma_{z} $.

The results confirm that parameter updates exhibit similar convergence trends across runs, yielding final values of $ \theta_{x}=-0.70\pm0.12 $°, $ \theta_{y}=71.74\pm0.11 $°, $ \theta_{z}=0.45\pm 0.10 $°, $ d=48.32\pm0.17 $ mm, $ \gamma_{x}=9.25\pm1.3 $°, $ \gamma_{y}=2.28\pm 0.34 $°, and $ \gamma_{z} $ consistently below 0.1°. The update behavior of most parameters changed significantly before and after enabling scan position correction at the 1000th epoch. For example, $ d $ converges to approximately 49.5 mm before epoch 1,000, differing from its final convergence value. This reflects the coupling effect between pose parameters and scan position as discussed in Section S3, which justifies our segmented updating strategy. The corrected position errors ($ \Delta{x}_{j} $, $ \Delta{y}_{j} $) are presented in Fig. 6b, and the gray and pink bars represent the mean value and variation range of the position error across the nine runs. This visualization confirms that the scan position correction is consistent across the nine runs. In conclusion, our full-pose self-calibration method demonstrates strong stability with respect to initial parameter states and consistently converges to uniform calibration results across various scenarios.

We developed a table-top EUV reflection ptychography system for the simultaneous self-calibration of critical system parameters including the diffraction distance $ d $, three-dimensional illumination angle $ {\boldsymbol{\theta}} $, three-dimensional detection postures $ {\boldsymbol{\gamma}} $, and scan position. This framework demonstrates excellent imaging performance on both customized wafers and real chip samples characterized by robust and consistent self-calibration. The previously reported EUV reflective ptychography systems and their characteristics are summarized in Table 3. For the first time, we integrate the full-pose parameters of a reflection ptychography system into a unified self-calibration framework and demonstrate the universality of our imaging system using both simple (customized wafer) and complex (real chip sample) specimens. To provide an intuitive reference, Fig. 7 juxtaposes representative imaging results from the literature (drawn from the studies compared in Table 3) with those obtained in this study. We acknowledge that a direct quantitative comparison is challenging because of the differences in experimental hardware (e.g., driving laser sources and detectors) and reconstruction software. However, the visual comparison in Fig. 7 clearly demonstrates the significant advancements achieved by the proposed method.

Fig. 7  Representative imaging results of reported reflection ptychography systems based on EUV source. a1 and a2 show the full-field results with full-pose calibration in this work. b1 and b2 show the pseudocolor imaging results of a Siemens star sample without and with tilt-angle correction adapted from Senhorst et al. Opt. Express 32, 44017-44030 (2024)⁴¹.Copyright 2024 Author(s), licensed under a Creative Commons Attribution 4.0 License. c1 and c2 are pseudocolor imaging results adapted from Shao et al. Light Sci. Appl. 13, 196 (2024)³⁵. Copyright 2024 Author(s), licensed under a Creative Commons Attribution 4.0 License. d1–d3 are the amplitude and phase results adapted from Tanksalvala et al. Sci. Adv. 7, eabd9667 (2021)³³. Copyright 2021 Author(s), licensed under a Creative Commons Attribution 4.0 License. e1 and e2 show the amplitude and phase results adapted with permission from Poter et al. Optica 4, 1552-1557 (2017)³². Copyright Optica Publishing Group. f1 and f2 show the amplitude and phase results adapted from Zhang et al. Ultramicroscopy 158, 98-104 (2015)³¹. Copyright 2015 Author(s), licensed under a Creative Commons Attribution 4.0 License. g1 represent the pseudocolor result adapted with permission from Seaberg et al. Optica 1, 39-44 (2014)³⁰. Copyright Optica Publishing Group.

The successful implementation of this approach relies on two elements: comprehensive modeling of the physical system and advanced computational techniques, which include specialized loss functions and robust optimizers. Modeling a full set of system parameters enables synthetic data to progressively approximate real-world physical systems, reducing discrepancy between simulated and actual data. This advancement stems from deeper physical insights and continuous refinements in numerical methods. However, well-designed loss functions and optimizers enable algorithms to maintain strong performance even when synthetic and real-world data cannot be aligned perfectly. The former approach aims to minimize mismatches, whereas the latter enhances robustness against such discrepancies. Our study demonstrates that combining both strategies leads to superior outcomes. This intelligent computing paradigm bridges the gap between physical principles and numerical simulations, paving the way for transformative progress in practical implementations of complex imaging systems. Our research not only advances EUV imaging capabilities but also inspires adaptations across broader imaging modalities and spectral regimes.

Compared to established techniques such as SEM and AFM, a key strength of EUV reflective ptychography is the chemical sensitivity inherent in its phase information. However, unlocking this capability requires angular scanning to acquire the phase-angle spectrum of the sample. Although experiments reported here were performed at a single angle, and thus, they do not fully exploit the material sensitive potential of the technique. We expect that the calibration strategy proposed in this work will still substantially improve the analysis efficiency and accuracy in multiangle scanning scenarios. This stems from the computational efficiency of the automatic differentiation based correction framework and high fidelity imaging enabled by full pose self calibration.

However, practical deployment in online settings is limited by the computational speed of phase retrieval algorithms. Gradient based optimization schemes converge more slowly than the projection based methods. Therefore, future work should focus on integrating full pose learning into accelerated projection based frameworks while leveraging graphics-processing-unit (GPU) accelerated parallel computing via compute unified device architecture. Such technical advances would enhance computational throughput, paving the way for EUV reflection ptychography to fulfill its potential in industrial metrology.

The mapping operator T transforms the pattern from an Ewald sphere into a real detection plane via fast interpolation. We establish the transformation relationship of the sampling grid from the real camera coordinate system ($ x_{rc} $, $ y_{rc} $, $ z_{rc} $) to the sample coordinate system ($ x_{s} $, $ y_{s} $, $ z_{s} $), as shown in Fig. 8 a. In a well-aligned setup, the camera should be orthogonal to the light emitted from the sample; however, achieving this in practice is challenging. We represent the sampling point $ {\bf{p}} $ in the actual camera coordinate system as $ [{\bf{p}}]_{rc}=[x, y, 0]^{\top} $. Considering possible three-dimensional rotation of the camera, its representation in the orthogonal camera coordinate system ($ x_{oc} $, $ y_{oc} $, $ z_{oc} $) is $ [{\bf{p}}]_{oc}={\bf{R}}_{{\boldsymbol{\gamma}}} [{\bf{p}}]_{rc} $, where $ {\bf{R}}_{{\boldsymbol{\gamma}}} $ represents the coordinate rotation operator as

Fig. 8  Sampling grid transformation in the mapping operation T. a Schematic of tilt detection in reflection ptychography. b Grid transformation from real camera plane to Ewald Sphere. The parameters are set to clearly demonstrate the effect of grid warping: wavelength = 17 nm, camera pixel size D = 2,200 $ \mu $m, diffraction distance d = 50 mm, $ \theta_{y}=70 $°, and grid size = 16 × 16. c Uniform source and warped target grids on Ewald Sphere. The wavevector range of the source grid is about [−120, 120] rad/$ \mu $m in both horizontal and vertical directions. d Zoom-in region marked by the dotted box in c illustrates the procedure of 2D interpolation and parameters used to calculate the Jacobian matrix. The Jacobian value at the ($ i $,$ j $)-th sampling point is determined by the coordinates of its four neighboring sampling points. e1–e6 show grid settings with different pose parameters.

Further, we represent $ {\bf{p}} $ using the global sample coordinate system. Given distance $ d $ between the origins of coordinate systems ($ x_{oc} $, $ y_{oc} $, $ z_{oc} $) and ($ x_{s} $, $ y_{s} $, $ z_{s} $), and a three-dimensional transformation with angle $ {\boldsymbol{\theta}} $, the representation of the sampling point in the sample coordinate system is $ [{\bf{p}}]_{s}={\bf{R}}_{{\boldsymbol{\theta}}} ([{\bf{p}}]_{oc} + {\bf{d}}) $, where $ {\bf{d}} $ = [0, 0, $ d $]$ ^{\top} $. The coordinate relationship for far-field diffraction is established between the internal coordinates of the sample and reciprocal wavevector coordinates. Therefore, we transform the camera point $ {\bf{p}} $ into the corresponding wavevector coordinate $ {\bf{k}}=k_{0} [{\bf{p}}]_{s}/|[{\bf{p}}]_{s}| $, where $ k_{0}=2\pi/\lambda $. The probe illuminates the sample at a grazing angle, and therefore, the tilt phase causes a shift in the diffraction pattern during the Fourier transform. We add the illumination wavevector $ {\bf{k}}_i=[{\rm{cos}}\theta_{x}{\rm{sin}}\theta_{y}, {\rm{sin}}\theta_{x}, {\rm{cos}}\theta_{x}{\rm{cos}}\theta_{y}]^{\top} $ in the wavevector coordinate transformation to counteract this shift: $ {\bf{k}}=k_{0} [{\bf{p}}]_{s}/ |[{\bf{p}}]_{s}| + {\bf{k}}_{i} $. In summary, the coordinate transformation for a sampling point can be expressed as

where unit represents vector normalization $ \cdot $/$ |\cdot| $. When the sample thickness is negligible, we require only the first two dimensions ($ k_{x} $ and $ k_{y} $) of $ {\bf{k}} $ to construct a two-dimensional sampling grid. We can derive the warped grid on the Ewald sphere using Eq. 4 by denoting the sampling grid of the real camera $ G_{rc} $; we call this the target grid $ G_{t} $. An illustration of the grid transformation from $ G_{rc} $ to $ G_{t} $ is shown in Fig. 8b. On the Ewald sphere, we have a uniform source grid $ G_{s} $, as shown in Fig. 8c, which is determined by the sampling conditions on the sample plane. Thus, the complete mapping operator $ {\bf{T}} $ can be represented as

where ITP is a two-dimensional (2D) interpolation operator, $ {\bf{y}} $ and $ {\bf{I}} $ represent the images on tilt detection plane (target) and Ewald sphere (source), and $ {\rm{det}}J $ represents the determinant of the Jacobian matrix defined by $ {\rm{det}}{J}=\dfrac{\partial k_{x}}{\partial {x}}\dfrac{\partial k_{y}}{\partial {y}}-\dfrac{\partial k_{x}}{\partial {y}}\dfrac{\partial k_{y}}{\partial {x}} $. To avoid the dimensionality issue and cumbersome differentiation in the above equation, we numerically calculate the determinant using $ G_{s} $ and $ G_{t} $ as

where $ \Delta_{x} $ and $ \Delta_{y} $ represent the center difference operations along the x- and y-directions, respectively, and $ \delta k_{x_{0}} $ and $ \delta k_{y_{0}} $ represent the sample intervals of the uniform grid $ G_{s} $. These parameters used to calculate the Jacobian matrix are shown in Fig. 8d. The multiplication operation with $ |{\rm{det}}J| $ ensures the conservation of energy between the source $ {\bf{I}} $ and target $ {\bf{y}} $. Based on the above derivation, we can easily present the sampling grid settings for different pose parameters. Fig. 8e1, e6 show the distributions of the source and target grids on the Ewald sphere for the different pose parameters.

In our forward model incorporating the coordinate mapping operator, sampling on the virtual detection plane requires an explicit definition. The coordinates of the virtual plane are represented in the wave vector $ k $ space. We employ an anisotropic grid with a sampling interval and dimensions $ \delta{\bf{k}} $, $ {\bf{M}} $ to accurately match the distorted diffraction patterns in the reflection geometry: $ \delta{\bf{k}} = 2{\text{π}} D\left[\cos\theta_{x_0},\ \cos\theta_{y_0}\right]^{\top}/(\lambda d) $, $ {\bf{M}} = \left[m,{\rm{round}}(m\cos\theta_{x_0}/ 2\cos\theta_{y_0})\right]^{\top} $ where $ D $, $ m $ and $ \theta_{x_0}=0^\circ $/$ \theta_{y_0}=70^\circ $ represent the camera pixel size, vertical dimension of the raw diffraction patterns, and default illumination angles, respectively. The resulting object-plane sampling interval is $ 2{\text{π}}/({\bf{M}} \odot \delta{\bf{k}}) $, which is defined as the standard sampling $ \delta{\bf{p}}_{{\rm{st}}} $. Such sampling results in a numerical aperture ratio of 2:1 between the y- and x-directions on the virtual plane, while the numerical aperture in the y-direction is consistent with the physical numerical aperture of the system in the y direction.

Three regularization losses are applied beyond the data fidelity term: (1) Probe support: $ \left\| {\bf{M}}_s \odot {\bf{P}} \right\|_2^2 $ with binary mask $ {\bf{M}}_s $, centering the probe and suppressing peripheral artifacts; (2) Total variation: $ \left\| \nabla {\bf{x}} \right\|_1 $ for image edge enhancement and noise suppression; (3) Pattern constraint: $ \displaystyle\sum\nolimits_j \left\| {\bf{M}}_p \odot {\bf{I}}_j \right\|_2^2 $, where $ {\bf{I}}_j $ represents the virtual-plane predicted pattern and $ {\bf{M}}_p $ represents a binary mask constraining the energy in unmeasured regions.

Our algorithm is implemented using the open-source Python package PyTorch and executed on a server with Nvidia A100 GPU. In the experiment, high-resolution reconstruction follows a three-stage framework with low resolution (LR), medium resolution (ML), and high resolution (HR) stages. In the LR stage, the images and system parameters are reconstructed using 512 × 512 data size with 2,000 epochs. All pose parameters ($ d $, $ {\boldsymbol{\theta}} $ and $ {\boldsymbol{\gamma}} $) are updated at the 200th epoch, whereas the scan positions remain fixed until the 1,000th epoch. In the ML and RL stages, the variables are further refined to data sizes of 1,024 × 1,024 and 1,900 × 1,900, each with 50 epochs. Camera edge pixels are discarded in the HR stage because of elevated noise levels. We employ the Adam optimizer⁵⁰ with 10 batches per iteration throughout, resetting the first-order and second-order momenta whenever new variables are activated. The learning rates for the variables and regularization weighting coefficients are listed in Tables 4, 5. The pose parameters do not significantly change during the ML and HR stages, indicating that they were successfully corrected during the LR stage.

National Key Research and Development Program of China (Grant No. 2021YFB3602604), National Natural Science Foundation of China (62131011), and Fundamental Research Funds for Central Universities (E3ET6201X2). The authors thank the technical support from Hongbo Xie, Zefan Huang, Guanbo Wang, Chen Li, Xingwang Xie, Huaqiu Liu, Xiaohui Zhang, Haoyu Wu, Peixiang Xiong, Tianlei Jia and Tianhao Zhang.