Hard X-ray projection imaging below 5 nm resolution

Hard X-ray projection imaging below 5 nm resolution | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Hard X-ray projection imaging below 5 nm resolution Wenhui Zhang, Jan Lukas Dresselhaus, Holger Fleckenstein, Mauro Prasciolu, and 12 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4093473/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract High-resolution X-ray imaging of non-crystalline objects is often achieved through the approach of scanning coherent diffractive imaging known as ptychography. The imaging resolution is usually limited by the scattering properties of the sample, where weak diffraction signals at the highest scattering angles compete with background noise and parasitic scattering. Here we demonstrate that X-ray multilayer Laue lenses of high numerical aperture (NA) can be used to create a strong reference beam that holographically boosts that weak scattering from the sample over a large range of scattering angles, enabling high-resolution imaging that is tolerant to such noise. An imaging resolution of about 5 nm was achieved at a photon energy of 17.4 keV with lenses of 0.014 NA from a Siemens star test object and a sample of hierarchical nanoporous gold, recording projection holograms at a magnification of more than 30,000 directly on a pixel-array detector. A numerical study compared this approach to far-field ptychography, indicating significant advantages for using high-NA lenses in the presence of background noise. This imaging modality is particularly fast and efficient to record high-resolution transmission phase-contrast images over large fields of view in a facile manner. Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Introduction High-resolution characterization of materials is essential for the understanding and improvement of functional and active materials, such as catalysts, solar cells, batteries, storage media, and integrated circuits. Imaging of such materials at the nanometer scale is required to understand relationships between microstructure and various mechanical, electronic, or magnetic properties. The rise in the brightness of synchrotron radiation sources over the last several decades has brought extensive developments 1–8 in X-ray imaging and tomography in the photon energy range of about 5 keV to 20 keV to fulfil this need. Given the weak interaction of X-rays with matter in this spectral range, phase-contrast and diffraction-based imaging methods offer the highest sensitivity for the investigation of nano-structures in bulk samples (thicker than around 1 µm). However, for this same reason, it is difficult to efficiently deflect X-rays by significant angles, which has limited the creation of high-resolution lenses. To overcome this restriction, the method of ptychography is often used 3 . By measuring diffraction patterns from several localized and overlapping regions of the sample as it is stepped across a focused beam, it is possible to recover the complex-valued exit wave of the sample at a resolution dependent on the angular extent of the measured diffraction rather than the numerical aperture (NA) of the focused x-ray beam (the probe). In this way, a full-period resolution of 10 nm has been obtained at a soft X-ray wavelength of 1.7 nm with a probe size of 150 nm (ref. 9), and 20 nm resolution was accomplished at 2 nm wavelength—35 times better than the information limit of the probe (700 nm in size) 10 . The imaging resolution in these examples was ultimately limited by the strength of the diffraction signal. This is generally the case, since the diffraction intensities tend to decrease with the fourth power of the resolution length d (ref. 11). Improving the resolution requires ever increasing incident fluence and hence radiation dose to the sample (which it might not be able to tolerate) 12 . However, it is often not possible to extract the full potential signal for a given dose since the weakest diffraction (at highest scattering angles) may be dominated by background noise such as scattering from air, beamline slits, or the focusing optic itself. Thus, the high resolutions mentioned above were accomplished with samples of high scattering power and with measures to carefully eliminate all sources of noise. Furthermore, the diffraction signals usually vary over a dynamic range of many tens of thousands—from single photons to that of the direct beam—often requiring multiple exposures. Thus, the gain in resolution achieved by X-ray ptychography usually comes at a higher cost of imaging speed, dose, and radiation damage than ultimately necessary. With the recent development and use of multilayer Laue lenses (MLLs) 13,14 , it has become possible to focus hard X-rays with a high convergence angle (or NA) to give diffraction-limited spot sizes smaller than about 4 nm (ref. 15). These diffractive lenses are fabricated layer by layer by magnetron sputtering—a process in which errors in layer placement can accumulate to result in aberrations that may be several waves in magnitude 15 . Although such errors can be compensated, for example by use of a refractive corrector 16,17 they do not necessarily restrict the large angular divergence of the probe beam. In a ptychography measurement, this large divergence overcomes the problem of high dynamic range and sensitivity to weak diffraction signals by providing a strong zero-order reference beam that interferes coherently with the scattering from the sample. When placed in the focal plane, the sample would need to be stepped by distances that are even smaller than the probe size to obtain a ptychographic dataset. This leads to datasets with a large number of steps of the scanned object, coupled with modest requirements of the number of pixels required to record the diffraction pattern 18 . A certain field of view can be covered in far fewer steps (and detector frames) by placing the sample considerably out of focus. In this case, the diffraction pattern must be sampled more finely to match the larger width of the probe. However, given that direct-detection pixel array detectors with large pixel numbers are widely available for X-rays, this approach—known as near-field ptychography 19 may have practical benefits of speed and less demanding requirements for the sample scanning hardware. When a weakly-scattering object is placed out of the focus of a converging beam, a magnified projection image is formed in the far field of that focus, with a magnification equal to the ratio \(M=L/{\Delta }f\) of the distance from focus to detector, \(L\) , to the defocus distance \({\Delta }f\) . It is well known that this projection image is equal to a magnified version of the in-line hologram that would be obtained by illuminating the object with a plane wave, measured at approximately the defocus distance 20–22 , as shown schematically in Fig. 1 (a). It is also equivalent to the defocused coherent image recorded in a transmission X-ray microscope under plane-wave illumination and an objective of the same NA 23 . In all three cases, the image is equal to the Fresnel transform of the transmission function of the object. For small defocus, the contrast of such images of pure phase objects increases with \({\Delta }f\) (ref. 24). Returning to the holographic interpretation of the projection image, the reference wave is provided by the beam diverging from the focus and which does not interact with the object. Obviously, at the detector, this reference wave \(R\) only extends to scattering angles covered by the diverging beam, as given by the NA of the objective lens. Within this range, the interference between \(R\) and the diffraction \(O\) from an object can be considered as \({\left|O+R\right|}^{2}={\left|R\right|}^{2}+2\left|O R\right|\text{cos}\varphi +{\left|O\right|}^{2}\) . The relative phase \(\varphi\) between the object and reference is encoded in the hologram. In the hard X-ray regime, most (microscopic) objects are weakly scattering such that the reference-beam intensity \({\left|R\right|}^{2}\) is many thousands of times greater than the far-field diffraction signal \({\left|O\right|}^{2}\) , which can therefore be neglected in the region of the reference. It is seen in the term \(2\left|O R\right|\text{cos}\varphi\) that the diffraction signal is greatly amplified by the strength of the reference amplitude. For complete constructive or destructive interference ( \(\varphi =0\) or \(\pi\) ), this boost factor is equal to \(2\left|O R\right|/{\left|O\right|}^{2}=2\left|R\right|/\left|O\right|\) and the contrast of the hologram is \(2\left|O\right|/\left|R\right|\) . Both the non-holographic diffraction signal \({\left|O\right|}^{2}\) and the boosted holographic signal \(2\left|O R\right|\text{cos}\varphi\) exhibit Poisson noise. When photon counting is the only noise source, the advantage in signal to noise with and without the reference is actually only a factor of \(2\text{ cos}\varphi\) , with a root-mean-square value of \(\surd 2\) . Although studies have shown that coherent mixing of the signal with a strong reference does not necessarily improve the signal to noise ratio of reconstructed images 25 , the signal boost and the large offset \({\left|R\right|}^{2}\) will dominate any extraneous scattering. Additionally, compared to the diffraction signal \({\left|O\right|}^{2}\) without a reference, the boosted (low-contrast) holographic signal covers a much lower dynamic range, easing the requirements of the detector, and the in-line hologram generally requires a lower degree of spatial coherence to encode the transmission function than does the far-field pattern 26 . There have been several challenges in realizing high-resolution X-ray imaging in the holographic regime. One has been the production of a strong reference that covers a suitably large range of scattering angles. Objects such as gold particles 27 , pinholes 28,29 , and aerogel 30 have been used or suggested. These tend to provide reference beams that are not much stronger than the scattering from the object, and which vary considerably in intensity. A strong reference was created by the focused beam of a soft X-ray zone plate 31 although the usable angular range was restricted by the annular pupil. Another challenge is how best to reconstruct the transmission function of the object (its exit wave) from the hologram, and in particular how to overcome the twin-image problem for the in-line geometry. Near-field ptychography 19 using MLLs overcomes both these challenges, although when framed as a series of holographic images recorded for various shifts of the object relative to the illumination, it is not immediately clear how this approach recovers the object’s transmission function. For a perfect (unaberrated) wavefield, stepping the object across the beam at a fixed defocus would simply shift the projected hologram across the illuminated region of the detector without providing new information. For a perfect spherical wave, only changing defocus gives the needed diversity 24,31 . A structured wavefront created by a diffuser enables image reconstruction without changing focus 19 . The illumination phase at any point \({r}_{0}\) can be approximated through a truncated Taylor expansion as \(\phi \left(r\right)= {\phi }_{0}\left({r}_{0}\right)+\left(r-{r}_{0}\right) {\phi }_{1}\left({r}_{0}\right)+{\left(r-{r}_{0}\right)}^{2} {\phi }_{2}\left({r}_{0}\right)\) . As a feature in the object is moved from point to point, variations of \({\phi }_{2}\) lead to the needed change of the illumination condition. In our experiments using a pair of MLLs, we find that the inherent low-order wavefront errors, as well as higher-frequency errors due to sub-ångström variations in lens layer thicknesses 32 , adequately provide the necessary diversity in the illumination beam, without generating zeroes in the far-field reference, and no additional diffuser is required. Finally, the fact that near-field ptychography records holograms, rather than far-field diffraction patterns, means that the lateral object positions in the scan are evident in the holograms themselves, providing the means to account for errors due to incorrect stage motion. Here, we evaluate the performance of high-resolution imaging via near-field ptychography using high-NA MLLs by quantifying the resolution of a Siemens star test object and of a hierarchical nanoporous gold structure, demonstrating imaging over large fields of view with reasonable efficiency. A numerical study is then made to compare this scheme to ptychography with a low-NA probe where the high-resolution information is recorded as “dark-field” scattering beyond the angular extent of the reference beam. Results A schematic diagram of our projection imaging experiment is shown in Fig. 1 (b), where a pair of MLLs with NAs of 0.014 (both horizontal and vertical) focused an X-ray beam of 0.071 nm wavelength, recorded on a pixel-array detector with 75 µm pixels, placed L = 2.37 m downstream of the focus. We first characterized the high-NA near-field ptychography performance by using a Siemens star test object, made of gold of 750 nm thickness on a silicon nitride membrane. The object was placed at a defocus distance of 200 µm where the beam width was 5.6 µm, giving a magnification of 11,850 corresponding to a 6.3 nm pixel size in the object space. A ptychographic dataset of 21 × 21 points was obtained with a step size of 0.3 µm in each direction, giving an overlap of 94.6% between neighboring holograms and an overall field of view of about 11.6 µm. The exposure time per step was 1 s, giving 497 photons per pixel per frame within the lens pupil at the detector and a highest scan-accumulated exposure of the object of 4470 ph/nm 2 in the central region of the scan (of width 630 nm). One of the recorded projection holograms, uncorrected for the reference beam intensity, is shown in Fig. 1 (b). Starting from random initial guesses for both the probe wavefront and object transmission, maps of these functions were recovered using a schedule of iterative phasing algorithms as described in “Materials and Methods”. The results are shown in Fig. 2 including the 5.6-µm wide incident wavefront at the sample plane in Figs. 2 (c) and 2(d). Although this wavefront aberration spans many waves, resulting in a point spread function at the focal plane 15 that is wider than the diffraction-limited spot size (equal to \(\lambda /\text{N}\text{A}= 5.1 \text{n}\text{m})\) , this does not hamper the image resolution obtained. The innermost spokes of the Siemens star object can clearly be resolved in Figs. 2 (a) and (b)—evidently the resolution is much better than the smallest spoke width of 50 nm and may be limited only by the Nyquist frequency (twice the pixel size, i.e., 12.6 nm). To obtain quantitative estimates of the resolution, we computed it via three different criteria: the edge criterion, which is a measure of the edge width 33 , the Fourier ring correlation (FRC) 34 , and the phase retrieval transfer function (PRTF) 35,36 . Figure 3 shows edge profiles of spokes, between the two rings, from the phase image of the Siemens star shown in Fig. 2 (a) along four different directions: vertical, horizontal and the two orthogonal directions at 45º to the horizontal. The widths listed in the figure were determined by the distance between points corresponding to 25% and 75% of the total step height of the fitted error function (erf). It is seen that the widths along the four directions are different. This may indicate an anisotropic resolution, but it is more likely a measure of the actual projected transmission function of the spokes, which are much thicker than they are wide. Scanning electron micrographs of the spokes have shown that the walls of the spokes are not all parallel, and the plane of the object might not have been exactly normal to the beam. The average of the widths along the four directions was 12.6 nm, and the smallest was 7.1 nm. To calculate the FRC, we first split the full dataset into two half-datasets by selecting every other frame of the full dataset. With these two half-datasets, each with double the step size in the fast-scan (vertical) direction, we obtained two independent reconstruction results, A and B , each with an average incident fluence of 1200 ph/nm 2 . We smoothed the edges of each image, and computed the FRC as 34 , $${\text{F}\text{R}\text{C}}_{\text{h}\text{a}\text{l}\text{f}}\left(q\right)=\frac{\sum {F}_{A}\left(q\right){F}_{B}^{*}\left(q\right)}{\sqrt{{\sum \left|{F}_{A}\left(q\right)\right|}^{2}{\sum \left|{F}_{B}\left(q\right)\right|}^{2}}}$$ 1 where \({F}_{A}\left(q\right)\) and \({F}_{B}\left(q\right)\) are the Fourier transforms of these two images as a function of the spatial frequency \(q=(\text{sin}\theta )/\lambda\) , and * denotes the complex conjugate. Perfect imaging would give unity correlation over the full spatial frequency range. In practice, the FRC decreases with increasing q because high spatial frequency components generally have a lower power and are thus more easily affected by noise 11 . Figure 4 (a) shows a plot of FRC half in green, giving a half-period resolution of 9.2 nm by the half-bit criterion 34 . However, we noticed that the image reconstructed from the full dataset appears to have a better resolution than either of those from the half datasets. Even though two independent full datasets are not available, the FRC of the full dataset (with an average fluence of 2400 ph/nm 2 ) can be estimated from the so called “CC-star” formula 37 as $${\text{F}\text{R}\text{C}}_{\text{f}\text{u}\text{l}\text{l}}=\sqrt{\frac{2{\text{F}\text{R}\text{C}}_{\text{h}\text{a}\text{l}\text{f}}}{1+{\text{F}\text{R}\text{C}}_{\text{h}\text{a}\text{l}\text{f}}}}$$ 2 . This metric, plotted in purple in Fig. 4 (a), indicates a half-period resolution of 6.5 nm by the half-bit criterion, a value that is consistent with the edge criterion estimation. Finally, we used a modified version of PRTF to evaluate the stability of the iterative phase retrieval process, which is defined as follows, $$\text{P}\text{R}\text{T}\text{F}\left(q\right)=\left|⟨\text{e}\text{x}\text{p}\left[i\phi \left(q\right)\right]⟩\right|=\left|⟨\frac{G\left(q\right)}{\left|G\left(q\right)\right|}⟩\right|$$ 3 , where \(G\left(q\right)\) is the Fourier transform of the reconstructed phase image and \(⟨\dots ⟩\) represents the average over a large number of independent reconstructions. If the Fourier phases fluctuate, the PRTF tends to zero and if they are stable the PRTF tends to one. Figure 4 (b) shows the radial average of the PRTF, obtained from 18 independent reconstructions. We can see that even at the highest spatial frequency of 80 \({\mu }\) m − 1 , the PRTF value is about 0.4, indicating a stable phase retrieval process over the entire frequency range of the phase image. Together, the three resolution criteria are consistent with a half-period resolution that is not appreciably worse than the Nyquist sampling obtained with the particular choice of magnification of 11,850 (giving an image pixel width of 6.3 nm). Next, we imaged a nanoporous gold (NPG) sample with a hierarchical porous structure that confers a high specific strength and low density 38 . The finest pores have a width of about 10 nm. A 1-µm wide column of this material was created by focused ion-beam milling—an SEM image of the object is shown in Fig. 5 (a). A near-field ptychographic dataset was recorded at an even higher magnification factor than the Siemens star object, of 32,470 obtained at a defocus distance of 73 µm, giving a sampling size of 2.3 nm in the object space over a hologram width of 2.0 µm. The dataset was acquired by stepping the object in a grid consisting of 3 × 21 locations (horizontal × vertical) with step sizes of 0.25 µm and 0.1 µm in the horizontal and vertical directions, and recording holograms with an exposure time of 5 s at each position, giving 2240 photons per pixel per frame within the lens pupil at the detector and a highest scan-accumulated exposure of 26,800 ph/nm 2 at the object plane. The recovered phase image of the NPG object is shown in Fig. 5 (b). The small pores in the structure are well resolved across the entire field of view. An enlarged view of the image is given in Fig. 5 (c), which can be compared with the similar region of the object as imaged by SEM (see insert). Note that the X-ray image is a projection, whereas the SEM image shows only the surface details. Applying the same resolution metrics as for the Siemens star object, the edge profiles along horizonal and vertical directions of the edges of a pore are shown in Fig. 6 . Although the images of pore structures are not as high contrast as the bars of the Siemens star, the edge profiles indicate an edge width of about 8 nm. FRC curves calculated from two half datasets (Eq. 1 ) and as estimated for the full dataset (Eq. 2 ) are shown in Fig. 7 (a) in green and purple, respectively, and with an average incident fluence of 10100 ph/nm 2 for the half datasets. The latter indicates a half-period resolution of 4.1 nm, which is consistent with the 8 nm edge width. Figure 7 (b) shows the phase-image PRTF curve, which remains above 0.6, indicating a high stability of the reconstruction. The resolution of the reconstructed image is slightly worse than the 5.1 nm diffraction-limited resolution of the lens pair, which ultimately limits the spatial frequency content captured in the dataset. Compared with the Siemens star dataset, the higher magnification gave a pixel size of 2.3 nm that matched this information capacity of the holograms. This suggests that the resolution of the image of the Siemens star was limited by the lower magnification. A possible reason why the evaluated resolution of the image of the NPG object was slightly worse than the diffraction limit is that the NPG object is three dimensional with a thickness comparable to the depth of focus. For a coherent image (spatial frequencies recorded on the Ewald sphere), the depth of focus can be quantified as \(\lambda /\text{N}{\text{A}}^{2}=360 \text{n}\text{m}\) , compared with the 1 µm thickness of the object. Numerical study and discussion As mentioned in the Introduction, the projection holographic imaging modality using a high NA objective, as shown in Fig. 1 (b), may have several advantages over conventional diffraction-based imaging depicted in Fig. 1 (c) and which can be characterized as ptychography with a low-NA objective. Although it was not feasible for us to carry out such conventional low-NA ptychography under similar conditions with the same sample, some published results do hint that high resolution requires additional care. At the PETRA III (DESY, Hamburg) P06 beamline, for example, to avoid background noise due to scattering from air, optics and apertures, the hard X-ray ptychographic microscope PtyNAMi 39 performs measurements in vacuum to achieve high resolution. In the soft x-ray spectrum, it has also been observed that background noise lowers the imaging contrast, and therefore the resolution 40 . To further compare these imaging modalities, we carried out a numerical study of ptychographic imaging with high-NA and low-NA reference beams. Similar comparisons of simulated experiments have been performed previously, including the comparison of holography with coherent diffractive imaging 41 and near-field ptychography with far-field ptychography 25 . While these studies differ somewhat in their conclusions, neither compared the performance of ptychographic imaging using a high-NA reference to that using a low-NA probe. Our near-field ptychographic set-up differs from that considered by Du et al. (ref. 25) who simulated the incident illumination as unity magnitude and random phases. This does correspond to a divergent reference wave of high scattering angle (dependent on the real-space pixel size used in the simulation) but with an intensity distribution in the far field that follows a negative binomial distribution where the most common value is zero 42 . That would require measurements of high dynamic range and would not boost the signal as uniformly as the divergent beam from a lens, which might possibly explain the finding that near-field ptychography performed slightly worse. To compare the imaging performance in terms of resolution, sensitivity to noise and background scattering, and dose efficiency, we simulated datasets with different scan-accumulated fluences (See Supplemental Information for details). We considered the ideal case of Poisson noise only (due to photon counting statistics) as well as Poisson noise plus a Gaussian-distributed background of 0.5% of the signal power. Such background is typically caused by scattering from the MLL optics and various apertures, scattering from air, and incoherent scattering such as fluorescence and Compton scattering. We modelled the background to have a uniform angular distribution across the detector, based on measurements made at various detector distances. Detector readout noise or detector non-linearities were not considered. In both high-NA and low-NA cases, we simulated two datasets, one with only Poisson noise and the other with Poisson noise plus background. For each of these datasets, we obtained three reconstructions starting from random independent initializations. During the iterative phase retrieval reconstruction (see Methods), background correction 43 was used for the dataset with background. We calculated the FRC of each reconstruction against the ground truth. Finally, we computed the average image of each set of three reconstructions and the average of their FRCs. In Fig. 8 , we show the reconstruction results for an incident fluence of 1055.7 photons/nm 2 . The phase images and metrics in Fig. 8 , for the high-NA and low-NA probes at a fixed fluence (and hence fixed dose), indeed do show advantages of using the high-NA reference, especially in the presence of background noise. The high-NA configuration results in a better resolution no matter whether there is background noise. Whereas the resolution obtained in the low-NA case degrades with the addition of background, as apparent from the FRC plots and reconstructed images. The low-NA images of Fig. 8 (c) and (f) appear somewhat smoother than their high-NA counterparts, but a comparison with the ground truth shows that the high-frequency details have a reduced contrast in the low-NA images. Further reducing the fluence from 1055.7 photons/nm 2 has a larger effect on the high-NA than the low-NA case, as can be seen in the plot of the resolution of reconstructed images as a function of the simulated fluence, shown in Fig. 9 . The reconstructions for this plot were computed via the same procedure as above, and the resolution (in units of the Nyquist spatial frequency of 225 µm –1 ) was quantified from FRC curves using the half-bit criterion. At the lowest fluence of 31.7 photons/nm 2 the low-NA case outperformed the high-NA case. At this fluence there were only 2.5 photons per pixel on average recorded on the detector in the high-NA case. The high-NA reference beam causes a near-even distribution of photons across the detector, whereas in the low-NA case the photons are concentrated at low scattering angles, into fewer pixels where the signal to noise ratio is thus higher. We find for this particular object and defocus settings that the FRC is roughly equal at a fluence of about 120 photons/nm 2 where the high-NA hologram has a signal of about 10 photons per pixel. At higher fluences than this we see that the images using the high-NA objective improve in resolution and quality much faster than the low-NA case (see Supplemental Information for reconstructed images). Even though the SNR does improve with increasing fluence in the low-NA case, given the rapid decrease in diffraction signal with scattering angle, this improvement is mainly at low resolution. And when background is considered there is a significant deterioration of the resolution in low-NA case. In contrast, there is almost no impact of background on resolution for high-NA case, which, again, illustrates its tolerance to background. Note that the experimental diffraction data from the nanoporous gold object was recorded with 2240 ph/pixel per frame and a highest fluence of 26,800 ph/nm 2 , but the resolution was not as good as achieved by these simulations. This could be due to several reasons. The simulation considered a 2D object but the nanoporous gold in the experiment was a 3D object. Since the thickness was not taken into account this might have had an effect on the imaging resolution. In the experimental data, diffraction is not recorded in the gaps between detector panels, resulting in missing data. Conclusion We have demonstrated hard X-ray imaging at a resolution better than 5 nm by near-field ptychography using multilayer Laue lenses. This result was demonstrated on a hierarchical nanoporous gold object, which contains a network of pores with sizes of about 10 nm in diameter. We find that high-magnification point-projection imaging, phased within the framework of near-field ptychography, provides a robust and dose efficient way to obtain images—no special precautions were taken to reduce background or scattering sources. The measurements are equivalent to near-field holograms, or defocused images obtained in a microscope with coherent illumination. The reference wave diverging from the lens focus interferes with the scattering from the object and boosts that scattering signal by a large factor. While this boost also amplifies the effect of photon counting noise, and hence does not necessarily improve the signal to noise ratio of the measurement, it does place the signal on a large bias that renders it insensitive to background noise and avoids the need for measurements of high dynamic range. This requires a sufficiently large divergence of the reference wave, which is achieved by our multilayer Laue lens system with a numerical aperture of NA = 0.014, giving a diffraction-limited resolution of 5 nm. The lenses are not perfect and exhibit both low-order wavefront aberrations and variations in the intensity across the wavefront. These have the effect of distorting and modulating the projection holograms. But far from degrading the images, these defects provide the diversity needed for ptychographic phase retrieval. Another benefit using lenses in near-field ptychography is that, unlike scattering from an object such as a pinhole or from a random diffuser, there are no zeroes in the reference wave in the far field, so all spatial frequencies of the object are transferred more or less equally. Furthermore, and common to our previous work of imaging via speckle tracking 44 , projection images are recorded directly on pixel-array detectors to avoid dose inefficiencies of optical imaging of a scintillator 23 . Through numerical simulations, we compared the performance of near-field ptychography using aberrated high-NA lenses against far-field ptychography with a focused probe of much lower numerical aperture. In this case the high-resolution information is recorded as a dark-field signal and is more susceptible to background noise, as compared with the high NA case, at least for exposures greater than 120 ph/nm 2 . The large probe size in near-field ptychography requires far fewer steps of the sample to achieve a given field of view, enabling high-speed imaging when detector read-out and stage motion are limiting factors. Near-field ptychography with high-NA optics may thus provide an optimal way for imaging at highest resolutions and speeds at current and upcoming synchrotron radiation facilities. Materials and Methods Experiment Multilayer Laue lenses, consisting of 10,854 bi-layers of SiC and WC were fabricated in our laboratory by masked-layer deposition using magnetron sputtering onto flat silicon substrates 14,15 . At a photon energy of 17.4 keV (0.071 nm wavelength), the lenses had focal lengths of 1.15 mm (horizontally focusing) and 1.25 mm (vertically focusing), and NAs of 0.014. Measurements were carried out at the P11 beamline of the PETRA III synchrotron radiation facility where a collimated and monochromatized beam of 17.4 keV X-rays was brought to a common focus by the lenses. A pixel-array detector (Eiger, Dectris) with square pixels of width 75 µm was placed L = 2.37 m downstream of the focus. We refer to the measurement without a sample as the reference beam, white field, or map of the lens pupil. This reference beam covered an area of 900 × 900 pixels and raw holograms were cropped to 1000 × 1000 pixels. Signal beyond the lens pupil was not used because it was much weaker than the holographic signal and corrupted by parasitic scattering. The MLLs, as all diffractive optics, are chromatic, with a focal length that is inversely proportional to the wavelength. A change in wavelength thus changes the focus to object distance and so too the magnification. This must not change the extent of the full hologram by more than about half the width of a pixel, limiting the tolerable relative bandwidth to the inverse of the number of pixels across the beam at the detector. The relative bandwidth of \(1.3\times {10}^{-4}\) produced by the monochromator was indeed smaller than this and also smaller than the inverse of the number of bi-layers in each lens (1/10,854). The flux in the focused beam was 3.85 × \({10}^{8}\) ph/s. Samples were placed downstream of the focus as determined by speckle tracking 45 and were raster scanned in the transverse plane to obtain a dataset for near-field ptychography at a particular defocus distance \(\varDelta f\) . Reconstruction Prior to reconstruction, bad pixels of the detector were masked out. The recovery of the complex-valued image of the object transmission and the lens wavefront were carried out using the open-source software package PyNX 46 , providing several iterative phase retrieval algorithms used here: the difference map (DM) 47 , alternating projection (AP) 43 and maximum likelihood gradient descent (ML) 48,49 . An image of the Siemens star object was reconstructed using 1500 iterations of DM, 2000 iterations of AP, followed by 500 iterations of ML. The NPG sample had a more complex structure than the Siemens star and we obtained reliable reconstructions by using the probe recovered from the Siemens star dataset as the initial probe—iterations with a random initial probe often converged slowly or even did not converge. The transmission image was still initialized from random values and reconstructed by running 2000 iterations of AP and then 500 iterations of ML. During reconstruction, we often observed a phase vortex appearing in the reconstructed probe, as also reported by others 50 . Such reconstructions were discarded. The equal-spaced raster scan can introduce periodic artefacts in the reconstructed image. To reduce these, we performed a smoothing of the object with a gaussian kernel during the DM and AP iterations and carried out the final ML iterations without smoothing. Simulations In the numerical study, near-field (high-NA) ptychography data was simulated using the Fresnel scaling theorem 51 to avoid the need for large array sizes. Diffraction patterns in the low-NA case were simulated via Fourier transform based Fresnel diffraction 52 . Reconstructions were carried out by applying 2000 iterations of AP and 200 iterations of ML. This schedule was repeated three times with uncorrelated random initializations. Declarations Acknowledgements We acknowledge Sabrina Schneider, Harumi Nakatsutsumi, Tjark Delmas (DESY) and Martin Domaracký for their technical support, and Johanna Hakanpää, Sofiane Saouane, and Guillaume Pompidor for assistance at beamline P11 of the PETRA III facility at DESY. We thank Patrick Huber (DESY and TUHH) for helpful suggestions. We acknowledge support from DESY (Hamburg, Germany), a member of the Helmholtz Association HGF. This research was supported in part through the Maxwell computational resources operated at DESY. Additional support was provided by the Cluster of Excellence CUI: Advanced Imaging of Matter of the Deutsche Forschungsgemeinschaft (DFG) – EXC 2056 – project ID 390715994. Author contributions Measurements (J.L.D., H.F., M.Z., O.Y., H.N.C., S.B.), optics fabrication and characterisation (S.B., M.P., J.L.D., M.Z.), instrumentation (H.F), analysis (W.Z., T.L., N.I., C.L., J.H., H.N.C.), control and data acquisition software (D.E., I.D.G.A., P.M.), sample preparation (M.P., S.S.), concept (H.N.C., S.B.). W.Z. and H.N.C. wrote the manuscript with input from all other authors. Data availability All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplemental Information. Additional data related to this paper may be requested from the authors. Conflict of interest The authors declare no conflict of interest. References Michelson, A. et al. Three-dimensional visualization of nanoparticle lattices and multimaterial frameworks. 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Ptychographic X-ray speckle tracking with multi-layer Laue lens systems. J. Appl. Crystallogr. 53 , 927–936 (2020). Ivanov, N. et al. Robust ptychographic X-ray speckle tracking with multilayer Laue lenses. Opt. Express 30 , 25450–25473 (2022). Favre-Nicolin, V. et al. PyNX: high-performance computing toolkit for coherent X-ray imaging based on operators. J. Appl. Crystallogr. 53 , 1404–1413 (2020). Elser, V., Rankenburg, I. & Thibault, P. Searching with iterated maps. Proc. Natl. Acad. Sci. 104 , 418–423 (2007). Odstrčil, M., Menzel, A. & Guizar-Sicairos, M. Iterative least-squares solver for generalized maximum-likelihood ptychography. Opt. Express 26 , 3108–3123 (2018). Thibault, P. & Guizar-Sicairos, M. Maximum-likelihood refinement for coherent diffractive imaging. New J. Phys. 14 , 63004 (2012). Stockmar, M. et al. X-ray near-field ptychography for optically thick specimens. Phys. Rev. Appl. 3 , 14005 (2015). Paganin, D. Coherent X-ray optics . (Oxford University Press on Demand, 2006). Zhang, W., Zhang, H., Sheppard, C. & Jin, G. Analysis of numerical diffraction calculation methods: from the perspective of phase space optics and the sampling theorem. J. Opt. Soc. Am. A 37 , 1748–1766 (2020). Table 1 Table 1. Parameters used in the numerical simulations High-NA case Low-NA case Experiment (NPG) Wavelength (nm) Aperture of lens (μm) 0.071 40 0.071 40 0.071 35 Focal length (mm) 1.43 143 1.2 NA of lens 0.014 0.00014 0.014 Defocus distance (μm) 37 10 73 Detector distance (m) 1.2 1.2 2.37 Magnification 32,432 / 32,465 Scan step size (nm) 110 44 250 (H) × 100 (V) Scan points 50 500 63 Additional Declarations (Not answered) Supplementary Files SupplementalFINAL.docx Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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for the understanding and improvement of functional and active materials, such as catalysts, solar cells, batteries, storage media, and integrated circuits. Imaging of such materials at the nanometer scale is required to understand relationships between microstructure and various mechanical, electronic, or magnetic properties. The rise in the brightness of synchrotron radiation sources over the last several decades has brought extensive developments\u003csup\u003e1\u0026ndash;8\u003c/sup\u003e in X-ray imaging and tomography in the photon energy range of about 5 keV to 20 keV to fulfil this need. Given the weak interaction of X-rays with matter in this spectral range, phase-contrast and diffraction-based imaging methods offer the highest sensitivity for the investigation of nano-structures in bulk samples (thicker than around 1 \u0026micro;m). However, for this same reason, it is difficult to efficiently deflect X-rays by significant angles, which has limited the creation of high-resolution lenses. To overcome this restriction, the method of ptychography is often used\u003csup\u003e3\u003c/sup\u003e. By measuring diffraction patterns from several localized and overlapping regions of the sample as it is stepped across a focused beam, it is possible to recover the complex-valued exit wave of the sample at a resolution dependent on the angular extent of the measured diffraction rather than the numerical aperture (NA) of the focused x-ray beam (the probe). In this way, a full-period resolution of 10 nm has been obtained at a soft X-ray wavelength of 1.7 nm with a probe size of 150 nm (ref. 9), and 20 nm resolution was accomplished at 2 nm wavelength\u0026mdash;35 times better than the information limit of the probe (700 nm in size)\u003csup\u003e10\u003c/sup\u003e. The imaging resolution in these examples was ultimately limited by the strength of the diffraction signal. This is generally the case, since the diffraction intensities tend to decrease with the fourth power of the resolution length \u003cem\u003ed\u003c/em\u003e (ref. 11). Improving the resolution requires ever increasing incident fluence and hence radiation dose to the sample (which it might not be able to tolerate)\u003csup\u003e12\u003c/sup\u003e. However, it is often not possible to extract the full potential signal for a given dose since the weakest diffraction (at highest scattering angles) may be dominated by background noise such as scattering from air, beamline slits, or the focusing optic itself. Thus, the high resolutions mentioned above were accomplished with samples of high scattering power and with measures to carefully eliminate all sources of noise. Furthermore, the diffraction signals usually vary over a dynamic range of many tens of thousands\u0026mdash;from single photons to that of the direct beam\u0026mdash;often requiring multiple exposures. Thus, the gain in resolution achieved by X-ray ptychography usually comes at a higher cost of imaging speed, dose, and radiation damage than ultimately necessary.\u003c/p\u003e \u003cp\u003eWith the recent development and use of multilayer Laue lenses (MLLs)\u003csup\u003e13,14\u003c/sup\u003e, it has become possible to focus hard X-rays with a high convergence angle (or NA) to give diffraction-limited spot sizes smaller than about 4 nm (ref. 15). These diffractive lenses are fabricated layer by layer by magnetron sputtering\u0026mdash;a process in which errors in layer placement can accumulate to result in aberrations that may be several waves in magnitude\u003csup\u003e15\u003c/sup\u003e. Although such errors can be compensated, for example by use of a refractive corrector\u003csup\u003e16,17\u003c/sup\u003e they do not necessarily restrict the large angular divergence of the probe beam. In a ptychography measurement, this large divergence overcomes the problem of high dynamic range and sensitivity to weak diffraction signals by providing a strong zero-order reference beam that interferes coherently with the scattering from the sample. When placed in the focal plane, the sample would need to be stepped by distances that are even smaller than the probe size to obtain a ptychographic dataset. This leads to datasets with a large number of steps of the scanned object, coupled with modest requirements of the number of pixels required to record the diffraction pattern\u003csup\u003e18\u003c/sup\u003e. A certain field of view can be covered in far fewer steps (and detector frames) by placing the sample considerably out of focus. In this case, the diffraction pattern must be sampled more finely to match the larger width of the probe. However, given that direct-detection pixel array detectors with large pixel numbers are widely available for X-rays, this approach\u0026mdash;known as near-field ptychography\u003csup\u003e19\u003c/sup\u003e may have practical benefits of speed and less demanding requirements for the sample scanning hardware.\u003c/p\u003e \u003cp\u003eWhen a weakly-scattering object is placed out of the focus of a converging beam, a magnified projection image is formed in the far field of that focus, with a magnification equal to the ratio \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(M=L/{\\Delta }f\\)\u003c/span\u003e\u003c/span\u003eof the distance from focus to detector, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(L\\)\u003c/span\u003e\u003c/span\u003e, to the defocus distance \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\Delta }f\\)\u003c/span\u003e\u003c/span\u003e. It is well known that this projection image is equal to a magnified version of the in-line hologram that would be obtained by illuminating the object with a plane wave, measured at approximately the defocus distance\u003csup\u003e20\u0026ndash;22\u003c/sup\u003e, as shown schematically in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(a). It is also equivalent to the defocused coherent image recorded in a transmission X-ray microscope under plane-wave illumination and an objective of the same NA\u003csup\u003e23\u003c/sup\u003e. In all three cases, the image is equal to the Fresnel transform of the transmission function of the object. For small defocus, the contrast of such images of pure phase objects increases with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\Delta }f\\)\u003c/span\u003e\u003c/span\u003e (ref. 24).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eReturning to the holographic interpretation of the projection image, the reference wave is provided by the beam diverging from the focus and which does not interact with the object. Obviously, at the detector, this reference wave \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(R\\)\u003c/span\u003e\u003c/span\u003e only extends to scattering angles covered by the diverging beam, as given by the NA of the objective lens. Within this range, the interference between \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(R\\)\u003c/span\u003e\u003c/span\u003e and the diffraction \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(O\\)\u003c/span\u003e\u003c/span\u003e from an object can be considered as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\left|O+R\\right|}^{2}={\\left|R\\right|}^{2}+2\\left|O R\\right|\\text{cos}\\varphi +{\\left|O\\right|}^{2}\\)\u003c/span\u003e\u003c/span\u003e. The relative phase \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\varphi\\)\u003c/span\u003e\u003c/span\u003e between the object and reference is encoded in the hologram. In the hard X-ray regime, most (microscopic) objects are weakly scattering such that the reference-beam intensity \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\left|R\\right|}^{2}\\)\u003c/span\u003e\u003c/span\u003e is many thousands of times greater than the far-field diffraction signal \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\left|O\\right|}^{2}\\)\u003c/span\u003e\u003c/span\u003e, which can therefore be neglected in the region of the reference. It is seen in the term \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(2\\left|O R\\right|\\text{cos}\\varphi\\)\u003c/span\u003e\u003c/span\u003e that the diffraction signal is greatly amplified by the strength of the reference amplitude. For complete constructive or destructive interference (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\varphi =0\\)\u003c/span\u003e\u003c/span\u003e or \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\pi\\)\u003c/span\u003e\u003c/span\u003e), this boost factor is equal to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(2\\left|O R\\right|/{\\left|O\\right|}^{2}=2\\left|R\\right|/\\left|O\\right|\\)\u003c/span\u003e\u003c/span\u003e and the contrast of the hologram is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(2\\left|O\\right|/\\left|R\\right|\\)\u003c/span\u003e\u003c/span\u003e. Both the non-holographic diffraction signal \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\left|O\\right|}^{2}\\)\u003c/span\u003e\u003c/span\u003e and the boosted holographic signal \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(2\\left|O R\\right|\\text{cos}\\varphi\\)\u003c/span\u003e\u003c/span\u003e exhibit Poisson noise. When photon counting is the only noise source, the advantage in signal to noise with and without the reference is actually only a factor of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(2\\text{ cos}\\varphi\\)\u003c/span\u003e\u003c/span\u003e, with a root-mean-square value of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\surd 2\\)\u003c/span\u003e\u003c/span\u003e. Although studies have shown that coherent mixing of the signal with a strong reference does not necessarily improve the signal to noise ratio of reconstructed images\u003csup\u003e25\u003c/sup\u003e, the signal boost and the large offset \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\left|R\\right|}^{2}\\)\u003c/span\u003e\u003c/span\u003e will dominate any extraneous scattering. Additionally, compared to the diffraction signal \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\left|O\\right|}^{2}\\)\u003c/span\u003e\u003c/span\u003e without a reference, the boosted (low-contrast) holographic signal covers a much lower dynamic range, easing the requirements of the detector, and the in-line hologram generally requires a lower degree of spatial coherence to encode the transmission function than does the far-field pattern\u003csup\u003e26\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eThere have been several challenges in realizing high-resolution X-ray imaging in the holographic regime. One has been the production of a strong reference that covers a suitably large range of scattering angles. Objects such as gold particles\u003csup\u003e27\u003c/sup\u003e, pinholes\u003csup\u003e28,29\u003c/sup\u003e, and aerogel\u003csup\u003e30\u003c/sup\u003e have been used or suggested. These tend to provide reference beams that are not much stronger than the scattering from the object, and which vary considerably in intensity. A strong reference was created by the focused beam of a soft X-ray zone plate\u003csup\u003e31\u003c/sup\u003e although the usable angular range was restricted by the annular pupil. Another challenge is how best to reconstruct the transmission function of the object (its exit wave) from the hologram, and in particular how to overcome the twin-image problem for the in-line geometry. Near-field ptychography\u003csup\u003e19\u003c/sup\u003e using MLLs overcomes both these challenges, although when framed as a series of holographic images recorded for various shifts of the object relative to the illumination, it is not immediately clear how this approach recovers the object\u0026rsquo;s transmission function. For a perfect (unaberrated) wavefield, stepping the object across the beam at a fixed defocus would simply shift the projected hologram across the illuminated region of the detector without providing new information. For a perfect spherical wave, only changing defocus gives the needed diversity\u003csup\u003e24,31\u003c/sup\u003e. A structured wavefront created by a diffuser enables image reconstruction without changing focus\u003csup\u003e19\u003c/sup\u003e. The illumination phase at any point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({r}_{0}\\)\u003c/span\u003e\u003c/span\u003e can be approximated through a truncated Taylor expansion as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\phi \\left(r\\right)= {\\phi }_{0}\\left({r}_{0}\\right)+\\left(r-{r}_{0}\\right) {\\phi }_{1}\\left({r}_{0}\\right)+{\\left(r-{r}_{0}\\right)}^{2} {\\phi }_{2}\\left({r}_{0}\\right)\\)\u003c/span\u003e\u003c/span\u003e. As a feature in the object is moved from point to point, variations of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\phi }_{2}\\)\u003c/span\u003e\u003c/span\u003e lead to the needed change of the illumination condition. In our experiments using a pair of MLLs, we find that the inherent low-order wavefront errors, as well as higher-frequency errors due to sub-\u0026aring;ngstr\u0026ouml;m variations in lens layer thicknesses\u003csup\u003e32\u003c/sup\u003e, adequately provide the necessary diversity in the illumination beam, without generating zeroes in the far-field reference, and no additional diffuser is required. Finally, the fact that near-field ptychography records holograms, rather than far-field diffraction patterns, means that the lateral object positions in the scan are evident in the holograms themselves, providing the means to account for errors due to incorrect stage motion.\u003c/p\u003e \u003cp\u003eHere, we evaluate the performance of high-resolution imaging via near-field ptychography using high-NA MLLs by quantifying the resolution of a Siemens star test object and of a hierarchical nanoporous gold structure, demonstrating imaging over large fields of view with reasonable efficiency. A numerical study is then made to compare this scheme to ptychography with a low-NA probe where the high-resolution information is recorded as \u0026ldquo;dark-field\u0026rdquo; scattering beyond the angular extent of the reference beam.\u003c/p\u003e"},{"header":"Results","content":"\u003cp\u003eA schematic diagram of our projection imaging experiment is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(b), where a pair of MLLs with NAs of 0.014 (both horizontal and vertical) focused an X-ray beam of 0.071 nm wavelength, recorded on a pixel-array detector with 75 µm pixels, placed \u003cem\u003eL\u003c/em\u003e = 2.37 m downstream of the focus. We first characterized the high-NA near-field ptychography performance by using a Siemens star test object, made of gold of 750 nm thickness on a silicon nitride membrane. The object was placed at a defocus distance of 200 µm where the beam width was 5.6 µm, giving a magnification of 11,850 corresponding to a 6.3 nm pixel size in the object space. A ptychographic dataset of 21 × 21 points was obtained with a step size of 0.3 µm in each direction, giving an overlap of 94.6% between neighboring holograms and an overall field of view of about 11.6 µm. The exposure time per step was 1 s, giving 497 photons per pixel per frame within the lens pupil at the detector and a highest scan-accumulated exposure of the object of 4470 ph/nm\u003csup\u003e2\u003c/sup\u003e in the central region of the scan (of width 630 nm). One of the recorded projection holograms, uncorrected for the reference beam intensity, is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(b).\u003c/p\u003e \u003cp\u003eStarting from random initial guesses for both the probe wavefront and object transmission, maps of these functions were recovered using a schedule of iterative phasing algorithms as described in “Materials and Methods”. The results are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003e including the 5.6-µm wide incident wavefront at the sample plane in Figs.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003e (c) and 2(d). Although this wavefront aberration spans many waves, resulting in a point spread function at the focal plane\u003csup\u003e15\u003c/sup\u003e that is wider than the diffraction-limited spot size (equal to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\lambda /\\text{N}\\text{A}= 5.1 \\text{n}\\text{m})\\)\u003c/span\u003e\u003c/span\u003e, this does not hamper the image resolution obtained.\u003c/p\u003e \u003cp\u003eThe innermost spokes of the Siemens star object can clearly be resolved in Figs.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003e (a) and (b)—evidently the resolution is much better than the smallest spoke width of 50 nm and may be limited only by the Nyquist frequency (twice the pixel size, i.e., 12.6 nm). To obtain quantitative estimates of the resolution, we computed it via three different criteria: the edge criterion, which is a measure of the edge width\u003csup\u003e33\u003c/sup\u003e, the Fourier ring correlation (FRC)\u003csup\u003e34\u003c/sup\u003e, and the phase retrieval transfer function (PRTF) \u003csup\u003e35,36\u003c/sup\u003e. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e3\u003c/span\u003e shows edge profiles of spokes, between the two rings, from the phase image of the Siemens star shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003e(a) along four different directions: vertical, horizontal and the two orthogonal directions at 45º to the horizontal. The widths listed in the figure were determined by the distance between points corresponding to 25% and 75% of the total step height of the fitted error function (erf). It is seen that the widths along the four directions are different. This may indicate an anisotropic resolution, but it is more likely a measure of the actual projected transmission function of the spokes, which are much thicker than they are wide. Scanning electron micrographs of the spokes have shown that the walls of the spokes are not all parallel, and the plane of the object might not have been exactly normal to the beam. The average of the widths along the four directions was 12.6 nm, and the smallest was 7.1 nm.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTo calculate the FRC, we first split the full dataset into two half-datasets by selecting every other frame of the full dataset. With these two half-datasets, each with double the step size in the fast-scan (vertical) direction, we obtained two independent reconstruction results, \u003cem\u003eA\u003c/em\u003e and \u003cem\u003eB\u003c/em\u003e, each with an average incident fluence of 1200 ph/nm\u003csup\u003e2\u003c/sup\u003e. We smoothed the edges of each image, and computed the FRC as\u003csup\u003e34\u003c/sup\u003e,\u003c/p\u003e\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$${\\text{F}\\text{R}\\text{C}}_{\\text{h}\\text{a}\\text{l}\\text{f}}\\left(q\\right)=\\frac{\\sum {F}_{A}\\left(q\\right){F}_{B}^{*}\\left(q\\right)}{\\sqrt{{\\sum \\left|{F}_{A}\\left(q\\right)\\right|}^{2}{\\sum \\left|{F}_{B}\\left(q\\right)\\right|}^{2}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003cp\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F}_{A}\\left(q\\right)\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F}_{B}\\left(q\\right)\\)\u003c/span\u003e\u003c/span\u003e are the Fourier transforms of these two images as a function of the spatial frequency \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(q=(\\text{sin}\\theta )/\\lambda\\)\u003c/span\u003e\u003c/span\u003e, and * denotes the complex conjugate. Perfect imaging would give unity correlation over the full spatial frequency range. In practice, the FRC decreases with increasing \u003cem\u003eq\u003c/em\u003e because high spatial frequency components generally have a lower power and are thus more easily affected by noise\u003csup\u003e11\u003c/sup\u003e. Figure\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e(a) shows a plot of FRC\u003csub\u003ehalf\u003c/sub\u003e in green, giving a half-period resolution of 9.2 nm by the half-bit criterion\u003csup\u003e34\u003c/sup\u003e. However, we noticed that the image reconstructed from the full dataset appears to have a better resolution than either of those from the half datasets. Even though two independent full datasets are not available, the FRC of the full dataset (with an average fluence of 2400 ph/nm\u003csup\u003e2\u003c/sup\u003e) can be estimated from the so called “CC-star” formula\u003csup\u003e37\u003c/sup\u003e as\u003c/p\u003e \u003cp\u003e \u003c/p\u003e\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$${\\text{F}\\text{R}\\text{C}}_{\\text{f}\\text{u}\\text{l}\\text{l}}=\\sqrt{\\frac{2{\\text{F}\\text{R}\\text{C}}_{\\text{h}\\text{a}\\text{l}\\text{f}}}{1+{\\text{F}\\text{R}\\text{C}}_{\\text{h}\\text{a}\\text{l}\\text{f}}}}$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e.\u003cp\u003e\u003c/p\u003e \u003cp\u003eThis metric, plotted in purple in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e(a), indicates a half-period resolution of 6.5 nm by the half-bit criterion, a value that is consistent with the edge criterion estimation.\u003c/p\u003e \u003cp\u003eFinally, we used a modified version of PRTF to evaluate the stability of the iterative phase retrieval process, which is defined as follows,\u003c/p\u003e\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\text{P}\\text{R}\\text{T}\\text{F}\\left(q\\right)=\\left|⟨\\text{e}\\text{x}\\text{p}\\left[i\\phi \\left(q\\right)\\right]⟩\\right|=\\left|⟨\\frac{G\\left(q\\right)}{\\left|G\\left(q\\right)\\right|}⟩\\right|$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e,\u003cp\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(G\\left(q\\right)\\)\u003c/span\u003e\u003c/span\u003e is the Fourier transform of the reconstructed phase image and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(⟨\\dots ⟩\\)\u003c/span\u003e\u003c/span\u003e represents the average over a large number of independent reconstructions. If the Fourier phases fluctuate, the PRTF tends to zero and if they are stable the PRTF tends to one. Figure\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e(b) shows the radial average of the PRTF, obtained from 18 independent reconstructions. We can see that even at the highest spatial frequency of 80 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mu }\\)\u003c/span\u003e\u003c/span\u003em\u003csup\u003e− 1\u003c/sup\u003e, the PRTF value is about 0.4, indicating a stable phase retrieval process over the entire frequency range of the phase image. Together, the three resolution criteria are consistent with a half-period resolution that is not appreciably worse than the Nyquist sampling obtained with the particular choice of magnification of 11,850 (giving an image pixel width of 6.3 nm).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eNext, we imaged a nanoporous gold (NPG) sample with a hierarchical porous structure that confers a high specific strength and low density\u003csup\u003e38\u003c/sup\u003e. The finest pores have a width of about 10 nm. A 1-µm wide column of this material was created by focused ion-beam milling—an SEM image of the object is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e(a). A near-field ptychographic dataset was recorded at an even higher magnification factor than the Siemens star object, of 32,470 obtained at a defocus distance of 73 µm, giving a sampling size of 2.3 nm in the object space over a hologram width of 2.0 µm. The dataset was acquired by stepping the object in a grid consisting of 3 × 21 locations (horizontal × vertical) with step sizes of 0.25 µm and 0.1 µm in the horizontal and vertical directions, and recording holograms with an exposure time of 5 s at each position, giving 2240 photons per pixel per frame within the lens pupil at the detector and a highest scan-accumulated exposure of 26,800 ph/nm\u003csup\u003e2\u003c/sup\u003e at the object plane. The recovered phase image of the NPG object is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e(b). The small pores in the structure are well resolved across the entire field of view. An enlarged view of the image is given in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e(c), which can be compared with the similar region of the object as imaged by SEM (see insert). Note that the X-ray image is a projection, whereas the SEM image shows only the surface details.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eApplying the same resolution metrics as for the Siemens star object, the edge profiles along horizonal and vertical directions of the edges of a pore are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e. Although the images of pore structures are not as high contrast as the bars of the Siemens star, the edge profiles indicate an edge width of about 8 nm. FRC curves calculated from two half datasets (Eq.\u0026nbsp;\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) and as estimated for the full dataset (Eq.\u0026nbsp;\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e(a) in green and purple, respectively, and with an average incident fluence of 10100 ph/nm\u003csup\u003e2\u003c/sup\u003e for the half datasets. The latter indicates a half-period resolution of 4.1 nm, which is consistent with the 8 nm edge width. Figure\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e(b) shows the phase-image PRTF curve, which remains above 0.6, indicating a high stability of the reconstruction.\u003c/p\u003e \u003cp\u003eThe resolution of the reconstructed image is slightly worse than the 5.1 nm diffraction-limited resolution of the lens pair, which ultimately limits the spatial frequency content captured in the dataset. Compared with the Siemens star dataset, the higher magnification gave a pixel size of 2.3 nm that matched this information capacity of the holograms. This suggests that the resolution of the image of the Siemens star was limited by the lower magnification. A possible reason why the evaluated resolution of the image of the NPG object was slightly worse than the diffraction limit is that the NPG object is three dimensional with a thickness comparable to the depth of focus. For a coherent image (spatial frequencies recorded on the Ewald sphere), the depth of focus can be quantified as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\lambda /\\text{N}{\\text{A}}^{2}=360 \\text{n}\\text{m}\\)\u003c/span\u003e\u003c/span\u003e, compared with the 1 µm thickness of the object.\u003c/p\u003e "},{"header":"Numerical study and discussion","content":"\u003cp\u003eAs mentioned in the Introduction, the projection holographic imaging modality using a high NA objective, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(b), may have several advantages over conventional diffraction-based imaging depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(c) and which can be characterized as ptychography with a low-NA objective. Although it was not feasible for us to carry out such conventional low-NA ptychography under similar conditions with the same sample, some published results do hint that high resolution requires additional care. At the PETRA III (DESY, Hamburg) P06 beamline, for example, to avoid background noise due to scattering from air, optics and apertures, the hard X-ray ptychographic microscope PtyNAMi\u003csup\u003e39\u003c/sup\u003e performs measurements in vacuum to achieve high resolution. In the soft x-ray spectrum, it has also been observed that background noise lowers the imaging contrast, and therefore the resolution\u003csup\u003e40\u003c/sup\u003e.\u003c/p\u003e\u003cp\u003eTo further compare these imaging modalities, we carried out a numerical study of ptychographic imaging with high-NA and low-NA reference beams. Similar comparisons of simulated experiments have been performed previously, including the comparison of holography with coherent diffractive imaging\u003csup\u003e41\u003c/sup\u003e and near-field ptychography with far-field ptychography\u003csup\u003e25\u003c/sup\u003e. While these studies differ somewhat in their conclusions, neither compared the performance of ptychographic imaging using a high-NA reference to that using a low-NA probe. Our near-field ptychographic set-up differs from that considered by Du \u003cem\u003eet al.\u003c/em\u003e (ref. 25) who simulated the incident illumination as unity magnitude and random phases. This does correspond to a divergent reference wave of high scattering angle (dependent on the real-space pixel size used in the simulation) but with an intensity distribution in the far field that follows a negative binomial distribution where the most common value is zero\u003csup\u003e42\u003c/sup\u003e. That would require measurements of high dynamic range and would not boost the signal as uniformly as the divergent beam from a lens, which might possibly explain the finding that near-field ptychography performed slightly worse.\u003c/p\u003e\u003cp\u003eTo compare the imaging performance in terms of resolution, sensitivity to noise and background scattering, and dose efficiency, we simulated datasets with different scan-accumulated fluences (See Supplemental Information for details). We considered the ideal case of Poisson noise only (due to photon counting statistics) as well as Poisson noise plus a Gaussian-distributed background of 0.5% of the signal power. Such background is typically caused by scattering from the MLL optics and various apertures, scattering from air, and incoherent scattering such as fluorescence and Compton scattering. We modelled the background to have a uniform angular distribution across the detector, based on measurements made at various detector distances. Detector readout noise or detector non-linearities were not considered.\u003c/p\u003e\u003cp\u003eIn both high-NA and low-NA cases, we simulated two datasets, one with only Poisson noise and the other with Poisson noise plus background. For each of these datasets, we obtained three reconstructions starting from random independent initializations. During the iterative phase retrieval reconstruction (see Methods), background correction\u003csup\u003e43\u003c/sup\u003e was used for the dataset with background. We calculated the FRC of each reconstruction against the ground truth. Finally, we computed the average image of each set of three reconstructions and the average of their FRCs. In Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e, we show the reconstruction results for an incident fluence of 1055.7 photons/nm\u003csup\u003e2\u003c/sup\u003e.\u003c/p\u003e\u003cp\u003eThe phase images and metrics in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e, for the high-NA and low-NA probes at a fixed fluence (and hence fixed dose), indeed do show advantages of using the high-NA reference, especially in the presence of background noise. The high-NA configuration results in a better resolution no matter whether there is background noise. Whereas the resolution obtained in the low-NA case degrades with the addition of background, as apparent from the FRC plots and reconstructed images. The low-NA images of Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e(c) and (f) appear somewhat smoother than their high-NA counterparts, but a comparison with the ground truth shows that the high-frequency details have a reduced contrast in the low-NA images.\u003c/p\u003e\u003cp\u003eFurther reducing the fluence from 1055.7 photons/nm\u003csup\u003e2\u003c/sup\u003e has a larger effect on the high-NA than the low-NA case, as can be seen in the plot of the resolution of reconstructed images as a function of the simulated fluence, shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e. The reconstructions for this plot were computed via the same procedure as above, and the resolution (in units of the Nyquist spatial frequency of 225 µm\u003csup\u003e–1\u003c/sup\u003e) was quantified from FRC curves using the half-bit criterion. At the lowest fluence of 31.7 photons/nm\u003csup\u003e2\u003c/sup\u003e the low-NA case outperformed the high-NA case. At this fluence there were only 2.5 photons per pixel on average recorded on the detector in the high-NA case. The high-NA reference beam causes a near-even distribution of photons across the detector, whereas in the low-NA case the photons are concentrated at low scattering angles, into fewer pixels where the signal to noise ratio is thus higher. We find for this particular object and defocus settings that the FRC is roughly equal at a fluence of about 120 photons/nm\u003csup\u003e2\u003c/sup\u003e where the high-NA hologram has a signal of about 10 photons per pixel. At higher fluences than this we see that the images using the high-NA objective improve in resolution and quality much faster than the low-NA case (see Supplemental Information for reconstructed images). Even though the SNR does improve with increasing fluence in the low-NA case, given the rapid decrease in diffraction signal with scattering angle, this improvement is mainly at low resolution. And when background is considered there is a significant deterioration of the resolution in low-NA case. In contrast, there is almost no impact of background on resolution for high-NA case, which, again, illustrates its tolerance to background.\u003c/p\u003e\u003cp\u003eNote that the experimental diffraction data from the nanoporous gold object was recorded with 2240 ph/pixel per frame and a highest fluence of 26,800 ph/nm\u003csup\u003e2\u003c/sup\u003e, but the resolution was not as good as achieved by these simulations. This could be due to several reasons. The simulation considered a 2D object but the nanoporous gold in the experiment was a 3D object. Since the thickness was not taken into account this might have had an effect on the imaging resolution. In the experimental data, diffraction is not recorded in the gaps between detector panels, resulting in missing data.\u003c/p\u003e"},{"header":"Conclusion","content":"\u003cp\u003eWe have demonstrated hard X-ray imaging at a resolution better than 5 nm by near-field ptychography using multilayer Laue lenses. This result was demonstrated on a hierarchical nanoporous gold object, which contains a network of pores with sizes of about 10 nm in diameter. We find that high-magnification point-projection imaging, phased within the framework of near-field ptychography, provides a robust and dose efficient way to obtain images\u0026mdash;no special precautions were taken to reduce background or scattering sources. The measurements are equivalent to near-field holograms, or defocused images obtained in a microscope with coherent illumination. The reference wave diverging from the lens focus interferes with the scattering from the object and boosts that scattering signal by a large factor. While this boost also amplifies the effect of photon counting noise, and hence does not necessarily improve the signal to noise ratio of the measurement, it does place the signal on a large bias that renders it insensitive to background noise and avoids the need for measurements of high dynamic range. This requires a sufficiently large divergence of the reference wave, which is achieved by our multilayer Laue lens system with a numerical aperture of NA\u0026thinsp;=\u0026thinsp;0.014, giving a diffraction-limited resolution of 5 nm. The lenses are not perfect and exhibit both low-order wavefront aberrations and variations in the intensity across the wavefront. These have the effect of distorting and modulating the projection holograms. But far from degrading the images, these defects provide the diversity needed for ptychographic phase retrieval. Another benefit using lenses in near-field ptychography is that, unlike scattering from an object such as a pinhole or from a random diffuser, there are no zeroes in the reference wave in the far field, so all spatial frequencies of the object are transferred more or less equally. Furthermore, and common to our previous work of imaging via speckle tracking\u003csup\u003e44\u003c/sup\u003e, projection images are recorded directly on pixel-array detectors to avoid dose inefficiencies of optical imaging of a scintillator\u003csup\u003e23\u003c/sup\u003e. Through numerical simulations, we compared the performance of near-field ptychography using aberrated high-NA lenses against far-field ptychography with a focused probe of much lower numerical aperture. In this case the high-resolution information is recorded as a dark-field signal and is more susceptible to background noise, as compared with the high NA case, at least for exposures greater than 120 ph/nm\u003csup\u003e2\u003c/sup\u003e. The large probe size in near-field ptychography requires far fewer steps of the sample to achieve a given field of view, enabling high-speed imaging when detector read-out and stage motion are limiting factors. Near-field ptychography with high-NA optics may thus provide an optimal way for imaging at highest resolutions and speeds at current and upcoming synchrotron radiation facilities.\u003c/p\u003e"},{"header":"Materials and Methods","content":"\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003eExperiment\u003c/h2\u003e \u003cp\u003eMultilayer Laue lenses, consisting of 10,854 bi-layers of SiC and WC were fabricated in our laboratory by masked-layer deposition using magnetron sputtering onto flat silicon substrates\u003csup\u003e14,15\u003c/sup\u003e. At a photon energy of 17.4 keV (0.071 nm wavelength), the lenses had focal lengths of 1.15 mm (horizontally focusing) and 1.25 mm (vertically focusing), and NAs of 0.014. Measurements were carried out at the P11 beamline of the PETRA III synchrotron radiation facility where a collimated and monochromatized beam of 17.4 keV X-rays was brought to a common focus by the lenses. A pixel-array detector (Eiger, Dectris) with square pixels of width 75 \u0026micro;m was placed \u003cem\u003eL\u003c/em\u003e\u0026thinsp;=\u0026thinsp;2.37 m downstream of the focus. We refer to the measurement without a sample as the reference beam, white field, or map of the lens pupil. This reference beam covered an area of 900 \u0026times; 900 pixels and raw holograms were cropped to 1000 \u0026times; 1000 pixels. Signal beyond the lens pupil was not used because it was much weaker than the holographic signal and corrupted by parasitic scattering. The MLLs, as all diffractive optics, are chromatic, with a focal length that is inversely proportional to the wavelength. A change in wavelength thus changes the focus to object distance and so too the magnification. This must not change the extent of the full hologram by more than about half the width of a pixel, limiting the tolerable relative bandwidth to the inverse of the number of pixels across the beam at the detector. The relative bandwidth of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(1.3\\times {10}^{-4}\\)\u003c/span\u003e\u003c/span\u003e produced by the monochromator was indeed smaller than this and also smaller than the inverse of the number of bi-layers in each lens (1/10,854). The flux in the focused beam was 3.85 \u0026times; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({10}^{8}\\)\u003c/span\u003e\u003c/span\u003e ph/s. Samples were placed downstream of the focus as determined by speckle tracking\u003csup\u003e45\u003c/sup\u003e and were raster scanned in the transverse plane to obtain a dataset for near-field ptychography at a particular defocus distance \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\varDelta f\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003eReconstruction\u003c/h2\u003e \u003cp\u003ePrior to reconstruction, bad pixels of the detector were masked out. The recovery of the complex-valued image of the object transmission and the lens wavefront were carried out using the open-source software package PyNX\u003csup\u003e46\u003c/sup\u003e, providing several iterative phase retrieval algorithms used here: the difference map (DM)\u003csup\u003e47\u003c/sup\u003e, alternating projection (AP)\u003csup\u003e43\u003c/sup\u003e and maximum likelihood gradient descent (ML)\u003csup\u003e48,49\u003c/sup\u003e. An image of the Siemens star object was reconstructed using 1500 iterations of DM, 2000 iterations of AP, followed by 500 iterations of ML. The NPG sample had a more complex structure than the Siemens star and we obtained reliable reconstructions by using the probe recovered from the Siemens star dataset as the initial probe\u0026mdash;iterations with a random initial probe often converged slowly or even did not converge. The transmission image was still initialized from random values and reconstructed by running 2000 iterations of AP and then 500 iterations of ML.\u003c/p\u003e \u003cp\u003eDuring reconstruction, we often observed a phase vortex appearing in the reconstructed probe, as also reported by others\u003csup\u003e50\u003c/sup\u003e. Such reconstructions were discarded. The equal-spaced raster scan can introduce periodic artefacts in the reconstructed image. To reduce these, we performed a smoothing of the object with a gaussian kernel during the DM and AP iterations and carried out the final ML iterations without smoothing.\u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eSimulations\u003c/h3\u003e\n\u003cp\u003eIn the numerical study, near-field (high-NA) ptychography data was simulated using the Fresnel scaling theorem\u003csup\u003e51\u003c/sup\u003e to avoid the need for large array sizes. Diffraction patterns in the low-NA case were simulated via Fourier transform based Fresnel diffraction\u003csup\u003e52\u003c/sup\u003e. Reconstructions were carried out by applying 2000 iterations of AP and 200 iterations of ML. This schedule was repeated three times with uncorrelated random initializations.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003eAcknowledgements\u003c/p\u003e\n\u003cp\u003eWe acknowledge Sabrina Schneider, Harumi Nakatsutsumi, Tjark Delmas (DESY) and Martin Domarack\u0026yacute; for their technical support, and Johanna Hakanp\u0026auml;\u0026auml;, Sofiane Saouane, and Guillaume Pompidor for assistance at beamline P11 of the PETRA III facility at DESY. We thank Patrick Huber (DESY and TUHH) for helpful suggestions. We acknowledge support from DESY (Hamburg, Germany), a member of the Helmholtz Association HGF. This research was supported in part through the Maxwell computational resources operated at DESY. Additional support was provided by the Cluster of Excellence CUI: Advanced Imaging of Matter of the Deutsche Forschungsgemeinschaft (DFG) \u0026ndash; EXC 2056 \u0026ndash; project ID 390715994.\u003c/p\u003e\n\u003cp\u003eAuthor contributions\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eMeasurements (J.L.D., H.F., M.Z., O.Y., H.N.C., S.B.), optics fabrication and characterisation (S.B., M.P., J.L.D., M.Z.), instrumentation (H.F), analysis (W.Z., T.L., N.I., C.L., J.H., H.N.C.), control and data acquisition software (D.E., I.D.G.A., P.M.), sample preparation (M.P., S.S.), concept (H.N.C., S.B.). W.Z. and H.N.C. wrote the manuscript with input from all other authors.\u003c/p\u003e\n\u003cp\u003eData availability\u003c/p\u003e\n\u003cp\u003eAll data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplemental Information. Additional data related to this paper may be requested from the authors.\u003c/p\u003e\n\u003cp\u003eConflict of interest\u003c/p\u003e\n\u003cp\u003eThe authors declare no conflict of interest.\u0026nbsp;\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eMichelson, A. \u003cem\u003eet al.\u003c/em\u003e Three-dimensional visualization of nanoparticle lattices and multimaterial frameworks. \u003cem\u003eScience\u003c/em\u003e \u003cstrong\u003e376\u003c/strong\u003e, 203\u0026ndash;207 (2022).\u003c/li\u003e\n\u003cli\u003eHoller, M. \u003cem\u003eet al.\u003c/em\u003e Three-dimensional imaging of integrated circuits with macro-to nanoscale zoom. \u003cem\u003eNat. Electron.\u003c/em\u003e \u003cstrong\u003e2\u003c/strong\u003e, 464\u0026ndash;470 (2019).\u003c/li\u003e\n\u003cli\u003ePfeiffer, F. X-ray ptychography. \u003cem\u003eNat. 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Analysis of numerical diffraction calculation methods: from the perspective of phase space optics and the sampling theorem. \u003cem\u003eJ. Opt. Soc. Am. A\u003c/em\u003e \u003cstrong\u003e37\u003c/strong\u003e, 1748\u0026ndash;1766 (2020).\u003c/li\u003e\n\u003c/ol\u003e"},{"header":"Table 1","content":"\u003cp\u003eTable 1. Parameters used in the numerical simulations\u003c/p\u003e\n\u003cdiv\u003e\n \u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"534\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd width=\"31.401869158878505%\" valign=\"top\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.439252336448597%\" valign=\"top\"\u003e\n \u003cp\u003eHigh-NA case\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.186915887850468%\" valign=\"top\"\u003e\n \u003cp\u003eLow-NA case\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"28.97196261682243%\" valign=\"top\"\u003e\n \u003cp\u003eExperiment (NPG)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"31.401869158878505%\" valign=\"top\"\u003e\n \u003cp\u003eWavelength (nm)\u003c/p\u003e\n \u003cp\u003eAperture of lens (\u0026mu;m)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.439252336448597%\" valign=\"top\"\u003e\n \u003cp\u003e0.071\u003c/p\u003e\n \u003cp\u003e40\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.186915887850468%\" valign=\"top\"\u003e\n \u003cp\u003e0.071\u003c/p\u003e\n \u003cp\u003e40\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"28.97196261682243%\" valign=\"top\"\u003e\n \u003cp\u003e0.071\u003c/p\u003e\n \u003cp\u003e35\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"31.401869158878505%\" valign=\"top\"\u003e\n \u003cp\u003eFocal length (mm)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.439252336448597%\" valign=\"top\"\u003e\n \u003cp\u003e1.43\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.186915887850468%\" valign=\"top\"\u003e\n \u003cp\u003e143\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"28.97196261682243%\" valign=\"top\"\u003e\n \u003cp\u003e1.2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"31.401869158878505%\" valign=\"top\"\u003e\n \u003cp\u003eNA of lens\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.439252336448597%\" valign=\"top\"\u003e\n \u003cp\u003e0.014\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.186915887850468%\" valign=\"top\"\u003e\n \u003cp\u003e0.00014\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"28.97196261682243%\" valign=\"top\"\u003e\n \u003cp\u003e0.014\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"31.401869158878505%\" valign=\"top\"\u003e\n \u003cp\u003eDefocus distance (\u0026mu;m)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.439252336448597%\" valign=\"top\"\u003e\n \u003cp\u003e37\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.186915887850468%\" valign=\"top\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"28.97196261682243%\" valign=\"top\"\u003e\n \u003cp\u003e73\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"31.401869158878505%\" valign=\"top\"\u003e\n \u003cp\u003eDetector distance (m)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.439252336448597%\" valign=\"top\"\u003e\n \u003cp\u003e1.2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.186915887850468%\" valign=\"top\"\u003e\n \u003cp\u003e1.2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"28.97196261682243%\" valign=\"top\"\u003e\n \u003cp\u003e2.37\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"31.401869158878505%\" valign=\"top\"\u003e\n \u003cp\u003eMagnification\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.439252336448597%\" valign=\"top\"\u003e\n \u003cp\u003e32,432\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.186915887850468%\" valign=\"top\"\u003e\n \u003cp\u003e/\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"28.97196261682243%\" valign=\"top\"\u003e\n \u003cp\u003e32,465\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"31.401869158878505%\" valign=\"top\"\u003e\n \u003cp\u003eScan step size (nm)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.439252336448597%\" valign=\"top\"\u003e\n \u003cp\u003e110\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.186915887850468%\" valign=\"top\"\u003e\n \u003cp\u003e44\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"28.97196261682243%\" valign=\"top\"\u003e\n \u003cp\u003e250 (H) \u0026times; 100 (V)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"31.401869158878505%\" valign=\"top\"\u003e\n \u003cp\u003eScan points\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.439252336448597%\" valign=\"top\"\u003e\n \u003cp\u003e50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.186915887850468%\" valign=\"top\"\u003e\n \u003cp\u003e500\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"28.97196261682243%\" valign=\"top\"\u003e\n \u003cp\u003e63\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-4093473/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4093473/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eHigh-resolution X-ray imaging of non-crystalline objects is often achieved through the approach of scanning coherent diffractive imaging known as ptychography. The imaging resolution is usually limited by the scattering properties of the sample, where weak diffraction signals at the highest scattering angles compete with background noise and parasitic scattering. Here we demonstrate that X-ray multilayer Laue lenses of high numerical aperture (NA) can be used to create a strong reference beam that holographically boosts that weak scattering from the sample over a large range of scattering angles, enabling high-resolution imaging that is tolerant to such noise. An imaging resolution of about 5 nm was achieved at a photon energy of 17.4 keV with lenses of 0.014 NA from a Siemens star test object and a sample of hierarchical nanoporous gold, recording projection holograms at a magnification of more than 30,000 directly on a pixel-array detector. A numerical study compared this approach to far-field ptychography, indicating significant advantages for using high-NA lenses in the presence of background noise. This imaging modality is particularly fast and efficient to record high-resolution transmission phase-contrast images over large fields of view in a facile manner.\u003c/p\u003e","manuscriptTitle":"Hard X-ray projection imaging below 5 nm resolution","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-04-02 17:43:03","doi":"10.21203/rs.3.rs-4093473/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"d79898ce-040b-4e0d-ba50-d50034bece17","owner":[],"postedDate":"April 2nd, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-04-25T06:45:19+00:00","versionOfRecord":[],"versionCreatedAt":"2024-04-02 17:43:03","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4093473","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4093473","identity":"rs-4093473","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}